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Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University be can developed uncertainty highly not of are type them with This al. deal Jeremić et to knowledge. tools mathematical of the lack other but data the reduced. our more On be to collecting by uncertainty. not due reduced aleatory can arise with uncertainty uncertainties dealing of for epistemic type available This hand, is theory nature. mathematical of developed variabilities Highly inherent the with associated are through uncertainties of propagation (b) equations. and the differential uncertainties all governing of quantification and and data classification (a) the steps: of mean the against neglected. calibrated is exhibit usually uncertainties generally are about sets information data models models experimental the material those Although modeling, distribution, data. constitutive statistical experimental deterministic of traditional set In against calibrated are simulation. and modeling in them different. very be 3104 could of material 401 same page: the material of material of behavior in the influence uncertainties them, anticipated on between Depending of interaction and advanced schematic behavior. properties stress-strain using a elastic-plastic shows by bi-linear 105.1 gained a on Fig. advantages uncertainties example, the For ELASTO-PLASTICITY outweigh uncertainties. PROBABAILISTIC point-wise could models. and properties 105.2. constitutive spatial material inherent to in due uncertainties uncertain is These behavior etc.) bone powder, concrete, Notes ESSI h netite a ebodycasfidit laoyadeitmctps laoyuncertainties Aleatory types. epistemic and aleatory into classified broadly be can uncertainties The two involves properties material uncertain with structures and solids of simulation and modeling The for account to best is it and materials real in inevitable are properties material in uncertainties The iue151 niiae nuneo aeilFutain nSrs-tanBehavior Stress-Strain on Fluctuations Material of Influence Anticipated 105.1: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeaie tanand strain generalized eeice l nvriyo aiona Davis California, of University al. Jeremić et boundary the elastic–plastic). with or dealing issue (elastic starts The properties one material (2004) when uncertain al. pronounced with et problems very II value becomes (1999a,b), Stokoe properties Fenton and ( 2004), material ( 1999a,b), Hiltunen Kulhawy uncertain and and of Marosi Phoon ( 1995), ( 2000), (1998), Popescu Rackwitz al. ( 1993), (2000b), et Duncan Baecher Popescu and (1996), applications, works DeGroot Nadim engineering recent and (1993), geotechnical note Lacasse Mayerhoff for We (1977), properties continuum. Vanmarcke (soil) their (1966), in material location Lumb in of to uncertainties function specialized the a are quantifying or they continuum in whether their on in depending modeled location 1983) are system fixed (Vanmarcke, parameters the a fields material in random uncertain uncertainties or theory, variables the probability random 3104 that of one of as assumes framework uncertainties, 402 but the aleatory page: system, Under for the uncertainties irreducible. in epistemic are uncertainties trading-off mathematical total in advanced the that using reduce note equations doesn’t to uncertainties governing important aleatory the is for through It arithmetic uncertainties propagation epistemic interval tools. their trade (1990), facilitate to ELASTO-PLASTICITY Elishakoff PROBABAILISTIC to useful and order proves 105.2. Ben-Haim it in Hence, models convex etc.). (1983), (1979) Moore Zadeh logic fuzzy (e.g. Notes ESSI .MneCromto svr popular very is method Carlo Monte Schüeller( 1997). by technique described Carlo is Monte of problem formulation mechanics of stochastic aspects for different of descriptions Nice coefficient. stochastic with for conditions. unsolvable) boundary is possibly and approach (and geometries intractable functional irregular more with characteristic problems even This and becomes problems and nonlinear field. problems wave linear the random satisfied for applied equation the complicated FPK ( 1974) very of a was derived Lee functional operator and characteristic media Later, stochastic random the in with by propagation approach. problems wave of functional the problem characteristic the of to using solution methodology ( 1952) Exact Hopf uncertainty. by operator attempted to due purely is Langtangen e.g. researchers of number by boundary ( 2005). for described Bergman method and is and solution initial (FEM) numerical Masud The (1991), method appropriate variable. element With response finite of by PDF equation . for FPK solved 1994) be can (Soize, PDE equation FPK the differential conditions partial FPK the satisfy is parameter will random only the where force problems for external developed been has theory mathematical Rigorous Bu 1 eeaie tesis stress Generalized nmcais h qiiru equation, equilibrium the mechanics, In ot al iuaintcnqei natraiet nltclslto fprildffrnilequation differential partial of solution analytical to alternative an is technique simulation Carlo Monte system the of stochasticity the when is work, this in interest of is which case, extreme other The = n h osiuieequation, constitutive the and , φ ( t ) nti ae h rbblt itiuinfnto PF ftersos variable response the of (PDF) function distribution probability the case, this In . A , σ B , and , φ ( t ) D sgnrlzdfre htcnb iedependent, time be can that forces generalized is r prtr hc ol elna rnon-linear. or linear be could which operators are σ Aσ = D = r sufficient are , φ ( t ) oehrwt h tancmaiiiyequation, compatibility strain the with together , 1 odsrb h eairo h solid. the of behavior the describe to u sgnrlzddisplacements, generalized is eso:2.Ags,22,15:04 2021, August, 26. version: is Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. properties Jeremić et uncertain with used structures be and can solids method of perturbation simulations probabilistic the for that statistical accuracy) claims order Closure reasonable requirement lower (with COV calculate to small parameters. order The material in (2003). moments the statistical Kavvas order for moments ”closure higher requirements for inherits need variation” it the to of solution, refers closed-form coefficient problem no ”small difficulty having the computational problems advantage and from for the problem” suffer method took doesn’t Carlo Authors method Monte the this with behavior used Although associated was mean expansion) the approximation. the computing media perturbation of In bounding term The of references. first 2000). the mentioned only (1999, above (considering Hori the behavior in and mean Anders stochastic the e.g. random First at recently, considering expansion only equation. equations published rate constitutive was constitutive elastic–plastic modulus elastic– elastic–plastic Young’s through general the uncertainties to in propagate SFEM coupling to of non–linear attempt formulations high available the material the is uncertain extending problem few in with plastic only difficulty problems exist major (elastic–plastic) there The non–linear Similarly, material parameters. (2003)). to Keese related and 2002) ( references Matthies published and Keese problems. elastic 1991); ( that linear Kiureghian note for Der to are important references mentioned is above It the (2002). in Chatzipantelidis described was and formulations SFEM Babuska the was of and of SFEM formulations (2001) most of different al. formulations regarding et issues different Deb Mathematical of by addressed disadvantages (1997). are al. and methods et advantages (2000)) Matthies Hori on by and review provided Anders al. nice 2003); ( A et Matthies al. and Mellah et popular. Keese Debusschere (1988); very (1991); 2003); ( Spanos Ke Karniadakis and and (Ghanem and Spectral Xiu Kiureghian and (2002); Der ( 1999)) SFEM, Borst (1992); of De formulations Hien and several Gutierrez and exist (2000); There (Kleiber Finite method. perturbation Stochastic such problems which popular value among most dif- boundary the stochastic is stochastic (SFEM) of For Method solution Element the coefficient. for random method with numerical equation of ferential development for instigated technique especially Carlo high Monte very be could properties. it become material with variable uncertain solution multiple associated the with cost until problems non–linear computational model deterministic The the of al. of disadvantage significant. 3104 et use major of statistically Mellah The repetitive 403 (1997), the (2003). page: Nobahar is al. (2002), analysis et al. Carlo Popescu et Monte Koutsourelakis 2005). (2001), ( 2003, al. et Griffiths al. Lima and et De Paice Fenton (2000), (2002); of e.g. problems, number al. a value et deterministic by boundary Griffiths whose used geotechnical (1996); been problem of has solution any ELASTO-PLASTICITY technique probabilistic for PROBABAILISTIC Carlo obtaining Monte obtained in 105.2. known. researchers be is can numerical) or solution analytical accurate (either solution that advantage the with tool Notes ESSI Luand Liu problems, non–linear geometric to related available also is references of number limited A with associated cost computational high the and solutions analytical finding in difficulties Various eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et as written be can equation constitutive elastic-plastic spatial-average of form incremental Formulation The General Elasto-Plasticity: publications. Probabilistic future in general reported a be 105.2.2 into will incorporation and and underway is three-dimensions work to This extension framework. method- allows element The finite it stochastic examples. that illustrative enough with Drucker–Prager general along to is In described, corresponding ology is equation model. model FPK material material the hardening hardening of linear linear methodology associative associative solution 1D the Drucker–Prager the paper govern 1D companion that particular the equation a FPK to of formulation specialized form general then This EL parameters. is the material of uncertain with made models is material one- derivation elastic–plastic elastic–plastic probabilistic scale paper, hardening this point-location linear In particular associative model. a Drucker–Prager material on namely demonstrated equation, is constitutive methodology dimensional developed the of Application in applications several with designed is methodology solution to namely The equation mind, FPK the coefficient. constitutive of elastic–plastic random form incremental with one-dimensional EL equation general, developed a method applied for Carlo The authors formulation Monte probabilistic the of obtain paper drawbacks forcing. to this the random from In suffer and technique. doesn’t perturbation coefficients equation FPK and random the of with form equation EL using differential approach ordinary 2003) ( corresponding non–linear Kavvas second-order, to Recently, any exact equation, to FPK properties. of form uncertain (EL) Eulerian–Lagrangian with generic 3104 a materials of obtained 404 elastic–plastic page: of behavior constitutive variable(s). internal of evolution of considers direction(s) one and/or when variable(s) behavior internal mean in the computing uncertainties in arises ELASTO-PLASTICITY difficulty approximation, PROBABAILISTIC media bounding 105.2. with below rarely are COVs COV their if only Notes ESSI • • h ou fpeetwr so eeomn fmtoooyfrtepoaiitcsmlto of simulation probabilistic the for methodology of development on is work present of focus The nu neti aeilpoete otecntttv qainaerno variables. random are equation when constitutive equation, the constitutive to properties scale material point-location uncertain input from behavior stress-strain probabilistic obtain fields; random are equation constitutive the to and properties material constitutive uncertain of input form) when (upscaled form equation, average spatial from behavior stress–strain probabilistic obtain dσ ij dt ( x t t , ) = < D 0% 20 ijkl 0% 20 ( x t .Frsisadohrntrlmaterials, natural other and soils For 2000). Kiureghian, Der and (Sudret t , ) Furthermore, (1999a,b)). Kulhawy and Phoon (1996); Nadim and (Lacasse ) d kl dt ( x t t , ) eso:2.Ags,22,15:04 2021, August, 26. version: (105.1) Jeremićet al., Real-ESSI otgnrlfr ficeetlcntttv qaini em fisprmtr a ewitnas written be can parameters its of terms the in Therefore, equation constitutive hardening). incremental distortional of (for form tensor general ( fourth-order most variables translational or (for internal hardening) tensor The second-order kinematic models), rotational variables). hardening and isotropic internal and and perfectly-plastic stress (for scalar(s) of be function a (also function potential f n where and eeice l nvriyo aiona Davis California, of University al. Jeremić et function the with replaced is The (105.4) forcing. Eq. stochastic of and side coefficient hand stochastic with right equation differential ordinary non-linear a is which constitutive linear elastic derivation. general 1-D non-linear for of formulation case probabilistic special the a as that, obtained to formu- is probabilistic addition equation the stochastic In follows, and what coefficient derived. In stochastic is with case. equation 1D forcing incremental a elastic–plastic to constitutive case 1-D 3D for general lation a from shifted is focus response, in coefficients results stochastic (105.3) with forcing. Equation equation stochastic of differential and function ordinary linear/non-linear forcing gener- a and be becoming properties can (105.3) material Equation This in randomnesses forcing. that stochastic so with alized, equations differential ordinary linear/non-linear ( term becoming co- forcing stochastic he with in equations randomness differential Similarly, ordinary linear/non-linear efficients. a becomes (105.1) Equation the that ( variables ( internal constants elastic in randomnesses to Due 3104 of 405 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. tensor stiffness continuum the where Notes ESSI steyedfnto,wihi ucino tes( stress of function a is which function, yield the is switnas written is (105.3) Eq. the behavior, 1-D on Focusing on forcing and parameters material random of effects the of understanding better gain to order In η D dσ dσ ( ijkl ,D ,r r, q, D, σ, ij ( dt x dt ( t x D = t , t ijkl t , el )                    ) = = D D steeatcsins tensor, stiffness elastic the is β r ; ijkl ijkl el el ∗ ( β ,t x, h aeilsins operator stiffness material the ) ,D ,r q, D, σ, ijkl − = ) ( σ ∂σ ij D ∂f β D , ijmn el ( rs ; ,D ,r q, D, σ, x D ijkl t t , rstu el ∂σ q , ∂U ) d mn ∗ ∂σ D ∂U r , ( ; dt x ijkl ∂σ tu x ∗ t ∂f ; t t , t , x pq − ( D ) t x ) t , D ijkl t el ∂q d ∂f t , ) pqkl el d ( ∗ ) n nenlvrals( variables internal and ) dt x D r a eete lsi relastic-plastic or elastic either be can kl t ∗ ijkl t , ep dt ( x ) kl β t t , ijkl steeatcpatccniumsins tensor, stiffness continuum elastic–plastic the is eut nEuto (105.3) Equation in results (105.3) Equation of ) σ ; ; ij ) f < f n nenlvrals( variables internal and ) eoe tcatc tfollows It stochastic. becomes (105.3) Eq. in 0 = η 0 as ∨ ∨ df ( f 0 = 0 = ∧ q ∗ n/rrt feouinof evolution of rate and/or ) f< df eso:2.Ags,22,15:04 2021, August, 26. version: 0) q ∗ ), U steplastic the is q ∗ (105.2) (105.5) (105.4) (105.3) could ) Jeremićet al., Real-ESSI ,dsrbdb density a cloud by a described now 105.2), Considering Fig. to (105.6)). (refer (Eq. points (ODE) initial equation of differential ordinary stochastic non-linear that condition the in initial point given the and for point, initial velocity the determines (105.6) Eq. condition, initial with eeice l nvriyo aiona Davis California, of University al. Jeremić et Liouville stochastic Kubo’s using terms, the mathematical in in points 1963): expressed these , (Kubo be all equation of can conservation equation the continuity expresses which This equation continuity a to according 3104 of 406 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. as written be can (105.4) Eq. now that so Notes ESSI iha nta condition, initial an with iue152 oeet fCodo nta ons ecie ydensity by described Points, Initial of Cloud of Movements 105.2: Figure nteaoeE.(105.6) Eq. above the In h hs density phase The σ ρ ∂ρ ∂σ ( ( σ, x, ( ( ∂t σ x )= 0) )= 0) ( t ∂t t , ,t x, ) ) δ = σ t , ( 0 σ ) η ( − = ,D ,r r, q, D, σ, σ − ρ 0 ) ∂σ ∂ of η σ [ σ σ ( ,t x, ( ; ,t x, a ecniee orpeetapiti the in point a represent to considered be can ,t x, ) ) ) D , )vre ntime in varies (105.6)) Eq. by dictated point any of (movement ( σ x 0 ) sataetr htdsrbstecrepnigslto of solution corresponding the describes that trajectory a as , q , ( x ) r , ( x ) , ( ,t x, σ )] ρ sae hsmyb iulzd rmthe from visualized, be may This -space. ( .ρ σ, [ σ 0) ( ,t x, nthe in ) t , ] σ eso:2.Ags,22,15:04 2021, August, 26. version: -space. ρ ( σ, σ saeadhne the hence, and -space 0) nthe in , σ -space σ (105.7) (105.6) (105.8) (105.9) -space. Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et of time covariance by the defined of function order covariance the ordered (to order second exact to variable state the of density symbol the 3104 where, of 407 page: probability the is density phase Kampen’s a Van of average recall ensemble to the density useful that proves states ELASTO-PLASTICITY it which PROBABAILISTIC Here 1976), , Kampen (105.7)). (Van 105.2. (Eq. Lemma condition initial deterministic original the of where Notes ESSI as 2003) Kavvas, 1996 ; Karakas, and for (Kavvas (105.8) by Eq. derived of recently form was average average ensemble the of derivation the necessitates density phase σ lnkeuto FE lokona owr–omgrvEuto rFokker–Planck–Kolmogorov or Equation Forward–Kolmogorov as known also (FPE, equation Planck ti eesr ooti h eemnsi ata ieeta qain(D)o the of (PDE) equation differential partial deterministic the obtain to necessary is it , nodrt bantedtriitcpoaiiydniyfnto (PDF) function density probability deterministic the obtain to order In n eragn h em ilstefloigFokker- following the yields terms the rearranging and (105.10) and (105.11 ) Eqs. combining By Cov ρ < ∂ δ h + − − stepoaiitcrsaeeti the in restatement probabilistic the is (105.9) Eq. and function delta Dirac the is ρ ( 0 ( ,t σ, σ [ η ∂σ ∂σ ( Z η ∂t ∂ ∂ ∂η x ( ) 0 ( ,t x, ρ < t t σ > t , ( dτCov ( σ = x ) 1 ( t , < ( ) t Z x ,t σ, − η , P 0 ) t η τ · − i t ( ( ( t , > dτCov ,t σ, τ ) ,t x, σ 0 = t , ( − > eoe h xetto prto,and operation, expectation the denotes x η ) 2 − t ( τ ]= )] t , ) This (105.9)). and (105.8) (Eqs. system PDE stochastic linear the from σ ) σ τ D , 0 ( ) ) x D , ftecntttv aeeuto E.(105.4)). (Eq. equation rate constitutive the of [ D , η t h ( t , ( η x ( σ ( ( ) x t x ( ,t x, D , − t x t ) τ − t q , ) t , ( τ 1 q , x ) ) ( ) η q , x t ∂σ D , ( ) ( x t q , ,t x, ( ) t x r , − ( ( t x τ x − ( 2 t ) x t ) τ ) r , ) h − i q , t ) r , ) r , ( , ( x ( x ( x ( t x t − η x t ) t ) ( τ t r , − , ,t x, t , ) τ , ( ( ) )) x x 1 ( ( t x t ) ) x t , h · i t , t − − )); ( τ η τ x η t , .I q (105.11), Eq. In ). P t , t ( t , ,t x, ( − − ,t σ, )); τ τ 2 ))] ) ) ) i eoe vltoayprobability evolutionary denotes h ∂ eso:2.Ags,22,15:04 2021, August, 26. version: ρ ρ < ( h ( ,t σ, ρ σ ( ( ( σ x ) ,t σ, ( ∂σ t x ftesaevariable, state the of t , t ) Cov ) t , t , > ) ) t , hsensemble This . i 0 ) σ σ [ i · paespace -phase saemean -space ] stetime the is (105.11) (105.10) (105.12) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et ( variable function density probability the Once location rate, unknown strain the the relate using can by one theory, strain location small space assumes real one the while that fact the (105.4)). (Eq. equation is constitutive (105.4)) incremental (Eq. the of equation density behavior probability original the the unknown, its while of that terms variable in note state linear to is (105.13 )) important (Eq. is FPE the It non-linear, (105.4)). (Eq. variable equation state the rate of PDF density the probability yield the will of conditions terms linear in deterministic this (non–linear, (105.13)), of inelastic (Eq. solution of The FPE behavior probabilistic equation. for constitutive relation incremental general stochastic most 1-D the elastic–plastic) is This order. second exact to 3104 of 408 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. Schüeller(1997)): 2004), ( Gardiner (1989), (Risken Equation) FPK Notes ESSI .I hscs tosohsi ieeta equation differential Itô stochastic case this In 2004 ). (Gardiner, equation differential Itô stochastic and nte osbewyt bantema fsaevral st s h qiaec ewe FPE between equivalence the use to is variable state of mean the obtain to way possible Another samxdElra-arnineuto.Ti tm from stems This equation. Eulerian-Lagrangian mixed a is (105.13) Eq. that note also should One x σ < ∂P t − − + + σ τ ( ( σ yuulepcainoperation expectation usual by ) t (1 = ) ( ∂t ∂σ ∂σ Z x η η ∂ ∂ > σ 0 1 t ( 2 t , t σ = ( hslnaiy ntr,poie infiatavnae nteslto fteprobabilistic the of solution the in advantages significant provides turn, in linearity, This . 2 σ dτCov − ( ) x ( Z t , x τ t ˙ − ) t Z σ − η τ ) 0 = ( τ x ( t , t t σ 0 t , t dτCov ) ( − P ˙ x − ∂η (= ( t τ σ t , τ ) ( ( d/dt D , ) ) σ t D , 0 D , )) ( x ( dσ η x ( t ( t , x ( x t as, ) ( σ − t t ) t − ) ( τ D , ) q , x τ ) P q , ) t ( q , t , ( ( x x ( ,t σ, ) x ( t t D , ) x ) t x r , ∂σ − q , t t ) − τ ( ( ( τ ttime at ) x sotie tcnb sdt bantema fstate of mean the obtain to used be can it obtained is x x r , ) t t r , t t , ) ) σ ( r , x ( ) ( x fteoiia - o-iersohsi constitutive stochastic non-linear 1-D original the of q , x t ( − t t x − ( t , τ t t x τ ) ) )) P t , ) skon h location the known, is t , , ( ( ( x ,t σ, ) x ( t r , x t t , − t ( ) − )) τ x ne prpit nta n boundary and initial appropriate under , τ t , t ; t , t , − ) − , τ τ ) ( x )) x t − t t , τ P )); eso:2.Ags,22,15:04 2021, August, 26. version: rmtekonlocation known the from ( P σ ( ( σ x x ( t t t , − x τ t ) t , t , sa nnw.If unknown. an is ) ) t , ) P ( ,t σ, (105.14) (105.15) (105.13) ) fthe of x t Jeremićet al., Real-ESSI cl osiuiemdln:a - ser iereatccntttv eair n )1D(shear) 1-D b) and behavior, constitutive behavior. elastic constitutive linear point-location hardening of linear (shear) types associative 1-D the Drucker-Prager particular In a) elastic-plastic two to form. modeling: specialized general is constitutive most relation scale in general forcing developed stochastic the and section following coefficients stochastic with equation incremental accordingly. treated be to needs within and appearing variable random state is the (105.18) that Eq. note of to important is It (105.14). Eq. h location the eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 409 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. is (105.13) Eq. to equivalent Notes ESSI where, 2003)) (Kavvas, (e.g. ( as, variable space state and of time mean in of evolution the ( describes independent process the which Wiener of term advantage the last taking the of By in property (105.16). together Eq. increment lumped of are right-hand-side (105.4)) the (Eq. of equation term) original increment the (Wiener of stochasticity the all that note and, hscnldstedvlpeto eainfrpoaiitcbhvo f1Deatcpatcconstitutive elastic–plastic 1-D of behavior probabilistic for relation of development the concludes This sannoa ner-ieeta qaini ueinLgaga om ic,although since, form, Eulerian-Lagrangian in equation integro-differential nonlocal a is (105.18) Eq. dW dσ b dσ < 2 ( ( ,t σ, ( ,t x, t ) ( dt 2 = ) ,t x, x = ) sa nrmn fWee rcs with W process Wiener of increment an is t ttime at ) + + > η b Z η ( Z ( = + 0 σ ( ,t σ, 0 t σ ( t t η dτCov x ( dτCov ( x skon h arninlocation Lagrangian the known, is ) t Z σ η dW − t 0 ( − ( η τ t σ x τ ( t , dτCov ( t σ t , ( t , x 0 t ( − 0 ) t x − ) − D , t ∂η τ τ η t , τ ) t , ( D , ) 0 ) σ ( ( D , D , σ x − ( x t ( ∂η ( ) x τ t x ( ( q , t , x t x ) t ( t , − D , t t σ ( ) − ) τ x D , ) ( q , dW < τ ) D , t x ( q , ) ) x t ( q , r , ( t , x t x ( ( − x t x ( ( ) t ) x τ x ) D , t t r , q , − ) ) ( t t − q , q , ) t ∂σ τ ( ) , ( ( τ ) x x ( x ( r , ) > ( t x x r , t t ) x ) ) t t ( 0 = , r , − ) q , t ∂σ x ( σ t , r , x τ ( t ftenniercntttv aeequation rate constitutive nonlinear the of ) dW < ( ( − x ) t x )) x ( x − r , ,oecndrv h ieeta equation differential the derive can one ), τ t x t t t t , τ − ) ) ) t ( , ) , ) τ r , )) x , , ( t sa nnw hc sdtrie by determined is which unknown an is ( ( − x ( ( x ( x x t τ x t t t ) − t ) t , t ) − , t , τ , > )); τ t , ( )) t , ( 0 = x x − ; t t − − t , eso:2.Ags,22,15:04 2021, August, 26. version: τ τ τ ti loitrsigto interesting also is It . )) )) η t , )) ( ; · − ) nteright-hand-side the on τ )) dt (105.18) (105.16) (105.17) Jeremićet al., Real-ESSI o function flow where eeice l nvriyo aiona Davis California, of University al. Jeremić et where, and, read to 405), Page (from (105.2) Eq. in given tensor constitutive tangent the of parts expand can one ( angle I 3104 of 410 page: written be can surface yield the cohesion), (without as: criteria yield Drucker-Prager obeying ELASTO-PLASTICITY materials PROBABAILISTIC For Constitutive 105.2. 1-D Probabilistic Elastic–Plastic Elasto-Plasticity: Probabilistic 105.2.3 Notes ESSI 1 2 = oedtie eiaino hspoaiitcdffrnito sgvni h Appendix. the in given is differentiation probabilistic this of derivation detailed more A yasmn soitv o ue ota h il ucinhstesm eiaie steplastic the as derivatives same the has function yield the that so rule, flow associative assuming By σ B A f ∂σ J α ∂f ii K kl = 2 ij sin( 2 = stefis nain ftesrs esr and tensor, stress the of invariant first the is = + + = = and p = nrmna Equation Incremental ∂σ J 1 2 ∂f S 2 ∂σ ∂σ G ∂U ∂f pq ij − K ∂I ∂ ∂f ij φ rs s r h lsi ukmdlsadteeatcsermdlsrespectively. modulus shear elastic the and modulus bulk elastic the are √ D ∂f αI ij ) 1 − D / J pqkl ( 1 stescn nain ftedvaoi testensor stress deviatoric the of invariant second the is 2 rstu 3 2 p 2 G

(3)(3 = 2 2 ∂σ

∂f G 2 tu + +

G ∂σ ∂I − = ∂ ∂σ ij 1 sinφ A √ ∂σ ∂ ∂I δ √ ∂f kl K J ij ij 11 1 2 J ∂I ∂f − )) 2 ∂ ∂σ 2 1 √ where , 2 3 2 J 2 ij + G G 2 2 G + ∂ ∂σ ∂σ ∂σ √ ∂I ∂I φ ∂σ J ij K 22 ∂I cd 1 1 1 1988b)). , Han and (Chen (e.g. angle friction the is δ 11 δ 1 − ik cd 2 δ δ δ 2 3 1 + jl kl l G δ α + 1 k nitra aibe safnto ffriction of function a is variable, internal an , ∂σ ∂I + K 33 ∂ 1 ∂σ ∂σ ∂I √ − 22 J ij 1 2 2 3 2 ! δ δ G 2 ij l δ 2 k ∂ 2 ∂σ !! √ + ab J ∂σ eso:2.Ags,22,15:04 2021, August, 26. version: s ∂I 2 ij δ 33 ab 1 = δ δ kl 3 σ l δ ij 3 k − 1 / 3 δ ij σ (105.19) (105.21) (105.22) (105.20) kk and , 2 Jeremićet al., Real-ESSI soitdpatct.B ouigoratnino n iesoa on-oainsaeserconstitu- shear scale η point-location with dimensional material one between hardening on relationship linear attention for tive isotropic our tensor Drucker-Prager focusing stiffness By case continuum plasticity. this probabilistic associated in a model, represents material 105.24) elastic–plastic (Eq. an equation above The spectively. tensor where (105.23) and sense. 105.22), ( deterministic a in (105.21), out ( Eqs. carried tensor in stress be appearing to not respect differentiations can with Therefore, invariants stress random. will the tensor become of stress derivatives will resulting the the hence random, be and to random assumed become are also properties material since that noted be should It e eeice l nvriyo aiona Davis California, of University al. Jeremić et of behavior (pre–yield) elastic for one shear recognized, elastic-plastic are cases scale two particular, point-location In 1-D equation. hardening, rate linear constitutive associative Drucker-Prager of behavior hardening) (linear substitute angle can friction of modulus change (shear of properties rate material the both considering By 3104 of 411 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. Notes ESSI eq p ( ,D ,r r, q, D, σ, 3 1988b ) Han, and (Chen e.g. assumption, common fairly a is This 2 = hnbecomes then (105.1) Eq. in tensor parameter The strain plastic equivalent of function a is variable internal of evolution the that assuming further By η D K / = ijkl P ep 3 e =                ij p = e − ; ij p      2 A ,t x,              G ∂q η ∂f ij 2 hnoecnwrite can one then G d ooti h atclrFKeuto o h probabilistic the for equation FPK particular the obtain to (105.13 ) Eq. in derived as ) n 2 2 dt n scalars and Gδ Gδ oread to 405 ) Page (on (105.5) Eq. in defined as 12 − r n ik ik 4 = G δ δ jl jl − 2 σ + + 12 √ B 1 ∂ 3 and √ ∂f + B K K ∂α ∂f J K 2 − − and ∂σ P √ 12 de 2 2 3 3 dα J G G 12 eq p o rce-rgrmtra oe,oecnsmlf h function the simplify can one model, material Drucker-Prager for K 2 P ∂ 2 δ δ √ ∂f ij ij re- (105.23) and (105.22), (105.21), Eqs. by defined are      δ δ J kl kl d 2 dt − 12 B α A 0 + ij n h tanrt ( rate strain the and ) A ; ; K f < f kl P 0 = 0 G ∨ ∨ ukmodulus bulk , ; ; ( df < f f f 0 = 0 = 0 = 0 ∨ ∨ ∧ ( df f< df f eso:2.Ags,22,15:04 2021, August, 26. version: d 0 = 0 = 12 K 0) /dt ∧ rcinangle friction , f< df ( t ) srno,one random, as ) 0) (105.23) (105.24) (105.25) α and , σ ij 3 ) , Jeremićet al., Real-ESSI where ( Eq. as ( equation same the is this that noting eeice l nvriyo aiona Davis California, of University al. Jeremić et the on coefficient first the of term of covariance independent the is in simplification (105.27) process Further equation random the (105.27). first of r.h.s equation the that FPK noting the by of possible simplification is in result substitution, after which, becomes kernel elastic–plastic probabilistic the differentiations, the out of values and later) uses other have also will (but 3104 of 412 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. ( material Notes ESSI ) ihntepoaiitceatcpatckernel elastic–plastic probabilistic the within (105.27)), (Eq. f 0 = ti motn ont httedffrnitosapaigi h offiin em fteFKPDE FPK the of terms coefficient the in appearing differentiations the that note to important is It a G ∂P G sue values assumes G ∨ ep ep + − + + ep df σ ( ( < f ( σ ( a 12 a a ∂t 12 = ) ) 0 = ∂P ∂σ ∂σ ∂σ Z ) | n ec hs ieetain a ecridoti eemnsi es.Atrcarrying After sense. deterministic a in out carried be can differentiations those hence and ∂ ∂ ( σ 0 ∂ 0 sdfie spoaiitceatcpatckre n sitoue osotntewriting the shorten to introduced is and kernel elastic–plastic probabilistic as defined is t 12 2 2 t 12 12 12 2 2 ) ( sdsrbdb h olwn rbblsi equation probabilistic following the by described is )      ∨ dτCov t , → σ ∂t 12 2 ( ) const. f G ( = 2 Z Z t 0 = − ) G t 0 0 t , 0 t t or = d 4 dτCov dτCov ) dt ∧ G ∂σ 12 =

t 2 f< df ∂ − 2 ( 12 − t G τ ) ∂ B ∂σ 0 0 . − √ ∂ ( 0 G P a 12 2 G ∂f )) J G ep G + ) ( 2 ep σ ( ( 9 + d a ( 12 t dt t ) ) K 12 ) ( d ∂ t d Kα ∂σ dt P ( ) ( dt √ 12 G t , ( t 12 a ;2 ); 12 J ep ( ) 2 ?? ) 2 ( t G ( + t ( ( ) a G ); .I diint ht h aeo lsi–lsi behavior elastic–plastic of case the that, to addition In ). t a 2 ) σ )) ) √ d G 12 ; 1 dt 3 G d ep 12 utemr,snetecvrac fzr ihany with zero of covariance the since Furthermore, . 2 dt I ep 1      12 ( ( ( t ( t a ( t − − ) t − ) α τ τ 0 ) τ ! ) d ) dt d G 12 dt 12 P ep ( t ( ( ( σ t − a 12 ) − τ ( ) r o fixed for are (105.28)), Eq. (i.e. ) τ t ) ) t , ) P eso:2.Ags,22,15:04 2021, August, 26. version: P ( σ ( 12 σ 12 ( t ( ) t t , ) t , ) ) (105.27) (105.26) (105.29) (105.28) Jeremićet al., Real-ESSI h D eciigtepoaiitcbhvo fcntttv aeeutoscnb rte nthe in written be can equations rate constitutive of behavior form: general probabilistic following the describing PDE Prob- The the for Conditions Boundary and Initial Elasto-Plasticity: Probabilistic 105.2.4 u h couepolm ilapa.Hne nti td h vlto of evolution the study this in Hence, mean appear. to will respect problem” with ”closure of accordingly. the perturbation mean treated but be be 2000) could Hori to , differentiations and need these (Anders and obtaining differentiations to random approach are possible (105.29)) One Eq. the ker- through elastic–plastic probabilistic defined the nel with (105.31) (Eq. equation integro-differential Eulerian-Lagrangian as (105.18)) Eq. in material Prager kernel elastic–plastic probabilistic the where eeice l nvriyo aiona Davis California, of University al. Jeremić et ( Eqs. ( of advection right–hand–side the the called on are terms respectively These terms second and first the of where, 3104 of 413 page: ELASTO-PLASTICITY PROBABAILISTIC read to 105.2. simplified further is (105.27) equation FPK the zero, is process random Notes ESSI 4 nie nbakt r o sdi ne umto convention. summation index in used not are brackets in Indices ti motn ont httedrvtvsapaigi h enadcvrac emo h above the of term covariance and mean the in appearing derivatives the that note to important is It stress shear of value mean a of evolution The ∂P ∂P N − + σ (1) ( ( 12 ∂t + σ σ dσ < ∂t blsi lsi–lsi PDE Elastic–Plastic abilistic 12 12 and ∂σ ∂σ Z ilb bandb h xetto prto ntePF(q (105.15)). (Eq. PDF the on operation expectation the by obtained be will ∂ t , ( 0 ∂ t t 2 12 12 2 ) ) dt 12 dτCov t , N ( ) t (2) = = = ) = G Z > r coefficients are 0 ep 0 − − − t = dτCov ( ∂σ ∂σ ∂σ t ∂σ ∂ζ ) ∂ ∂ ∂ d 12 12 12 G 12 dt 12 ep 0 P ( P ( t G t ( ) ( ) G σ σ ep d ep dt 12 12 4 ( P 12 ( t N t , t , ftePEadrpeetteepesoswti h ul braces curly the within expressions the represent and PDE the of ) t ( ( ) (1) d σ ) ) t d dt N N ) 12 dt 12 n iuin( diffusion and ) (1) 12 (1) ( t ( G ) ( t

− t , ) t ep ); + ) ( ∂σ G ; a σ ∂σ ∂ G ) 12 ep ∂ 12 ep sgvnb h q (105.29). Eq. the by given is 2 ( 12 2 sotie ysubstituting by obtained is t ( t − P − P ( τ σ N ( τ ) 12 σ d ) (2) 12 dt d t , 12 dt oﬃinsa h omo q (105.32) Eq. of form the as coeﬃcients ) t , ) 12 N ( ) t ( N (2) t − (2) −

τ ) ) P eso:2.Ags,22,15:04 2021, August, 26. version: ?? ( σ ,ad(105.27) and (105.26), ), 12 η ( t drvdfrDrucker- for (derived ) t , ) (105.30) (105.31) (105.32) Jeremićet al., Real-ESSI rgrmtra oe ihlna adnn,adb sn P rnfr ecie bv,cnb solved be ( can stress above, shear described of transform Drucker– FPK densities associated using probability by to for and case hardening, this linear with in model specialized material elasto–plasticity, Prager for strain) random and properties terial from extend could domain stress the theory, In time at ( of mass (Eq. probability value equation the rate constitutive all elastic that linear assume of can behavior probabilistic For problem. of type the eeice l nvriyo aiona Davis California, of University al. Jeremić et probability of evolution function the forcing describing the (PDE) assuming equation (i.e. differential only partial deterministic), properties as material rate) of (strain randomness the to attention focusing By Prob- for Equation Fokker–Planck–Kolmogorov Elasto-Plasticity: Probabilistic 105.2.5 choice. preferred the 2004) no (Gardiner, be as will i.e. condition boundaries conserved this the is express at system can barrier the one reflecting within term, a mathematical mass boundaries, probability In the the at that allowed required is is leaking it Since simulation. the ( of ( mass domain probability the This into diffuse and with. advect begin to (105.26)), of (Eq. ior distribution a be will there where, 3104 of 414 page: of equation density. continuity probability a is is equation which the (105.32), of Eq. ELASTO-PLASTICITY variable from PROBABAILISTIC state follows the This 105.2. and current. probability the symbol be The to 2004). considered (Gardiner, equation advection–diffusion resembles closely Notes ESSI 5 σ pcaie oDukrPae soitdlna adnn model. hardening linear associated Drucker-Prager to Specialized ihteeiiiladbudr odtos h rbblsi ieeta qain(ihrno ma- random (with equation differential probabilistic the conditions, boundary and initial these With to, translates this term, mathematical In o h otyedbhvo fpoaiitceatcpatccntttv aeEquation rate constitutive elastic-plastic probabilistic of behavior post–yield the For densities probability for (105.32) Eq. solve can one conditions, boundary and initial introducing After 12 ζ P ζ iheouino ie h nta odto ol edtriitco tcatcdpnigon depending stochastic or deterministic be could condition initial The time. of evolution with ( δ ( ( −∞ σ ( σ σ · 12 12 ) 12 blsi lsiiyadEat–lsiiyi 1-D in Elasto–Plasticity and Elasticity abilistic t , steDrcdlafunction. delta Dirac the is , t , fteewr oeiiilsrse obgnwt eg vrudnpesr nasi mass). soil a on pressure overburden (e.g. with begin to stresses initial some were there if )= 0) ) = ) | AtBoundaries ζ δ ( ( σ ∞ 12 t , ) 0 = ) 0 = σ 12 orsodn oteslto ftepeyedpoaiitcbehav- probabilistic pre-yield the of solution the to corresponding , σ 12 t , σ 12 pc)o h ytmtruhu h vlto i time/strain) (in evolution the throughout system the of space) si vle ihtm/ha tan( strain time/shear with evolves it as ) −∞ t 0 = to ∞ scnetae at concentrated is ota onaycniin r then are conditions boundary that so P ( σ 12 ( t ) t , ) ,will (105.13), Eq. by dictated ), σ eso:2.Ags,22,15:04 2021, August, 26. version: 12 ζ 0 = a be can (105.32) Eq. in 12 ra oeconstant some at or ). 5 ?? (105.30), (105.34) (105.35) (105.33) ) one )), Jeremićet al., Real-ESSI ( probabilistic problems. describing example in three (105.37)) following the and using verified (105.36) is (Eqs. behavior, equations elasto-plastic FPK proposed of Statements applicability Problem The Example Elasto-Plasticity: Probabilistic (105.37) 105.2.6 strain. and shear (105.36) with Eqs. stress solve shear can of one PDF (2007a), of evolution al. Jeremićfor et in described as (105.38), conditions equation boundary previous the in where where as 1D in stress of PDF of evolution for PDE the eeice l nvriyo aiona Davis California, of University al. Jeremić et shear probabilistic The ( component. modulus elastic–plastic hardening linear isotropic associative Drucker–Prager ( ratio Poisson’s follows: as are and pressure deterministic considered are parameters ( strain shear of deviation 3104 of 415 page: elastic linear for and case, 1D modulus for shear ELASTO-PLASTICITY case particular, PROBABAILISTIC this In in PDE 105.2. properties, following material simplified. probabilistic be with still can (but stress material of (PDF) function density Notes ESSI G ie yanra itiuina on–oainsaewt enof mean with scale point–location a at distribution normal a by given ) rbe I. Problem write can one rate, strain in randomness the neglecting by again state, elastic–plastic for Similarly, rbe II. Problem G ∂P G ep + ep I ( ( 1 G ( σ a ∂t a 0 = 12 2 = ) sgvntruhanra itiuina on–oainsaewt enof mean with scale point–location a at distribution normal a through given is ) ∂P ) 0 ( 12 Z . ( 2007a)) al. Jeremić et in (given stiffness, tangent elastic–plastic probabilistic the is t 707 . 0 )) ( o ipaeetcnrle etwt eemnsi ha taniceet h other The increment. strain shear deterministic with test displacement-controlled a for ) 03 t σ suetemtra slna lsi,poaiitc ihpoaiitcsermodulus shear probabilistic with probabilistic, elastic, linear is material the Assume ∂t G dτCov sueeatcpatcmtra oe,cmoe flna lsi opnn and component elastic linear of composed model, material elastic–plastic Assume 12 = MPa. P.Teami ocluae h vlto fPFo ha tes( stress shear of PDF of evolution the calculated to is aim The MPa. − ( t )) G − + 0 = 9 + G − 2 Z Kα ep G 0 t ( ( d t dτCov 2 G ) dt G + 12 d ep dt 2 ( 12 √ t 1 )) ; 3 ∂P 0 G I d 1 ep dt 2 ∂σ ( ( 12 G σ a a ( t ) 12 12 d α sue values assumes − dt ( 0 12 t )) τ ∂P 2 ; ) d ∂σ ( G dt σ 12 d 12 12 dt ( 12 t )) ∂ 2 P ∂ t ∂σ ( 2 or σ P 12 2 12 ∂σ ( t σ ( t 12 2 12 − )) ( τ t )) ihaporaeiiiland initial appropriate With . eso:2.Ags,22,15:04 2021, August, 26. version: ν 2 . 5 0 = G P n standard and MPa n a rt the write can one ) . 2 n confining and , σ 2 (105.37) (105.38) (105.36) 12 . 5 with ) MPa Jeremićet al., Real-ESSI ( parameter yield probabilistic The ( component. elastic–plastic hardening linear isotropic associative Prager ratio Poisson’s follows: as are and pressure deterministic considered are parameters other The ( eeice l nvriyo aiona Davis California, of University al. Jeremić et software statistical available 1988b )) commercially Han, and variables, (Chen random (e.g. basic variables from random III) of and functions of become II variances Problems coefficients and for diffusion means of and (e.g. estimations advection For the variables. within random appearing of variances terms covariance the of scale, rate location simulation strain For of stress. value of arbitrary PDF an of problems, out evolution example cancel strain to will three converted which the is value, all stress arbitrary solution of can of any in PDF strain be help of could with to evolution rate time dimension stress strain the intermediate of of once an value PDF as numerical of simulation the this evolution hence, describes in and the brought rate process, two, been strain the similarly, has Combining while, Time time, obtained. time. with be with stress strain of (105.36) of PDFs (Eqs. of evolution equation FPK evolution the the the that noted describes be (105.37)) should It PDF. or of evolution the of simulation entire containing the terms for hence, and assumed is rate strain constant d a simplicity, of sake coefficients For diffusion problems. and three advection all the for III, and II, I, Problems solve To Fokker–Planck– for Coefficients of Determination Elasto-Plasticity: Probabilistic or case 105.2.7 elastic linear for case. elastic-plastic method, for transformation simulations variable type either Carlo using Monte verified repetitive will solution the that, to modulus shear follows: pressure as confining are and deterministic considered are of deviation 3104 of 416 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. of deviation standard and Notes ESSI α σ 12 12 12 sgvntruhanra itiuina on-oainsaewt enof mean with scale point-location a at distribution normal a through given is ) 7 6 h lsi slope plastic The parameter yield The rbe III. Problem tsol lob oe htsnetemtra rprisaeasmda admvralsa point- a at variables random as assumed are properties material the since that noted be also should It addition In approach. equation FPK proposed the using solved be will problems three above The o ipaeetcnrle etwt eemnsi ha taniceet h te parameters other The increment. strain shear deterministic with test displacement-controlled a for ) /dt ihsersri ( strain shear with ) a esbtttdb osatnmrclvalue numerical constant a by substituted be can (105.37) and (105.36) Eqs. of coefficients in I 1 0 = omgrvEquation Kolmogorov 0 . 1 . h i st aclt h vlto ftePFo ha tes( stress shear of PDF the of evolution the calculate to is aim The . 03 sueeatcpatcmtra oe,wt iereatccmoetadDrucker– and component elastic linear with model, material elastic–plastic Assume I α P,yedparameter yield MPa, 1 0 α sart fcag ffito nl oenn ierhardening. linear governing angle friction of change of rate a is 0 = sa nenlvral n safnto ffito angle friction of function a is and variable internal an is 12 . 03 0 o ipaeetcnrle etwt eemnsi ha tanincrement. strain shear deterministic with test displacement-controlled a for ) . 707 P,adtepatcslope plastic the and MPa, P.Teami ocluaeteeouino h D fserstress shear of PDF the of evolution the calculate to is aim The MPa. 6 α 0 = . 071 lsi slope plastic , α 0 5 = . 5 G . 7 2 = φ α ie y( by given N d 0 . 5 5 = (1) 12 P,Pisnsratio Poisson’s MPa, /dt and . eso:2.Ags,22,15:04 2021, August, 26. version: 5 . α 0 = N sin( 2 = (2) . σ 0541 12 utb determined be must 0 ihserstrain shear with ) ν φ . 52 /s ) 0 = / ( p n standard and sassumed. is . 2 (3)(3 confining , ν − 0 = sinφ . 2 )) , Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 417 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. cients used. was (2002) Smith and Rose mathStatica Notes ESSI rbe II Problem rbe III Problem I Problem usiuigtevle fdtriitcadrno aeilpoete n h tanrt,coeffi- rate, strain the and properties material random and deterministic of values the Substituting .Frps-il lsi-lsi aetececet are coefficients the case elastic-plastic post-yield For I. coefficients the case, elastic linear pre-yield For N (1) and N N N N (1) (2) (2) (1) N (2) = = 0 = = 0 = = 0 = 4 = 0 = 2 = = fteFKeutoscnb bandfralproblems: all for obtained be can equations FPK the of * t Z d . . t . . dt 0 d 2 0058 4 MPa 147 MPa 27 00074     t 12 dt G d τ ar dτV 12 2 dt d * d G 12 dt dt h t 12 2 G 12 − t G (MPa i 2 G (MPa − 2 / ar V / s ar V 2 9 + s G G / d     s) / Kα [ 9 + dt G 2 s) 12 2 G ] 2 G 2 Kα − 2 + G G 2 √ 1 2 + 9 + 3 I √ 1 1 Kα α N 3 0 I (1)     1 G 2 α 2 + and d 0 dt + 12 √ 1 N + 3 I (2) 1 α ilb h aea hs o Problem for those as same the be will 0     eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI sn h ento fsri ae h bv qaincnb rte ntrso time of terms in written be can equation above the rate, strain of definition the Using .Ti ehdi plcbea o aeidpnetlna lsi aete1 ha constitutive shear 1D form, the the case of elastic equation algebraic linear linear rate-independent a for to as simplify applicable equation is Runger, and high method (Montgomery with variables This approach random equation of 2003). FPK method transformation of a use using the solution, through (exact) obtained accuracy solutions comparing 2002). by validation al., performed and et is (Oberkampf verification tion work that, simulations to and modeling addition any In in included verification. be for always used should efforts been published previously a have no plays could are approach) there FPK which as on elasto–plasticity solutions solutions based probabilistic are developed of (that development of presented solutions verification in developed role based verify crucial Carlo to effort Monte The the paper presented. that, companion is the to (results) in addition results and In The sections previous 2007a). problem. in al., 1D described (Jeremić approach probabilistic et equation elastic–plastic FPK and using elastic by for obtained are presented are results Problems Example section of this Verifications In and Results Elasto-Plasticity: Probabilistic 105.2.8 deterministic. is case eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 418 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. Notes ESSI o iereatccntttv aeeutos(rbe- n r-il aeo rbe-I h verifica- the Problem-II) of case pre-yield and (Problem-I equations rate constitutive elastic linear For elastic pre-yield the deterministic, is modulus shear the since III, Problem for that noted be should It σ σ coefficients the simulation elastic-plastic post-yield For 12 12 2 = 2 = N N G G (2) (1) (0 12 . 054 0 = = = 0 = = = u t t * = ) d ( . . dt , G, 0001 MPa 2365     12 d 2 dt v * G 12 12 ( ,t G, t 2 ) − G (MPa 2 G ) − ar V 9 + / G s /     9 + s) Kα 2 2 G Kα G 2 − 2 + G 2 G √ 2 + 1 9 + 3 √ I 1 1 3 Kα α I 0 1     α G 2 0 2 + d + dt 12 √ 1 N + 3 (1) I 1 α and 0     N (2) eso:2.Ags,22,15:04 2021, August, 26. version: are t as, (105.40) (105.39) Jeremićet al., Real-ESSI iuaino P,Drcdlafnto siiilcniinwsapoiae ihaGusa ucinof function Gaussian a with approximated was condition initial as function delta Dirac FPK, of simulation at hc a sda h nta odto o h P.Oemynt htat that note function, may delta One Dirac FPK. the the represent for to condition stress initial used shear figure the approximation comparison as the the the used over-predicted from of was slightly seen because which it be is exactly, can This behavior It mean deviation. the predicted standard 105.6. approach Fig. FPK in the shown even-though are that deviations standard and mean of eeice l nvriyo aiona Davis California, of University al. Jeremić et 105.4. and using 105.3 obtained Figures were in and shown modulus, shear is random strain with shear approach material and elastic time linear for with are stress PDFs shear Presented of PDF of evolution The I Problem 105.2.9 simulation this for variable number random large constant relatively material A each purpose. for computed. generated easily were resulting then (1,000,000) are of points step characteristics data strain) probabilistic of (or The time each for above. variable generated stress data random elastic-plastic each deterministic for the material equation of for solution rate data repeating constitutive sample by generating and distribution by normal technique standard simulation from Monte-Carlo properties using done is verification the transformations. and respectively G of PDF of evolution the predicting for allow will which 3104 of 419 page: of variables random ( of values the between random tions continuous the ELASTO-PLASTICITY given PROBABAILISTIC and, modulus) 2003), (shear Runger, 105.2. variable and (Montgomery variables random of method mation where, Notes ESSI σ 12 = σ iial,cmaio fteevolution the of comparison Similarly, 105.5. Fig. in compared are PDFs of evolution of contours The o o-iereatcpatccntttv aeeutos(otyedcsso rbesI n III) and II Problems of cases (post-yield equations rate constitutive elastic-plastic non-linear For of PDF of evolution the predict will (105.42) Eq. ), 12 u P P P − 0 0 = ( ( . 1 ( σ σ 054 σ ( Fg efiueEatcD)and reffigure:ElasticPDF) (Fig. 12 12 σ 12 ssc,i ol ebs ecie yteDrcdlafnto.Hwvr o numerical for However, function. delta Dirac the by described best be would it such, As . 12 σ = ) = ) ) / steabtaysri-aeasmdfrti xml rbe.Acrigt h transfor- the to According problem. example this for assumed strain-rate arbitrary the is 1/s 12 , as, 12 hudtertclybe theoretically should g g J ( ( ) v u = − or − 1 1 du ( ( G σ σ − 12 12 = G 1 t , , ( ihPDF with , σ u )) 12 12 − )) | 1 , J ( | | σ 12 J 12 | ) /dσ t , ) 1 g 12 r h nes ffunctions of inverse the are ..altepoaiiyms hudtertclyb concentrated be theoretically should mass probability the all i.e. ( G rnfrainmethod transformation and ) soet-n transforma- one-to-one as (105.40) or (105.39) Eqs. and J = G σ dv and 12 − σ with 1 12 ( σ σ 12 12 with t n a bantePFo ha stress shear of PDF the obtain can one , t , ,functions (105.42), and (105.41) Eqs. In . ) /dσ 12 105.4). (Fig. 12 or, σ 12 r hi epcieJcbasof Jacobians respective their are = eso:2.Ags,22,15:04 2021, August, 26. version: u ( 12 , G, 0 = 12 ) h rbblt of probability the , or σ 12 = (105.42) (105.41) v ( ,t G, FPE ) Jeremićet al., Real-ESSI skp h aefrsmlct ae npeetdeape,t rprycpueteapoiaeinitial approximate of the size capture step properly uniform to a examples, presented Fig. but In in PDF, shown of uniform sake. (as evolution condition simplicity fine, of calculation for that of same that stages the noted later in kept is expensive) is It quite is domain. (and needed entire not the is throughout discretization all uniformly discretization fine same at) (or to close domain stress prxmt nta odtos-alhvn eoma u tnaddvain of deviations standard but mean zero and having all - conditions initial approximate eeice l nvriyo aiona Davis California, of University al. Jeremić et at PDFs the at at with PDF stress actual compared shear the was figure of this PDF In cost). the 105.7. of computational Fig. condition higher in be initial at could the (but error approximating condition This of initial process. effect delta evolution The Dirac the the of approximating simulation better the by with minimized domain the into diffused and advected of deviation standard and zero mean model material elastic approach. linear equation for FPK using time) obtained (or I) strain (Problem versus modulus stress shear shear random of with PDF of Evolution 105.3: Figure 3104 of 420 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. Notes ESSI n a lont htfie prxmto fiiilcniinncsiae nrdsrtzto of discretization finer necessitates condition initial of approximation finer that note also may One 0 . 00001 MPa. 0 . 000005 ?? ,tesrs oanbetween domain stress the ), σ 12 P n ec hr sattlof total a is there hence and MPa 12 0 h nt ieec iceiainshm dpe eeue the uses here adopted scheme discretization difference finite The =0. 0 0 = . 00001 . 46% 0426 P,a hw nFig. in shown as MPa, 12 0 = banduigteFKapoc ihtredifferent three with approach FPK the using obtained . 46% 0426 − 0 . banduigtetasomto method transformation the using obtained 1 P and MPa 40 ?? , 000 hserri h nta condition initial the in error This . +0 oe.Ti o nyrequires only not This nodes. eso:2.Ags,22,15:04 2021, August, 26. version: . 1 P a iceie with discretized was MPa 12 0 . 0 = 01 . MPa, 46% 0426 0 . 005 sshown is MPa Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University It . 105.10 al. Fig. Jeremić et in (time) shown strain at is shear occurred surface versus material stress–strain stress this of shear of PDF of yielding the PDF the to the that view of noted Another is evolution 105.9. of Fig. surface in the shown to of corresponds is condition view initial A this that part. noted ( elastic be the stress may of It shear part of 105.8. elastic-plastic of PDF Fig. post-yield part in the the elastic shown for pre- is The condition and the initial part. random to The is elastic-plastic problem Problem–I. corresponding post-yield to one the identical to equations, is corresponding problem FPK other this two the solving and the part involves elastic yield problem this to solution The II Problem adaptive 105.2.10 an formulating problem. in on of going type is this will work of Current technique solution discretization problem. the this adaptive for solving algorithm An to approach sensitive. better memory much very a also be is but effort computational large model material elastic linear method. for transformation using time) obtained (or I) strain (Problem versus modulus stress shear shear random of with PDF of Evolution 105.4: Figure 3104 of 421 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. Notes ESSI P ( σ 12 )a il bandfo h ouino P qaino h pre-yield the of equation FPK of solution the from obtained yield at )) t 0079scn wihi qiaetto equivalent is (which second =0.00789 eso:2.Ags,22,15:04 2021, August, 26. version: 12 = Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University post-yield the in earlier, discussed reasons over- of al. Jeremić approach et because equation deviations FPK standard the of response pre–yield evolution the the behavior)/Monte-Carlo in predicted (pre-yield Although method transformation behavior). and (post-yield approach approach equation FPK completely using as obtained modeled were material the if same the to compared as elastic. uncertain) less is prediction (i.e. at of stress shear extension of fictitious PDF with where t the compared 105.12 Comparing was Fig. PDF. stress from shear of viewed of evolution easily PDF elastic be of much can evolution as This elastic-plastic uncertainty did. post-yield initial equation the the rate process amplify constitutive evolution process elastic not the evolution pre-yield in did the the words, equation as other during rate In rate constitutive mean. zone. higher elastic–plastic of elastic-plastic a post-yield evolution post-yield at the the the increases in to zone that respect elastic with with compared pre-yield deviation when the standard to in of aspect evolution slope interesting the the relative Another expected, of The yielded. as slope material that, relative the figure the after that is slope from changes seen note stress be shear can of It mean 105.11. of Fig. evolution in shown are deviations standard and 0.0426 FPE for (Problem–I) Modulus Solution. Shear Method Random Transformation for with Variable Function Equation and Density Rate Solution Probability Constitutive of Evolution Elastic Strain) for (or Stress Time Shear of Contours of Comparison 105.5: Figure 3104 of 422 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. Notes ESSI 0 = cmae h vlto fmasadsadr eitoso rdce ha stress shear predicted of deviations standard and means of evolution the compares 105.13 Fig. . 01489 % .Teeouincnor o D fsersrs esssri tm)aogwt h mean the with along (time) strain versus stress shear of PDF for contours evolution The ). s ,oecncnld httevrac fpeitdeatcpatcsersrs smc smaller much is stress shear elastic-plastic predicted of variance the that conclude can one ), 12 0 = . 0804% eso:2.Ags,22,15:04 2021, August, 26. version: wihi qiaetto equivalent is (which Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University equation FPK elastic-plastic post-yield at al. of Jeremić stress et solution shear the of for PDF condition This initial deterministic. the be be to to assumed assumed is was and parameter yield variable stress) yield random confining in the mean of distribution or PDF assumed tensor the stress to to distribution corresponds due a stress stress is shear there shear in distribution yield in The at deviation) however, standard deterministic, small is part very elastic (with linear pre–yield the problem, this In III Problem 105.2.11 elastic-plastic post-yield predict to approach behavior FPK elastic the pre-yield use the behavior. then obtain and method to FPK the predict transformation elastic-plastic be better the probably post-yield to through way would of One behavior, solution equation. elastic-plastic FPK for yielding probabilistic elastic condition the pre-yield overall of initial to solution the close the from that solution) obtained fact Carlo was equation the (Monte difference to one larger attributed verification somewhat is the The region and region. yielding solution the equation from FPK further between regions at closely matched it response Constitutive Elastic Transformation for Variable and Stress Solution Shear FPE for of Solution. (Problem–I) Method Deviation Modulus Standard Shear Random and with Mean Equation Rate of Comparison 105.6: Figure 3104 of 423 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: αI 1 fis nain fthe of invariant (first α . Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et simulation Carlo Monte the obtained from 105.17 . stress obtained Fig. shear those in of with shown deviations compared is standard was and approach and equation mean clearly FPK of 105.15 advected the evolution Fig. just from The much. stress diffusing process. shear without advection of process this evolution function shows elastic–plastic density the yield probability during in yield) domain uncertainty the initial (at into the initial amplify The didn’t process significantly. evolution strength elastic-plastic the that conclude versus can stress one shear for PDF of 105.16. evolution Fig. the in of shown are deviation) (time) standard strain and mean (including contours the 105.14 . Fig. in shown is and Yield at Solution). Stress Method) of Transformation (Variable PDF Actual : with Condition Condition Initial Initial of Delta Approximation Dirac Different of for Function Approximating of Effect 105.7: Figure 3104 of 424 page: ELASTO-PLASTICITY PROBABAILISTIC 105.2. Notes ESSI adcmaigtesoe feouino enadsadr deviation, standard and mean of evolution of slopes the comparing and 105.16 Fig. at Looking that to addition In 105.15. Fig. in shown is (time) strain versus stress shear for PDF of evolution The eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University on (2005)) Griffiths and al. Jeremić et and Fenton Fenton (2003), by Griffiths those and were Fenton papers (2002), published few Griffiths the and Among (Fenton Griffiths problem. constitutive probabilistic the probabilistic is on emphasis increasing an design. not reliability-based seen results subsequent has (2000b)). and safety Duncan community soil Duncan of geotechnical of characterization factors (cf. the structures large years, unsafe recent of in in use sometimes, also, Hence, However, but design, safety. over-expensive of in in factor only uncertainties (large) with applying deals geostructures, community by of engineering behavior geomaterial the geotechnical Traditionally, also but geomaterials. geomaterials, uncer- with of These characteristics made failure (1999a)). Kulhawy the and affect Phoon only (Lacasse Kulhawy not uncertainty) and tainties Phoon of (point (1996 ), variability errors Nadim and natural transformation Lacasse from and Nadim and stem testing uncertainties and These uncertainty), (spatial uncertain. geomaterials inherently is geomaterials Loading of Cyclic Modeling and Yielding Probabilistic 105.3 3104 of 425 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI n fteipratapcso rbblsi emcaissmlto hthsrcie esattention less received has that simulation geomechanics probabilistic of aspects important the of One iue158 nta odto o P qainfreatcpatczn (Problem–II). zone elastic–plastic for equation FPK for condition Initial 105.8: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University non-linear and linear both for (2007d)) al. et Sett (2007c), al. al. Jeremić et et (Sett al. yielding, and et stress equation Sett mean rate by assuming discussed loading, constitutive monotonic were elastic–plastic under to responses to stress-strain corresponding applied probabilistic equation, be simulated strategies differential easily Solution partial can models. is hence FPK constitutive elastic–plastic and for also different elasto–plasticity, but of of simulation techniques, and theory modeling simulation incremental probabilistic probabilistic the other with with not compatible elasto–plasticity associated fully probabilistic drawbacks to the approach overcomes FPKE only elasto–plasticity. probabilistic modeling to for (2003)) simulation Kavvas Eulerian–Lagrangian Kavvas and proposed (cf. ( 2003)) (2007b) approach Keese (FPKE) al. Jeremić Keese equation et Fokker–Planck–Kolmogorov (1997), al. of perturbation Jeremić al. form et recently, and et those, Matthies with Carlo Mises dealing al. Monte von in et and Both of (Matthies drawbacks simulation inherent probabilistic technique. their on have perturbation ( 2000)) techniques using Hori and material Anders plastic (1999), elastic-perfectly Hori and (Anders Hori and c- random spatially with of simulation material probabilistic elastic-plastic for (time) strain versus stress 1. View shear (Problem–II). of modulus PDF shear random of Evolution 105.9: Figure 3104 of 426 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI φ oluigMneCrotcnqe n hs yAnders by those and technique, Carlo Monte using soil eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University (Lacasse al. Jeremić et uncertainty) (point errors transformation and testing and uncertainty), (spatial moments. geomaterials statistical its were and responses (PDF) Simulated function density discussed. probability is of yielding terms probabilistic numerical in with The discussed cyclically considered. FPKE are solving model of material hardening-type technique and plastic hardening elastic–perfectly Both linear terials. for even observed for even was observed behavior was nonlinear domains Further, plastic and models. models. elastic between plastic transition addition perfectly smooth In elastic realistic, behavior. deterministic of very from (mean) a differ average possibility that, ensemble (mode) to the probable elastic hence, most of the and or influence strains) upon possibilities and large the strain depending (at all low possibility, domain very plastic a very into always at far is starts continues behavior there behavior plastic (stress), that function uncertainty, of yield magnitude constitutive in the on uncertainty was It effect to ( 2009b). its due Sett Jeremić and that and Sett Jeremićshown introduced and by discussed was was yielding loading monotonic probabilistic under of simulation concept The models. with hardening material elastic-plastic for (time) strain versus 2. stress View shear (Problem-II). modulus of shear PDF random of Evolution 105.10: Figure 3104 of 427 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI oeigo emtrasi neetyucran hsucranysesfo aua aiblt of variability natural from stems uncertainty This uncertain. inherently is geomaterials of Modeling geoma- of simulations cyclic 1–D to extended is yielding probabilistic of concept the paper, this In eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. probabilistic Jeremić et on emphasis increasing an design. not reliability-based seen results subsequent has (2000b)). and safety Duncan community soil Duncan of geotechnical of characterization factors (cf. the structures large years, unsafe recent of in in use sometimes, also, Hence, However, but design, safety. over-expensive of in in factor only uncertainties (large) with applying deals geostructures, community by of engineering behavior geomaterial the geotechnical Traditionally, also but geomaterials. geomaterials, uncer- with of These characteristics made failure (1999a)). Kulhawy the and affect Phoon only Kulhawy not and tainties Phoon (1996 ), Nadim and Lacasse Nadim and material elastic-plastic for (Problem–II). (time) modulus strain versus shear stress random shear for with PDF of evolution of Contour 105.11: Figure 3104 of 428 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University material elastic al. extended Jeremić et and modulus. material shear elastic-plastic random for for PDF cases of evolution of Comparison 105.12: Figure 3104 of 429 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI 0.008 Time (Sec) 0.01 0.01 0.012 0.012 0.014 0.014 0.01489 0.0426 0 0.002 Strain (%) 0.004 Hardening Plastic Linear Stress (MPa) Elastic Extended 0.006 0.008 eso:2.Ags,22,15:04 2021, August, 26. version: 0.0804 0 0 200 200 400 400 600 600 Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University non-linear and linear both for (2007d)) al. et Sett al. Jeremić et (2007c), al. et (Sett al. yielding, and et stress equation Sett mean rate by assuming discussed loading, constitutive monotonic were elastic–plastic under to responses to stress-strain corresponding applied probabilistic equation, be simulated strategies differential easily Solution partial can models. is hence FPK constitutive elastic–plastic and for also different elasto–plasticity, but of of simulation techniques, and theory modeling simulation incremental probabilistic probabilistic the other with with not compatible elasto–plasticity associated fully probabilistic drawbacks to the approach overcomes FPKE only elasto–plasticity. probabilistic modeling to for (2003)) simulation Kavvas Eulerian–Lagrangian Kavvas and proposed (cf. ( 2003)) (2007b) approach Keese (FPKE) al. Jeremić Keese equation et Fokker–Planck–Kolmogorov (1997), al. of perturbation Jeremić al. form et recently, and et those, Matthies with Carlo Mises dealing al. Monte von in et and Both of (Matthies drawbacks simulation inherent probabilistic technique. their on have perturbation ( 2000)) techniques using Hori and material on Anders plastic (1999), ( 2005)) elastic-perfectly Hori Griffiths and and (Anders and Fenton Hori Fenton (2003), and c- by Griffiths random those spatially and were of Fenton papers simulation (2002), published probabilistic few Griffiths the and Among (Fenton Griffiths problem. constitutive probabilistic the is simulation rate Carlo constitutive Monte and plastic solution for equation stress FPK solution. for shear (problem-ii) of modulus deviation shear standard random with and equation mean of Comparison 105.13: Figure 3104 of 430 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI n fteipratapcso rbblsi emcaissmlto hthsrcie esattention less received has that simulation geomechanics probabilistic of aspects important the of One φ oluigMneCrotcnqe n hs yAnders by those and technique, Carlo Monte using soil eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et moments. statistical its were and responses (PDF) Simulated function density discussed. probability is of yielding terms probabilistic numerical in with The discussed cyclically considered. FPKE are solving model of material hardening-type technique and plastic hardening elastic–perfectly Both linear terials. for even observed for even was observed behavior was nonlinear domains Further, plastic and models. models. elastic between plastic transition addition perfectly smooth In elastic realistic, behavior. deterministic of very from (mean) a differ average possibility that, ensemble (mode) to the probable elastic hence, most of the and or influence strains) upon possibilities and large the strain depending (at all low possibility, domain very plastic a very into always at far is starts continues behavior there behavior plastic (stress), that function uncertainty, of yield magnitude constitutive in the on uncertainty was It effect to ( 2009b). its due Sett Jeremić and that and Sett Jeremićshown introduced and by discussed was was yielding loading monotonic probabilistic under of simulation concept The models. hardening strength yield random with material elastic–plastic for equation (Problem–III). FPK for condition Initial 105.14: Figure 3104 of 431 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI nti ae,tecneto rbblsi iligi xeddt – ylcsmltoso geoma- of simulations cyclic 1–D to extended is yielding probabilistic of concept the paper, this In eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et where, al. Jeremić et al. (Jeremić et as written be can equation rate (2007b)): constitutive corre- 1–D (2003)) generalized Kavvas to Kavvas sponding (cf. equation Fokker–Planck–Kolmogorov form Eulerian–Lagrangian Elasto–Plasticity The Probabilistic to Approach Fokker–Planck–Kolmogorov 105.3.1 strength yield random with shown). material is elastic–plastic zone for plastic stress (only shear for (Problem–III) PDF of Evolution 105.15: Figure 3104 of 432 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI aibe bandb olcigtgte l h prtr n aibe nterhso h generalized the of r.h.s the on variables and operators the all together collecting by obtained variable, ∂P ( + σ ∂σ ∂ ( P ∂t ∂σ x ∂ ( t 2 σ t , 2 ( ) x t , t η t , ) Z ( ,D D, σ, 0 ) = t , t dτCov ) stepoaiiydniyo tes( stress of density probability the is ; x t 0 t , ) η ( ,D D, σ, + Z 0 t ; dτCov x t t , ); 0 η ( ,D D, σ, ∂η ( ,D D, σ, ; ∂σ x t − ; τ σ x t , t(suo time (pseudo) at ) t t , − ) τ ; η ) ( ,D D, σ, P ( σ ; ( x x eso:2.Ags,22,15:04 2021, August, 26. version: t t − t , τ ) t , t t , and , − ) τ η steoperator the is P ( σ ( x (105.43) t t , ) t , ) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et (105.44), Eq. In equation: rate constitutive random with material elastic–plastic for (time) (Problem–III). strain strength versus yield stress shear for PDF Contour 105.16: Figure 3104 of 433 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI dσ ( dt x t t , ) = η stesri,and strain, the is ( ,D D, σ, ; x t t , ) D stetnetmdls hc ol eeatco elastic–plastic: or elastic be could which modulus, tangent the is eso:2.Ags,22,15:04 2021, August, 26. version: (105.44) Jeremićet al., Real-ESSI o(n)priua osiuiemdl h eutn PEcnbe can FPKE resulting the model, constitutive particular (any) to (105.46 ) Eq. specializing By cific. eeice l nvriyo aiona Davis California, of University al. Jeremić et where, form: compact more a in written be can (105.43)) (Eq. equation This space. density bility respectively. variable(s) internal of evolution of rate(s) and variable(s), where, Carlo Monte and Con- Solution FPE Elastic-Plastic for for (Problem-III) Stress Strength Shear Solution Yield of Simulation Random Deviation with Standard Equation and Rate Mean stitutive of Comparison 105.17: Figure 3104 of 434 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI stems eea omo lsi–lsi osiuiert qain rte nproba- in written equation, rate constitutive elastic–plastic of form general most the is (105.43) Eq. D ∂P N D = (1) ( el σ                  , ( ∂t and x f t D D , t , el el U ) N t , − , (2) ) q ∂σ ∂f ∗ = r deto n iuincecet epciey n r aeilmdlspe- model material are and respectively, coefficients diffusion and advection are and , D ∂σ D ∂ el el ∂U ∂σ ∂U ∂σ N r ∗ (1) ∂σ ∂f − r lsi ouu,yedsrae lsi oeta ufc,internal surface, potential plastic surface, yield modulus, elastic are P D ∂q ∂f ( σ el ∗ ( x r ∗ t t , ) t , elastic elastic-plastic )

+ ∂σ ∂ 2 2 N (2) P ( σ ( x t t , ) t , ) eso:2.Ags,22,15:04 2021, August, 26. version: (105.45) (105.46) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et price. option and the price calculate stock to price with strike multiplied Scholes option are of of and functions, exercise density value Black of present cumulative probabilities Black–Scholes from where obtained famous option, option, European of (European) of the strategy modeling engineering solution financial the in equation to (1973) similar is yield coefficients (probabilistic) diffusion uncertain model properly to order in space) deterministic probability the strength. in (or, introduced written is is criteria yield criteria probabilistic The yield criteria. yield deterministic the of ment the and elastic being function density material (stress). cumulative function the of from yield probabilities calculated of be The easily can elastic–plastic being elastic–plastic. material of being probabilities material of probability the where to yielding: elastic stress from mean using switch controlled The be can equation). (solution) constitutive region elastic–plastic 3104 elastic–plastic of to 435 corresponding page: coefficients (with fusion equation constitutive elastic–plastic (with to equation corresponding constitutive other regions elastic elastic-plastic to and corresponding elastic one . in - N . . behavior twice CYCLIC AND FPKE material YIELDING prop- of in PROBABILISTIC material solution difference in 105.3. However, necessities uncertainties given strain. response, driving stress of and function erties density probability the obtain to solved Notes ESSI elc h osblte featcpatcbhvo nteeatcrgo n ievra h ocp of advection concept equivalent with The once, versa. (105.46) coefficients, Eq. vice solves diffusion and it and region as elastic limitation, this the will overcomes in yielding yielding behavior mean probabilistic such elastic–plastic example, of For then possibilities criteria material. the of yield yielding mean neglect probabilistic the complete as the uncertain, for are account parameter(s) not does yield material the if arises difficulty However, (2) el ti lovr neetn ont htpooe prahfrcluaigeuvln deto and advection equivalent calculating for approach proposed that note to interesting very also is It restate- probabilistic represents (105.48)) (Eq. criterion yield probabilistic the that noting worth is It h deto n iuincecet orsodn oeatccntttv qain n the and equation) constitutive elastic to corresponding coefficients diffusion and advection the , (1 if N if or, N (2) (1) eq eq − ( ( P σ σ [Σ (1 = ) (1 = ) y ≤ h h σ − − f f ]) i i P P ersnstepoaiiyo aeilbigeatc while elastic, being material of probability the represents < 0 = [Σ [Σ N 0 y y (1) eq ∨ ∨ ≤ ≤ ( d and σ σ h h f ]) ]) f i N N i 0 = N 0 = (2) (1) el el (2) eq + + ∧ (2009b)): Sett Jeremić and Sett (Jeremić and d P P h [Σ [Σ f i y y < ≤ ≤ 0) σ σ ] ] N N (2) (1) ep ep s lsi FPKE elastic use s lsi–lsi FPKE elastic–plastic use N (1) ep and N eso:2.Ags,22,15:04 2021, August, 26. version: (2) ep h deto n dif- and advection the , P [Σ y ≤ σ ] represents N (105.48) (105.47) (1) el and Jeremićet al., Real-ESSI n a oeta,i eiigteeatcadeatcpatcavcinaddffso coefficients ( diffusion properties and material advection field random elastic–plastic spatial and that elastic assumed the was deriving it in (105.51)), (Eq. that, note may One (105.46)). becomes: (Eq. (105.46)), elastic–plastic FPKE (Eq. the FPKE governing and governing elastic the the the of of model, coefficients solution constitutive diffusion shear the and plastic advection by elastic–perfectly given Mises is von 1–D distribution For its and ( (105.46)) stress ( (Eq. The elastic–perfectly stress For constant. yield remains functions. and of density distribution, distribution probability respective the their material, by plastic described of are stress and yield uncertain a are at occurs yielding The tensor stress deviatoric where, eeice l nvriyo aiona Davis California, of University al. Jeremić et material, plastic elastic–perfectly Mises von for becomes: (105.48)) then, Eq. (refer coefficients diffusion and advection time, where, for use we model such one just is Mises von that and general model is 3104 material development of Mises presented any 436 purposes. that von with page: illustration noted Only used be however, be may, material. to It plastic enough considered. elastic–perfectly been of has behavior model simulate cyclic material to applied strain) . is . . stress–shear yielding, CYCLIC probabilistic AND (shear of YIELDING concept PROBABILISTIC 1–D the with along 105.3. FPKE–approach, the section, this In Material Plastic Elastic–Perfectly 105.3.2 Notes ESSI h o ie il rtracnb rte as: written be can criteria yield Mises von The | p h·i σ N N N N k G − | J (1) (2) (1) (1) el eq eq ep 2 ersnsepcainoeainand operation expectation represents samtra aaee yedsrnt ie and like) strength (yield parameter material a is stesermodulus, shear the is − ( ( σ = 0 = σ σ y k (1 = ) (1 = ) 0 = d 0 = dt xy h − − G or i P P s ij [Σ [Σ = y y ; ; σ ≤ ≤ σ = ij σ σ N N ± − ]) ]) d (1) t (2) ep σ d el 1 xy dt y / xy 0 = 3 = d σ dt ste(eemnsi)iceetlserstrain, shear incremental (deterministic) the is h xy kk t G ± δ i σ ij d 2 y dt becomes: (105.49) Eq. shear, 1–D For . t oee,i motn ont htboth that note to important is however, It, . ar V xy σ ar V ,hwvr vle codn otegvrigFPKE governing the to according evolves however, ), 2 [ G σ ar V y ] [ sgvnb t xeietlymaue initial measured experimentally its by given is ) · ] [ ersnsvrac prto.Teequivalent The operation. variance represents G ] J 2 3 = / 2 s ij s ij eso:2.Ags,22,15:04 2021, August, 26. version: stescn nain of invariant second the is G and , t σ stepseudo the is y ol be would ) σ (105.50) (105.49) (105.52) (105.51) y and σ Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et in shown as distribution 0.067) ( of modulus parameter either shear scale be The to and assumed 3.31 105.18. was of Fig. strength parameter shear shape yield (with the Weibull Also, or normal ( 1996)). Nadim (2002), and Administration Highway Lacasse Federal Nadim Administration and Highway Lacasse (Federal clay for typical values 30%, of iteration. functional and method ADAMS utilizes which (2005), simultane- al. conditions, et boundary Hindmarsh incorporating a after obtaining solved, thereby then with incrementally, was and and ODEs differences, ously Fokker–Planck– of central the series by The of grid ODE. domain uniform of a stress series solved on The was first lines. (105.52), discretized of was Eq. method PDE by using Kolmogorov given steps coefficients time diffusion pseudo and with advection incrementally with (105.46)), (Eq. FPKE The Function Density Probability and Ghanem (1938), 105.3.2.1 Wiener (Wiener expansion chaos (1991)). sought polynomial Spanos if and using behavior, Ghanem material assembled Spanos 3104 local–average of be 437 The then page: behavior. can material material plastic for, local–average elastic–perfectly coeffi- the Mises diffusion von not and scale and advection point–location behavior, with represents FPKE, (105.52), of exam- Eq. solution for by the given words, . tools, cients . other . appropriate CYCLIC In AND by YIELDING (1991)). PROBABILISTIC Spanos points, Spanos and and Gauss 105.3. Ghanem Ghanem Loève at Loève(1948), (1947), example Karhunen for (Karhunen Karhunen–Loève expansion variables, ple random into discretized first Notes ESSI h il ha tegh( strength shear yield The iue151:Eatcpretypatcpoaiitcmdl D fyedstress yield of PDF model: probabilistic plastic Elastic–perfectly 105.18: Figure σ n y suotm tp,uigasadr pnsuc D ovr SUNDIALS solver, ODE open–source standard a using steps, time pseudo ftemtra a sue ohv envleo 0kawt COV a with kPa 60 of value mean a have to assumed was material the of ) G a loasmdt eete omlo ebl itiuin but distribution, Weibull or normal either be to assumed also was ) eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University elastic–perfectly Mises is al. Jeremić von et It for (105.52)). response deterministic (Eq. the coefficients even–though diffusion that, and note advected advection also origin) the to stress–strain by important at very governed from function domain, delta seen the Dirac be into deterministic can diffused (a As and stress branches. initial reloading for and PDF unloading 105.19 , Fig. and unloading, and loading between transition of clarity for unloading 150 of of junction value the a from at view truncated (b) are and plot) stress plot) shear the of the shear densities of of (probability clarity branches densities for (probability reloading 1500 branches and value unloading of a and at PDF loading truncated evolutionary of are junction stress loading: the cyclic from view under (a) model stress probabilistic shear plastic Elastic–perfectly 105.19: Figure 3104 of 438 the page: on focusing shown, are cycle loading–unloading–reloading the of views different Two 105.19. elastic–perfectly Fig. . ( in Mises, . . strength CYCLIC von shear AND yield probabilistic YIELDING stress), both cyclic shear PROBABILISTIC where of The case (PDF) the 105.3. 25%. function for density of probability (evolutionary COV response a strain and stress–shear shear MPa plastic 100 of value mean a with Notes ESSI b) a) σ y n ha ouu ( modulus shear and ) G r omlydsrbtd sshown is distributed, normally are ) eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University with modeled nicely successfully fairly somewhat is be only (1991)), al. Jeremić Dobry can et feature and this Vucetic Dobry models, and deterministic Vucetic using If (cf. cyclic captured. soil with modulus in (secant) observed of the reduction commonly of example, some strain, For captures behaviors. geomaterial model, of elastic–perfectly features simplest important very the even properties, geomaterial in equation. uncertainties implicit rate linearly constitutive implement elastic-plastic to to forward underway corresponding the is FPKE Work to the FPKE. inherent solving the issue, for evolution solving size rule the in step mid-point to used in due been oscillations are has The strain that shear method 105.20(a). with Euler Fig. stress in shear increasing of shown ( deviation with also strength standard cycles is shear of of distribution, yield 105.20(a)), Weibull couple the as first (Fig. both modeled the mean when are for evolutionary response, – mean the stress The – shear moments loops. of strain statistical 105.20(b)) its (Fig. of deviation (refer material terms standard plastic and in elastic–perfectly Mises plotted von is for stress 105.19) shear Fig. of PDF evolutionary the 105.20 , Fig. Loops In Strain Increasing of Case 105.3.2.2 full to transitions and it gradually 105.18), while (Fig. elastic, strength linear, shear more initially of) elasto–plasticity. probability again branch is the loading response branch, (positive, probabilistic of reloading the toward PDF hence subsequent the the response governed in loading the again this, the are transition to to weights gradually Similar similar very and and 105.19(b)). initially direction), increase (Fig. are opposite weights elasto–plasticity 105.18) the probability in (Fig. elastic–plastic (loading strength the unloading shear of branch, of stages (negative) later image During mirror small. of PDF governing the from the gradually increases, response strain stress as 105.19(a)). probabilistic However, (Fig. the elastic–plastic response. and more elastic increases becomes linear, behaving to material (ensemble fully) elastic–plastic response not stress of (but probability probabilistic closer initial is the realizations) hence, of probability all and the small of strains, very small of is at probability branch, elastic–plastic the loading being on the material in based Initially, response elastic–plastic. stress or of elastic realizations being material 3104 the of to ( simulation 439 weights during coefficients probability page: implicitly diffusion assigns consideration been coefficients and into has taken advection possibility and equivalent This strength the yield loading. using the of of possibility beginning PDF (small) very is a the the is from response there . from quantified . strength, probabilistic . elasto–plastic CYCLIC yield AND the becomes in YIELDING uncertainty material PROBABILISTIC yielding, to the in due 105.3. that is, uncertainties That introduced beginning. the to from due non–linear bi–linear, is material plastic Notes ESSI a sta,i n consider one if that, is 105.20(a) Fig. using made be can that observation important very The weights probability elastic–plastic since elastic (mostly) as behaves material the unloading, Upon N (1) eq and N (2) eq σ y ) Those (105.52)). Eq. refer , n h ha ouu ( modulus shear the and ) eso:2.Ags,22,15:04 2021, August, 26. version: G ) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University shear mean the 105.22(a), Fig. from observed be can al. Jeremić As et 105.22). plastic Fig. perfectly (refer Mises shown von probabilistic also this is of behavior material monotonic the comparison, of completeness For Loading Monotonic al. et 105.3.2.4 Sett al. elastic– et Mises (Sett von cycles as of number modeled with of when degradation degradation clay, (2008)) the modulus stiff capture exhibited Japanese to material, example, able plastic For be perfectly however, modulus. could, shear properties (secant) material of mean set different a deviation with standard and model mean ( the modulus both shear of function both is of behavior material elastic–plastic mean material Mises The von stress). the shear to of (mean up response repeatedly probabilistic cycled such repeatedly was shows cycled 105.21 (a) is material Fig. the strain. when (1988)), same Dobry the and at Vucetic Dobry and commonly Vucetic degradation, (cf. modulus clay (secant) in exhibit observed didn’t however, material, elastic–plastic Mises von Loops This Strain Constant of Case 105.3.2.3 modeling. to capabilities new our significant for added modulus have that to shear remark seems of to functions) important density is (probability It ( distributions parameters. statistical more (only) many modeling, require probabilistic which models, complex fairly stress shear of strain deviation increasing standard (b) with and loading mean cyclic (a) under of evolution model probabilistic loops: plastic Elastic–perfectly 105.20: Figure 3104 of 440 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI G n ha tegh( strength shear and ) a (b) (a) G σ n il ha tegh( strength shear yield and ) y ,aenee.Epnino lsi–lsi oeigit rbblt space probability into modeling elastic–plastic of Expansion needed. are ), 0 . 2 tan nyfis he ylsaeson ti motn ont that note to important is It shown. are cycles three first Only strain. % σ y .Tesm o ie lsi–efcl plastic elastic–perfectly Mises von same The ). eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI vrg falsc eair.I hscnet ti motn ont httebhvospeetdin effect scale presented The behaviors effect). the (scale that particles note soil between to correlation important the ensemble is account the it into represents take context, not hand, this do other In behav- paper the such this behaviors. on all such mean, of all The ensemble of the would weights. average represents particle probability then each respective behavior then their response PDF, with the strength respective of iors, of and their case PDF by modulus in The governed the governed, example, differently. if is for behave However, RVE different, a a are strength. in particle example, and particle each for modulus of soil its (RVE), individual by element an material, volume of plastic representative behavior elastic–perfectly a The in particles specimen. soil soil laboratory of number infinite of haviors eeice l nvriyo aiona Davis California, of University al. Jeremić et (a) of evolution mean (c) loading: and monotonic deviation, under standard model (b) probabilistic mean, plastic Elastic–perfectly 105.22: Figure state. plastic perfectly the reaching before strain shear with increases non–linearly stress loops: equal stress shear all of with deviation loading standard (b) cyclic and under mean (a) model of probabilistic evolution plastic Elastic–perfectly 105.21: Figure 3104 of 441 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI hscly n a iulz h rbblsi olcntttv epnea nesml ftebe- the of ensemble an as response constitutive soil probabilistic the visualize may one Physically, a (b) (a) a (b) (a) ± tnaddvaino ha stress shear of deviation standard eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University in effect simulation The for obtained. were that al. responses Jeremić and et different 30% completely was a that 105.21(a) observed Fig. is in It simulation 300%. was for 105.23(a) strength Fig. yield yield of of COV different COV with except strength. parameters, material same with model material plastic loading. elastic–perfectly monotonic to subjected was strength yield of COV uncertainty large the with by material influenced predominantly ( is strength 105.23 (b)) yield shear and (Fig. in of 300%), response (assumed COV strength deviation the yield standard of material, the COV this the hence, for to compared because, non–significant, is is 30%) This (assumed previous 105.22 (b))). modulus in and observed was 105.21(b) what 105.20 (b), from different (Figs. completely case is which increasing ( always is strength here shown yield response of COV that except model, ( modulus shear both in present response ( uncertainties deviation amount standard the The on dependent material. much plastic very elastic–perfectly is Mises von all to generic not is response ( ( modulus is shear modulus This in shear uncertainties in decreases. the then both and elastic, ( increases mostly strength yield is first material deviation, the where standard when 105.22 (b), the initially, Fig. 105.21 (b)), in because, shown, and as is well 105.20(b) response as (Figs. ...), monotonic figures re–unloading, the above all re–loading, the and unloading, from loops (loading, branch observed increasing any with be inside responses can cyclic As for respectively. 105.19)), (Fig. loops, stress equal confidence shear of important moment PDF second the of evolutionary of (square-root analysis the one stress of sensitivity shear show of 105.21(b) like deviation and standard application evolutionary 105.20 (b) the derivative Figs. parameters, some measuring derived. probability or as obtained the well easily example, as be for solution, can PDF, information, probable the engineering From most useful information. exceedance, other of at of moments, amount fail enormous often statistical they contains the as PDF than geomaterials, full other of A simulation PDF). failure of in (tails important probabilities very low is obtain elasto– accurately to Ability probabilistic stress of 105.19). based (Fig. PDF FPKE stress the shear of geomaterials. PDF evolutionary of accurate second-order behavior for solves simulated plasticity 3104 the of in 442 plastifies. confidence material page: the our as quantifies approach, FPKE updating the by using problems points elastic–plastic integration to Gauss (1991)) finite at Spanos properties stochastic and material to . Ghanem approach the . . Spanos spectral CYCLIC AND and the YIELDING Sett Ghanem extending PROBABILISTIC Sett by (cf. method technique. 105.3. element element element finite finite such elastic–plastic one stochastic proposed using (2007) others, among for, accounted be can Notes ESSI σ y b,sospoaiitcrsos fcci eairo h aematerial same the of behavior cyclic of response probabilistic shows 105.23(b), Fig. example, For ). a.Bt r enrsosso o Mises von of responses mean are Both 105.23(a). and 105.21(a) Figs. compare to interesting also is It also approach probabilistic modeling, geomaterial to approach alternate an of promise the to Further σ y r oenn.A aeilbcmsmsl lsi–lsi,teiflec funcertainty of influence the elastic–plastic, mostly becomes material As governing. are ) σ G y b,weethe where (b), 105.24 Fig. in observed be can response deviation standard Similar ). erae.Hwvr ti motn ont htti yeo tnaddeviation standard of type this that note to important is it However, decreases. ) σ y ,i o sue ob 0% h tnaddeviation standard The 300%. be to assumed now is ), eso:2.Ags,22,15:04 2021, August, 26. version: G n il strength yield and ) G and ) Jeremićet al., Real-ESSI ag il netit) vlto f()ma n b tnaddvaino ha stress shear of deviation standard very (b) with and but loops mean 105.21, equal (a) Fig. in of all simulation evolution with for uncertainty): used loading yield as large cyclic same the under exactly are model parameters probabilistic model (probabilistic plastic Elastic–perfectly 105.23: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et section previous kinematic. the or in isotropic discussed – and rule as hardening isotropic example, appropriate of with same responses but the used cyclic end, is on 105.3.2) this evaluated (Section To is yielding materials. probabilistic hardening of kinematic influence the section, this In Material Hardening 105.22(a) Figs. in 105.3.2.5 compared be similarly, can, behavior mean monotonic 105.24(a). on and strength yield of COV of mean (c) evolution and deviation, uncertainty): yield standard large (b) very with mode, but mean, 105.22, (a) Fig. in of parameters simulation for (model used loading as same monotonic the under exactly are model probabilistic plastic Elastic–perfectly 105.24: Figure 3104 of 443 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI freatcpretyplastic elastic–perfectly for 105.3.2 Section in shown simulations the between difference main The a (b) (a) a (b) (a) ± tnaddvaino ha stress shear of deviation standard eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI n iuincecet o h nenlvariable, advection plastic internal equivalent elastic–perfectly the the solved, for for be coefficients response will diffusion FPKE stress and scale probabilistic point–location of since 105.3.2), case Section the (refer for material explained As variable. internal where, eeice l nvriyo aiona Davis California, of University random strength yield the with al. incrementally, Jeremić et solved be to by for also given FPKE need coefficients the diffusion ( turn, with variable and in simultaneously advection Those, done with be (105.46), (105.55 )). to (Eq. Eq. needs stress shear solution of ( 105.54)) This Eq. evolution by probabilistic incrementally. given solved coefficients diffusion be and to advection with needs (105.53), (Eq. variable internal of lution elastic–perfectly the unlike However, ( case. those plastic case, elastic–perfectly plastic the like coefficients just diffusion components, and plastic advection and variable equivalent internal The ( of only. rule stress strain evolution shear of the for component probability because plastic is of the This by contributions absent. governed the are is elastic (105.54)), being (Eq. material the coefficients that diffusion weights and advection equivalent above where, 3104 of 444 page: of function ( a variable be internal as: of form, to evolution general assumed probabilistic the here the most . is govern in . . that variables CYCLIC AND FPKE internal YIELDING The PROBABILISTIC of strain. (change) 105.3. plastic evolution ( variables Such internal the plastifies. material material hardening a for that is material Notes ESSI ooti h rbblsi epneo o ie adnn aeil h PEfrpoaiitcevo- probabilistic for FPKE the material, hardening Mises von of response probabilistic the obtain To ∂P N N r N (2) (2) eq eq Σ (1) stert feouino nenlvral ( variable internal of evolution of rate the is ( eq q y IV ( ( IV N en pae fe ahiceetlstep. incremental each after updated being ( 105.55) and (105.54) Eqs. in ) ∂t σ x N ( = ) t (1) eq q t , and (1) eq = ) IV ) N t , ( t ( (1) σ eq ) q N N P = ) = ) d = (2) (1) [Σ eq and eq dt xy IV ∂q y ∂ d P and dt ≤ N [Σ xy r h qiaetavcinaddffso offiins epciey o the for respectively, coefficients, diffusion and advection equivalent the are n 2 (2) σ eq N     y N     ( (1 q (1) o adnn–yemtras ilhv otiuin rmbt elastic both from contributions have will materials, hardening–type for ) ≤ eq (1 (2) eq )] IV − σ t ilcnantehreigterms: hardening the contain will ) − ( P P q P d )] ( [Σ dt [Σ q d xy ( dt y x y xy ≤ t ≤ t , 2 * σ ) ar V σ t , ]) G ]) ar V ) o h + G Gr     N + √ i G [ 1 (1) G eq + 3 ∂q ∂ + q + ] r IV Gr P 2 ihpatcsri.Oemynt hti the in that note may One strain. plastic with ) 2 + √ [Σ n 1 and P 3 N y [Σ r (2) ≤ eq     y N q IV σ ≤ (2) ∗ eq ] )wl vlea the as evolve will (105.45)) Eq. refer , P * σ IV ( ] G q ar V a ewitnas: written be can , ( − x t t , G     ) G eso:2.Ags,22,15:04 2021, August, 26. version: t , + G − ) √ o 2 1 G 3 r + + G √     2 q 1 a ewritten, be can ) 3 r         (105.53) (105.54) (105.55) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University increases. strength yield when in words, uncertainty al. other Jeremić the et In cycles, 105.18 ) loading–unloading kurtosis. (Fig. through low strength cycled yield having is of distributions material PDFs non-Gaussian probability the initial dispersed in distributed change much normally the into The note evolved to interesting strength. the is in yield model), it material hardening of However, isotropic plastic distributions the isotropically. branch elastic–perfectly by (grew) loading prescribed for evolved (and for assumed strength expected (positive yield as As strength same 105.18). yield Fig. the (refer of are 105.3.2 PDFs branch) Section initial unloading The for negative . 105.26 and Fig. in shown are unloading) stress shear evo- of loops: deviation increasing standard (b) with and loading mean cyclic (a) under of model lution probabilistic hardening Isotropic 105.25: Figure are section parameters previous material the in other material All plastic elastic–perfectly 10. of rate simulation of 105.3.2). for case) (Section non–dimensional used as this a same in the with strength, be material to (yield assumed hardening variable internal isotropic of Mises evolution von of with for accordance cycles in loading–unloading evolves of hand, (105.55). Eq. other 3104 by the of diffusion given 445 on coefficients and page: diffusion stress, advection and shear with advection The with (105.53), (105.46), (105.54). Eq. Eq. Eq. following by strain, given plastic coefficients with . . probabilistically . CYCLIC ( AND evolve strength YIELDING yield PROBABILISTIC will the material, 105.3. hardening isotropic Mises von For Hardening Isotropic 105.3.2.6 Notes ESSI h vle Dso il teghatrec rnh(odn,ulaig elaig n re– and re–loading, unloading, (loading, branch each after strength yield of PDFs evolved The couple first during stress shear of deviation standard and mean evolutionary the shows 105.25 Fig. a (b) (a) σ y steitra aibe il strength Yield variable. internal the is ) eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et loading. cyclic of case the predictions in stress–strain obvious in becomes difference difference real models noted the probabilistic a but both (with model), with hardening similar a look for do hardening 105.28) pronounced more (Fig. of model probabilistic monotonically hardening loaded isotropic linear when and and 105.27) (Fig. level same isotropic shown. of to also behavior are cycled the 105.28) was comparison, (Fig. of it stress–strain completeness when for of) However, (PDF material, way. captures hardening realistic 105.20) more much (Fig. a model in probabilistic loops poorly. plastic expected, elastic–perfectly as the performed, 105.25) is, (Fig. That model hardening isotropic geomaterials, of 105.53)). ( behaviors (Eq. cyclic variable internal for of equation evolution formulation the in in variable nonlinearity and state the (105.54)), the Eq. to (refer is, due that is , strength yielding shear probabilistic of uncertainty in increase Mathematically, re– (d) and branch, re–loading (c) branch, unloading (b) branch, loading branch (a) unloading evolved after loops: stress increasing yield with loading of cyclic PDF under model probabilistic hardening Isotropic 105.26: Figure 3104 of 446 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI ) 105.22 (Fig. model probabilistic plastic perfectly both for curves loading monotonic that noted is It simulating in that, see clearly can one , 105.20 and 105.25 Figs. between made is comparison When q per nbt deto n iuinequations diffusion and advection both in appears eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et α where, write: kinematic ( can linear stress one simple back criteria, a of on evolution focus on now based we rule hardening models, probabilistic hardening elastic–plastic on Expanding Hardening Kinematic 105.3.2.7 mean, (a) mean of (c) evolution and loading: deviation, monotonic standard under (b) model probabilistic hardening Isotropic 105.28: Figure stress evolution shear of loops: deviation equal standard with (b) loading and cyclic mean under (a) model of probabilistic hardening Isotropic 105.27: Figure 3104 of 447 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI mdfidscn nain fdvaoi testno ( tensor stress deviatoric of invariant second –modified | p σ k J − α saanmtra aaee yedsrnt ie and like) strength (yield parameter material again is α − − | k 0 = σ y 0 = a (b) (a) or σ = α ± a (b) (a) ± tnaddvaino ha stress shear of deviation standard σ y α .B nrdcn aksrs ( stress back introducing By ). s ij becomes: (105.56) Eq. shear, 1–D For ). J α 3 = / 2( eso:2.Ags,22,15:04 2021, August, 26. version: s ij − α α ij ovnMssyield Mises von to ) )( s ij − α ij (105.57) (105.56) ) sthe is Jeremićet al., Real-ESSI .Oemynt htteyedsrnt random strength with yield (105.46), the that Eq. note to may One according ( evolves (105.55). variable Eq. stress by Shear advection given with coefficients (105.54). (105.53), diffusion hardening and Eq. Eq. advection to isotropic by according given for occur coefficients strength will diffusion stress yield back and the the of of evolution evolution Probabilistic probabilistic material. to similar probabilistically evolve ( stress back the evolving. starts incremental once each material, FPKEs, after governing stress the yield equivalent of the step estimating while (105.57), Eq. in keeping addition/subtraction of advantage The stress. yield equivalent uncertain an and kPa, 60 eeice l nvriyo aiona Davis California, of University al. Jeremić et hardening linear isotropic (ii) 105.21), (Fig. plastic, elastic–perfectly (i) for level, strain significantly. same the not to although cycles different, and are mean 105.29) the Fig. material, (refer hardening stress kinematic shear evolutionary for the material, responses, of plastic hardening mode elastic–perfectly response kinematic the the material Like and plastic plastic hardening similar. elastic–perfectly stress elastic–perfectly are isotropic from the shear than much those, cyclic Qualitatively, realistic differ the 105.20). didn’t more that (Fig. it was noted however, 105.29), is 105.25), model (Fig. It (Fig. hardening material material strength. kinematic hardening yield kinematic model, (equivalent) hardening of the of as isotropic response PDFs increased kurtosis the strength high unlike yield in but (equivalent) resulted 105.31. through, in Fig. uncertainty in cycled shown the was are case branch material hardening loading evolved isotropic each The after the (105.57)) to 105.30. Eq. elastic–perfectly Fig. Similar (refer the in stress shown for yield are equivalent as re–unloading) of same and ( PDFs stress re–loading, the back unloading, the be (loading, of to PDFs branch evolved each assumed The are 105.3.2. parameters Section in material material and plastic other mode, All mean, of material loops. terms hardening in strain kinematic stress shear a of when evolution deviation, probabilistic standard the shows 105.29 Fig. (105.57). Eq. will stress from yield uncertainty equivalent initial of the 3104 distribution transfers of 30%, 448 one same of if page: the isotropic COV obtained, However, and a 105.3.2 be with Section 105.3.2.6. kPa in Section material 60 in plastic of elastic–perfectly material stress the hardening yield for stress equivalent yield in assumed resulting the kPa, as . same . 20 . CYCLIC of AND deviation YIELDING PROBABILISTIC standard Initially, a 105.3. with stress. of yield stress equivalent a at the occurs as yielding the material, hardening kinematic for Hence, Notes ESSI 8 ihnndmninlrt feouino aksrs ihpatcsri f10. of strain plastic with stress back of evolution of rate non–dimensional with nti td,tebc tes( stress back the study, this In iial,we n oprsrsos ma n tnaddvaino ha tes o loading for stress) shear of deviation standard and (mean response compares one when Similarly, Σ y ,i h qiaetyedsrnt n sgvnby given is and strength yield equivalent the is (105.55), and (105.54) Eqs. in appearing ), α fzr enadasadr eito f2 P ilrsl ntesame the in result will kPa 20 of deviation standard a and mean zero of α sasmdt vlewt lsi tanadhne twould it hence, and strain plastic with evolve to assumed is ) α szr,and zero, is σ y eemnsi sta twl ipiyteprobabilistic the simplify will it that is deterministic α σ ,teitra aibefrknmtchardening kinematic for variable internal the ), y σ 8 y a yldcul ftmswt increasing with times of couple cycled was , sasmdt aeama au f6 kPa 60 of value mean a have to assumed is to α nohrwrs deterministic a words, other In . α α ttebgnigadedof end and beginning the at ) ± eso:2.Ags,22,15:04 2021, August, 26. version: σ y emdi h following the in termed , σ y of Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et (c) Branch, unloading (b) branch, loading loops: (a) of increasing branch end re–unloading with and (d) loading beginning and the branch, cyclic at re–loading under stress back model of probabilistic PDF hardening evolved Kinematic 105.30: Figure loops: stress increasing shear of with deviation loading standard (b) cyclic and under mode mean, model (a) probabilistic of hardening evolution Kinematic 105.29: Figure 3104 of 449 page: . . . CYCLIC AND YIELDING PROBABILISTIC 105.3. Notes ESSI a (b) (a) eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Method Element Finite Elastic-Plastic Stochastic hardening kinematic 105.4 linear hardening. (iii) of rate and in differences 105.28), expected (Fig. with hardening similar, qualitatively linear are isotropic 105.33)), (Fig. (ii) 105.22), (Fig. tic, hardening kinematic and (i) plastic can elastic–perfectly one between (iii). models, similarity responses material qualitative probabilistic 105.32), the (Fig. observe hardening easily kinematic linear (iii) and 105.27), (Fig. re–loading (c) branch, unloading (b) loops: branch, increasing loading branch with (a) re–unloading loading after (d) stress and cyclic branch, yield under equivalent model of probabilistic PDF hardening evolved Kinematic 105.31: Figure 3104 of 450 page: . . . FINITE ELASTIC-PLASTIC STOCHASTIC 105.4. Notes ESSI oooi odn ae,hwvr o l rbblsi aeilmdl ()eatcpretyplas- elastic–perfectly ((i) models material probabilistic all for however, cases, loading Monotonic eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI f()ma n b tnaddvaino ha stress shear of evolution deviation loops: standard equal (b) with and loading mean cyclic (a) under of model probabilistic hardening Kinematic 105.32: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et mean mean, (a) (c) of and evolution deviation, loading: standard monotonic (b) under model probabilistic hardening Kinematic 105.33: Figure 3104 of 451 page: . . . FINITE ELASTIC-PLASTIC STOCHASTIC 105.4. Notes ESSI a (b) (a) ± a (b) (a) tnaddvaino ha stress shear of deviation standard eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI I olbrto ihD.Za Cheng) Zhao Dr. with collaboration (In (1996-2004-) Elasto-Plasticity Deformation Large 106 Chapter 452 Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et position initial The position initial equation: its deformation from the particle by a defined of displacement the shows 106.1 Figure representations of sets two using be will we so, doing In must one deformations. finite, large namely: are of rotations theory and strains the however to If, apply resort can problems. we these If enough, solving small in deformation. are theory rotations, elastic deformation those and small the translations to both than primarily including greater applicable deformation, considerably total is resulting deformations theory the inelastic plasticity experience solids, can 3104 of that of behavior 453 bodies nonlinear page: material the modeling In Deformation Kinematics 106.2.1 Preliminaries: Mechanics HIGHLIGHTS AND Continuum SUMMARY CHAPTER 106.1. 106.2 Highlights and Summary Chapter 106.1 Notes ESSI 1 • • (1969). Malvern See iue161 ipaeet tec n oaino aeilvector material of rotation and stretch Displacement, 106.1: Figure X x coordinates Spatial coordinates Material i I = = x X i ( I X ( x 1 X , 1 x , X 2 2 I X , x , ftepril o cuyn h position the occupying now particle the of 3 3 ntedfre ofiuain localled also configuration, deformed the in t , t , nteudfre ofiuain localled also configuration, undeformed the in ) ) X I d X I u i d x i x i x i ueincoordinates Eulerian sgvnb the by given is arnincoordinates Lagrangian X dX eso:2.Ags,22,15:04 2021, August, 26. version: I I otecretposition current the to onwposition new to ueinequation Eulerian . , dx (106.1) (106.2) i . x 1 i : , , Jeremićet al., Real-ESSI at gradient deformation The eeice l nvriyo aiona Davis California, of University al. Jeremić et The vector material infinitesimal gradient deformation the to Similarly The thus derivatives: coordinates, partial the Cartesian are rectangular letters the only. case to indices lower work covariant and our the to indices limit notation coordinate We spatial, tensor material the the indices. simplifying the and for coordinate setting letters spatial Lagrangian capital the material, for use the will between we difference setting, the Eulerian emphasize To (106.2). and 3104 of The 454 page: Gradient Deformation 106.2.2 . . . PRELIMINA MECHANICS CONTINUUM 106.2. Notes ESSI h Jacobian The X h w oiin r once ytedisplacement the by connected are positions two The h eomto rdeti enda the as defined is gradient deformation The eomto gradients deformation pta eomto gradient deformation spatial gradients deformation spatial the I dx x dX ( F F J oascaei ihavector a with it associate to F i kK iJ = Kk k = K ( = det F = ) X ( = − Jk F I 1 ∂X a erpeetdas: represented be can (106.4) mapping the of ( ∂x kK ) + F F = − Kk kK 1 K k u dX ∂X i = ∂x ) = ) = − K δ K k 1 ik x = dx k,K 6 1 = ; e ∂X k ∂x F ijk and X r h rdet ftefntoso h ih–adsd feutos(106.1) equations of side right–hand the on functions the of gradients the are = kK dx K,k K k e QR P ∂X X i ∂x dX rnfrs(ovcs na rirr nntsmlmtra vector material infinitesimal arbitrary an on (convects) transforms at I ( ( K k F = K F F dx x Ij dx Kk iP F i x = ) i kK i oascaei ihavector a with it associate to F r esr eerdt h eomd ueinconfiguration: Eulerian deformed, the to referred tensors are k − ) x − at − jQ 1 = k,K , 1 F u pta eomto gradient deformation spatial x F X jK at i i kR dX : K,k w–on tensor two–point x = i K dx steivret h w–on tensor two–point the to inverse the is δ IK k u I : hs etnua atsa components Cartesian rectangular whose dX I ( F at Kk X eso:2.Ags,22,15:04 2021, August, 26. version: ) − I : 1 prtso narbitrary an on operates F kK at X (106.9) (106.4) (106.6) (106.3) (106.7) (106.5) (106.8) I : dX I Jeremićet al., Real-ESSI ralternatively: or at element volume infinitesimal epciey and respectively, where eeice l nvriyo aiona Davis California, of University here. presented al. example Jeremić et decomposition polar the for only applied be will system coordinate yields: differentiation of rule chain the system, coordinate Cartesian rectangular the position reference fixed the If 3104 of 455 page: as: defined is and . . . PRELIMINA MECHANICS CONTINUUM 106.2. gradient deformation relative The Notes ESSI 2 • • • • • • referring dmntaeta h oinaddfraino an of deformation and motion the that demonstrate 106.2.2 Figure as well as (106.13), Equation h oa decomposition polar The ii oyrtto by rotation body rigid a to translation body rigid a to translation body rigid a by rotation body rigid a by stretch a tec by stretch a R F f ξ ∂X U ∂ξ km ij ik = kj k R I = , χ = v jk x = t R ik i ( ξ x = ik k,m and ∂x r oiiedfiiesmerctnos called tensors, symmetric definite positive are ∂ξ i U τ , δ m R k kj ij ≡ X ) U v kj ∂x ∂X ik i = kj ∂x ∂ξ otesm eeec xsaduiglwrcs nie o oh hsrfrnet h same the to reference This both. for indices case lower using and axes reference same the to , san is m , v I m k ik n also and R or rhgnltensor orthogonal kj hoe emt h nqerepresentation unique the permits theorem X R R F X I kj ik h urn position current the , kI x x i , f , i i oss fcneuieapiain of: applications consecutive of consist km = f stegain o h eaiemto function: motion relative the for gradient the is R km ki F R mI kj uhthat: such = δ ij x i n h aibeposition variable the and ih tec tensors stretch right 2 : eso:2.Ags,22,15:04 2021, August, 26. version: and ξ etsrthtensors stretch left i r l eerdto referred all are (106.14) (106.13) (106.12) (106.11) (106.10) , Jeremićet al., Real-ESSI n o h ueinformulation: Eulerian the for and ofiuain.TeGendfraintensor deformation Green The configurations. vector material tensors Stretch strain and The Tensors Deformation Tensors, Strain 106.2.3 eeice l nvriyo aiona Davis California, of University al. Jeremić et ( tensor deformation Cauchy The length squared new 3104 of 456 page: . . . PRELIMINA MECHANICS CONTINUUM 106.2. Notes ESSI dS 4 3 nte aefor name Another called Also h eomto tensors deformation The ) 2 ( ( ( ( fa element an of ds dS ds ds ) ) ) ) 2 2 2 2 − − = = ih acyGentensor Cauchy–Green right ( ( dX dx dS dS dX i I ) ) c iue162 lutaino h equation the of Illustration 106.2: Figure C 2 2 I b ij E o h arninfruainw write: we formulation Lagrangian the For . ij ( IJ 2 = 2 = dx ds dx IJ is dX ) j i igrdfraintensor deformation Finger dx dX 2 and dnie ntedfre configuration: deformed the in identified J fteeeetit hc h ie element given the which into element the of i I e E C ij e ij IJ dx IJ X c dX r ends htte ietecag ntesur egho the of length square the in change the give they that so defined are j I ij and oeie lodntdas denoted also sometimes , d J . X c ij I r oncigtesurdlntsi arninadEulerian and Lagrangian in lengths squared the connecting are 3 or C u etCuh–re tensor Cauchy–Green left IJ i eerdt h neomdcngrto,gvsthe gives configuration, undeformed the to referred , F ij 4 d = x ( b i x ij R i ) ik − . U 1 dX ie h nta qae length squared initial the gives , kj I = eso:2.Ags,22,15:04 2021, August, 26. version: sdeformed: is v ik R kj . (106.16) (106.15) (106.18) (106.17) Jeremićet al., Real-ESSI x 106.20): ( and (106.19 ) equations tensor deformation for the expression the and (106.7) equation using by Similarly, tensor deformation the between F connection the obtained have we so eeice l nvriyo aiona Davis California, of University al. Jeremić et tensor strain Lagrangian the for expression general the express 3104 can of we 457 (106.5) page: equation using By obtain: we (106.16) into (106.18) equation . . . substituting PRELIMINA similarly, MECHANICS and CONTINUUM 106.2. yield: (106.15) into (106.17 ) equation Substituting Notes ESSI i kI 5 arninfra ilb eoe by denoted be will format Lagrangian ,rfrne otesm xsfrboth for axes same the to referenced (106.3), equation displacement the from starts one If description Eulerian and Lagrangian in tensors strain the for expressions The nteform: the in C x L: ( ( 2 2 2 1 X x e E I IJ ( k,I ij K,i IJ δ = E KI = ( = dX X IJ dx = δ δ I KJ F I ij i C = )( )( + kI IJ − X x 2 1 + u F k,J c − I K,j (( kJ ij δ 1 2 ( KI F dS dX δ = ) ( dx 1 2 IJ kI ( ; δ c ds u IJ ) (( J ij J F 2 K,J x = ) δ ) = ) X kJ = + 2 n h eomto gradient deformation the and KI k,I I ) = + u dS dX = dX − dx + I,J dx E u K x δ u i K,I I IJ I ( IJ I dX + k x K,I ( F dx F − k,J 1 2 ) δ = Ki u L: kI K KJ ( ) J,I ( k u dX F u hl ueinfra by format Eulerian while 2 1 F I ( = ( = δ I,J Kj kJ + + KJ (( J F ( ) ) F F u u ; + ds dx dX KI + K,I K,I kI Ki c u ) ij k dX u F K dx 2 J,I u u K,J = KJ ( = as: K,J K,J = i I E: + )( dx )( ) ) dX F F u F − − − − i Ki Kj K,I e c kJ ij ij I δ δ δ δ E ) C IJ IJ IJ IJ dx dX dx − u IJ F = ( IJ 1 K,J dS = ) = ) = ) = ) Ki j k F J = ) E: 2 1 dX ntrso ipaeet is: displacements of terms in ⇒ = ) Kj ) − ) . as: 2 1 δ K ecnetbihtecneto between connection the establish can we ij − ( C F IJ Ki ) n h eomto gradient deformation the and − eso:2.Ags,22,15:04 2021, August, 26. version: 1 ( F Kj ) − 5 1 sotie from obtained is (106.23) (106.21) (106.22) (106.24) (106.25) (106.20) (106.19) X I and Jeremićet al., Real-ESSI sosdffrn arninsri esrsotie o atclrcoc fparameter of choice particular a for obtained measures strain Lagrangian different shows 106.1 Table eeice l nvriyo aiona Davis California, of University al. Jeremić et is: tensor strain corresponding the integer, an be to stretch of function monotonic smooth, any is function Scale descriptions: Eulerian and Lagrangian the in stretch of form Cartesian the obtain we direction denoting By as configuration. undeformed or and deformed configuration undeformed either in element, gradients. an displacement of of components the tensor. in strain terms finite complete quadratic the and represent linear (106.26 ) only and involve (106.25) They equations that noting worthwhile is It 3104 of 458 page: . . . PRELIMINA MECHANICS CONTINUUM 106.2. tensor strain Eulerian the for expression general the Similarly, Notes ESSI cl ucini fe ae nteform the in taken often is function Scale Λ nteElra etn,gnrlzdsri esri endas defined is tensor strain generalized setting, Eulerian the In a considering by defined be can tensors strain General h stretch The ( N e E f L: N 2 1 ) ij ( IJ I ( λ ftoeeeet htiiildirection initial whit elements those of δ n = = ) Λ ij = i by (106.16) and (106.15) equations dividing By . ( 2 ; − N dX dS δ ) λ U δ ij = I ki 2 1 ∈ IJ 2 2 samaueo xeso fa nntsmleeetadi ucino direction of function a is and element infinitesimal an of extension of measure a is − m δ ( m dX 2 [0 2 1 kj e δ dS v m − ij ij , ( ij 2 + δ ∞ m I − δ = ij C and IJ δ ) δ − IJ ki 2 1 ij ujc to subject u ( dX ; + k,j δ dS δ ki ij where u J 2 1 n + F − i,j − i ( IJ n u u ntvco ntedfre ofiuain ednt aeilstretch material denote we configuration, deformed the in vector unit a u ( + i k,i i,j F k,i = = ki f u δ ( ) + 1 0 = (1) j,i and kj R F ) dx ds − δ IJ u IK − − 1 kj i j,i ( ( λ = U u u F − − 2 KJ k,i k,i kj m R f , u u ) u u IK k,j − N − k,i E: = 0 k,j k,j 1 1 = (1) 1 I 1) )= )) U u v = ) = ) n pta stretch spatial and k,j KJ / IK λ = 2 ( 2 ) n m R ) = where , KJ = v IK dx ds scale f R ( i ds e ( c KJ ij λ ij m ) ) 2 dx ntrso ipaeet is: displacements of terms in ds o h stretch. the for ) 1978 (Hill, function uhthat: such a aeayvle fw choose we If value. any have may and j λ ( ( n dS ) ftoeeeet ihinitial with elements those of ) eso:2.Ags,22,15:04 2021, August, 26. version: 2 epcieyadb using: by and respectively N I ntvco nthe in vector unit a (106.31) (106.30) (106.28) (106.27) (106.26) (106.29) m m . Jeremićet al., Real-ESSI atceat particle v eeice l nvriyo aiona Davis California, of University al. Jeremić et dx The Tensor Deformation of Rate 106.2.4 3104 of 459 page: . . . PRELIMINA MECHANICS CONTINUUM 106.2. Notes ESSI i i at 6 /dt localled Also tedse ie ersnstetaetre fparticles of trajectories the represents lines dashed the 106.3 Figure In aeo eomto tensor deformation of rate iue163 eaievelocity Relative 106.3: Figure p dv v h pta oriae are: coordinates spatial The . i and k = = q v i v eaiet h atceat particle the to relative ∂x ( ∂v tec tensor stretch i x + 1 m k x , dv dx 2 i x , m Hencky Biot Almansi Green–Lagrange name measure Strain at 3 = t , q v ) or al 0.:Dffrn arninsri measures. strain Lagrangian Different 106.1: Table r agn otetotaetre.Terltv eoiycomponents velocity relative The trajectories. two the to tangent are k,m P d eoiystrain velocity X dx i 6 m Q ecie the describes = L dv p km . u i r ie by: given are i dx fparticle of parameter m agn motion tangent 1/2 -1 0 1 d Q p x q i m tpoint at E E E E for expression IJ IJ IJ IJ H B A GL v i i + q n( ln = ( = = ntrso eoiycomponents velocity of terms in d v = eaiet particle to relative v i U δ U IJ IJ U IJ 2 − IJ − d − P v E ) U i δ eso:2.Ags,22,15:04 2021, August, 26. version: δ IJ and IJ IJ m − IJ 2 ) / Q / 2 2 h eoiyvectors velocity The . P tpoint at (106.33) (106.32) p . dv v i i of = Jeremićet al., Real-ESSI natraewyo eiigtert fdfraintno osa olw.Tert fcag of change of rate The follows. as goes tensor deformation of rate the length squared deriving of way alternate An eeice l nvriyo aiona Davis California, of University al. Jeremić et length squared the of position change relative of infinitesimal rate any the occupying that follows it thus and since becomes: (106.37) equation the then and (106.33) equation From get: we differentiation of order the interchanging since with and since 3104 of 460 page: where: . . . PRELIMINA MECHANICS CONTINUUM 106.2. tensor deformation The Notes ESSI pta rdeto h velocity the of gradient spatial dx ( d L D d d ∂X d d d d ds ∂v ( km ( ( ( ( ( ( km dX k dt dt dt dt dt dt ds ds dx ds dx ds ) dx dx k m 2 = ) ) ) ) = k k m 2 = = dX m 2 2 2 2 k ) ) ) 1 2 ( = 2 = 2 = 2 = 2 = dx 2 1 = = ≡ /dt m dx ( ( ( ds L k dv dx L d = ∂x dx dx k d d d km ≡ km ) D k D m 2 ( ( ( L dt k k k ∂X dt dt dx ds dx km ∂x 0 = /∂X dx sgvnas: given is km km D + + eas h nta eaiepsto vector position relative initial the because , tfollows: it ) m k dt k k km dx d k L ds dx n kwsymmetric skew a and , L ) ) dv dX mk dx dx and m mk dx m m k ∂X dt ) ∂x dx m k k + ) dX = ) = m M 2 = k W k L m = 2 + 2 1 km D km dX dx ⇒ ( tfollows: it mk d L dx dx sse ymti uhthat such symmetric skew is k m km v L m ∂X k k,m dt ∂x = km W − tflosthat: follows it (106.39) equation and m k d dx km ∂X L ∂v and a edcmoe stesmo h symmetric, the of sum the as decomposed be can ( dx mk dt dx m dX dx k m dx k = ) k m dX m ) pntensor spin tpoint at k dx + = 2 = m D k W dv d km 2 = km ( dx k dX dt + = k dx p = where L m W L sdtrie ytetensor the by determined is km k W ) 2 1 km D km ∂X ( ∂x km dx km L dx km W m k m dx sfollows: as ( dX m dx ds km = − m = v ) k m k dx 2 L d = v = = mk osntcag ihtm.By time. with change not does ftemtra instantaneously material the of k,m k − ∂X dt ∂x ∂x = ) W eso:2.Ags,22,15:04 2021, August, 26. version: dt dx m k k mk m − dX ial eobtain: we Finally . W mk m D km tpoint at (106.35) (106.36) (106.40) (106.39) (106.41) (106.42) (106.38) (106.37) (106.34) aeof rate p . Jeremićet al., Real-ESSI ieetaei ihrsett time: to respect with it differentiate time: to respect with eeice l nvriyo aiona Davis California, of University al. Jeremić et inversely: or inversely: or that: follows it (106.44) and (106.43) equations from and that: 3104 of 461 page: since . . . PRELIMINA MECHANICS CONTINUUM 106.2. Notes ESSI and (106.4) equations from start we gradient deformation the of change of rate the obtain To (106.15) equation differentiate we deformation, of rate the to rate strain the compare to order In ( v D dF d dE d dS k,m dt ( km dt dt ds ( kK IJ ) ds 2 = ) ( = 2 ) and = = 2 dx dt 2 = − F F = d dt Ik Ik k,K dX ( dx dS d ) D ∂X dt − ∂x X I k km 1 ( ) K k D dt ds 2 tfollows it (106.43) and (106.42) equations the From time. through constant are K,m dE F km = dt ) mJ 2 = 2 IJ = dx F = ˙ kK 2 = ( ∂ m d F dF ∂X ( d mJ dt ( dX dX dx kK F ( 2 = dt K Km ) k I − I ( E dt 1 d F ) IJ = dX ( − Km E dt 1 dX ∂X IJ I ∂v = ) F − ) J Ik k K L 1 dX ) km ) = = = D J km ∂x ∂v ( m k F ∂X mJ ∂x dX m K = J 2 = ) v k,m = dX x L m,K km I ( F F = Ik mK eso:2.Ags,22,15:04 2021, August, 26. version: dx D dt km = k,K F F ˙ mJ kK = ) dX J (106.48) (106.46) (106.45) (106.44) (106.43) (106.47) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et as: defined is tensor stress Kirchhoff spatial, the and is: that the using by theory, mechanical pure the to function, component and restriction potential with strain elastic the 106.49, a of equation to form in general described respect most is with The derivatives component. whose stress corresponding tensors, the deformation determines 3104 of of strain 462 of page: function scalar the called called is material . A . . HYPEREL RELATIONS: CONSTITUTIVE Introduction 106.3. 106.3.1 Hyperelasticity Relations: Constitutive 106.3 Notes ESSI tensors: yuigthe using By 8 7 • • • p 194. pp. (1983) Hughes 190. and Marsden pp. See (1983) Hughes and Marsden See yasmn yeeatcrsos,tefloigaetecntttv qain o h aeilstress material the for equations constitutive the are following the response, hyperelastic assuming By .PoaKrho testensor stress Piola–Kirchhoff 1. tensor: stress Mandel W W .PoaKrho testensor: stress Piola–Kirchhoff 2. xo fetoyproduction entropy of axiom = = taneeg ucinprui oueo h neomdconfiguration undeformed the of volume unit per function energy strain W W P T S IJ IJ iJ xo fmtra rm indifference frame material of axiom ( ( X X = = 2 = K K S C C , F , IJ ∂C IK hyperelastic ∂W kK IJ ( S F IJ ) KJ iI ) ) t 2 = or: 2 = C ∂C or ∂W W IK IJ re elastic Green 7 ∂C = : ∂W ( F W KJ iI ( ) X t K c , fteeeit an exists there if , ij 8 ecnld that conclude we , ) lsi oeta function potential elastic W eed nyon only depends eso:2.Ags,22,15:04 2021, August, 26. version: hc ersnsa represents which , xo flocality of axiom X K and (106.51) (106.52) (106.53) (106.49) (106.50) W also , C IJ , Jeremićet al., Real-ESSI nain clrfntos tstse h relation: the satisfies It functions. function energy scalar strain invariant the isotropy, material of case the In Hyperelasticity Isotropic 106.3.2 rhgnlrtto rnfrain endb h oa eopsto hoe neuto (106.13), equation in theorem decomposition polar the then: by defined transformation, rotation orthogonal gradient: deformation the eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 463 page: from: defined is relation stiffness tangent Material . . . HYPEREL RELATIONS: CONSTITUTIVE 106.3. Notes ESSI where where • h pta agn tffestensor stiffness tangent spatial The W W E L tensor stress Kirchhoff dS ijkl IJKL Q IJ ( ( KI X X = 2 = K K τ ij 4 = stepoe rhgnltasomto.I echoose we If transformation. orthogonal proper the is F C , C , iI 2 = ∂C KL KL F ∂C jJ IJ = ) = ) ∂ ∂b ∂W ( 2 IJ F ∂ ∂C W ij kK 2 W W ∂C W 2 = KL ) ( t KL ( X X F F dC K K lL iA U , Q , ) KL t ( L F KL KI IJKL jB = = ) C ) t 2 1 IJ ∂C E ∂W L W ijkl ( Q IJKL AB ( JL X sotie ytefollowing the by obtained is = ) K t dC v , F iA kl KL ) ( F jB W ) t S ( AB X K C , IJ ) Q eog otecaso isotropic, of class the to belongs KI push–forward eso:2.Ags,22,15:04 2021, August, 26. version: = R KI where , prto with operation R KI (106.57) (106.58) (106.56) (106.55) (106.54) (106.59) sthe is Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et where and spectral The as: respectively (106.23), and (106.22) equations by defined were tensors Cauchy–Green right and Left 3104 of 464 page: . . . HYPEREL RELATIONS: tensor: CONSTITUTIVE deformation of 106.3. invariants function energy Notes ESSI tsol entdta osmaini mle vridcsi parenthesis in indices over implied is summation no that noted be should It fidca oaini hswr,w hl aea ffr orpeetaltetnoileutosi niilform. indicial in equations tensorial the all represent to effort an make shall we work, this in notation indicial of equation where: 11 10 13 12 9 acyGentensor Cauchy–Green (1991). Taylor and Simo See stretches. Principal o xml,i h rsn case present the in example, For that So ih n etsrthtensors, stretch left and Right n sgvnby given is and (106.5) equation from deduced be can eigenvectors the of mapping The N R F P C I W λ C I I I k 3 2 1 ( iJ IJ IJ iJ n ( C A r h eigenvectors the are λ = i ( IJ ) A = A 2 def def def = ( = = k e ( det = = = n ) N ) W = ≡ k i ( 10 λ A X λ A I def = λ A F k =1 3 A 2 ) ( C A 2 eopsto hoe o ymti oiiedfiietensors definite positive symmetric for theorem decomposition 2 1 kI X 1 = = W − 1 II n n ) K h pcrldcmoiinof decomposition spectral The . N N I λ i ( F t i ( . A 1 F 2 λ , A I C A 6 I a erpeetdi em fpicplsrths rsmlryi em fprincipal of terms in similarly or stretches, principal of terms in represented be can ( ( iJ A ) A kJ ) − IJ N ) N + 1 ) C N N λ , N J C IJ = ) ( J ( J I ( A J A ( A J IJ ( 1 2 A ) A ) ) o example. for λ , λ ) ; ) C 6 1 A 12 A A 6 3 JI e A , a lob rte as written be also can − IJK of = ) N ( I c I ( C − 2 A U e ) W IJ 1 where λ QR P KL ) sthe is A 4 ij Values . ( + X , = C v K I A IP kl b 3 I , ij hegnetrwt members with eigenvector th 0 = C aetesm rnia values principal same the have 1 A = JQ I , λ = F A 2 2 C I , C iK r h ot ftecaatrsi polynomial characteristic the of roots the are 1 KR F IJ 3 , ( iJ ) 3 F = , = jK P R J ) iJ A A t 2 =1 =3 and λ ( 2 A ) b N ij I ( A ste ie by given then is N ) N 1 ( A J ( A ) , ) nodrt olwteconsistency the follow to order In . N 9 13 eso:2.Ags,22,15:04 2021, August, 26. version: 2 ( A λ 11 . ) i and ttsthat: states ; i N = 3 ( A ) 1 ota h actual the that so , , 3 otestrain the so (106.66) (106.61) (106.63) (106.62) (106.67) (106.65) (106.64) (106.60) Jeremićet al., Real-ESSI a hw that shown has tensors (1986) strain generalized for representation useful a devise eeice l nvriyo aiona Davis California, of University al. Jeremić et eigendiad Lagrangian The eigendiad Eulerian the that follows it as (106.68 ) written and (106.71) equations comparing By where is the of them function between a difference be the will if perturbation equal Our these are performed. of are number calculation ( two precision in view, machine involved the of than arithmetics point smaller precision numerical from equivalence finite the about example the From certain for of all never numbers. results because are or we and numbers, two However, of numerical possible two case tests. theoretically compression of The is isotropic distinct. equal or 3104 them tests being of make triaxial 465 stretches standard to page: principal order in of stretches values principal three for values numerical the to . . . HYPEREL RELATIONS: i.e. CONSTITUTIVE 106.3. Notes ESSI oget: to (106.72 ) equation into (106.65), eigenvectors, the of 15 14 (1994). 106.31 (). Parisch and and (106.30) Schellekens equations and by Schellekens Defined and ( 1986) Morman also See aeue ernsrpeetto hoe nodrto order in theorem representation Serrin’s used have ( 1986) Morman and (1985) Ting Recently, h hrceitcpolynomial characteristic The stretches, principal non–equal of case the for valid is (106.63) equation from decomposition Spectral λ 1 b b n arccos = Θ N λ ij m ij 6= A i ( I A ( A = = ) = λ ) n 2 N λ j ( λ √ A 6= J A 2 ( 1 A 2 ) A m 3 λ ) = s   n 3 = ftoo l he rnia tece r qa,w hl nrdc ml perturbation small a introduce shall we equal, are stretches principal three all or two If . i ( 2 I ( A ( b 1 I λ b 2 ) 2 1 3 2 2 + ( 2 n ) q A ) − ij j ( ij ) A b 2 C − q ij m ) 9 I − 2 λ I IJ 1 2 N λ ( 4 A 1 I a esae as stated be can A − ( 4 I 1 I 2 I ( A − ) ( 2 A 1 I − ) 3 − ) 27 + 1 − − I N 3 2 2 − I I λ J I 1 λ ( I 1 P A 2 3 λ ( 2 1 λ ( 4 − I A A ) λ cos macheps ( ( 2 ( 2 3 ,cnb eie,i n usiue mapping substitutes one if derived, be can (106.63), equation from , A ) A λ ) ( 2 λ A ) ) A 2 − ( 2 ) b + A ) ij + I b ) 2 + Θ a esolved be can (106.64) equation from 1 I ij + λ I 3 δ λ 3 + ( 2 IJ I A λ pcfi otecmue ltomo hc computations which on platform computer the to specific ) ( − 3 3 A ) ( − I 2 λ A πA ) + 3 + 2 ( − ) λ A I 2 ( − I ) A 3 3 δ 2 λ ) λ ij δ ( − ( − ij A A 2 2 ) )   ( C A 15 macheps − 1 E ) IJ IJ and . e ij through eso:2.Ags,22,15:04 2021, August, 26. version: 14 C for IJ m λ and A n : i ( A ) b n ij m j ( Morman . A (106.70) (106.69) (106.73) (106.71) (106.72) (106.68) ) a be can Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et as defined are stretches principal isochoric the where tensor deformation Finger the of part isochoric the while as eigenvectors. and where 3104 of 466 page: . . . HYPEREL RELATIONS: CONSTITUTIVE 106.3. that: used was it where Notes ESSI ecnobtain: can we (106.68) and (106.63) equations to Similarly that (106.77) equation in as: written indices be where can ( 106.73) and (106.72) equations in denominator the that noted be should It tpoe sflt eaaedfraini ouercadiohrcprsb utpiaieslto a of split multiplicative a by parts isochoric and volumetric as in gradient deformation deformation separate to useful proves DeformationIt of Decomposition Volumetric–Isochoric 106.3.3 h reenergy free The h scoi ato h re eomto tensor deformation Green the of part isochoric The x β ˜ W λ b C C δ ( 2 ( ( F ˜ ˜ x C C b ij IJ λ A iI → IJ IJ − β ( 4 − − = A 1 = = ( = ersnsa nemdaecngrto uhta deformation that such configuration intermediate an represents X 1 1 ) = ( = ) x ) ) ij J − K J i IJ IJ F ˜ J − F that (106.80) equation from follows also It isochoric. purely is = − λ , iβ F ,B C B, A, I − iI 3 2 = ( = 1 3 1 iI λ b ( 3 2 vol λ λ ) A ij ) A − − C λ A ( 2 − ) F F A 2 1 A − IJ = 1 ) ( = iI βI W 2 b = ( + ij ) n λ ˜ = b − r ylcpruain of permutations cyclic are i ( 2 N A 2 λ iso ste eopsdadtvl as: additively decomposed then is I ( A 1 ) λ ˜ 3 F 1 I ij ) ( W δ λ λ A 2 n A where λ jJ n ij ( − 2 ( ) j ( 1 A i ( A λ F N ) 2 A ( ) N − 6= ) 3 jJ X F ) J ( ) = t n I A ( jJ − A K ) λ A j ( ) − A 3 1 2 ) ) , t N λ ) − λ λ ˜ A 6= A F t ( 2 ( J ˜ ( A A A A iβ λ ) ) ) 3 − = A + ⇒ λ F ( 2 vol B iI D ) W J ( A − ) ( 3 1 X λ 1 0 6= , ( 2 K A 2 ; , ) J , 3 − C ob valid. be to (106.73) and (106.72) equations for tflosdrcl rmtedfiiinof definition the from directly follows It . ) vol IJ λ b ij ( 2 F a edefined be can (106.63) equation in defined , C βI ) a edfie iial as similarly defined be can = def = J D 3 1 δ ( A βI ) X eso:2.Ags,22,15:04 2021, August, 26. version: I F ˜ → βI x and β sprl volumetric purely is F iI aetesame the have (106.84) (106.83) (106.82) (106.80) (106.81) (106.74) (106.77) (106.75) (106.76) (106.78) (106.79) D ( A ) Jeremićet al., Real-ESSI that: (106.63) equation eeice l nvriyo aiona Davis California, of University al. Jeremić et that: follows it (106.78) equation from also and introduce we quantities these E function energy strain isotropic the stretches: of principal form of general of most terms the in presented have we (106.3.2 ) Section In Formulation Simo–Serrin’s 106.3.4 3104 of 467 page: . . . HYPEREL RELATIONS: CONSTITUTIVE 106.3. Notes ESSI hw httemtra agn tffestensor stiffness tangent gradient material the the calculate that tensor to shown stress necessary Piola–Kirchhoff is it 2. that obtain (106.3.1) to Section in shown also was It where ijkl 16 (1996). Runesson See eursscn re eiaie fsri nryfunction energy strain of derivatives order second requires ) C ( M W C D IJ IJ ( − = A ( A 1 = ) ) ) W IJ λ .With (106.77). equation by defined was def ( = = = A 4 ( = X M K M iue164 ouerciohrcdcmoiino deformation. of decomposition isochoric Volumetric 106.4: Figure λ D F λ , ( − IJ IJ (1) ( A 1 ( iI A 2 X A 1 ) ) ) I ) λ , + N − A 1 I 2 M ( A C λ , ) IJ IJ (2) n 3 16 N i ( , A − ) J + ( eododrtensor order second a ) A n ) M j ( I A 1 IJ (3) ) vol − F β λ I ( F ( 2 A jJ S x ) β IJ ) − F δ t i IJ I n codnl te tesmaue.Lkws,i was it Likewise, measures. stress other accordingly and L + IJKL I 3 M λ M ( − A iso IJ IJ ( ( a ela h pta agn tffestensor stiffness tangent spatial the as well (as 2 A A ) F ( ) i ) β C ,w e from get we (106.86), equation by defined − 1 x ) ∂ i IJ 2 / W ( rm(106.73) from ∂C IJ eso:2.Ags,22,15:04 2021, August, 26. version: ∂C KL ) nodrt obtain to order In . W/∂C ∂W IJ (106.87) (106.86) (106.88) (106.85) norder in W Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et energy measures: free stress defined the previously of the derivatives all of decompose terms appropriately in measures stress various defined function have we (106.3.1) Section In Measures Stress 106.3.5 of derivation Complete the define to position a in now are We 3104 of 468 page: eigenvectors of properties orthogonal the from . since, . . HYPEREL RELATIONS: CONSTITUTIVE 106.3. that: concluded be also can It Notes ESSI • • adlsrs tensor: stress Mandel tensor: stress Piola–Kirchhoff 2. M δ δ IJ IJ IJKL ( W A = = ) ecan we (106.84) equation in defined as decomposition, function energy free the With . T S λ A X IJ IJ =1 (1) 2 3 def M + − = N = = = 2 = I IJ ( (1) A ) I λ D ∂M ∂C iso C iso + 3 N ( − 1 ( λ ∂C A IK M T A S ∂W 2 λ J ( ( − KL ) IJ ( IJ A ) A IJ (2) 2 A 2 I S IJKL IJ ) ) 3 ) KJ + M = + I = 2 = ( IJKL vol IJ vol C (2) ( 2 = C N sgvni pedx(704.2). Appendix in given is T − S ∂ − + ∂C IJ 1 I IJ ( iso A 1 C ) − IJ ) λ ) IK IJ IJ W (3) 2 δ ( A KL C Simo–Serrin ∂C M M 2 + ∂W − δ KL IJ ( 1 (3) N KJ A IJ ) ∂ KL ) J ∂C ( A vol = + + 2 = ) IJ W + λ M λ A A 2 ( 2 A C IJ 2 1 ( A ) orhodrtensor order fourth IK M ) ( ( δ C ∂ ∂C IJ C ( IJ A − iso − ) 1 KJ M W 1 ) ) IK A KL KL ( A 2 + ( ) C + − C − 1 M IK ) D JL IJ ( ( 0 A ∂ ∂C A ) M vol ( + ) δ KJ M W KL IJKL C eso:2.Ags,22,15:04 2021, August, 26. version: IJ ( A − ) 1 + ) M as: IL KL ( ( A C ) − 1 ) JK (106.93) (106.92) (106.91) (106.90) (106.89) − Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University choose. we model al. Jeremić material et hyperelastic which of independent is rest the while then: is energy free the of derivative The is tensor stress Piola–Kirchhoff 2. decomposed The function energy free the of part isochoric the of derivative the while yields used, was (704.9) equation where 3104 of 469 page: . . . HYPEREL RELATIONS: CONSTITUTIVE 106.3. Notes ESSI a sdand used was (704.7) equation where • • h eiaieo h ouercpr ftefe nryfnto is function energy free the of part volumetric the of derivative The ti biu htteol aeildpnetprsaedrvtvsi h form the in derivatives are parts dependent material only the that obvious is It w tensor stress Kirchhoff tensor stress Piola–Kirchhoff 1. S ∂ ∂W ∂ IJ vol iso A ∂C ∂C ∂C W W = ( λ IJ IJ = = τ P ( ( ( ∂ IJ ab λ A J ˜ iJ iso ( ) ) A ) vol W ∂ ∂ = ) = = 2 = vol = = ) λ S ˜ ( = = B = ∂J ∂ IJ λ W vol ( iso F iso S A ∂ + ( ∂e ∂W aI ∂J W IJ ) 2 ∂ 1 τ J iso P ) ab vol iso ( ) ij iJ ( ∂λ ∂ F ( W ∂λ F J J + ∂C W S ∂ vol bJ = + iI ) λ IJ ˜ ( ( ∂J ( ( A vol W ) λ ( ˜ ) B ∂C A IJ C vol F tiso λ t ) ( ∂J ) τ A ( aI − ( 2 = P ab A J IJ ) λ ˜ S 1 ) ( ) iJ ) ( ) IJ ) F A IJ ∂λ ∂C = ∂C J bJ ) + ∂W + = + ) ( ( 2 1 IJ w ∂ IJ t C A F S w − iso ) A ∂ − aI IJ ( A F = vol 3 1 ∂C W 1 as: 704.5 Appendix in derived is ( ) iI ( F 2 = ∂J IJ W ∂ M ( 1 2 IJ ) bJ λ iso t ∂ ( + IJ ( ( ) 2 = F J A iso W A tvol ∂ ) aI ) ) 2 1 ∂λ λ ˜ ) W ) ( A J B ( S λ ˜ ∂ w ∂C F ( ( IJ ( iso A ( A λ A bJ C ) ( ) IJ W ( A ) ) − M t ) 1 λ ˜ ∂ ) ( ∂C ) IJ F ( B λ iso IJ A iI ( ) A IJ W + ) ) ) A t ( 2 + M ∂ 2 + iso IJ ( A ∂ W ∂ F ∂C ) λ ˜ vol aI ) ( ( A A λ ˜ IJ W ( ) F ( = A bJ ( eso:2.Ags,22,15:04 2021, August, 26. version: ) F 2 1 ) ) iI w t λ ˜ ∂ ) ∂C A ( t vol A ( ) M IJ W ∂ IJ ( A vol ) /∂J W ) A (106.100) (106.95) (106.94) (106.97) (106.96) (106.98) (106.99) and w A , Jeremićet al., Real-ESSI h ouercpart volumetric The eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 470 page: decomposition made operator definitions stiffness appropriate tangent the the . with toward . . together (106.3.1) HYPEREL used RELATIONS: section is CONSTITUTIVE in (106.84) decomposition 106.3. function energy free The Operator Stiffness Tangent 106.3.6 Notes ESSI r eddi diint h needn esr,frtedtriaino aiu tesadtangent and stress various of determination the for tensors, independent tensors. the ( stiffness stretches to principal addition isochoric to in to possible needed respect is are with it function that energy in strain consequence of practical derivatives a has observation This a create model. material the of choice the are n h opeedrvto sgvni h pedx(704.4). appendix the in given is derivation complete the and 704.3 . appendix in given again is derivation complete the and Y ial,oecnwietevlmti n scoi at ftetnetsins esr as: tensors stiffness tangent the of parts isochoric and volumetric the write can one Finally, part isochoric The nasmlrmne otesrs entosi scerta h nymtra oe eedn parts dependent model material only the that clear is it definitions stress the to manner similar a In AB J L L 1 ∂ 4 ∂C 2 IJKL iso IJKL 2

iso and ∂ epaederivations template J 2 IJ W ∂J∂J vol 2 ∂C ∂ w ( W = = 2 λ ∂J∂J A KL ( Y vol ( h eann eodadfut re tensors order fourth and second remaining The . A vol J AB ) ) L ) W ( IJKL = C ∂ ( M 2 − ∂ 1 4 + 1 2 KL ( vol ) B Y J + KL iso ) W AB ) ∂ iso B W ( C o aiu yeeatciorpcmtra oes nyfis n second and first Only models. material isotropic hyperelastic various for ( / ∂J vol ( L M ( M − / IJKL ∂C W 1 ( KL ( IJ ) ( ∂C B A IJ IJ ) ) ! ) ) IJ B + A 4 = ∂C ( C ( 2 + J ∂C M KL − ∂ ∂C 1 IJ vol KL ( ∂ ) A w ) KL ∂J 2 W ) IJ A a ewitnas: written be can ) ) A vol ( ( ( a ewitni h olwn form: following the in written be can ∂C C J M + W ) − ( KL IJKL ( 2 1 1 A C ) ) w IJ − 4 + A 1 + ) ) KL A ( M 2 1 ∂C ( J ∂ IJKL ( C A ∂C 2 ∂ IJ − ) M ∂ 1 iso ∂J IJ vol ∂C ) 2 ) IJ IJ W vol ( W A A ∂C KL W ) 2 + and KL I IJKL ( J C eso:2.Ags,22,15:04 2021, August, 26. version: − ∂ M = 1 vol ) IJKL ( ∂J W vol A ) λ ˜ ( L A J IJKL n aoin( Jacobian and ) ) r needn of independent are I IJKL ( C − = 1 ) (106.101) (106.102) (106.103) (106.104) (106.105) J ) Jeremićet al., Real-ESSI h euaiycniinthat condition regularity The of powers in series models infinite hyperelastic an compressible as of 1984) family exploited widely A definition. function nwa olw,w ilpeetanme fwdl sdsri nryfntosfriorpcelastic isotropic for functions energy strain used widely of number a present solids. will we follows, what In where: setting iff provided free met is stress configuration is reference configuration the in vanishes energy that requirement The nainso stretches of invariants function energy strain The eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 that: of requires 471 condition page: configuration that is restriction only The as: represented is stretches principal of . . terms . HYPEREL in RELATIONS: solid CONSTITUTIVE isotropic for 106.3. function energy strain The Models Hyperelastic Isotropic 106.3.7 Notes ESSI 18 17 sdmil o ubrlk materials. rubber–like for mainly Used (106.61). equation also See lgtymr eea omlto sotie yuigpicplsrthsi h tanenergy strain the in stretches principal using by obtained is formulation general more slightly A W I W I I W W 3 2 1 c pqr (1 ======vol iso , p,q,r iso 1 W ( 0 = N W W , X →∞ W )=0 = 1) λ λ λ ( λ =0 1 2 2 2 1 2 = = 1 λ λ + + λ , 2 2 3 2 N r N p,q c λ X X r vol λ + →∞ →∞ pqr 1 = 2 =0 3 2 =0 2 2 λ , W 17 λ + 3 2 3 , : ( λ c ) c I 2 λ and pq 00 1 1 2 , 3 2 W 3 − + 0 r ... , W ( ( W 3) λ a ihrb eadda ucino rnia tece rteprincipal the or stretches principal of function a as regarded be either can I I c 1 2 ) 1 3 100 p sasmercfnto of function symmetric a is λ and scniuul ieetal nifiieynme ftmsi satisfied. is times of number infinitely an differentiable continuously is − − ( 2 2 ∂W I 2 + 2 3) 1) − c p (1 ∂λ r pqr c ( ( 3) , I 010 I i 1 2 1 q ( 0 = , − − ( 1) + I 3 3) 3) 0 = c − q 001 , ,q p, ( 1) I 2 r 0 = − 1 = 3) scoi eitrcdculn spsil by possible is decoupling deviatoric Isochoric . , 2 and , 3 λ .. , 1 ( λ , I ) 3 2 oobtain: to − λ , 1) 3 lhuha prpit natural appropriate an although , as: eso:2.Ags,22,15:04 2021, August, 26. version: 18 c r end(Ogden, defined are 000 0 = Reference . (106.108) (106.109) (106.111) (106.110) (106.107) (106.106) Jeremićet al., Real-ESSI hl h ouercpr a edfie ychoosing by defined be can part volumetric the while hr h olwn a used was following the where as: written be can function energy strain isochoric The eeice l nvriyo aiona Davis California, of University al. Jeremić et where can model elastic isotropic Neo–Hookean of part selecting isochoric by the The obtained contains be stretches cases. principal special of invariants as of model terms in Neo–Hookean defined model hyperelastic isotropic general The Model Neo–Hookean 106.3.7.2 3104 of 472 page: expressed is energy strain The ( 1984). as: Ogden stretches by principal . defined of . . was function HYPEREL RELATIONS: models a CONSTITUTIVE as hyperelastic of 106.3. set general very A Model Ogden 106.3.7.1 Notes ESSI eiaie eddfrbidn tensors building for needed Derivatives eiaie eddfrbidn tensors building for needed Derivatives vol iso iso W ∂ ∂ ∂ ∂ ∂ ∂ G ∂ ∂ ∂ ∂ 2 iso 2 2 iso 2 W W W λ λ ˜ ˜ λ λ ˜ ˜ ∂ ∂ = and A A W W A A iso iso iso iso λ λ ˜ ˜ ∂ ∂ = = = A 2 A 2 N W W W W λ λ ˜ ˜ X r = = →∞ =1 K B B N G K 2 X r 2 →∞ b N G =1 b X r = 0 = = 0 = r h ha n ukmdl respectively. moduli bulk and shear the are →∞ =1 λ ˜ λ ˜ µ c λ N G 1 2 r A r 1 2 X r µ c + →∞ λ =1 r r c ( 2 2 λ r λ ˜ λ 1 µ 2 2 N 3 2 r λ ˜ c + − λ ˜ r 1 µ + 1 = A r λ ˜ ( 1 λ + µ λ ˜ 3 2 2 µ µ i r 2 , − r r λ ˜ = − q = − 2 µ + 3 1 r 0 = J 1) K λ + 2 − 3 µ b 1 3 , r λ ˜ λ λ c ˜ 3 µ − J p i r A . 00 2 3) − w w − µ = A A 1 r 1 − and and G/ 2 2 2 oget: to , Y Y AB AB N 2 = r ie ytefloigformulae: following the by given are r ie ytefloigformulae: following the by given are , c 001 0 = , c 002 eso:2.Ags,22,15:04 2021, August, 26. version: = K b / 2 as: , (106.115) (106.117) (106.113) (106.112) (106.121) (106.120) (106.119) (106.116) (106.114) (106.118) Jeremićet al., Real-ESSI .Bt onyadRvi taneeg ucin eoesmlr foecosst set to chooses one if similar, equation become from functions representation energy isotropic and strain general Rivlin the and of Mooney part Both isochoric the (106.111). to similar quite actually is which c b, a, with eeice l nvriyo aiona Davis California, of University al. Jeremić et form: the in invariants stretch principal isochoric of set alternative an choosing By Model Logarithmic 106.3.7.4 3104 of 473 page: form: the of . . behavior . HYPEREL isochoric RELATIONS: for CONSTITUTIVE function energy 106.3. strain a proposed Mooney Model Mooney–Rivlin 106.3.7.3 Notes ESSI eiaie eddfrbidn tensors building for needed Derivatives c a 10 I iso I iso iso ∂ ∂ ∂ r ylcpruain of permutations cyclic are ˜ ˜ n ∂ 2 1 ∂ ln ln iso 2 2 W W W = λ ˜ λ ˜ and ∂ A W A iso iso λ ˜ C ∂ = 4 = = 2 = = A 2 W W 1 λ ˜ = = = = 2 = b B N and p,q n X →∞ =0 en h aeilprmtr n ouepeevn constrain preserving volume a and parameters material the being λ λ ˜ ˜ 0 = 2 = + N N C n n X X 2 ln 1 ln ln ln C C →∞ →∞ c =0 =0 1 a 01 1 1 c λ λ n ˜ ˜ λ ˜ 2 2 C pq 2 1 = A + I λ 1 ˜ ˜ 1 λ ˜ 1 2 ( − 2 2 6 + C a a I 3 ln λ − ˜ + n n 2 + 1 2 λ ˜ 2 2 ln − 3 λ 2 ln ˜ λ oobtain: to ˜ 2 C C λ λ ˜ ˜ 3 2 2 λ ˜ + 3) 2 1 2 1 2 2 2 + 3 + n n ln − λ λ ˜ ˜ p + (1 + + 2 C A − A − λ 2 ˜ ( λ λ ˜ ˜ n I 3 , 3 2 4 2 4 + 3 ln 2 λ λ ˜ ˜ 2 + 2 λ ˜ − 2 2 2 2 , − 2 3 ln n n I ˜ 3) 2 3 2 2 + + + λ 3) ˜ oegnrlfr a rpsdb Rivlin: by proposed was form general more A . ln − 2 3 λ ˜ q λ λ + ˜ ˜ λ ˜ 1 ln w λ 3 ˜ 3 2 3 2 1 n n 3 C A ln 2 − − − 2 2 and λ ˜ + 2 n 3 3 3 λ ˜ ln + 1 − 2 λ ˜ Y + 2 λ ˜ 1 ln AB + 1 λ a ˜ n 1 2 λ ˜ 2 λ ˜ r ie ytefloigformulae: following the by given are 2 − 4 + 2 λ λ ˜ ˜ 2 1 − 2 ln + − 2 2 n n λ ˜ 2 ln + 3 − − λ ˜ 2 λ ˜ 1 3 − 2 − 2 2 3 n + ln λ ˜ λ ˜ eso:2.Ags,22,15:04 2021, August, 26. version: 3 − 2 2 n 2 − 3 λ a 1 = / ( λ (106.128) (106.123) (106.122) (106.124) (106.127) (106.126) (106.125) b λ N c and , 1 = Jeremićet al., Real-ESSI hl h ouercpr ssgetdi h form: the in suggested is part volumetric the while be can function energy strain isochoric simpler somewhat form: A the (1992). in presented Miehe and Simo by proposed was eeice l nvriyo aiona Davis California, of University al. Jeremić et in ( 1984) Pister and Simo by proposed was function energy strain form: of the part volumetric a or form Another Model Simo–Pister 106.3.7.5 3104 of 474 page: . . . HYPEREL RELATIONS: CONSTITUTIVE 106.3. stretch logarithmic isochoric the where Notes ESSI h rtadscn eiaie ihrsetto respect with derivatives second and first The tensors building for needed Derivatives of terms in function energy strain the of part isochoric the of representation general The vol iso λ W ˜ d ∂ ∂ ∂ d d ∂ ∂ i ln d vol 2 2 2 iso W W λ vol ˜ λ ˜ 2 ∂ vol dJ dJ dJ A W = vol W A vol iso iso λ ˜ dJ ( ∂ W = = J A 2 W 2 W √ ( W W λ ˜ J 2 = = ) 2 B G K ln 2 ( ) 2 J = b = = = 0 = ) G 4 1 (ln λ K ˜ = K ln − i K b − = J 2 J b λ b ˜ J λ ˜ 2 ) + 2 J 1 G A − 2 √ 2 + − 1 1 J 2 2 2 4 ln − 2 J + 4 − λ ˜ ln 2 1 J − 2 J A K λ ˜ 1 ln K K i 2 − b − b b 2 λ ˜ J ln 2 2 − 2 2 ln J + λ ˜ J i ln ln w a used: was A λ ˜ 3 and 2 Y J AB r hngvnas: given then are r ie ytefloigformulae: following the by given are eso:2.Ags,22,15:04 2021, August, 26. version: I ˜ 1 ln (106.130) (106.133) (106.132) (106.139) (106.138) (106.137) (106.134) (106.131) (106.129) (106.136) (106.135) and I ˜ 2 ln Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et body a of in borhood locked are particles when large relatively be could and specimens. grains, density soil high of deformations elasticity reversible however, the plastic, from predominantly develop are could soils in soils deformations that in argued deformation be plastic certainly bending view, plastic of and yielding point crushing, micromechanical slipping, the from arises From material. multiplicative the in of interest nature early particulate an stirred (1969) Lee and ( 1968) Fox dislocations ( 1967), crystal computations, Liu elastoplastic decomposition. of deformation and micromechanics large Lee of on by context Kröner work (1960) the to the and In back (1957), modeling. traced continuum al. be to et application can and Bilby decomposition of multiplicative works the decomposition for early multiplicative motivation the the The is decomposition gradient. strain displacement additive of the of generalization appropriate An Kinematics 106.4.2 algorithm. presented algorithmic the the and with algorithm consistent numerical tensor the present stiffness to tangent proceed then We relations. constitutive pertinent algorithm proposed the limit, counterpart. the strain in small that but scheme the shown developments, solution also to is numerical reduces earlier It for of equations. method extension constitutive Newton’s developed pertinent an utilizes algorithm integrating not consistently for strain which is infinitesimal development algorithm the novel The a from rather idea (1997). material the anisotropic Sture 3104 as Jeremić of on and well 475 as by based isotropic page: earlier with is used derivation integration be to Consistent new designed is a plastic it models. and present that elastic in we into novel is Here, gradient algorithm deformation The the area. of parts. research for decomposition a multiplicative algorithms the still on elastic–plastic is based deformation algorithm, models is large material elasto–plasticity of anisotropic deformation development and small However, isotropic classical of established. analysis well numerical generally and structure mathematical The . . . DEFORMATION FINITE Introduction 106.4. 106.4.1 Hyperelasto–Plasticity Deformation Finite 106.4 Notes ESSI 20 19 1993). ( Sture and 1979) ( Whitman and Lambe also See particles. clay like plate For h esnn eidmlilctv eopsto sarte ipeoe fa nntsmlneigh- infinitesimal an If one. simple rather a is decomposition multiplicative behind reasoning The the from justified be may soils for technique decomposition multiplicative of appropriateness The and gradient deformation the of decomposition multiplicative the introduce briefly we follows, what In x i x , i + dx i na nlsial eomdbd sctotadulae oa unstressed an to unloaded and cut–out is body deformed inelastically an in 19 fgaue opiigteassembly the comprising granules of eso:2.Ags,22,15:04 2021, August, 26. version: 20 tcan It . Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et relieved position piece The every between then discontinuous. relationship and and linear pieces a incompatible assume small thus may in and, is we direction, cut–out configuration and arbitrary, stress be unstressed other must The an body in stresses. the loading of stresses, to lead residual will are unloading there effect, if Baushinger’s strong a with rials displacement into body mapped be would it configuration, 3104 of 476 page: . . . DEFORMATION FINITE 106.4. Notes ESSI 22 21 eerdt aeCreincodnt system. coordinate Cartesian same to referred rotation. and Translation d x ˆ k ( = iue165 utpiaiedcmoiino eomto rdet schematics. gradient: deformation of decomposition Multiplicative 106.5: Figure F ik e ) − 21 1 dx Ω n ueyeatculaig h lsi nodn safittosoe ic mate- since one, fictitious a is unloading elastic The unloading. elastic purely and i 0 σ X X d X Configuration Reference x ˆ i , x ˆ i F F + p d x ˆ Configuration Current i h rnfrainwudb opie farigid a of comprised be would transformation The . Ω Configuration Intermediate Ω x dx F e i and x d F x e d -1 x ˆ d x i u x nteform the in , eso:2.Ags,22,15:04 2021, August, 26. version: 22 : (106.140) x ˆ i is Jeremićet al., Real-ESSI that When density energy free the propose We Relations Constitutive small 106.4.3 of body. continuum unloading or elastic assembly body, fictitious particulate the a deformed of a rather of state but outs free deformation cut stress continuous a compatible, toward a unloading necessarily elastic a not linear of a deformation macroscopically plastic assembly, irreversible particulate the part, macroscopically elastic and The dislocation body. crushing slipping, of process define: can one that so eeice l nvriyo aiona Davis California, of University al. Jeremić et where becomes inequality dissipation pertinent C where 3104 of 477 page: body a of is configuration configuration reference . current . . the DEFORMATION the FINITE considering to By connection 106.4. the body. a of deformation discontinuous where Notes ESSI ¯ ij e 24 23 whereas , 106.3.1. section See 106.2.2. section See h lsi ato h eomto gradient, deformation the of part plastic The enwdfieeatcdomain elastic define now We T ¯ ρ F dx B D ij ( Φ W T ¯ 0 F kj p ij W = k = ssmercladw a replace may we and symmetrical is ik siorpcin isotropic is e e def ( = steMne stress Mandel the is ) ( = { ( C T − ¯ ¯ C T ¯ ¯ ij ij F e 1 W ij ij e ) ki sntt eudrto sadfraingain,snei a ersn h incompatible, the represent may it since gradient, deformation a as understood be to not is F L , κ , ¯ p K ersnsasial yeeatcmdli em fteeatcrgtdfraintensor deformation right elastic the of terms in model hyperelastic suitable a represents ik dX ij p e ¯ ( α κ ) α + = ) − α i | 1 ) ⇒ X Φ( F ,ta the that 1996), (Runesson, elsewhere shown been has It hardening. the represents α ρ T ij ¯ F d T 0 ij ¯ K x ki W ˆ ij ¯ e ⇒ k wihi h aehr)i ojnto iheatciorp,w a conclude can we isotropy, elastic with conjunction in here) case the is (which α , ersnsmcomcaial ueeatcrvra fdfrainfrthe for deformation of reversal elastic pure a micro–mechanically represents e K ( = F ¯ ( κ ˙ C α ij α ¯ 24 ij ) e F ≥ def = + ) ≤ and ik e B 0 ) 0 F W − } as ρ ki 1 L e ¯ 0 hc sdfie in defined is which , F F ij p W ij kj p stepatcvlct rdetdfie on defined gradient velocity plastic the is p dX ( κ T ¯ α j 23 ij ) : by τ F ij kj p in ersnsmcomcaial,teirreversible the micro–mechanically, represents Φ . Ω ¯ sfollows as , eso:2.Ags,22,15:04 2021, August, 26. version: Ω ¯ . dX (106.142) (106.145) (106.141) (106.143) (106.144) i then , Jeremićet al., Real-ESSI eobtain we (106.152) equation and hr eue that used we where decomposition multiplicative the using By hl edaigwt ml lsi eomtos ee h e–oka lsi a saotd The adopted. is law as elastic tensor stretch Neo–Hookean right the Here, is deformations. situation elastic generic small with dealing be shall eeice l nvriyo aiona Davis California, of University al. Jeremić et (106.147) equations evolution of system (106.149): the and by governed are flow plastic and deformation incremental The Algorithm Integration Implicit 106.4.4 where 3104 of 478 page: . . . DEFORMATION FINITE 106.4. Notes ESSI a eitgae ogive to integrated be can (106.150) equation from rule flow The as written be now can relations constitutive The we since convenience of matter a largely is this that emphasized is it law, elastic of choice the to As n n F κ L n K T κ F ¯ ¯ ¯ ˙ ˙ ¯ F ¯ ˙ +1 +1 +1 β β ij ij ik ij p p α ik p F F F ˙ = ˙ = = ¯ ¯ = := = ( = ij ik ij e e,tr p F µ µ T F K ¯ ¯ exp = F ¯ jk ij p ik ˙ F ∂K ∂K e ∂ ∂ = = α ¯ ik = p ( li Φ Φ ¯ ( e U F ¯ ˜ − κ ) n ∗ ∗ β β kl kj − e p β 1 +1 n n F 1 ) , T +1 +1 ˙ = ¯ F jk ⇒ J F ∆ p ij ¯ , , F F lk e im ¯ µ µ U ) ¯ ik im = F e,tr − M ¯ n stepatcpr ftedfraingradient. deformation the of part plastic the is kl e ¯ ik κ κ +1 1 e T ij β β ¯ n = ˙ = exp M ij F = 0 0 = (0) 0 = (0) n ¯ F km ( p U µ ik ¯ ˜ U F mk ¯ ˜ p kl ∂ e ∂ ij kl e − n T Φ − ¯ , ( F ij − ∆ J 1 J ∗ ¯ kj ¯ p 1 F e e µ ˙ = ) ) exp kj p n 1 hr ehv sdteiohrcvlmti pi fteelastic the of split isochoric/volumetric the used have we where , / +1 µ 3 M − . M ¯ ¯ 1 − ij kj ∆ µ n +1 M ¯ kj eso:2.Ags,22,15:04 2021, August, 26. version: (106.153) (106.155) (106.154) (106.152) (106.146) (106.148) (106.151) (106.149) (106.147) (106.150) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et deformation). 106.4.3 Remark solids. path–dependent when for increments, path small rium for case the indeed is This 106.4.2 Remark solutions of family solids. decomposition isotropic spectral the to with restricted contrasts are This which solid. anisotropic general a for valid tensor nonsymmetric general the 106.4.1 Remark obtain we (106.156), equation the 3104 in of expansion 479 order page: first the using by and series Taylor in expanded be can tensor a of exponent the that recognizing By . . . DEFORMATION FINITE 106.4. then is deformation elastic The Notes ESSI 27 26 25 tsol ecle aLui’ eisepnin ic xaso saotzr lsi o tt n nrmna plastic incremental (no state flow plastic zero about is expansion since expansion, series MacLaurin’s called be should It 1992). ( Simo See (1974). Pearson example for See exp n n +1 +1 C C ¯ ¯ ij ij e e − ∆ = = = def µ exp = exp = = n +1 ntelmt hntedslcmnsaesffiinl ml,teslto (106.158) solution the small, sufficiently are displacements the when limit, the In expansion series Taylor’s for approximation proper a is (106.157) equation from expansion series Taylor’s The +∆ n M δ +1 n n ¯ ir +1 +1 C lj ¯ µ − C ij F e,tr ¯ 2 ¯ − − il e,tr = im ∆ e n ∆ ∆ +1 − µ δ µ µ T − lj n M ∆ ¯ n n +1 − +1 +1 ∆ ir n µ M +1 ¯ M M µ n M ∆ n ¯ ¯ ¯ +1 ir F n +1 ¯ µ ir ir lj T T +1 mj e M n C is (106.158) equation by given solution approximate the is, That . ¯ ¯ n M +1 rl e,tr +1 ¯ ir n M +1 ir n C ¯ n ¯ +1 C +1 lj 27 n rl n e,tr ¯ +1 +1 rl F e,tr C + ¯ spoe o sal auso lsi o tensor flow plastic of values “small” for proper is ¯ rk M C e,tr rj e,tr ¯ ¯ 1 2 rl e,tr δ lj exp lj ∆ T − ∆ − µ n ∆ µ − ∆ n +1 δ +1 µ → ∆ lj µ F ¯ M n n µ − ¯ kl e,tr +1 0 +1 n ls +1 ∆ hc r eurdfrfloigteequilib- the following for required are which M C ¯ ¯ µ M il exp e,tr lj ¯ ∆ n lj +1 µ n M n +1 − ¯ +1 lj M ∆ M ¯ ¯ µ lj sj n +1 eso:2.Ags,22,15:04 2021, August, 26. version: + M ¯ · · · lj 25 ∆ (106.157) (106.156) (106.158) µ n +1 M ¯ lj 26 . Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et section. next the in algorithm solution iterative an build to used corrector 106.4.4 Remark 3104 of 480 page: . . . DEFORMATION FINITE 106.4. to collapses Notes ESSI a eitgae ogive to integrated be can (106.151) rule hardening The out working that In used was space. it strain (106.159) in counterpart equation deformation corrector small predictor–plastic the elastic deformation small a is which eomto tensor deformation yngetn h ihrodrtr with term order higher the neglecting By ,adtecntttv relations constitutive the and ( 106.162), (106.161), equations by defined is problem incremental The F n F n n +1 +1 +1 ij ij lim lim → → κ C S ¯ qainfrsaldfraineatcpatciceetlaayi.Ta eebac ilbe will resemblance That analysis. incremental elastic–plastic deformation small for equation ¯ δ δ α IJ ij e ij ij = n δ = 2 = ij +1 n n κ 2 + C 2∆ +1 ¯ α ij ∂C e eebe the resembles (106.161) equation that note to interesting is It ∂W C ∆ + ⇒ ¯ n µ n = ij +1 e,tr +1 IJ n n C δ +1 ¯ µ +1

ij ij − ij n e +1 il tr 2 + + = + = ∂K ij ∆ ∂ a ewitnas written be can Φ n µ +1 n ∗ α ∆ + − − ∆ + − − = = +1 M

∆ n ¯ n +1 ij +1 lj µ n δ ∆ ∆ δ ∆ ∆ δ ij ij ij +1 M µ µ µ µ µ µ ¯ 2 + 2 + 2 + 2 n 2 n ir n ij tr +1 +1 n n +1 δ n +1 +1 − M M n n n il +1 M ¯ ¯ n 1 +1 +1 +1 ¯ M M ∆ +1 2 + ij ir ¯ ¯ C ij ¯ ∆ il ir µ M ij e,tr ij e,tr ij e,tr rj − e,tr − ¯ µ n n n δ ij 2 2∆ +1 +1 +1 2∆ rj − δ ,teslto o h ih elastic right the for solution the (106.158), equation in + rl M M 2 + il e,tr µ ¯ 2∆ ¯ µ n 2 + n lj ij +1 +1 n µ n 2∆ + +1 C +1 M n ¯ n ¯ n +1 il e,tr +1 +1 il tr ir rj e,tr n M M rl e,tr µ n ¯ ¯ +1 n +1 2 lj +1 ij n M rj tr M ¯ +1 ¯ n lj lj M +1 ¯ M ir ¯ lj n +1 eso:2.Ags,22,15:04 2021, August, 26. version: rl tr n +1 lsi predictor–plastic elastic M ¯ lj (106.162) (106.163) (106.160) (106.161) (106.159) Jeremićet al., Real-ESSI sue sasatn on o etnieaieagrtm npeiu qain ehv introduced have we equation, previous In algorithm. iterative Newton a for point tensor starting a as used is tensor 106.4.5 Remark where h prpit ulbc to pull–back appropriate the given a For section. conditions (KKT) Karush–Kuhn–Tucker the and eeice l nvriyo aiona Davis California, of University al. Jeremić et the to residual applied new be the can change, expansion iterative the series Taylor obtain order to first order in The (106.172) tensor process. deformation equation iteration elastic the right during trial fixed The tained tensor. deformation elastic right Euler ward Tensor 3104 of 481 page: . . . DEFORMATION FINITE 106.4. Notes ESSI eitoueatno fdfrainresiduals deformation of tensor a introduce We h lsi rdco,patccretrequation corrector plastic predictor, elastic The hssto olna qain ilb ovdwt etntp rcdr,dsrbdi h next the in described procedure, type Newton a with solved be will equations nonlinear of set This R n n n T Φ( = Φ n ∆ n R ¯ Z S +1 +1 +1 +1 +1 ¯ R ij ij ij new < µ kj ij ij C C τ S K = ¯ ¯ = ij IJ osotnwiig h ra ih lsi eomto esri endas defined is tensor deformation elastic right trial The writing. shorten to ij ij n h ih lsi eomto tensor deformation elastic right the and e,tr e α ersnstedffrnebtentecretrgteatcdfraintno n h Back- the and tensor deformation elastic right current the between difference the represents = C = current 0 ¯ = = T |{z} ik ¯ e C R ¯ = = = ij n − ij e +1 S ij old K , ¯ ; n kj F +1 h adlsrs tensor stress Mandel The ∂κ ∂W n n ¯ − + α iI +1 +1 e F n ) n α | +1 d iI C C n +1 p n n ¯ ¯

n C +1 ¯ +1 +1 n ij ij F e,tr e,tr +1 Φ − ij +1 e ij S F F ¯ 1 ¯ C ≤ ¯ IJ or , + BackwardEuler rM ri − − e,tr ij e,tr n B 0 d +1 ∆ ∆ (∆ 0 n n S − µ µ ¯ T +1 +1 n rps–owr to push–forward or {z IJ F µ ; n ∆ C F iM ¯ ) +1 p n jJ ij n µ e,tr e n +1 +1 Z +1 n ∆ n − ij +1 M F h prddquantities upgraded the , − +1 Z ¯ µ ¯ 1 1 rj F e,tr ir ij Z n jJ T p +1 ij ∆ + n T +1 ¯ } 0 = Φ ij − n C ¯ T µ +1 a eotie rmtescn il–ichffstress Piola–Kirchhoff second the from obtained be can rj e,tr ∂ F ∂ n rS C ¯ T +1 B ¯ + ik e mn Z ilgive will n as ij n +1 F d C jS p ¯ T ¯ il e,tr mn − 1 n n ∆ + n +1 +1 +1 S M S ¯ ¯ IJ µ lj IJ ∂ and n ∂K and +1 Z eso:2.Ags,22,15:04 2021, August, 26. version: α n R ij n +1 +1 ij new dK τ K ij α α rmteold the from a efud then found, be can n +1 C ¯ ij e,tr (106.171) (106.169) (106.170) (106.168) (106.166) (106.165) (106.167) (106.164) (106.173) (106.172) smain- is R ij old Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et as 3104 rewritten of be 482 can page: (106.173) equation the and write can we . . . DEFORMATION FINITE 106.4. that using By Notes ESSI R d T ¯ T mn ¯ ij new mn = = = = C = = = ¯ mk e ∆ + + ∆ + ∆ + + ∆ + ∆ + ∆ + d d d S C C C ¯ ¯ ¯ ¯ R R R kn 2 1 2 1 mk mk mk e e e ij ij ij old old old µ µ µ µ µ µ ∆ ∆ ∂ ∂ ∂ ∂ ∂ ∂ S S µ µ ¯ ¯ ∂ ∂ + + + n n n n n n ∂K ∂K ∂K kn kn C ∂ ∂ ∂ ¯ T T +1 +1 +1 +1 +1 +1 ⇒ ¯ ¯ ∂ sk d d d ∂ T e n n mn mn ¯ Z Z Z Z Z Z + + C C C T +1 +1 α α α T ik ¯ ¯ ¯ ¯ ¯ ij ij ij mn ij ij mn mn ij ij ij − e e e Z Z C 2 1 ¯ 1 pq ij + + + mk d dK dK dK e C ¯ C C T ¯ ¯ ¯ d d d d mk C e C C sn ¯ mk sk e e ¯ α α α ¯ d C (∆ (∆ (∆ ¯ mk e mk sj e S e ¯ mk e + L − kn ¯ µ µ µ − knpq e 1 L L C ) ) ) ¯ 2 1 ¯ ¯ 1 knpq e n n n knij e T sk e C ¯ C ¯ +1 +1 +1 ¯ T sn ¯ sk e mk e d sk − Z Z Z C = d d ¯ 1 ij ij ij − d pq C C e L ¯ ¯ ¯ T C S 1 + + + ¯ ¯ ¯ ij pq knpq e e e sn kn ij e T ¯ + + sn + + rm(106.55) from + d C ¯ 2 1 pq e um nie rearrangement indices dummy um nie rearrangement indices dummy C ¯ mk e rm(106.174) from L ¯ knpq e d C ¯ pq e eso:2.Ags,22,15:04 2021, August, 26. version: + (106.175) (106.174) (106.176) Jeremićet al., Real-ESSI for (106.177) solve can we notation introducing Upon eeice l nvriyo aiona Davis California, of University al. Jeremić et (106.180) from follows it that using By indices rearranging by or, 3104 of 483 page: . . . DEFORMATION FINITE 106.4. Notes ESSI h oli ohave to is goal The d T = d dK d = = 0 C C C mnij ¯ ¯ ¯ ij pq pq e e e α + + ∆ + ( + ∆ + ∆ + + ∆ + ( = = ( = ( = = ∂K R R d R 1 1 2 2 ∂κ T δ T T δ C ¯ ij ij ij im old old old mnij µ im µ µ µ mnpq mnpq ∆ ∆ ij β e α ∂ ∂ ∂ ∂ δ δ µ µ + + + + n n n n ∂K ∂K nj nj dκ ∂ ∂ ∂ ∂ +1 +1 +1 +1 ) ) ) − d d d ∂ ∂ T T n n ¯ ¯ − − ∆ + β Z Z Z Z ∆ + (∆ (∆ C +1 +1 1 α α T T ik ik ¯ ¯ ¯ 1 1 R mn ij mn ij = mn mn ij e Z Z µ µ ij new − µ pq pq + dK dK − − − µ ) ) d R ∂ R R ∂ C n n d d ¯ C C C C n mn old +1 +1 ¯ ¯ α α 0 = (∆ ¯ ¯ n (∆ mn mn ij old old ∂ e +1 ∂ mk mk +1 sj sj e e e e + T Z Z T ¯ ¯ Z µ − µ Z ik ij ij ik − − − − ) mn ooecnwrite can one so L L ) mn ¯ ¯ d 1 1 ∆ + n d d knij e knij e ∂K ∂κ (∆ +1 (∆ (∆ T T ¯ ¯ sk sk β Z α µ C µ C ¯ µ µ d d ¯ ij ) sj e d d ∂ ) ) C C sj e ∂K ¯ ¯ n ∂Q C C + n ∂K n n ¯ ¯ ij ij e e +1 +1 − +1 +1 ij ij − e e β + 1 Z 1 Z Z Z + + α mn = T ij mn mn T ¯ ¯ sk sk − dK − ∆ + − + d + (∆ ∆ α ∆ 1 2 1 2 µ + µ µ µ ∆ ∆ ∂ ) ∂ ∂ n µ µ n n ∂K H +1 ∂K ∂K +1 +1 ∂ ∂ αβ Z n n Z Z ∂ α +1 ∂ +1 mn α α T mn mn T ¯ ∂K ¯ Z pq Z ∂Q pq mn mn dK d dK β (∆ C α C ¯ α ¯ µ pk e pk e ) L H L ¯ eso:2.Ags,22,15:04 2021, August, 26. version: ¯ kqij e kqij e αβ ∂K ) ∂Q d β C ¯ ij e (106.177) (106.178) (106.182) (106.181) (106.180) (106.179) Jeremićet al., Real-ESSI n ihteslto for solution the with and eeice l nvriyo aiona Davis California, of University al. Jeremić et notation following the introducing Upon becomes (106.183) equation (106.181), using By 3104 of 484 page: . . . DEFORMATION FINITE 106.4. Notes ESSI rtodrTyo eisepnino il ucinyields function yield a of expansion series Taylor order first A new F new pq + − Φ( Φ( = T T F new d ¯ ¯ ∂ (∆ ij ij pq Φ( K , K , Φ( µ ∂ T ¯ ) T α α T ij ¯ ( ¯ = ) = ) pn T ∂ ij K , mnpq Φ( = = = K , α ∂K T ¯ α ) ij − + + + + + + + + + + d ) = ) − K , C α ¯ C 1 ¯ pq e d old old old old old ∂ ∂ 2 1 ∂ ∂ ∂ ∂ ∂ sq e α old (∆ Φ( Φ( Φ( Φ( Φ( Φ( Φ( ) − ∂ ∂ becomes (106.184) (106.182), from Φ( Φ( Φ( Φ( Φ( ∂ Φ( − Φ( Φ( ∂ ∂ H R ∂ Φ( ∂K ∂K ∂K ∂K µ T T T T T T T ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 T T T T T T T T ¯ ¯ ) ¯ ¯ ¯ ¯ ¯ mn old ij ij ij ij ij ij ij ∂ ∂ αβ ¯ T ∂ T T ¯ mn mn ¯ ¯ ij ij ij ij ij T pn T T T ¯ ij K , K , K , K , K , K , K , ¯ T ∂ ij ij α α α α ¯ ¯ ¯ ij K , K , K , K , K , sn pn pn mn K , Φ( K , K , − ∂K ∂ K , α α α α α α α Φ + α α α α α ∂K d T ) ) ) ) ) ) ) α ¯ α α ∗ β + ) + ) + ) + ) + ) (∆ α ij + ) ) ) d dK dK dK dK 2 1 ) K , α T C ¯ ¯ µ d C ∂ mn ¯ sq C C α α α α e C ) ¯ ¯ Φ( ¯ α mk e sq sq mk n e e e ) ∂ − +1 T ¯ T H L 1 ¯ ij − − ¯ Z mn knpq e αβ 1 1 K , T mn C ¯ ¯ sn T T sk e ¯ ¯ α ∂K ∂ sn sn + ) d Φ d − C + + ¯ C d C ∗ β ¯ 1 ¯ pq e (∆ mk pq e e 1 1 2 2 T ¯ sn µ ∂ ∂ L ∆ ) ¯ Φ( Φ( + knpq e um nie rearrangement indices dummy ∂ ∂ T T µ 2 1 ¯ ¯ T T ¯ ¯ ij ij mn mn C ∂ K , K , ¯ n mk e ∂K +1 α α Z ) ) L eso:2.Ags,22,15:04 2021, August, 26. version: ¯ α mn C C knpq e ¯ ¯ mk mk e e H L L d αβ ¯ ¯ C knpq e knpq e ¯ pq e ∂K ∂ Φ ∗ β d d C C ¯ ¯ (106.185) (106.184) (106.186) (106.183) pq pq e e Jeremićet al., Real-ESSI eak106.4.6 Remark eeice l nvriyo aiona Davis California, of University al. Jeremić et gradient deformation relative the compute element, increment finite displacement a given in point quadrature specific a State Trial 1997 ). Sture, (Jeremić and counterpart strain small it’s with exactly compares residual that noting Upon small getting are deformations as limit, the in since parameter 106.4.7 Remark 3104 of 485 page: . . . DEFORMATION FINITE 106.4. setting After Notes ESSI h rcdr ecie eo umrzsteipeetto ftertr algorithm. return the of implementation the summarizes below described procedure The n d T d +1 (∆ (∆ mnpq R F Z f pq pq pq µ µ d ij F d = ) = ) (∆ (∆ pq = new ie h ih lsi eomtostensor deformations elastic right the Given → → → → µ µ δ ( = ) old n ij T ) Φ( mn mnpq F becomes + ntepretypatccs,teiceeticnitnyparameter inconsistency increment the case, plastic perfectly the In ntelmt o ml eomtos stoi epne h nrmn inconsistency increment the response, isotropic deformations, small for limit, the In Φ 2 2 δ 1 2 T pq pm ¯ m u F − ij E ∂σ pq i,j ) δ K , ∂ pq mnpq old ( − nq T Φ mn 1 pq mnpq Φ α ∆ + R n 0 = ) − +1 ( E pq T ) mnpq ( Z mnpq δ ∆ µ − n sdfie nsri pc,teiceeticnitnyparameter inconsistency increment the space, strain in defined is mp mn 1 n mn ∂m ecnslefrteiceetlicnitnyparameter inconsistency incremental the for solve can we +1 ∂σ δ n ) − qn +1 u − E mn ij i 1 ∆ Z n h ih eomto tensor deformation right the and , mnpq ∆ + mn R µ old E mn old F ijpq µ ) Φ pq ∂m F − ∂σ δ ( T pm ij pq mnpq pq δ E nq ( ijmn T ) ∆ + − mnpq 1 n ∂ µ C − ) ¯ n − ∂K 1 ∂m pq +1 e 1 ∂σ n Z n e fhreigvariables hardening of set a and R +1 α mn ij mn mn old m mn E H ijpq αβ + ∂K ∂ ∂K ∂ − Φ eso:2.Ags,22,15:04 2021, August, 26. version: Φ 1 ∗ β α R + H mn old αβ ∂K ∂ Φ ∂K α ∂ Φ d H (∆ ∗ β αβ µ d n (∆ ) ∂K ∂ +1 (106.191) (106.187) (106.190) (106.189) (106.188) is Φ f µ ∗ β ij n ) d K (∆ o a for α µ at ) Jeremićet al., Real-ESSI xtcntttv nerto procedure. integration constitutive Exit lsi tffestensor stiffness elastic tp2. step set vlaeteyedfnto n h residual the and function yield the evaluate eunAlgorithm Return procedure. integration constitutive exit and vlaeteyedfunction yield the Evaluate opt h ra lsi eodPoaKrho tesadtetileatcMne testensor stress Mandel elastic trial the and stress Piola–Kirchhoff second elastic trial the Compute eeice l nvriyo aiona Davis California, of University al. Jeremić et superscript omitting 3104 of 486 page: . . . DEFORMATION FINITE 106.4. Notes ESSI 28 rmse .t tp9 l ftevralsaei intermediate in are variables the of all 9. step to 3. step From tp3. step tp1. step n L R n n n n n n Φ n n n ¯ n +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 ijkl ( ij ( ( +1 hc o convergence, for Check k k k ∆ C K T C C K C ) T ) ) T ¯ ¯ ¯ ¯ ¯ κ ij ij ij e,tr ij ij ij ij e e,tr 28 e µ e α α α 4 = n ( k = Φ( = +1 k th ) fcnegnei o civd i.e. achieved, not is convergence If = = = = S ======¯ ∂ ij trto.Konvariables Known iteration. e,tr n n ; C n +1 ¯ +1 n n n n n n n n n 1 + ij n e +1 T K C +1 +1 +1 +1 +1 +1 2 = ( ¯ C +1 n ∂ ¯ ij k e,tr fyedciei a envoae ( violated been has criteria yield If ij +1 f α e, C ) 2 ∆ T K κ C il e,tr . T L ¯ ¯ ir ¯ ( W α ( ij ∂ ij κ ij ( k e,tr ij e ∂ µ α ijkl k e ( k ( ) C k α ( ( n ) n ¯ ( k ) k k F n ) − k ) +1 kl e ∂W ) ) +1 n ) rk , ( e +1 k C n S ¯ ) ¯ +1 n ; ij Φ e,tr T lj e,tr +1 Φ K tr C ( n n ¯ ( α ( k +1 T +1 k ij e,tr ¯ ) ) ij e,tr ) f ≤ K kl − α ( TOL NT K , k n ) n F +1 α lj e ) ; ∆ If . ( = µ and n ( k n +1 Φ n ) +1 F n T ( +1 k k rk Φ e ij ( ) R k Z tr ) TOL NT > ) ij ( T ij ( k n ≤ k ) +1 ) ≤ k n ; 0 +1 n Φ hr sn lsi o ncretincrement current in flow plastic no is there 1 + tr f n TOL NT ir +1 > ∆ ofiuain o h aeo rvt eare we brevity of sake the For configuration. or T 0 µ rce ose 1. step to proceed ) k ( n k R fcnegneciei ssatisfied is criteria convergence If . ) +1 ij ( k f ) kl k TOL NT > n eso:2.Ags,22,15:04 2021, August, 26. version: F lj e hncmuethe compute then (106.194) (106.195) (106.192) (106.193) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et T 3104 of 487 page: where . . . DEFORMATION FINITE 106.4. Notes ESSI ¯ mn ( k +1) tp6. step 4. step tp7. step 5. step dK F d C d dκ ∆ T H ¯ ¯ K (∆ T mnij T pq ¯ pq κ e, α µ ¯ α ( mn mn ( α ( ( α ( ( α ( k ( ( k k k k k k k µ k = +1) d +1) +1) +1) +1) +1) ) +1) +1) ( C T ¯ pae h ih eomto esr h adnn aibeadteMne stress Mandel the and variable hardening the tensor, deformation right the Updated parameter inconsistency incremental the Compute pae h nossec parameter inconsistency the Updated = k = mnpq ∂ paergtdfraintensor deformation right Update pq +1) e, ( = k Φ( = δ ( ∆ = H = ) k im = = = = = ) +1) d αβ d T − ¯ (∆ ∂ δ C ij ¯ ( − nj d ( µ T k ¯ mk k e, T K κ C = ) 1 (∆ ¯ µ F ¯ pn ( ) K , ¯ ( k α ( mn ∆ +

pq ( e, k ( α k ( mn ) k ( ∂ k +1) k ) µ k ( ) − + +1) ∂K Φ ) k α ) ( + ( + ) k + k R ∗ Z µ ) d +1) + , d ) ) ( β d mn ( (∆ ( mn d k k k ( ( C ∂ d ) ( ) κ ) ¯ T ) ¯ K ∂K ( Φ µ ( sk α ( e, C − mn ∂Z C ∂ k ¯ ( k H ¯ ( k ∗ ( α ) ( +1) sq T k e, k ; pq , k e, ¯ +1) d αβ ( ( − β +1) ) ik +1) ( mn k k ( ( ( (∆ k k k k ) ) ) ) ∆ ) ) − +1) F ) ¯ ) ∂ ) µ 1 mn µ − ∂K Φ ( ) ( 1 k T C ¯ k ∗ ¯ +1) ( sn , ) Φ ( k sj T ( e, β ¯ k k ) F sn ( ( ) ¯ ( ) k k ) = k mn + ) ) ) n − + F +1 C − 1 2 ( ¯ k pq 1 F Z pq e, ) 1 2 ¯ C ¯ ∆ ( ( mn mn ( ( T ∂Z k mk k ¯ e, k k ∂ ) µ sk +1) ) ) ( ( ∂K Φ( k k ( ∆ + mn R ) k ) +1) adnn variable hardening , T + T mn L ¯ α ∂ ¯ mnpq ( ij ( knpq e, k T k ¯ 2 1 µ ) ( ( mn ) k k ( K , H ∆ k ) ) ¯ ( ) k α µ d d α ) ∂Z ( ( ∂K ( k C (∆ k k ¯ ) ) − ) ) pq e, mn ( + 1 k µ α ∂Z ( C ∂ ) k ¯ ( +1) T k ∂ mk e, ¯ ∂K d mn +1) ( pq Φ ( k (∆ k ) ( ) k α ) C L µ ) ¯ ¯ eso:2.Ags,22,15:04 2021, August, 26. version: ( e, knpq pk e, H k K ¯ ( +1) ( k α k α ( ) ) ( k k +1) ) L ) ¯ H e, kqij ¯ ( α ( k n adlstress Mandel and k ) ) ! (106.199) (106.197) (106.200) (106.198) (106.201) (106.202) (106.196) Jeremićet al., Real-ESSI n aigtefis re alrsre xaso eobtain we expansion series Taylor order first the taking and trigfo h lsi rdco–lsi orco equation corrector predictor–plastic elastic the from Starting Tensor Stiffness Tangent Algorithmic 106.4.5 2. step to return and eeice l nvriyo aiona Davis California, of University al. Jeremić et where as or as written be can equation Previous 3104 of 488 page: . . . DEFORMATION FINITE 106.4. Notes ESSI tp9. step 8. step d n ∆ Φ T d d C ¯ +1 K C C C mnij T ( ¯ ¯ ¯ pq µ κ e, ¯ k ij ij ij mn e e e C α ( ( α ( ( ( +1) ¯ k k k k k ij ∆ + e ) ) ) ) ) ( Set residual the and function yield the evaluate = T = = = Φ( = mnij = = = = ∆ = δ µ n im k d − d − +1 ∂ C C = ) T K κ C = δ ∂Z T ¯ ¯ ∆ ∆ ¯ ¯ C ¯ T nj ¯ α ( mn ¯ ij ij e,tr e,tr ij pq ( e, µ e α k ( ij µ µ mn k e,tr k ( k +1) ij ( ( +1) k ∆ + +1) k d k 1 + +1) ∂K ∂ ∂Z +1) +1) − − ∂Z C ¯ − T ¯ ij e,tr mn d d C K , ij µ α ¯ ij ∆ (∆ (∆ sk ( e k dK µ − α ( ) k µ µ − n +1) ∂Z ∂ d ) ) = d α +1 1 (∆ T C Z Z ¯ ¯ Z d ik ) T mn ( ( ij ij mk ¯ e k k C µ sn ij ¯ ) ) ; − ij ) e,tr d R Z ∆ C C C ¯ ¯ ij ij ( ¯ − k µ sk mk e e sj e, +1) ∆ + d ( ∂ k ∂Z (∆ − ∆ + ) T ¯ 1 = mn d µ ij − µ T ¯ C 1 ) ¯ d µ sn (∆ ij d e, Z T ¯ T ( ¯ + ij sk (∆ k ( mn µ k +1) ∆ + ) ) 2 1 µ + ∂K ∂Z ) C − − ¯ 2 1 mk d µ ∂ e ∂Z ∆ ij α T ∆ ¯ C µ mn ¯ (∆ L H ij µ ¯ ij e,tr knpq e ∂K ∂Z ( αβ k µ 2 1 ) ) − ij α ∂Z ∂K ∂ C ∂ ∂K ∂Z ¯ d ∆ Φ T dK mk e C ¯ ¯ mn ( pq ∗ β µ ij k pq e α ) ( α L k ¯ H +1) C knpq e ¯ eso:2.Ags,22,15:04 2021, August, 26. version: αβ pk e, Z rm(106.175) from ( k ij ( ∂K ) d ∂ k C L Φ +1) ¯ ¯ pq e e, kqij ∗ β ( k ) (106.205) (106.204) (106.206) (106.203) (106.208) (106.207) Jeremićet al., Real-ESSI enx s h rtodrTyo eisepnino il function yield of expansion series Taylor order first the use next We eeice l nvriyo aiona Davis California, of University al. Jeremić et used have we where where 3104 of 489 page: . . . DEFORMATION FINITE then 106.4. is tensor deformation elastic right in increment the for solution The Notes ESSI yuigslto for solution using By Since eaenwi h oiint ov o h nrmna nossec parameter inconsistency incremental the for solve to position the in now are We F d = Γ d F d S (∆ C ¯ pq pq ¯ ∂ kn ∂ ij e T ¯ µ = Φ F ( pn = ) = T ( = pq mnpq ∂ ∂ 1 2 T ( ¯ T C Φ T pn ¯ F L mnij ∂ mnpq ¯ sq e ) pq ∂ ∂ knpq e T − ¯ Φ ∂ T pn 1 ¯ − C Γ ( Φ mn ) ¯ 1 T ) − sq e − d mnpq osotnwriting shorten to 1 T d 1 ¯ C C ¯ ¯ sn C − d n ¯ pq sq e e Γ C 1 +1 mn d e,tr ¯ d ) + C ij e C − Z T ¯ − ¯ ¯ mk 1 e sn ij 2 1 mn e,tr 1 w a write can we 106.209 from − d T ∂ + ¯ d C − sn ¯ − ∂ T (∆ ¯ C mn e,tr ¯ Φ 1 2 mn ∆ d sk d e µ (∆ ∂ C µ ¯ ) ∂ T C F pq − ¯ e ¯ Z Φ µ mn mk 1 e pq mn ) + T Z ¯ ( C L sn T 2 1 ¯ ¯ ij ∆ + mnpq knpq e mk e ∂ + ∆ + ∂ T F ¯ L d µ Φ 2 1 mn ¯ pq ) knpq e − d µ C ¯ − 1 (∆ d d mk C e C C ¯ ∂ ¯ ¯ ∂K (∆ mk e ∂ µ pq pq n e e ∂K L Φ +1 ) ¯ µ α − − ∂ knpq e L ∂Z Z ) ¯ ∂K ∂ T α knpq e ¯ d mn ∂K ∂K ∂K ∂Z Φ mn (∆ ∂ ∂ mn α Φ Φ H d ij α α α µ d C d ¯ αβ H ) T d d H ¯ pq C e ¯ H mn (∆ (∆ αβ d ∂K pq αβ ∂ e Φ( αβ Φ µ µ + + + ∂K ∂ ∗ T β ∂K ) ) ∂ ¯ Φ ∂K ∂ ij Φ H H ∂K ∂K ∂K + ∗ ∂ ∂ ∂ Φ β K , eso:2.Ags,22,15:04 2021, August, 26. version: ∗ β αβ αβ Φ Φ Φ ∗ β ∂K α α α ∂ α Φ ∂K ∂K 0 = ) ∂ ∂ dK dK dK α 0 = Φ Φ H ∗ ∗ β β α α α αβ d (∆ 0 = = = = = ∂K ∂ Φ µ ∗ β ) (106.211) (106.209) (106.214) (106.213) (106.210) (106.212) (106.215) Jeremićet al., Real-ESSI estensor ness 106.4.8 Remark Then eeice l nvriyo aiona Davis California, of University al. Jeremić et configuration reference 3104 of where 490 page: . . . DEFORMATION FINITE 106.4. write can we 106.209 using by and Notes ESSI ulbc oterfrneconfiguration reference the to Pull–back tensor stiffness tangent Algorithmic lim d n L P ¯ ¯ +1 C knvt S AT pqvt ¯ pq e L L ¯ ( d ∆ ijkl S AT T vtpq S AT = C L ¯ = ( = ( = mnpq µ pq ijkl S AT e P ¯ L F ¯ = pqvt = = knpq e ∆ + op ) n − becomes ntelmt o ml eomtos stoi epne h loihi agn Stiff- Tangent Algorithmic the response, isotropic deformations, small for limit, the In E +1 1 ( d vtpq TS AT T C P F T T ¯

¯ rsop µ im mnpq mnpq vt pqvt e,tr p Ω δ mv Γ F = = = = 0 ) n − op +1 ) ) 1 δ − − F nt ( 1 1 δ jn T p rv E R E E − rsop δ mv knpq knpq knpq n knvt δ δ F +1 st mv Γ ) op δ − F Υ nt δ ∂K

∂Z − 1 kr p nt ( vtpq − Υ Υ T L − δ ¯ n 1 ij n − α rv rsop mnpq − vtpq − ijkl +1 cd F 1 1 − H F δ R F Γ op st ) Ω αβ ab i nemdaeconﬁguration intermediate (in ls − − p cdvt E

0 1 ( ( knpq ∂K ∂Z n Υ T T δ ∂K ilsteagrtmctnetsins tensor stiffness tangent algorithmic the yields +1 ∂ δ Γ R mv vtab rsop Γ mrpq − Φ rv ij L α knmr 1 ¯ Υ ∗ β δ mnrs S AT Γ δ ) ) nt H ! mrpq − − st − n 1 αβ 1 1 − cd H d Z δ C E n mr mn ¯ n ∂K rv Z ∂ cd vt cdab e,tr cd Γ Φ mn Γ + E δ E ∗ β st cdab Υ cdab − Z vtab − ∆ 1 mn Υ Υ Γ µ H vtab − vtab − 1 1 mr ∂ n H H ∂K +1 ! mr mn eso:2.Ags,22,15:04 2021, August, 26. version: Z Ω α ¯ mn !! ste endas defined then is ) H αβ ∂K ∂ Φ ∗ β (106.220) (106.216) (106.221) (106.217) (106.218) (106.219) L ijkl in Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et it’s with exactly compares 106.221 by given 1997). tensor Sture, Stiffness (Jeremić and counterpart Tangent strain Algorithmic small the that noted is It 3104 of 491 page: . . . DEFORMATION FINITE 106.4. since Notes ESSI i lim = Γ lim lim H lim mn = = = F T ¯ mnpq ab = = lim = m n n n mn ab ab ab Υ = E R E − abpq abpq abmn F 1 2 mnpq n ∆ pq cd Υ Υ µ ( H E T mnpq − mnpq − ∂m ∂ = = ∂K mnpq ∂ mn 1 1 cdab e T ¯ Φ ad mn α m + ) − mn m H δ δ C ∂K ¯ pm pm 1 ∂ sb αβ mn e n Φ − δ δ +1 α ∂K − nq nq ∂ − H ∆ Z 1 Φ ∆ + ∆ + αβ mn T µn ∆ ∗ β ¯ sd µ ∂K ∂ ab − + ∂m Φ µ µ ∂K E ∆ ∂m ∂Z ∗ β ∂ 2 1 abpq ∂σ mn µ T α ∂ ¯ mn ∂ F pk mn rs T ¯ Υ Φ H pq cd mnpq − αβ E C ( C 1 ¯ T ¯ kspq e ck e ∂K sq mnpq e ∂ L Φ ∂m ¯ ∂K − kdab e ∗ β 1 ) mn − T α ¯ + 1 sk H ∂ + ∂K n αβ ∂ ∂K +1 Φ 1 2 ∂K α ∂ Z ∆ α Φ H mn µ ∗ β αβ ∂Z ∂ H + ∂K T ∂ eso:2.Ags,22,15:04 2021, August, 26. version: ¯ αβ mn rs Φ ∂K ∂ ∗ β ∂K ∂ Φ C ¯ Φ α rk e H ∗ β L ¯ αβ + kspq e ∂K ∂K ∂ ∂ Φ Φ ∗ β α H αβ (106.222) (106.223) (106.225) (106.224) ∂K ∂ Φ ∗ β Jeremićet al., Real-ESSI I olbrto ihD.Ya Feng) Yuan Dr. with collaboration (In (1994-2016-) Equations Equilibrium Static of Solution 107 Chapter 492 Jeremićet al., Real-ESSI where: eeice l nvriyo aiona Davis California, of University al. Jeremić et with (107.1) equation of intersection the finding at aimed They constant (1971). Wempner and (1979) of Equations forms Force Various Residual the Constraining 107.3 another. parameter not for control stage are single parameters a by control characterized Multiple be λ therefore may stages. and stage analysis each of in independently series varied a nonlinear complex performing a involves Processing generally steps. iterative problem and steps incremental stages, into solution” breakdown the hierarchical ”advancing of idea the in rooted by strongly are importance practical of procedures solution All of case the describes (107.1) fixed where nonlinear of set a in results methods element finite called by equations problems algebraic such 3104 of of Discretization 493 solids. page: of analysis (1993). Felippa on based is chapter Equations This Force Residual HIGHLIGHTS AND The SUMMARY CHAPTER 107.1. 107.2 Highlights and Summary Chapter 107.1 Notes ESSI tgsaeol ekyculdi h es htedslto foemypoietesatn point starting the provide may one of solution end that sense the in coupled weakly only are Stages . 2 1 continuation i ieetfr nta ti cldwt cln matrix scaling with scaled is it that in form different bit A called also npeiu hpesw aedrvdtebsceutosfr(aeilado emti)nonlinear geometric) and/or (material for equations basic the derived have we Chapters previous In external ds s r f ( int = u = λ , where ( Z u s = ) ) arc-length ds u loading r h nenlfre hc r ucin ftedisplacements, the of functions are which forces internal the are ψ ref 2 xeti eysml rbes h otnainpoesis process continuation the problems, simple very in Except . f u s 2 int path steaclnt endas defined , arc-length the is d ( u u T ) ehd ihvrosmtoso prxmtn h xc egho narc. an of length exact the approximating of methods various with methods etradtescalar the and vector S following − d u λ eiulfreequations force residual f + ext proportional dλ 0 = 2 methods ψ f 2 1 odn nwhich in loading Riks (1972), Riks of work original the from stemmed have λ 2 sa is : : load–level S 1984 (). Felippa by introduced , the aaee htmultiplies that parameter loading pattern eso:2.Ags,22,15:04 2021, August, 26. version: is multilevel u kept h vector the , fixed. f ext n involves and Equation . f (107.2) (107.3) (107.1) ext s sa is = Jeremićet al., Real-ESSI qiiru state. equilibrium eeice l nvriyo aiona Davis California, of University al. Jeremić et n parameter load ∆ and matrix Scaling 3104 of 494 length. arc page: where . . . FORCE RESIDUAL THE CONSTRAINING 107.3. form: incremental an with replaced be can (107.3) form Differential Notes ESSI u .W a ov h augmented the solve can We 107.4). ( equation constraint one the and (107.1 ) equations equilibrium 4 3 n (107.1). Figure See h anesneo h r-eghmtosi httela parameter load the that is methods arc-length the of essence main The u n scalar and nnw ipaeetvralsado xr nnw ntefr fla parameter. load of form the in unknown extra on and variables displacement unknown ref a ∆ (∆ = l sarfrnevlewt h ieso of dimension the with value reference a is sthe is iue171 peia r-eghmto n oainfroeDFsystem. DOF one for notation and method arc-length Spherical 107.1: Figure s ) Load 2 λ ∆ λ 0 aiso h eie intersection desired the of radius − ∆λ f λ aibew r eln with dealing are we variable ext (∆ 1 ∆λ are ∆λ f ext l 2 3 λ ) f f nrmna n not and incremental 2 ext f ext = S

u δλ 0 suulydaoa o-eaiemti htsae h tt vector state the scales that matrix non-negative diagonal usually is u ψ ref 2 0 u 2 f ( u ext ∆ 0 δλ δλ ∆ , δ u λ u u 1 2 T 1 0 0 f f S ext f ext ext ∆ u ) trtv aus n r trigfo h atconverged last the from starting are and values, iterative ∆ + n ∆ 1 + u δ 3 2 λ u ∆ 2 n ersnsa prxmto oteincremental the to approximation an represents and 1 √ ψ unknowns ( l ∆ u f 2 2 ! u , δ ∆ T λ u − ( u 2 u S 2 3 f 3 (∆ ∆ ext , λ u 4 l ) 3 nodrt ov hspolmw have we problem this solve to order In . ti motn ont httevector the that note to important is It . ) f 2 ext ) ( ( u u p 1 Displacement , , Hypersurface Constraint or λ λ p 1 Path Equilibrium eso:2.Ags,22,15:04 2021, August, 26. version: λ f f ext ext becomes ) ) u a aibe With variable. (107.4) ∆ u Jeremićet al., Real-ESSI h rvossse a ewitnas: written be can system previous the eeice l nvriyo aiona Davis California, of University al. Jeremić et the that mentioned be should It as: written be can (107.8) equation the the defining by or where 3104 of 495 page: . . . FORCE RESIDUAL THE CONSTRAINING 107.3. of system Notes ESSI 6 5 yuigatuctdTyo eisexpansion. series Or Taylor truncated a using By n a ov rvossse ftoeutosfor equations two of system previous solve can One   r   K   a augmented K new new = 2 δ δ δλ δλ t u u u = n ( ψ   ref 2 = u u 2 1 +     ∂ λ , K 2 a r ∆ = = ∂ ( u old t = ) u ψ u u ref 2 Jacobian. qain yapyn the applying by equations ,λ u 2 −K − T K 2 + augmented ) = 0 = S ∆   sthe is t − u u 2 1 T r r 2∆ ψ ref 2 u old old   − S ψ u 2 ref 2 u 2 f ψ λ ( ( tangent a r K ∆ ext u u ∆ old old 2∆ u λ , λ , t − f u 2 stiffness T f ψ λ T + ) + )   S ext   S augmented δ   u f 2 stiffness K ∂ 2∆ 2∆ + δ δλ r   t − u ( ∂ δ matrix u f ψ λ u u ext   λ , − δ ψ λδλ = ) Newton-Raphson f 2 arx h i st have to is aim The matrix. f stiffness δ − ext 6   u − K   + f 2 1 δλ as:   0 = a r ∂ old old = r a r arxrmisnnsnua vnif even non-singular remains matrix ( ∂λ old old u   λ , δ u   ) and δλ 5 n (107.4) and (107.1) equations to method = δλ : r new eso:2.Ags,22,15:04 2021, August, 26. version: ( u λ , 0 = ) and K t a new ssingular. is (107.10) (107.9) (107.6) (107.8) (107.7) (107.5) 0 = so Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et vector displacement the ( of load components to specialized further term be the can (107.4) equation general rather with A 1983). ( Dvorkin and S obtained is scaling 3104 of 496 page: various assigning By order. in are observations and comments further parameters to Some called numbers form. so the general reduce rather to a intended in is that equation constraining an drift introduced have we (107.3) section Control In Arc-Length Hyper–Spherical Generalized, 107.6 Control Displacement CONTROL LOAD 107.4. 107.5 Control Load 107.4 Notes ESSI If Load 10 = S 9 8 7 hti h yidia osrit rgnrldslcmn control. displacement DOFs. general (1971). or rotational Wempner Say constraint, and and cylindrical translational (1979) the both Riks is includes ( 1972), That model fRiks FEM of work if original example For the to reduces actually It Coefficients = K ψ ro nteiceetlnnierfiieeeetpoeue h osriigeuto a given was equation constraining The procedure. element finite nonlinear incremental the in error λ u f t u I ≡ i and . and displacement 0; ψ ψ ψ f u f f = ref ψ iue172 nuneof Influence 107.2: Figure ≡ ≡ ψ u u 8 1 = and 1 0 u tews opyia enn a eatiue oti cln ye With type. scaling this to attributed be can meaning physical no otherwise but n tt control state and n nsu ihsmtigvr iia othe to similar very something with up ends one ψ oto a entaiinlyascae ihtecs nwihol n fthe of one only which in case the with associated traditionally been has control ψ f ψ h ehdi aldthe called is method the ∆ f u < l , a o esmlaeul eo sflcocsfor choices Useful zero. simultaneously be not may ψ ψ u f , S and 9 u Displacement Hypersurfaces Constraint u ψ with ref f Path Equilibrium 10 << ψ sseie.Ti a ergre ihra ain fstate of variant a as either regarded be may This specified. is n a bandffrn osriigshmsfo (107.4). from schemes constraining different obtain can one ψ u ψ u u u and ≡ arclength 1; ψ f ψ Load f ntecntan ufc shape. surface constraint the on ≡ λ 0 method f and ψ S f 7 = = fw choose we If . external ψ I u ntefiieeeetliterature, element finite the In . eso:2.Ags,22,15:04 2021, August, 26. version: ∆ S l work are ψ f I >> S , Bathe of constraint K = ψ t u and diag Displacement Hypersurfaces Constraint Path Equilibrium diag λ ( control ) K u ψ t f ) ( K > nice t ψ ) u . Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et level becomes: level: load unknown (1993). Dhatt Felippa and and Batoz (1991) by proposed Crisfield approach by the described following as by (1979), (107.6) equation from constraint the introduce of eigenvalues non–convex. the of 3104 one of least 497 at matrix page: passed, stiffness using definite by non–positive equations of constraint scaling possibility of aspect important One control. defines one if point example limit For . . . unknowns. HYPER–SPHERICAL different as GENERALIZED, of taken functions 107.6. variable, is the parameter out control singles the that if norm a which in control, Notes ESSI tas: it where where an 11 iterative u aigi idthat mind in having But h trtv displacement directly may iterative one The (107.10) equations of system the to solution the of control better get to order In λ ∆ δ λ δλ old δ u new u u ihtesolution the With . t new a eue ee n yrewriting by and here, used be can (107.4) equation from constraint The unknown. only the is = = = = ∆ = hneta ol tmfo h tnadla-otoldNwo-aho,a xdload fixed a at Newton-Raphson, load-controlled standard the from stem would that change u K ∆ = λ − − − old t − → K K K 1 u f + t − t − t − 0 ext old 1 1 1 δλ ⇒ r stedslcmn etrcrepnigt h xdla vector load fixed the to corresponding vector displacement the is + r f int u u δλ δ ψ ref 2 old u u u 2 ( u ssilunknown! still is old 11 λ , ∆ = old λ , o the for ) δ ψ old = u − u f 2 old − ssltit w at,adwt h etncag ttenew the at change Newton the with and parts, two into split is eaelk state like behave (107.4) equation from constraint our makes that λ − K + λ old δ u t − δλ u δ i 1 u ¯ + ti,o ore osbet aetepeiu parameters previous the make to possible course, of is, It . K ,tenwiceetldslcmnsare: displacements incremental new the (107.12), from f ext + t δλ f int snnpstv,tu edrn h osrithypersurface constraint the rendering thus non–positive, is δλδ i ( hcmoeti sd ra ain fthe of variant a as or used, is component th = f u ext u old − t K K ) t − = tuulyhpesta fe h ii on is point limit the after that happens usually It . t − 1 λ − r new old K t − f + 1 ext δλ K f int t − u ( 1 ref u f S ext old ∆ = = ) = eso:2.Ags,22,15:04 2021, August, 26. version: − diag δ u λ u ¯ T old + S ( K f ∆ δλδ ext t u ) u or hncoet the to close then − t f S ext δλ = and , f ext λ K (107.11) (107.13) (107.12) t control sthe is δ u ¯ is Jeremićet al., Real-ESSI where: or: eeice l nvriyo aiona Davis California, of University al. Jeremić et for solved be can (107.17) equation scalar quadratic The terms: collecting by or, 3104 of 498 page: λ substituting by then . . . HYPER–SPHERICAL GENERALIZED, 107.6. Notes ESSI old + a δλ + 2 δλ a a a 1 u u

δλ 1 3 2 ψ ψ = ref 2 ref 2 n nsu ihtefloigqartcsaa equation: scalar quadratic following the with up ends one u u u 2 2 2 ψ = = 2 = δλ ref 2 2 + u 2 (∆ 1 ∆ = u u a u u ψ ψ ∆ 2

ref 2 ref 2 new old δλ u u 2 2 − u u ψ old ∆ δ a ref 2 ) + + u u 2 2 T ∆ u t T + + δ a S new δ u u S ¯ 3 u (∆ δ p old δ t T 0 = + u ¯ a u n yrcligthat recalling by and (107.14) equation into (107.13) equation from S a +2 1 u t + δλδ 2 2 T new + − δ ∆ S

u ψ ¯ u a u (∆ + ) u f 2 t 1 ψ old ∆ ref 2 T a u 2 T 3 u S + S old δ u λ δ ∆ t T ; u + ∆ ¯ new u S u δ old ∆ + u old ) ¯ δλ ∆ 2 + ψ u + − f =

2 λ old δ ! δ u old ¯ ∆ u δλ u ¯ ψ + ref 2 (∆ = l ψ u 2 2 + − 2 f δ 2 + δ = ! u δλδ ¯ ∆ u δλ l l t T − ) ∆ 2 ∆ + u S 2 : a + λ t δ 2 old u − λ + t ∆ old + p 2 λ a ψ ψ ψ ∆ old a 1 2 2 f f f 2 2 2 λ ! ! ! − old 2 a ψ + 1 δλ f 2 eso:2.Ags,22,15:04 2021, August, 26. version: δλ a δλ 3 2 0 = + + 2 ψ f 2 ! (∆ = l ) (107.16) (107.17) (107.18) (107.14) (107.15) 2 λ new = Jeremićet al., Real-ESSI o h rttocss h correct the cases, two first the For eeice l nvriyo aiona Davis California, of University al. Jeremić et at: happen can This positive: be step predictor the over expenditure work external the that requires rule simplest The Work External Positive Sense Positive 107.6.1.1 in Path Equilibrium Traversing 107.6.1 3104 of 499 page: (107.13): equation from defined is change complete the then and . . . HYPER–SPHERICAL GENERALIZED, 107.6. if or, Notes ESSI ffciea ii ons oee,i al when fails it However, points. limit at effective that is conclusion simple The can (107.18) from solutions the so possible, are rule as: that categorized hypersurface from be constraint exceptions the some intersect However, generally points. which two directions, at possible two has path equilibrium the on • • • • nabgiyi nrdcdi h ouinfor solution the in introduced is ambiguity An a f iterates. divergent or erratic point, Complex behavior. iteration erratic by triggered be or point Real point. limit Real below. proposed schemes the of one no is Real δλ ∆ ∆ 1 ext T W u 0 = = new K roots roots roots ii or limit = t − − then: , 1 ∆ = f 2 f a ext T ext a ot.Ti sa nsa aeto tmysga hr unn on,abifurcation a point, turning sharp a signal may It too. case unusual an is This roots. 3 2 of of of ∆ u 0 = unn on nlsdb h osrithprufc.Tero scoe yapplying by chosen is root The hypersurface. constraint the by enclosed point turning u equal opposite old equal = + f ext T δ sign sign u ¯ K + in.Ti cuswe h trto rcs ovre omlyadthere and normally converges process iteration the when occurs This signs. t − opposite same δλδ 1 δλ f ext u hudhv h inof sign the have should ∆ λ> δλ t as λ to a ecoe yapyn n ftefloigschemes. following the of one applying by chosen be can that 0 that of of ∆ f ext λ ∆ old δλ λ and old hsi nuuulcs.I a inlaturning a signal may It case. unusual an is This . .Tetneta h eua point regular the at tangent The (107.18). from hsuulyhpeswe on vra”flat” a over going when happens usually This . K t − f 1 ext T f ext K are t − 1 f ext orthogonal: hscniini particularly is condition This . eso:2.Ags,22,15:04 2021, August, 26. version: (107.19) (107.22) (107.21) (107.20) Jeremićet al., Real-ESSI asstept to path the causes (107.21) rule work positive the of application point turning a Near Criterion Angle 107.6.1.2 points Turning eeice l nvriyo aiona Davis California, of University al. Jeremić et solutions both compute to is idea The effective. structure the point turning encountered. a is Physically, passing point impassable. in turning becomes another because point until incorrect turning is the so rule reverses work it positive point a turning the crosses it When itself. 3104 of 500 page: . . . HYPER–SPHERICAL GENERALIZED, 107.6. Notes ESSI rvn l neesr upie” o odacuto oeo upie nmtra olna nlssoemgttk a take might one analysis nonlinear 270. material pp. in (1991) surprises Crisfield of in some examples of some account at good look a For surprises”. unnecessary all prevent 12 • • n ih s wytetn unn onsi aeilnnieraayi?.Tease srte ipe tyto ”try simple: rather is answer The analysis?”. nonlinear material a in points turning treating ”why ask might One h ramn fbfrainpit so ahrseilntr n slf o h erfuture. near the for left is and nature special rather a of is points bifurcation of treatment The ops unn on moigacniino h nl ftepeito etrpoe more proves vector prediction the of angle the on condition a imposing point turning a pass To unn points, Turning points, Bifurcation ∆ ∆ u u 1 new 2 new ∆ = ∆ = 12 a etaesdb oicto fapeiu ue sdsrbdi h etsection. next the in described as rule, previous a of modification a by traversed be can iue173 ipeilsrto fBfrainadTrigpoint. Turning and Bifurcation of illustration Simple 107.3: Figure u u old old + + δ δ u u ¯ ¯ Load λ + + f δλ δλ 1 2 δ δ u u Point Bifurcation t t Point Limit δλ Equilibrium Secondary Path Path Equilibrium Primary 1 and δλ 2 n hnboth then and Displacement Point Turning u eso:2.Ags,22,15:04 2021, August, 26. version: ∆ p 1 new eessetra work external releases and double ∆ u 1 new akupon back (107.23) (107.24) : Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et for solution The point: equilibrium 3104 of where 501 page: solution the finding by implemented be . can . between . procedure HYPER–SPHERICAL angle GENERALIZED, The minimum direction the step backing. 107.6. with incremental double from old solution the the to prevent closest lies that one The Notes ESSI where n obtains: one (107.14) equation constraint the into one applying by achieved is solution predictor The step Predictor point. limit at no precisely that 107.6.2 arrive report to (1991) never Crisfield appears and one (1979) because Dhatt occurred, and has Batoz problem However, point, such limit solved. the be at Precisely cannot equations introduced. the is limitation a (107.25) to (107.12) equations negative. is work external the so reversed be should n olwn h ehdthrough method the following and (107.6) equation from constraint the introducing directly By ∆ ∆ cos

δ K u ¯ λ u u t ψ p p ref 2 φ = stetnetsins arxa h einn ficeet usiuigeuto (107.26) equation Substituting increment. of beginning the at matrix stiﬀness tangent the is u 2 = = = − (∆ ± K K k ∆ r ∆ t − u t −

∆ 1 u new 1 u ∆ u λ u r old ψ old ref 2 ψ old p ref 2 u 2 q ) u 2 e k k T sraiyfound: readily is | T δ and ∆ = ∆ S ∆ u (∆ ∆ λ t T ∆ (∆ λ p 2 u S p 2 u δ l δ λ new δ u u

new u p u new t K t T u t = ψ k S | ref 2 ) t − + u 2 δ (∆ + ) 1 ∆ K = u f δ ψ ext u t t − u ! f 2 k old 1 ∆ t T ∆ f ∆ = (∆ + ext S u λ u and δ old new old u netetrigpithsbe rse,tewr criterion work the crossed, been has point turning the Once . λ t k k λ p + ) ∆ T p δ 2 ) p u ψ ψ ∆ 2 ∆ new t f f 2 2 ψ u u ! ! f 2 old old forward n ec h aiu oieo h angle: the of cosine maximum the hence and , + (∆ = = = + δ δ u ¯ u ¯ + + Euler, l ) δλδ δλδ 2 ∆ u u u explicit t old t k steoesuh.Ti should This sought. one the is eso:2.Ags,22,15:04 2021, August, 26. version: tpfo h atobtained last the from step K t ssnua and singular is (107.28) (107.27) (107.25) (107.26) Jeremićet al., Real-ESSI .Teie st n h new the find to is idea work The this In (1991). Crisfield size. by length advocated step strategy the the controlling follow for will strategies we different advocated have workers of number A Increments Automatic 107.6.3 advised size. is step smaller one with negative, solution be converged previously to happens pivot one than eeice l nvriyo aiona Davis California, of University al. Jeremić et applied. be an can of that introduction criteria the convergence for several calls scheme iterative an of Introduction Criteria Convergence 107.6.4 (1981). iterations of number desired where applying: by 3104 of 502 page: procedure of simplified occurrence the the In to topic. research vigorous a still . . . HYPER–SPHERICAL GENERALIZED, so i.e. definite matrix, 107.6. non–positive scaling a as chosen is matrix where Notes ESSI 17 16 15 14 13 • o xml,i oain n ipaeet r involved. are displacements and rotations if example, For 2–norm. called so The phenomena. bifurcation account (1991). into Say Crisfield takes see one details if work more For to guaranteed not is Which fw s ulda norm Euclidean use we if vector Absolute ∆ aetesm nt ee novoscoc o h cln arxis matrix scaling the for choice obvious an Here, unit. same the have matrix Scaling ∆ ∆ I l desired > l new l old k 0 u ∆ = steodiceetlfco o which for factor incremental old the is δ ≈ u esrdi naporaenr,sol o xedagvntolerance given a exceed not should norm, appropriate an in measured , stegvniceetlnt.Teaslt au of value absolute The length. increment given the is Displacement k 3 absolute . l old S n eaiepvtdrn atrzto ftetnetsins matrix stiffness tangent the of factorization during pivot negative one sue nodrt nueta o rbe novn ie units mixed involving problem a for that ensure to order in used is I desired = δ I u old q t T 15 S ( Convergence δ δ 16 n h parameter the and u u n t ) h emnto rtracnb rte as: written be can criteria termination the T ≤ S 0 ( h usino hoigtergtsign right the choosing of question The . δ u S ) ≤ = rtra h hnei h atcorrection last the in change The Criteria. K u t ic,atrpsiglmtpit h tffesmti is matrix stiffness the point, limit passing after since, , n I sstto set is old 14 trtoswr required, were iterations 13 oso h nlssadtyt etr from restart to try and analysis the stop to h eaiesign negative the 2 1 Ramm (1982) Ramm by suggested as iteration | δ u t T S δ eso:2.Ags,22,15:04 2021, August, 26. version: u termination S t | − = snee ftestiffness the if needed is diag scoe ihrespect with chosen is + I desired or incremental ( 17 K − u δ l parameters all , et hr are There test. t − o example, For . u is (107.28) in 1 steinput, the is ) ftestate the of K f nthe on If, . t fmore If . (107.29) (107.30) length Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 503 page: . . . HYPER–SPHERICAL GENERALIZED, 107.6. Notes ESSI • • n oetm,i sipratt oeta urnl sdwti E n h elES rga is program Real-ESSI the matrix FEI scaling and total within unit used with the currently criteria by that convergence absolute note divided an to is important is displacements it iterative time, more all One of be norm number will a a norm of DoFs. Hence, larger function of a DoFs, number features. be of then model will number a tolerance larger DoFs specified of since of of problem case values a the create absolute will in up This that sums calculated. essentially expected 107.32 is equation it Since displacements, discretized DoFs. all be 5,000,000 can model) and (simplest DoFs cantilever feature a 5 therefor example, (and using For elements DoFs). finite similar and of for nodes number of criteria different number convergence a different displacement with discretized of are objectivity that preserve models to (same) order in important is This Average displacement In iterative dominate. of displacement norm displacements total Euclidean few of of or ratio norm one of use where that, problems to for to criteria order convergence relax and to (meter order in system the of decisions. units convergence Relative basic and comparison to for converted used is is that tolerance and Newton) Supplied (DoFs). unit Freedom with of criteria convergence absolute an is matrix program scaling Real-ESSI the FEI and within used Currently vector state in variables is mixed matrix have don’t we hand other oa ubro DoFs: of number total vector convergence and comparison for used is that to and converted Newton) is and tolerance (meter Supplied decisions. an system is (DoFs). the Freedom program of of units Real-ESSI Degrees basic the different and for FEI units within matrix mixes used scaling criteria currently unit with that criteria note convergence to absolute important is it Again k k u δ δ k esrdi naporaenr,sol o xedagvntolerance given a exceed not should norm, appropriate an in measured , u u Displacement u Displacement S k k k average relative scaled = I S . = = ≤ I hsmasta bouetlrneciei ie nt o ieetDegrees different for units mixes criteria tolerance absolute that means This . q u Convergence Convergence ( δ DFs nDoF u ) k T u S k scaled ( δ u ) rtra ti eeca oueterltv ovrec criteria convergence relative the use to beneficial is It Criteria. rtra h hnei h atcorrection last the in change The Criteria. ≤ srecommended: is u S S = = u I I hsmasta eaietolerance relative that means This . hntesmls hiefrscaling for choice simplest the then hrfraeaetlrnecriteria tolerance average Therefor . eso:2.Ags,22,15:04 2021, August, 26. version: k δ u k scaled u δ iie ythe by divided , u n Euclidean and ftestate the of (107.32) (107.31) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 504 page: . . . HYPER–SPHERICAL GENERALIZED, 107.6. Notes ESSI • • • Average units. matrix mixed scaling feature unit will a criteria tolerance criteria, convergence is this matrix for program, scaling ESSI for choice obvious an Here, and comparison for Relative used is that and Newton) and (meter system decisions. the convergence tolerance of Supplied units (DoFs). basic Freedom and to FEI of converted Degrees within is matrix different for used scaling units unit currently mixes with criteria criteria, criteria tolerance absolute convergence convergence a based is program displacement Real-ESSI for the as same the is Much matrix scaling for choice obvious an Here, norm Euclidean compare tolerance: to predefined be some would with test residual convergence of appropriate an path, equilibrium the from and Newton) Absolute and decisions. (meter convergence system and the comparison of for units used basic is to that converted is tolerance Supplied units. mixes ee novoscoc o cln arxis matrix scaling for choice obvious an Here, residual the of norm on in DoFs) DoFs, of of (number number discretization account model the vector: a into force of take influence to the need reduce a to is order there path, equilibrium the from departure step: iterative the of beginning the (scaled) at divided forces is residual norm of force norm residual a absolute a defined) by (previously criteria, tolerance to used are hrfrrltv oeac rtrawl etr ie units. matrix mixed scaling feature will unit criteria a tolerance criteria, relative convergence therefor this for program, Real-ESSI the FEI and k k k r r r ( ( ( Residual Residual u u u Residual λ , λ , λ , ) ) ) k k k scaled scaled relative Force Force Force = = = q q Convergence Convergence Convergence q ( DFs nDoF ( q r r ( ) ) r ( T T r 0 ) ) S S T T ( ( S S r r ) ) ( ( r r ≤ ≤ 0 ) rtra nodrt rvd cln frsda ocsthat forces residual of scaling provide to order In Criteria. rtra ic h residual the Since Criteria. ) rtra ihteresidual the With Criteria. r ≤ average r S S = S = diag = diag diag ( ( K K ( t t ) K ) oee,aan sntdaoe within avove, noted as again, however, t ) oee,wti E n h Real- the and FEI within however, , r S ( r u = ( eso:2.Ags,22,15:04 2021, August, 26. version: λ , u I λ , ) sue,teeo relative therefor used, is S esrstedeparture the measures ) hc esrsthe measures which , = I hsmasthat means This . S = I (107.33) (107.34) (107.35) sused, is Jeremićet al., Real-ESSI where is it the and render undesirable that is ratios with units work physical to on convenient more dependency this iteration Newton–Raphson of implementation eeice l nvriyo aiona Davis California, of University al. Jeremić et inequalities following of either if diagnosed be can Divergence occur: cycle. iteration erroneous of an sort interrupt Some converge. to guaranteed for used be can approach similar The displacement iterative of norm Euclidean scaled of ratio using k by dimensionless rendered be can Criteria 3104 of 505 page: . . . HYPER–SPHERICAL GENERALIZED, 107.6. Notes ESSI δ u • nte motn hn ob osdrdis considered be to thing important Another Since k scaled rmequilibrium. from the in reached that: follows it (1991), Crisfield by out pointed As point. this at appropriate is caution of word A work Energy hr h trtv hnewas change iterative the where esr fte”tffes of ”stiffness” the of measure a k k k k g k k r r δ δ u u u predictor predictor u u k k and k k u k k scaled scaled n cldEciennr fttldisplacement total of norm Euclidean scaled and r r scaled scaled hnecriterion: change k k k k and Based ( ( scaled scaled g δ δ r u u k k are ) ) ≥ ≤ scaled scaled r T T sal aepyia nts od necessarily do so unites, physical have usually g ( ( Convergence dangerous r r u u current ) ) k k ≥ ≤ = = g r r k q iterative ( ( δ δ u rwhfcosta a estt,frexample for to, set be can that factors growth u ) rtra h rvoscnegneciei a ecmie nasingle a in combined be can criteria convergence previous The Criteria. ) T T ( K K r ( Residual direction, ) δ t − t hsmrl mle hta that implies merely This . u ≤ 1 divergence = ) K t u − ( r r Convergence K ) δ k u t − hscnocrwe h ouini tl a away far still is solution the when occur can This . iegne h etnRpsnshm snot is scheme Newton–Raphson The Divergence. 1 = eeto ceei hrfrncsayi re to order in necessary therefor is scheme detection r u give (107.37) equations that noted be should It . − k and ( δ r u k Criteria: dimensionless. u ) T k scaled K t ( stationary δ u : u ) and ≤ k eso:2.Ags,22,15:04 2021, August, 26. version: Displacement r o eea purpose general a For . u g energy u r = g r oiinhsbeen has position 100 = Convergence . (107.41) (107.36) (107.40) (107.39) (107.38) (107.37) Jeremićet al., Real-ESSI exhibit eeice l nvriyo aiona Davis California, of University al. Jeremić et proce- The (107.1). Figure the with relation solution in ∆ converged described, previously briefly a from be starts will dure scheme the of 3104 of progress 506 The page: Progress Algorithm The are number 107.6.5 iteration the to limits Typical cycle. iteration one in performed 20 . iterations . . HYPER–SPHERICAL of GENERALIZED, number the 107.6. to Notes ESSI ol edt h etpoint next the to lead would displacements the to added When ∆ constants where 107.17 equation quadratic fe hs ausaecluae,teudtn rcdr ol edto: lead would procedure updating the calculated, are values these After point next the to lead would displacements the to added When hudb optdwith computed be should procedure updating the calculated, u λ 20 19 18 to old 1 n (107.13) and (107.11) See (107.12). equation From (107.6.2). Section in explained As nsm ae,teNwo–aho trto ceewl ete ieg o ovre u rather but converge, nor diverge neither will scheme iteration Newton–Raphson the cases, some In weeconstants where 107.17 equation quadratic use again then would iteration next The , ∆ ∆ ∆ 50 ∆ = oscillatory λ λ λ . 1 3 2 sobtained is ∆ = ∆ = λ 1 ocalculate to , λ λ 2 1 eair oaodecsiebucn rud odpatc st u pe limit upper put to is practice good a around, bouncing excessive avoid To behavior. + + δλ δλ 18 2 1 n h etpitotie is obtained point next the and ∆ u δλ ( ( old u u and and 1 3 2 λ , λ , ∆ = and u u 20 3 2 0 0 f f 19 ext ext n odlevel load and n odlevel load and ol edto: lead would u 2 ) ) δ . . u and ∆ ∆ 1 u u a = 3 2 1 ∆ a , ∆ = ∆ = − λ 2 ( old K u and u u 0 λ λ t − λ , ∆ = 2 1 0 0 1 tteedo h rvosiceetti process this increment previous the of end the at , tteedo h rvosiceetti process this increment previous the of end the at , r + + 0 a ( f ( 3 u δ δ ext u λ u u 1 hudb optdwith computed be should 1 2 λ , 2 1 ) λ , ocalculate to , niceetl agnilpeitrstep predictor tangential incremental, An . 1 1 + ) f ext ) δλ h rtieainwudte use then would iteration first The . 1 K t − δλ 1 f eso:2.Ags,22,15:04 2021, August, 26. version: ext 3 and fe hs ausare values these After . δ u ∆ 3 u = old a δ u 1 ¯ ∆ = a , + 2 (107.43) (107.42) δλ u and 1 2 δ and u a t 3 . Jeremićet al., Real-ESSI I olbrto ihD.Nm aazl n rf José Abell) Prof. and Tafazzoli Nima Dr. with collaboration (In (1989-2006-2016-2018-2019-) of Equations Motion Dynamic of Solution 108 Chapter 507 Jeremićet al., Real-ESSI n h olwn ieec eain r sdt relate to used are unknowns, are basic relations quantities the difference as displacement following use to the is and algorithm Newmark using motion of equations integrating hs qaosgv eainhpbtenkon aibe ttm step time step at time variables next knowns between relationship give equatons These eeice l nvriyo aiona Davis California, of University al. Jeremić et then: are predictors The equations: parameters, two two following uses the 1959 ) by (Newmark, method integration time Newmark The Integrator Newmark 108.3.1 (1987). Hughes and (1991) Mlejnek and Argyris follows section Equi- This Dynamic of Equations the for Methods Integration Direct 108.3 3104 of 508 page: ). 94 Dynamics page in on 102.2 Displacements section Virtual (see of Principle HIGHLIGHTS AND The SUMMARY CHAPTER 108.1. 108.2 Highlights and Summary Chapter 108.1 Notes ESSI hr r eea osbeipeetto ehd o emr nertr n osbeapoc to approach possible One Integrator. Newmark for methods implementation possible several are There edt ert n mrv!BJ improve! and rewrite to need (1991). Mlejnek and Argyris by book a is subject this on reading Great n n n n n n +1 +1 +1 +1 +1 +1 x x x x x x ¨ ¨ ˙ ˙ ˙ librium ======n 1 + n n β β − − x x ˙ ∆ ∆ γ 1 ∆ + ∆ + β β ehdi ngnrla mlctoe xetwhen except one, implicit an general in is Method . t t 2 ∆ ∆ γ 1 n t t n +1 2 t t n +1 n x [(1 n x x x ˙ + x − − ∆ + − − n β γ 1 n x ∆ ) 1 x 2 − t n t + x ¨ [( n β γ − x ˙ + 2 1 β + 1 − γ n ∆ 1 − x ˙ n β t +1 ∆ + 1 n ) β γ x − ˙ n x ¨ x ¨ + ] t 2 n + 1 β x ˙ 1 β + 1 − − n n x +1 ¨ 1 2 2 γ 1 β β x ¨ − ] n 2 +1 γ n n β x ¨ x ¨ x ˙ n and x ¨ n +1 n β x ¨ oteukonvralsat variables unknown the to 0 = eso:2.Ags,22,15:04 2021, August, 26. version: to n and β +1 and x γ n h response the and 1 = γ n sdefined is and , / 2 . (108.2) (108.6) (108.4) (108.3) (108.1) (108.5) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et becomes 108.9 Equation 108.11, Equation of change the to addition an introducing by form residual alternative an using 1978b) ac- parameter Liu, of and order Hughes ( the and 1978a) degrade Liu, will integration method time (HHT) integration Hilber-Hughes-Tailor time The Newmark curacy. the in introduced damping Numerical Integrator HHT 108.3.2 ratio spectral as obtained, is damping numerical strongest family for method method integration acceleration time average Newmark or the rule of trapezoidal members Any include: Well-known accurate. damping. second-order numerical and introduce stable unconditionally is procedure the ). 1991 Mlejnek , and (Argyris procedure solving iterative an constitute 108.9 to 108.5 Equation 3104 of 509 page: becomes method integration Newton The . . . FOR METHODS INTEGRATION DIRECT 108.3. correctors: the and Notes ESSI o ceeainieainand iteration acceleration for Decreasing method. integration If time Newmark 108.4. same and α the 108.3 exactly or becomes 108.2 method integration and time 108.1 HHT formulas finite-difference Newmark the retaining but if only and for method 1987). (Hughes, dissipation numerical increase value h trto ehdfrHTtm nerto ssmlrt hto emr ieitgain Due integration. time Newmark of that to similar is integration time HHT for method iteration The parameters the If γ n n n M +1 +1 +1 β β ≥ M ∆ ∆ x R x 1 ¨ ˙ 1+ (1 + α 1 2 t t γ 2 2 β = β , oipoeteperformance: the improve to = = M 1 = + 1 = M + γC ∆ / n n ≥ α n / 2 +1 +1 t 1+ (1 +1 ) 6 .Frvle of values For (1987). Hughes second-order is accuracy the , γ 4 1 + γ x x ¨ ˙ and ∆ x β ¨ ( ∆ γ K tC 1+ (1 + + + and t α + ) γ β β ∆ 1+ (1 + C 1 2 ∆ ∆ γ 1 = γ 1 x ) 1+ (1 + 2 t t satisfy = 2 α n / n +1 ) 2 +1 − F α n epii)cnrldffrnemto for method difference central (explicit) and , x n ( ) x n +1 α β +1 ∆ ) R K x, t ˙ 2 K n ∆ +1 x ∆¨ x = ) x − − = αF n − +1 n ( R +1 n x, ˙ R α ρ n ∞ uhsand (Hughes 1977), al., et (Hilber -method x β ) 0 = − 1 = n ,pg 502) page (1987 ), (Hughes +1 / 4 f and γ γ eso:2.Ags,22,15:04 2021, August, 26. version: au rae hn05will 0.5 than greater value 1 = β β / 1 = 0 = 2 ieracceleration linear , and and γ γ 2 = α 1 = (108.10) (108.13) (108.12) (108.11) 0 = (108.9) (108.7) (108.8) / 3 / the , the , 2 If . Jeremićet al., Real-ESSI o page on 102.12.1.4 section in described when occurs as equation damping example displacements, viscous for differential natural (see have a 138 solid example, For porous soil. and and fluid fluid pore of interaction during created is that the of of and matrix damping damping solids viscous the of mimicking generating are analysis by damping, domain structures Caughey and time solids and for Rayleigh methods methods, damping These viscous structures. used, Structures commonly and are Solids here for Presented Damping Viscous Synthetic 108.4 1987). (Hughes, accurate second-order and stable unconditionally is it eeice l nvriyo aiona Davis California, of University boundary the of nodes and element al. Jeremić the et that Schanz assumed is and it (Nenning method infinity this at In boundaries (1997 )). model al. at et Kallivokas laws (2010); decaying using waves the absorb can elements approximate( 1991)). to (Givoli order waves in seismic boundaries for model condition the conditions. radiation at boundary the absorbing conditions is special method have used conditions commonly boundary the of Absorbing One (2003a)). al. boundaries. et mesh the (Bielak at reflection reflections of problem models the large other reduce for with to available order method also in is this element method coupling finite reduction on in Domain implemented done applications. been better has for Research methods numerical methods. numerical other model to the comparing of boundaries the from spectral motions or the elements (2010)). of al. reflection finite numerical et as or as (Semblat such elements disadvantage such finite also methods low-order but nonlinearities for using the dispersion of modeling or advantages geometries are complex methods, for There elements meshfree differences, the finite elements. through elements, propagation spectral finite wave and elements, seismic boundary the as simulate such to systems available soil-structure methods numerical different are ApproachesThere Damping Viscous Synthetic 108.4.1 3104 of 510 page: . . . FOR DAMPING VISCOUS SYNTHETIC 108.4. iteration. displacement for Notes ESSI hs ytei icu apn prahssol edsigihdfo aua icu damping viscous natural a from distinguished be should approaches damping viscous synthetic These nte ehdapial nfiieeeetmtosi ocle nnt lmn ehd These method. element infinite called so is methods element finite in applicable method Another the attenuate can directly which conditions boundary non-reflecting called so methods are There boundaries from motions the reflecting of issue the with better deal can method element Boundary parameters the If − 1 / 3 ≤ α ≤ 0 γ , α , β = and 2 1 (1 ?? γ − satisfy npage on 2 α ) β , ?? = ). 4 1 (1 − α ) 2 C sn asadsins matrices. stiffness and mass using eso:2.Ags,22,15:04 2021, August, 26. version: (108.14) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et factor (Semblat quality damping the of Caughey inverse for as: the dependence of attenuation/frequency relation The same the (1997)). involving model a matrix build damping the creating for used the are matrices they which mass equations, and in of stiffness system method the the well. classical Since solving as for a matrices. created stiffness is be and damping to mass have Caughey the of on based damping. built Rayleigh is a matrix damping as known also is method a with matrix. dampers damping viscoelastic Caughey added of the form of the in effects matrix the damping approximate methods these dampers. viscoelastic waves body as well formulations as waves PML (2009)). surface Basu several of 3104 (2003); are treatment of Nielsen There the 511 and allows (Festa page: which boundaries. methods model specific element the with finite laws at for attenuation proposed located on thickness based finite is layer and absorbing properties This (PML). and Layers boundaries Matched the . Perfectly . . at called FOR dissipate DAMPING VISCOUS to SYNTHETIC distance enough model. 108.4. the have to waves back seismic reflect the to case not this In infinity. in are Notes ESSI opyia age apn.I a lob sdfrdmigottersda ae oigotof out coming waves residual the with out them damping of for of rest used elements the layer. be the leave boundary also and all reduction can elements for domain particular It the damping for use used damping. to be Caughey need could physical a damping no case, be this not problem. In might the There model. in the damping elements. for different considered for be damping to of modes of number on depends used be to order the where osdrn h eainhpbtenitra rcinadfeunyfrdmig ti osbeto possible is it damping, for frequency and friction internal between relationship the Considering The Method. Layer Absorbing Caughey called so is here used be to method The added with structures in damping model to methods energy-based two proposed (2006) al. et Bilbao boundaries from waves the of reflection prevent to method developed recently a been has There age apn omlto ngnrlcnb xrse as expressed be can general in formulation damping Caughey h a ti mlmne nES iuao ie h potnt oteue ouedffrn types different use to user the to opportunity the gives Simulator ESSI in implemented is it way The Q C − [ = 1 ≈ M 2 ] ξ m X j =0 − 1 a j ([ M ] − 1 [ K ]) j Q − 1 n h apn ratio damping the and eso:2.Ags,22,15:04 2021, August, 26. version: 2 ξ nd a ewritten be can re fthis of order (108.15) (108.16) Jeremićet al., Real-ESSI switnsprtl o ahterm: each for separately written is eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 512 page: observed: with be can damping, equations following Rayleigh system a the of as formulation . known . and . FOR parameters also DAMPING dynamic is From VISCOUS SYNTHETIC (108.16). damping, Caughey 108.4. order second The Damping Caughey 108.4.2 Notes ESSI otedmigrtoo the of ratio damping the So Then coefficient: second the on based matrix damping the Writing (108.17), Equation on Based formulation matrix damping the if formulation, damping Caughey in terms two first the Considering o rttomdsi arxfr ed to: leads form matrix in modes two first for (108.23) Equation Presenting ξ a C a C K ξ 2 1 n n 1 0 n   = = = = = 2 = a = ω ω 1 1 1 a a j i a 2 2 ω 2 1 0 ω ξω a eotie as: obtained be can M ξ 0 K M n C 2 ω ω ω M n 1 n j i n = ω   n n + a   1 a a a ω 2 1 0 1 2   ω M n =   ξ ξ j i   n a th 0 2 nd a ewitnas: written be can oeo h ytmis: system the of mode re,aaRyeg Damping Rayleigh aka Order, eso:2.Ags,22,15:04 2021, August, 26. version: j 1 = nEquation in (108.18) (108.22) (108.21) (108.20) (108.19) (108.17) (108.23) (108.24) Jeremićet al., Real-ESSI fteebud r aifcoiynro,teconstants the narrow, satisfactorily are bounds these If coefficients eeice l nvriyo aiona Davis California, of University al. Jeremić et If ratio damping Actual 108.1: Figure value damping actual an compute to used be can and by bounded ratio damping a have will mode such Any range. frequency where 3104 of 513 page: where . . . FOR DAMPING VISCOUS SYNTHETIC 108.4. Notes ESSI dominates. (108.23) Equation in term last the as relation linear (2006)). (Hall damping Rayleigh ω sueu ocneinl eemn alihdamping Rayleigh determine conveniently to useful is 2006) ( Hall by given procedure following The hw that shows (108.1) Figure n sotieterange the outside is a a = ∆ ∆ > R 0 1 2 = 2 = eemnsbud ntedmigrto htaeipre otoemdswti h specified the within modes those to imparted are that ratios damping the on bounds determines ξ 1 a + 1 + 1 ξ ξ Compute . 0 ω ˆ ω 1 ˆ and + 1 + 1 R R 2 + − R a R 2 1 2 2 + eetadsrdaon fdamping of amount desired a Select . R 2 √ √ 2 + ∆ R R √ √ ω ˆ from: R R to ξ n R ω ˆ = then , ξ ξ n max fmode of ξ if n ω ξ > n n ˆ = max safnto ffrequency of function a as ω Above . or ξ a n ω 0 o mode for n and = R ξ R a ω ˆ 1 n rqec ag from range frequency a and ω ˆ , r hncluae from: calculated then are n that and , ξ n n rmEuto (108.23). Equation from nrae with increases ξ max eso:2.Ags,22,15:04 2021, August, 26. version: = ξ ω n ξ n +∆ = fmode of ξ min ω and n prahn a approaching , if ξ n min ω hnusing when n ω ˆ = = (108.27) (108.26) (108.25) to ξ √ − R R ∆ ω ω ˆ ˆ . , . Jeremićet al., Real-ESSI as: following: as written be can lation the as logic same the Following Damping Caughey well-chosen. not 108.4.3 is range frequency the if range, frequency prescribed the eeice l nvriyo aiona Davis California, of University al. Jeremić et The Damping Caughey 108.4.4 3104 of 514 page: . . . FOR DAMPING VISCOUS SYNTHETIC 108.4. Notes ESSI osdrn h atceceti h omlto,tedmigrtoo h ytmcnnwb shown be now can system the of ratio damping the formulation, the in coefficient last the Considering So outside motions for high unrealistically be could damping that out pointing worth It’s Note: ysligtefloigsto equations, of set following the solving By 4 th ξ a C C a 2 1 n a 2 3      2 = = offiin fteCuhydmigfruaincbotie as: obtained cab formulation damping Caughey the of coefficient = = = ω ω ω a eotie as: obtained be can 1 1 1 a a k j i a ω ω 2 2 2 2 3 0 ξ ξ 3 5 KM KM ω ω ω ω 1 n k j i + − − ω ω ω 1 1 KM K a j i k 3 3 3 2      1 ω =     n − a a a a + 2 1 0 1 2     K ω a 2 4 = 2 = M ω 2     nd n a 3 ξ ξ ξ 3 3 4 k j i ω rd th re,tels offiin fthe of coefficient last the order,     6 M Order Order 3 rd re age apn offiinscnb found: be can coefficients damping Caughey order 3 rd re age apn formu- damping Caughey order eso:2.Ags,22,15:04 2021, August, 26. version: (108.29) (108.28) (108.33) (108.31) (108.30) (108.32) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 515 page: . . . FOR DAMPING VISCOUS SYNTHETIC 108.4. Notes ESSI otedmigcecet a eotie ysligtefloigsto equations: of set following the solving by obtained be can coefficients damping the So ξ 2 1 n        = ω ω ω ω 1 1 1 1 k j i l a 2 0 ω ω ω ω ω 1 n k j i l + ω ω ω ω a i j l k 3 3 3 3 2 1 ω ω ω ω ω n i j l k 5 5 5 5 +               a 2 a a a a 2 3 2 1 0 ω        n 3 = +        a 2 3 ξ ξ ξ ξ k j ω i l        n 5 eso:2.Ags,22,15:04 2021, August, 26. version: (108.35) (108.34) Jeremićet al., Real-ESSI I olbrto ihD.Nm aazl,Po.JseAel r unFn,D.HxagWn ) Wang Hexiang Dr. Feng, Yuan Dr. José Abell, Prof. Tafazzoli, Nima Dr. with collaboration (In (1989-2001-2006-2016-2018-2019-2020-2021-) Aspects Interaction, Theoretical Structure Soil Earthquake 109 Chapter 516 Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et E E energy the of Part outside ways. domain of into number back a radiated in and dissipated reflected is be system will SSI system enters SSI that the enters System that SSI energy in Seismic Dissipation Energy Seismic SSI with dissipated 109.2.2 be to needs that energy kinetic incoming the region. represents then difference field the wave outgoing while from obtained be can energy kinetic Outgoing using calculated be can flux energy 122): Alternatively, page elements. 2002), of Richards, layer and that ((Aki to belonging nodes for model boundary the of outside just where Kinetic as: 2003a ) 109.7). , al. Figure et (see (Bielak Method structure Reduction the Domain and using system surface closed foundation through as flux well energy as volume soil (significant) ( waves source the at energy system amounts large SSI release into Earthquakes input 3104 energy of Seismic 517 page: 109.2.1 Dissipation and Jeremić(2010) Propagation Energy HIGHLIGHTS AND Seismic SUMMARY CHAPTER 109.1. 109.2 Highlights and Summary Chapter 109.1 Notes ESSI r r 1 4 = 6 = • • ehnclsimcwv nryetr h S ytmtruhacoe surface closed a through system SSI the enters energy wave seismic Mechanical o xml,sm ftercn ag atqaeeeg eessaelse:Nrhig,1994, Northridge, listed: are releases energy earthquake large recent the of some example, for S ytmoclae n mt,rdae ae akit h domain the into back waves radiates emits, and etc.). oscillates foundations, system layers, SSI soil/rock surface, (free boundaries impedance from reflection wave E . M . 8 8 ≈ flux × × be Ω+ 1 10 h E 10 . 0; , 6 flux 20 = 16 M × − J J ρAc adva hl,1960, Chile, Valdivia, ; eb M oaPit,1989, Prieta, Loma ; Ω+ 10 = be Ω+ − , Z 5 K 0 n ato hteeg ae tt h ufc hr S ytmi located. is system SSI where surface the to it makes energy that of part and ) u ¨ t be Ω+ e 0 u ˙ − i 2 , dt K K eb be Ω+ Ω+ Γ while , u r asadsins arcs epcieyfrasnl ae felements of layer single a for respectively matrices, stiffness and mass are e 0 ; M Γ M M Richter Richter nldsbt noigadoton ae n a ecalculated be can and waves outgoing and incoming both includes eb Ω+ u ¨ e 0 u ¨ b 0 and 6 = 9 = + K . . u 9 5 eb , Ω+ e 0 , E E r ceeain n ipaeet rmafe field free a from displacements and accelerations are r r u 7 = b 0 1 = i 1 i . × ato eesdeeg srdae smechanical as radiated is energy released of Part . 5 1 × × u 10 i 10 20 17 J J ; w uar-naa,2004, Sumatra-Andaman, ; i (2003a)), al. et Bielak DRM, (from , Γ eso:2.Ags,22,15:04 2021, August, 26. version: by Γ htencompasses that M M Richter Richter 6 = 9 = . . 7 3 , , Jeremićet al., Real-ESSI nrydsiainta sbigwdl sd eeaesm details: some are Here used. widely structure). being the is and that foundation dissipation (soil, energy system the of components plastic three of all result in a present is the is dissipation of proportional and part displacement dissipation major This, for earthquakes. responsible large probably for is dissipation structure energy and foundation soil, of deformation Elastic-plastic Plasticity by Dissipation Energy 1987). (Hughes, production 1991 ), or 109.2.2.1 , Mlejnek (damping) and (Argyris dissipation controlled energy carefully numerical be is, to has That damping) (negative means. numerical purely by produced eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 518 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI • • • • • • oeaotpatcdsiaini motn tti on.Teei icneto bu plastic about misconception a is There point. this at important is dissipation plastic about note A or dissipated be can energy the of part simulations, numerical in that note to important also is It system: SSI within mechanisms following the of one through dissipated is energy seismic of rest The ytrtcenergy hysteretic energy, damping where that stated was It displacement. dissipation incremental hysteretic times force as term energy defined The hysteretic simply energy. by was strain indirectly step elastic calculated and time is energy energy each absorbed hysteretic of paper, difference this with energy the compared In input taking were building. results absolute multi-story analysis on a Numerical based on discussed. methodology experiments and analysis presented energy was An demand) energy 2017 ). and (or al., Mezgebo 2008; 2014 ; et , al. , Mahmoud Deniz et and 2017; Symans Moustafa 2008 ; Lui, 2011 ; , 2007 , Moustafa (Léger Kunnath, 2011 ; researchers and Saravanathiiban, many Kalkan and by 1993; Gajan demand al., structural et of Cosenza measure 1992; a Dussault, as and energy using in work definitive the Misconception: the of Origins water) (air, fluids external and internal surrounding, with structure of water) coupling (air, Viscous fluid pore with solid porous of coupling Viscous structure the of behavior damage elasto-plastic, Inelastic, system foundation the of behavior damage elasto-plastic, Inelastic, rock and soil of behavior elasto-plastic Inelastic, E E i i = ste(boue nu energy, input (absolute) the is E k + E E E sircvrbe hc niae htti aaee a osdrdtesm as same the considered was parameter this that indicates which irrecoverable, is a ξ h . + steasre nry hc scmoe featcsri energy strain elastic of composed is which energy, absorbed the is or E lsi dissipation plastic a = E k + E a enconsidered been has 1990) ( Bertero and Uang by paper The ξ + E neuto o nryblnei given: is balance energy for equation An . E s + k ste(boue iei energy, kinetic (absolute) the is E h eso:2.Ags,22,15:04 2021, August, 26. version: bobdenergy absorbed E ξ steviscous the is E (109.1) s and of Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 519 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI • h lt,wihi la ilto ftemdnmc.Tecag fpatcfe nrywsnot was energy free plastic of change The thermodynamics. in of observed violation was clear dissipation) a ap- (plastic is dissipation energy which energy hysteretic plots, the incremental plastic Negative so results. used the was in (SDOF) material peared single-degree-of-freedom Inelastic a loading. of earthquake dissipation under energy oscillator in calculated true (2007) not Kunnath generally and is energy. Kalkan which dissipated constant, total remains the energy to free equal materials. plastic approximately elastic-plastic if total is valid the only motion that is ground stated statement of was This end It the loading. at earthquake energy energy under input perform buildings to multi-story ( 1990) Bertero of and analysis Uang response from 109.1 Equation number used vast (1992) systems. a Dussault Léger ESSI in and on of carried analysis been energy has on dissipation studies energy of of misconception the (1990), Bertero the thermodynamics. and of with violation Studies: contradiction a Other also in in was was damping Misconception result which Such viscous paper. (accumulated) periods, same non-negative that time the remains in plots certain derived it the during equation that of dropping ensure one clearly to in was used able appeared energy equation the be it the that However, should ignored fact, was In dissipation incrementally. it energy non-negative. But viscous be compute non-negative. also to should be always dissipation should energy term viscous incremental this that paper. stated this author in The in found is dissipation energy energy regarding issue energy another on studies Besides, following structures. all Unfortunately, and almost soils by earthquake period. magnified) of not time analysis (if any inherited during been non-negative has and misconception incrementally work plastic this be of must difference the energy) as free energy (defined free dissipation plastic plastic plastic and but work negative, plastic incrementally Both be energy. can free to plastic and energy dissipation hysteretic plastic renaming of by combination clarified be could negative misconception This of found indications paper. were the this, thermodynamics, of doing of sections principles After various basic dissipation in the plots. plastic violates these and which from dissipation, components energy deducted energy since incremental or other paper, of calculated this plots easily in were be energy) There (hysteretic can thermo- directly. dissipation of defined plastic correctly law not of to second was plot necessary the it direct is uphold no which to was energy, and There free materials plastic dynamics. elastic-plastic of of absence dissipation the energy is evaluate theory this of problem The o icu apn nry a acltddrcl sn apn offiin n velocity. and coefficient damping using directly calculated was energy) damping viscous (or Uang in used parameter key the was energy input Although eso:2.Ags,22,15:04 2021, August, 26. version: lsi work plastic icu damping Viscous hc sthe is which , Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 520 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI • ) thsbe elzdta hsassumption this that realized been has Dolinski It ( 1996), al. (2013)). et al. 1.0 et Zhou to Osovski (1991), (2010), 0.6 al. Li between et and Belytschko constant Ren (1984), a (2010 ), al. be al. et et to Clifton assumed (e.g. was studies ratio many this in data, (Quinney–Taylor experimental heating of into amount converted large work plastic as of as known denoted ratio is usually work The plastic coefficient), work. non-recoverable cold the of mechanical of input energy part the stored remaining 1934) ( of the The all, Quinney not heat. and but into Taylor part, converted large is 1925), ( a energy Taylor that proved and and metals Farren early on the colleagues experiments In performed his science. and material Taylor and century, engineering 20th mechanical Work): from researchers (Cold by Energy extensively Free studied Plastic on without Studies ignored Early significant, be or not should negligible energy evidences. be free experimental could or plastic realized mistake reasoning Nevertheless, not this plausible analyzed. still of case is influence This the dissipation the on energy that depending equation. plastic Note balance and energy researchers. work many wrong plastic by the of misconception using the obtained that plastic were considering means results without systems Misleading Lui ESSI and on energy. analysis Mezgebo energy free (2014); performed Mahmoud (2017) and al. Moustafa et (2011); Deniz (2017); Moustafa plastic studies of recent drops of large number since The A case, this dissipation. plots. in the energy significant in plastic was noticed thermo- system and were the of work work of violation plastic energy direct of free a plastic misconception in and is a change soil which was the this periods, both time Again, in certain dissipation ele- dynamics. during energy structural hysteretic decreasing and that was soil observed structure foundation be the in can dissipation It Energy calculated. experi- were system. centrifuge ments foundation and rocking simulations a numerical on both publications. ments performed other (2011) in Saravanathiiban appeared indicated dissipation and clearly plastic Gajan and statement work such plastic earthquake. observed, of of was misconception end same thermodynamics the cumulative the at of the energy) present violation that input ones direct stated (absolute the demand no was energy to Although It the close to (1992). very equal Dussault is was Léger and energy which and hysteretic equation, (1990) Bertero balance and energy Uang the consideration by in no energy was of There free application structures. plastic the of in of protection developments seismic for recent systems and dissipation practice energy publications. current passive these summarized in (2008) noticed al. be various et can on Symans misconceptions analysis similar energy and performed theory, authors same same the the using by structures papers Several study. this in considered β a enue nams l ae aeso hstpc ae on Based topic. this on papers later all almost in used been has , hsisehsbe one u and out pointed been has issue This eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI sosta tffca n es adhave sand dense and clay stiff for 109.1, that Figure shows in 109.1 shown Figure as While capacities, as dissipation clay cycle. energy soft loading-unloading-reloading different and predict single stiff send of loose energy models and recognizing elastic-plastic dense by simple design as example, well in For used soils. be can different soil for to capacity dissipation dissipation energy direct to possibility The structures. eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 521 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI dal,mjrt fteicmn nrywudb isptdi ol eoerahn onainand foundation reaching before soil, in dissipated be would energy incoming the of majority Ideally, eto lsi reeeg scue ypril eragmn ngaua sebyudrexternal under develop- assembly granular scale. in particle rearrangement on particle by analyzed loading. caused is a is that often energy example is free plastic conceptual which of work, a ment plastic through used and confusion. The is plasticity a latter to work. of the due cold while source dissipation as engineering, energy geotechnical concept and engineering. same physics mechanical the solid in is in energy popular more free is devel- plastic for term that former thermodynamics Note of law second models. Collins the constitutive (2002); Collins enforce Kelly oped to (1987); and (2007). to Collins Wehrli Dafalias (2002); used and and Collins Feigenbaum frequently Ziegler (2000); (2003); Puzrin are (1975); and thermodynamics Popov Houlsby (1997); of and Houlsby principles Dafalias and basic models the constitutive Italy. engineering, new the in civil slide derive model Vajont of to 1963 field used the is like the landslides, thermoporomechanics In large which presented in in increase is 2012), pressure engineering pore 2007, and geotechnical al., (impact, heating in et loading (Veveakis theory cyclic of this or papers of much monotonic in heat application rapid conducts in One air reasonable the earthquake). since is simply assumption vibration, condition, to This assumptions adiabatic some the metal. with is than issue assumption slower same used the widely on One (Hodowany 2002) colleagues al., his problem. et and Rosakis Ravichandran by 2000; validated papers al., follow-up is some is and et are properties, model There material This experiments. and metal. of dissipation in sets energy by of of evolution evolution the the calculating model of to capable thermoplasticity presented (2000) on al. based et model Rosakis during constitutive works. polymers a theoretical of 2003) and generation) al., experimental (heat been et both dissipation presenting Rittel has loading, energy 2000 ; there cyclic the Rabin, years, on and papers 20 Rittel insightful 2000; recent several (Rittel, the published Rittel effective In issue. only this the (heat). on all is dissipation developments almost it many energy (and because measure study material, this directly the in in to heat used distribution approach into was temperature converted imaging obtain work Infrared to plastic dependent. studies) of rate future fraction strain the and that strain showed both (1994) is al. et Quinney–Taylor Mason of later, evolution the Decades involve to models. complicated constitutive too thermomechanical simply in it’s but coefficient cases, all in valid not is eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI lsopatct fsldskeleton solid Soil/Rock of in Elasto-plasticity Nonlinearities by Dissipation Energy ). 1991 Mlejnek , and (Argyris 109.2.2.4 computations throughout system the of energy maintain to impossible is it eeice l nvriyo aiona Davis California, of University al. Jeremić et N: for that HHT: so for constants while proper changing by with models system elastic linear elastic for modes linear quency for preserving Hilber– ( energy and constants are 1959 ) of 1977) (Newmark, choice al. , (N) Newmark et example (Hilber For dissipation) (HHT) (energy Hughes–Taylor damping. positive the production) is (energy systems var- negative (elastic-plastic) in nonlinear and energy for calculated effect affects common motions Most of ways. equations ious nonlinear of integration numerical above, noted Production As and Dissipation Energy Numerical 109.2.2.3 using modeled solid realistically porous is of dissipation coupling viscous in particular, results In fluid dissipation. and energy components structural proportional or velocity foundation and/or for particles responsible soil and is water...) (air, fluid pore of coupling Coupling Viscous Viscous by Dissipation Energy larger 109.2.2.2 much undergo flexibility. can through soils capacity soft/loose dissipation that energy note increased to offering thus important deformation/strain, is it capacity, dissipation higher much 3104 of 522 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI icu culn)effects (coupling) Viscous iue109.1: Figure − 0 . 3 E 3 α ˙ vc ≤ 0 = nrydsiaincpct o n yl tvrossrisfrfu eei soils. generic four for strains various at cycle one for capacity dissipation Energy = α . ≤ 0; n 2 β 0 k γ , − 0 = 1 ( U ˙ . 0 = 25 i − γ , . u 5(1 ˙ u i 0 = ) − 2 − p nryls e ntvlm.I sntdta hstp of type this that noted is It volume. unit per loss energy . 5 2 − .Bt ehd a lob sdt ispt ihrfre- higher dissipate to used be also can methods Both ). α ) U β , 2008). al., (Jeremić et formulation 0 = . 25(1 − α ) 2 oee,frnnierproblems nonlinear for However, . γ eso:2.Ags,22,15:04 2021, August, 26. version: ≥ 0 . 5 β , 0 = . 25( γ +0 . 5) 2 , Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Models Empirical Motions: Seismic Production 109.2.3 and Dissipation Components Energy and Numerical Systems Structures, in 109.2.2.8 Nonlinearities by Dissipation Energy Isolators Seismic in 109.2.2.7 Nonlinearities by 3104 Dissipation of 523 Energy page: 109.2.2.6 . . . AND PROPAGATION ENERGY Zone SEISMIC Interface Foundation 109.2. – Soil/Rock Slip in Gap, Nonlinearities by Dissipation Energy 109.2.2.5 Notes ESSI Saturated Dry eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI ny aeeuto,ta ecie aepoaaini vertical, in propagation wave describes that equation, wave a only, respectively. ments eeice l nvriyo aiona Davis California, of University al. Jeremić et M (9 density homogeneous mass is layer h, Each thickness halfspace. the velocity a by horizontal wave represents the characterized compressional wave that in is 1C infinity bedrock and a to is isotropic layers extend is of and that 109.2 3104 layers, bottom Figure of horizontal the in 524 N At Shown page: of direction. consists 1996a). ( model Kramer The example, follow setup. for here propagation by, shown Developments found section. comporessional as this and/or in approach, shear described standard of is system . propagation . viscoelastic . vertical AND linear PROPAGATION with the ENERGY through associated SEISMIC waves is propagation 109.2. wave of theory The Modeling Propagation Wave 1D/1C 109.2.4 Notes ESSI KG = 3 2 opesoa aevlct slsdt bancntandmodulus constrained obtain to lused is velocity modulus wave shear Compressional shear obtain to used is velocity Wave Shear ouigo etclpoaaino ha ae,adwt rsneo ipaeet in displacements of presence a with and waves, shear of propagation vertical on Focusing displace- vertical of horizontal only cause will waves compressional or shear of propagation Vertical E ) ρ u / (1 (3 ∂ ∂t = − K 2 u 2 u ν + ) ( = / ,t z, g ( + ((1 ) G , ) E ∂ ∂z 2 ( = ν u 2 )(1 G + − (3 η iue192 rbe ecito:Wv Propagation Wave Description: Problem 109.2: Figure 01 0 1 0 1 0 1 0 1 0 1 0 1 0 1 bedrock . layer #m . layer #2 layer #1 surface M 2 ∂z ν ∂ )) − 3 2 ). V u ∂t 4 p G 3 )) n apn factor, damping and / ( M − G ) ). β . G , Vsm, Vpm, V s = ρ2, β2 Vs2, Vp2, ρ1, β1 Vs1, Vp1, V M p G/ρ p z = ietoscnb rte as written be can directions , ρ ρ p G βm m, ha aevelocity wave shear , M/ρ = eso:2.Ags,22,15:04 2021, August, 26. version: V , s 2 M ρ , = G V = p 2 ρ E/ , M 21+ (2(1 = x K V direction ν s (109.2) (109.3) 4 + )) 2 , and , G/ E = 3 , Jeremićet al., Real-ESSI n obtains one ( 109.3) Eq into (109.4) Eq Substituting layer z–direction. the downward and negative, direction the z in positive traveling in upward, wave the reflected in the traveling represents wave term incident second the represents term first of the where motion harmonic a for equation wave the obtains one (109.6) Eq. and frequency (109.4) Eq. combine By eeice l nvriyo aiona Davis California, of University al. Jeremić et β where which in solution: general the has which 3104 of 525 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI srltdt h viscosity the to related is , nrdcn oa oriaesse o ahlyr h ipaeet ttetpadbto of bottom and top the at displacements the layer, each for layers. Z rock system and coordinate soil local the a of Introducing each for valid is equation This ratio damping soil use use can one convenience, For frequency with displacements oscillation, Harmonic m ( u u u G ωη u k U k G 2 m m ( ( ∗ ( r : are ,t z, ,t z, stecmlxwv ubrand number wave complex the is z = ( ( + 2 = = = ) z z ω = ) = ) iωη G = ( = 0) = . G Gβ ρω Ee + h + ) U Ee m ∂ iωη ∂z iωη 2 ikz ( ( = ) 2 z i u ( 2 ) kz + E · = = = + e m E e F ωt iωt G ρω ρω m G + ) 1+2 + (1 − ∗ + 2 · 2 F ikz U e η m e F ik ) m by e iβ − h iωt m i ( ) kz + − F ωt m ) e G − ik ∗ m stecmlxsermdls h rtcldmigratio, damping critical The modulus. shear complex the is h m ) · e iωt ω β a ewitni h as: the in written be can , orpeettecmlxsermodulus. shear complex the represent to eso:2.Ags,22,15:04 2021, August, 26. version: (109.12) (109.11) (109.10) (109.4) (109.9) (109.8) (109.7) (109.5) (109.6) Jeremićet al., Real-ESSI utato n diinEs 5ad1 il h olwn euso omlsfrteamplitudes, the m: for formulas recursion following the yield 16 and and 15 Eqs. addition and Subtraction where eeice l nvriyo aiona Davis California, of University obtains al. it Jeremić et (109.15), Eq. to according addition, In zero. be must stresses shear E the surface, free the At 3104 of 526 page: form, another In . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI ,(109.12), (109.11), Eq by Hence, interfaces. are all coefficients at the (109.16), continuous and be (109.15) must displacements and Stresses layer of bottom and top the at stress shear the and 1 h ha teso oiotlpaeis plane horizontal a on stress shear The = F m E α F E τ τ E τ τ α F m m ( ( +1 m m m m m m ,t z, ,t z, 1 ( ( +1 +1 +1 +1 aey h mltdso h nietadrflce ae r laseula h free the at equal always are waves reflected and incident the of amplitudes the Namely, . z z = fteicdn n eetdwv nlyrm1 xrse ntrso h mltdsi layer in amplitudes the of terms in expressed m+1, layer in wave reflected and incident the of , stecmlxipdneratio impedance complex the is = ) = ) = = 0) = = − + = k m h F F k 1 2 1 2 G ikG m +1 m m m E E = ) · +1 +1 G G m m ∂u ik ∂z ∗ m ∗ m ∗ (1 1+ (1 ( = = ik Ee m +1 − + G m E k ikz ( = α m ∗ m G η α m k m +1 ∂z∂t m ( m ∗ e m − E ∂u ik ) ) ρ ( G G e e m Ee m m e F ik ik ρ m ∗ m ∗ h +1 − m m m m = ik +1 − h h G G F m ikz m + m G ( m h m ∗ m ∗ E + m F + ∗ ) ) +1 e m ∂u ∂z e m − iωt 1 2 iωt 1 2 e e ) F ik F 1 − e F / m m m ik 2 1+ (1 h (1 m − m ik h m − − m h α α F m m m m ) m ) e ) e e e iωt − − − r respectively: are ik ik ik m m m h h h m m m ) eso:2.Ags,22,15:04 2021, August, 26. version: (109.18) (109.19) (109.20) (109.17) (109.21) (109.13) (109.16) (109.14) (109.15) E m +1 Jeremićet al., Real-ESSI refil oin a eFuirsnhszdfo h oohoai solutions. monochromatic the from synthesized Fourier be can frequency motions angular field with free monochromatic 1953). be velocity The (Haskell, to phase wave them. considered horizontal between SH is and conversion incident waves to incident mode generality, applicable the of for also loss account and Without general and are wave below SV formulations and potential P wave of incidence the on focus we ( by represented be can and eeice l nvriyo aiona Davis California, of University al. Jeremić et are Modeling (1953) Propagation Haskell Wave (1950); Analytic Thomson 3D/3C Motions: Seismic 109.2.5 in computed be can motion the system, the in layer one any in layer. known other is any motion the if Hence, system. displacements the between n,m by: A defined function transfer is The m and functions. n m f level are and functions at transfer m Other e the formulas. from recursion obtained above easily the into condition this substituting by determined function transfer The 3104 of 527 page: the build to (109.20) . . . Eq AND PROPAGATION and ENERGY (109.19) SEISMIC Eq 109.2. of use repeated layer, field: surface wave the with Beginning surface. Notes ESSI xrse yteequaction: the by expressed m There 109.3. Fig. in shown as ground layered the in propagation wave (2020). inclined al. the et is Wang Considered on part in based is This the after summary, In function transfer the equations, these on Based n 1 = aeswt ae thickness layer with layers A A F E u ¨ ( m m n,m n,m , ,t z, 2 = . n .., , = ( ( = ) ω ω f e m = ) = ) m .Sneteicdneo out-of-plane of incidence the Since ). ( ( ∂ ∂t ω ω 2 ) e u ) u u 2 e E E m m n n = 1 1 ( ( ω ω e − + ) + ) m ω 2 and E f f ( m n Ee ( c and ( ω o nietwvswt rirr iesga n utpefrequencies, multiple and signal time arbitrary with waves incident For . ω f i ( ) d m ) kz m + F r ipyteapiue o h case the for amplitudes the simply are density , ωt r optdfrallyr ntesse,teaclrtosare accelerations the system, the in layers all for computed are )+ e F − i ( ρ kz m − opesoa velocity compressional , ωt ) ) A SH ( w ) a efudbtenaytolyr nthe in layers two any between found be can aei ipe n oecneso) here conversion), mode (no simpler is wave α m eso:2.Ags,22,15:04 2021, August, 26. version: E n ha aevelocity wave shear and 1 = F 1 1 = n a be can and , (109.24) (109.26) (109.25) (109.23) (109.22) β w m Jeremićet al., Real-ESSI ubr qasto equals number, reflected nitude eeice l nvriyo aiona Davis California, of University al. Jeremić et and with deformation expressed volumetric be can media elastic S linear in (109.27) Eq. propagation 3104 of 528 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI Interface Interface Interface Interface Interface Interface aevco potential vector wave Layer n Layer The hrfr,teukonvralsfor variables unknown the Therefore, ,tedslcmn fwave of displacement the 1999 ), Weber, and (Arfken theorem decomposition Helmholtz to According Layer 2 Layer 1 Layer Layer n Layer Layer 3 Layer Ψ u ρ φ φ u ¨ m m = m 0 P SV ( = reflected , [Ψ = [ = ∇ n-2: n-1: and 3: -1 2: 1: φ λ 0: aeptnilmagnitude potential wave φ + 2 + m 0 m 0 SV e e × ∇ ik ik µ w/c ( ( ) x ,where (109.29), Eq. as expressed be can potential wave x 훼 ∇· ∇∇ P − − Ψ n-1 훼 γ 훼 훼 γ And . 훼 αm aeptnilmagnitude potential wave βm n 1 3 iue193 Dlyrdgon n refil motion field free and ground layered 2D 109.3: Figure 2 , , , , Ψ , 훽 훽 z 훽 z 훽 u 훽 ) ) n-1 n 1

3 Ψ 2 + , − where 109.28, Eq. in shown as

Ψ + , ,

….. , cot 휉

휉 휉 sdvrec repr orsodn odvaoi deformation. deviatoric to corresponding part free divergence is φ 휉 µ n 휉 1 3 2 m 00 × ∇ × ∇ n-1 , m 00 − , , , d d d e d 1 e ik n 1 γ 3 , ik 2 d α ( ( x m n-1 x + + Ψ γ △ △ and γ △ △ αm △ βm m m 00 u 1 3 n n ’ 2 ’ z . th - ’ ’ z ’ ) 1 cot ) ] ω ω ] e ω ω ω ae r ipie noincident into simplified are layer e − 1 3 − − n n ’ 2 ’ iwt ’ - iwt ’ 1 1 φ γ ’ β m 00 ω ω m ω ω incident , ω n 1 r nietadrflce nlsfor angles reflected and incident are n 3 - ” 2 1 ” ” ” ” △ △ △ △ △ 1 n 3 ” 2 n ” ” ” - φ 휙 1 ” SV 푛−1 ′ stecr repr orsodn to corresponding part free curl the is 휙 휙 휙 aeptnilmagnitude potential wave 휙 휙 1 ′ 3 2 ′ ′ 푛 ′ 푛−1 ′′ 휙 휙 휙 P 휙 1 ′′ 3 2 ′′ ′′ 푛 ′′ aesaa potential scalar wave eso:2.Ags,22,15:04 2021, August, 26. version: 훹 훹 훹 훹 훹 푛−1 ′ 1 ′ 3 k 2 ′ 푛 ′ ′ P 훹 stehrzna wave horizontal the is 훹 훹 훹 훹 1 aeptnilmag- potential wave ′′ 3 2 푛 ′′ ′′ ′′ 푛−1 ′′ z z z z n-1 z 0 1 0 0 2 3 0 n 0 P x x x x x 1 (109.29) (109.28) (109.27) 2 3 Ψ and n-1 n m 0 φ and and SV Jeremićet al., Real-ESSI ,dlttoa aesolutions wave dilatational (1953), Haskell by formulation original the ∆ with consistent be To depth. the angle. eeice l nvriyo aiona Davis California, of University al. Jeremić et solutions wave rotational and wave dilatational the magnitudes φ 3104 of 529 page: . . . AND PROPAGATION factor ENERGY SEISMIC time hereafter. the omitted by 109.2. ?? characterized and is understood field be potential will the It of nature harmonic The respectively. wave, Notes ESSI m m .Nt htwhen that Note ). ,tedslcmn n tesfil of field stress and displacement the (109.28)-(109.31), Eqs. From displacements The and n oainlwv solutions wave rotational and σ +2 φ ω τ u u Ψ zx γ zz z m = ∆ x = α m Ψ +2 ikγ = m = 2 = = = m 2( = +[1 1 2 { ik +4 and {− φ − β ( ρ ikγ r eae to related are ∂u m ρ ∂x ∂u m ( ( ∂z and m β β − ( ik α { c β ω x w α w β m m γ x 4 β 2 α m m ω ( m 2( β m + m 2 m 2 ) γ − α m Ψ ) ) ( 2 ω {− β ) β 2 2 (1 m α [( ∂u 2 m . c ∆ ω ∂z ∂u ω eoecmlxnmes h ln aemgiueepnnilydcy along decays exponentially magnitude wave plane The numbers. complex become m α ∂x [( m ω ) [( m − γ m m 2 z ) ω m 0 ( ) z α ω [(∆ 2 u 2 m 0 2 m ) ][( or m 0 + [(∆ x β ( ∆ − c u , ω m 0 − β m α 2 2 ω m m 0 c m m 0 ω m y m 00 )[(∆ ∆ + ω ) m 00 + ) ) and m sgetrthan greater is − 00 cos 2 n nefca stresses interfacial and ) ) [(∆ cos ω ∆ cos m 00 m 0 ω m 00 ( ω m 00 ) m kγ ∆ + ) ( cos m 0 m ( ) cos kγ kγ r rtitoue sE.(109.30). Eq. as introduced first are cos β − sfollows: as m ( β β m 00 ( kγ m ∆ z ( kγ m ) kγ ) z m 00 z cos α + ) − β ) m ) α γ m cos − c z m α i ( z h niec from incidence the , ( ) kγ m i z ) ω i ( − ) ( ( ω − m 0 and α ω kγ − m 00 ∆ i m m 0 i − (∆ ( α i z m + ω (∆ + m ) ω γ m 0 m 0 z ω − and β m 0 ( ω m 00 ) m m σ 0 − − ) m 00 i − sin zz (∆ ) ∆ + a edtrie ySelslw(equation law Snell’s by determined be can ) ∆ sin ω ω sin τ , i m 00 (∆ m m 00 m 0 ( zx kγ ) ( m 0 ) ( sin − kγ sfollows: as sin m 00 ) kγ ) β sin m ∆ β ∆ + a eepesdi aepotential wave in expressed be can β ( P m ( z kγ m m 00 kγ )] z ( m or z ) kγ )] } β m 0 sin α )] m th e } m SV } ) α ikx e z sin e z m ikx ( ae a ecluae from calculated be can layer )] eso:2.Ags,22,15:04 2021, August, 26. version: ikx )] kγ z } aei eodtecritical the beyond is wave )] ( e α kγ ikx m α z m )] z )] (109.33) (109.32) (109.35) (109.34) (109.31) (109.30) e − iwt . Jeremićet al., Real-ESSI where 109.41, Eq. by solved be can and magnitude of wave component rotational fourth and and magnitude third wave the (i.e., free is traction magnitude potential wave onaycniin a eimposed. be can conditions boundary S ∆ eainbetween relation matrix transformation where 1953): (Haskell, notations matrix following d eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 530 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI m (0) m ) olwn onaycniin r noprtd 1 At (1) incorporated: are conditions boundary Following sn h eainbetween relation the Using 109.39 . Eq. to leads 109.38 Eq. applying Recursively en h ipaeetadsrs ouinat solution stress and displacement the Define ω , /c, n oe onay(.. ouin fwv nietlayer incident wave of solutions (i.e., boundary lower and ) ω n S S S S S 0   u m (0) ( ( ( ( ˙ is n m m m z ∆ bigstegpbtenteuprbudr ie,rsos tgon surface ground at response (i.e., boundary upper the between gap the bridges 109.40 Eq. , ω − ( ) ) − K z = 1) n 00 1) = = n 00 m 2   L w = = D D = [∆ 2 n = Y i / m m E =1 − d S (2 n 00 m m E [∆ 1 (   β m ∆ + G ) m [∆ − n 2 c σ /c, L L m 00 ) L i ) 1 and S . 41 31 m 00 ∆ + S n 0 ( = (0) ( , ∆ + zz m ∆ + + φ S n − Y i ( m 0 n 0 n 00 =1 − z ( 1) L L , m m E m 0 1 − and ∆ − G = 42 32 , m = ∆ m 00 1) ∆ i G ) n 0 m 00 Ψ and − d where 109.38, Eq. as established be can then − ω , m m L L n 0 1 − ∆ E ) S n 00 43 33 r ie as given are τ , m 0 ∆ D n ( − m ω , zx m 0 m + + − ω ω , ( 1) m 00 n 0 z L L r ie nApni (Eqs. Appendix in given are ω , m m 00 − 44 34 n 00 = − ω   + m 0 ω d − K ω , m m ω m 0 1 n 1 0 ω , )] th m 00 ]   T T and m 00 + nefc as interface S ( - (109.32) Eqs. . ( ( L L (0) + ω K 42 32 m 0 n ω s0.Teeoe h eetddilatational reflected the Therefore, 0). is 2 ] th m 0 T 2 ttegon ufc ( surface ground the At (2) ; − − ] T ae,teicdn in-plane incident the layer, L L ∆ 41 31 S n ( )∆ )∆ m and ) hc seulto equal is which , n 0 n 0 ?? eso:2.Ags,22,15:04 2021, August, 26. version: ω ?? ( + ( + n and a erdcdt the to reduced be can ) ,uo hc specific which upon ), L L G ?? 43 33 m .Terecurrence The ). ∆ − − = n 0 L L D is 44 34 m S − z P ( E ) ) ˙ [ K m 0 = u ω ω (109.40) (109.39) (109.38) (109.37) (109.36) (109.41) and m − − x 1 n n 0 0 ω ( 1) 1 ,the ), z   2 . m /α and SV = n 2 Jeremićet al., Real-ESSI n oainlwv magnitudes wave rotational and eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 531 page: computed easily be can field stress and displacement whole (109.32)-( the Eqs. . . following layer, . AND PROPAGATION each ENERGY of SEISMIC magnitudes 109.2. rotational Notes ESSI 0... einlSimcMto oeiguigSretn aePoaain SW4 Propagation, Wave Serpentine using Modeling Motion Seismic Regional Models Geophysical Scale 109.2.6.1 Large Motions: Seismic 109.2.6 where wave , 109.43 and modulus Eq. Lame in complex shown with as handled velocities be can viscosity (Chirit¸˘a2008), material al. , et coelastic cnb sdt rc akdlttoa aemagnitude wave dilatational back trace to used be can 109.42 Eq. relation recurrence Finally, nadto,vsoiycnas eicue ihsih oicto.CnieigKli-og vis- Kelvin-Voigt Considering modification. slight with included be also can viscosity addition, In        G ∆ ∆ ∗ ω ω = m m m 00 m 00 00 00 − − − − G 1 1 1 1 1+2 + (1 + − − ∆ + ω ω ∆ m m 0 0 m 0 m 0 − − ξi − − 1 1 1 1 )        = ?? β m ∗ D ). ' m − 1 β − m ω 1 m E 1+ (1 m o h rest the for        ξi ∆ ∆ ω ω ξ ) m 00 m 00 m m 00 00 stedmigratio. damping the is − ∆ + + − α m ∗ ∆ ω ω m m 0 0 ' m 0 m 0 n        α − m 1 1+ (1 aes ae nteesle iaainland dilatational solved these on Based layers. ξi ) eso:2.Ags,22,15:04 2021, August, 26. version: (109.43) (109.42) ∆ m Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et measured as waves seismic of correlation of lack the to points. contributing observation two sources at main Four 109.4: Figure correlation. of lack of sources main of illustration an shows 109.4 Figure to: due are ( 2009) Zerva few to last according the correlation, over spatial incoherence of lack motion of small seismic over sources researching motions main in ground The done of is decades. variability been it spatial has analysis in work domain results Significant time that for phenomena distances. analysis, a domain is correlation) frequency for of lack called called is it 3104 (as of incoherence 532 motion page: Seismic Incoherence Motion Seismic 109.2.8 . . . AND PROPAGATION ENERGY SEISMIC 109.2. ... is response Site Response Site 109.2.7 Notes ESSI 2. 1. • • • • hsi ato refil oin eto n utue refil oin o rdcn ieresponse. site producing for motions field free uses just and section motions field free of part is This ttolcto onsa h srae aetaes rpgtsfo rtt eodpoint. second to first from propagates travels, wave (surface) the as points location two at long for correlation effects significance. of much passage lack of Wave of not source is it significant NPPSSS a for is seismic however This that (bridges), dissipation) structures distance. energy that (damping, along losses and experiences wave points observation between distance the to effects Attenuation effects source Extended effects, Scattering effects, passage Wave effects, Attenuation r epnil o hnei mltd n hs fsimcmtosdue motions seismic of phase and amplitude in change for responsible are otiuet ako orlto u odffrnei eoddwv field wave recorded in difference to due correlation of lack to contribute Fault Attenuation Wave Front(s) 1 2 eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI noprto flc fcreaineet nsimcmtoscnb oeuigtefloigmethods: following the using done be can motions seismic in effects correlation of lack Simulation of and Incorporation Modeling Correlation of Lack for develop behavior 109.2.8.1 to incoherent used possible be model can to propagations. that used wave are events seismic that American seismic exist North models seismic east not are American there does North Rather, there east was Unfortunately, models. for that incoherence set set (2009). data data single Walling a a enough from locations motions at large source ground motions a and the seismic in site of variability different variability as same a over the that collected be assume will models combination the source – all site is That assumption. ergodic an Zervas and Zerva 1992b); (2009) 2005, Zerva (1992a, ( 2006); Abrahamson Zerva 1996); ( motions and number seismic al. Liao correlation) A et (2002); lacking Roblee above. (or (1991); incoherent mentioned al. of significant correlation, subject et of as the Abrahamson lack phenomena, on of available this of sources are of four references frequency excellent all nature in of to probabilistic decrease relation strong for in a present increased is is correlation uncertainty there that, Moreover, to addition motions. In observed decreases. points observation eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 533 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI 4. 3. • • • ti eyipratt oeta l urn oesfrmdln noeetsimcmtosmake motions seismic incoherent modeling for models current all that note to important very is It between distance separation the as increases motions of correlation the that concluded studies Early nfl C hr vr on fitrs i Dvlm fsi/ok a ton pcfid lack specified, own, it has soil/rock) of volume 3D a and (in correlation, interest of of correlation. lack point of same every share where planes, points horizontal 3C, the of full all set in where a plane, or horizontal planes a in vertical or of plane correlation, set vertical of a a lack in in are single usually that a directions, two share points in plane) the 2D, point, all along control where a directions, from horizontal 1D) the (in of distance one same usually the direction, one in 1D, along different have will and points. observation fault) it rupturing makes Seismic the it as (along fault. timing the points and along different path sources travel from seismic emitted generate thus and is propagates energy rupture ruptures, fault the as to effects contribute source that Extended features geologic subsurface enough) field. known wave not of or modification (unknown to due is tering effects Scattering r epnil olc fcreainb raigasatrdwv ed Scat- field. wave scattered a creating by correlation of lack to responsible are otiuet ako orlto ycetn ope aesuc field, source wave complex a creating by correlation of lack to contribute eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI fadltv olwihsiesu uigsaigdet etitditn odlt.I scerta dilative that clear is It frequencies. dilate. higher to of intent restricted amplification to significant due show shaking will during soil up stiffens which be soil is/should dilative it a (as of soil change no-volume of only response if shows assumed 109.6 Figure frequencies. different of cation stiffness). in in changes (increase in dilative resulting and constraint, stiffness), volume in full (reduction with change compressive no-volume for soil for shearing stiffness (right) where responses dilative three soil shows compressive and 109.5 for (middle) Figure opposite compressive the stress. (left), is confinement dilative of It for (confinement) reduction soil. a pressure example that in mean of For result in stiffness will the increase stress. increasing the by thus horizontal to resisted soil, soil) contribute the be will of loose will stress for direction horizontal decrease horizontal additional in and soil, change (dense) dense volume for intended (increase constraint any vertically be in that happen will change soil means only the can That horizontally change while soil. volume top, volume such on other However, example) its by for change loose). (foundation is constraint to no it try is if will there compress (since response soil or site dense the dimensional is waves) one it shear for if (SV) example, (dilate propagating For vertically soil. propagation, of wave constraints (1C volume to due response affect affect essential two are there soil very of is shearing behavior During soil that shear. behavior: emphasize during soil to response of important types volumetric is It of function behavior. a soil for much important very Volume is deformation. shearing data during change change volume soil/rock about information (crucial) important very Soil for Data of Use Change Volume of Lack 109.2.9 Abrahamson by developed Code purpose this models. for motions used incoherent is those develop (1992b) only to incoherence that that and used model fact motions are the translational to motions record from only used translational stems stations are This recording seismic motions incoherent. of majority perturbed/made translational vast not only currently are that motions (incoherent) rotational note uncorrelated the appropriate to while with effects, important enriched is than It are that components. waves), surface and body (inclined, eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 534 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI • • hne nsins fsi uigsern eomto iliflec aepoaainadamplifi- and propagation wave influence will deformation shearing during soil of stiffness in Changes by provided not is that response, volume soil The 3C full in DRM through obtained motions motions, seed called so the using is here used method The opesv loe ol ildces ouedet shearing to due volume decrease will soils (loose) shearing Compressive to due volume increase will soils (dense) Dilative G/G max G/G n apn uvsfrdsrbn n airtn aeilbhvo fsi smsiga missing is soil of behavior material calibrating and describing for curves damping and max n apn uvsaeaalbe ihn ouecag aa n response a and data) change volume no with available, are curves damping and G/G max n apn uvsdt a significantly can data curves damping and eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et history time frequencies. a different is at plot amplitudes Left shows soil. plot dilative soil. right for the dilative motions while and of motions, change frequency of in no-volume increase through (significant) propagation the wave note seismic Please dimensional One 109.6: Figure stiffness; in decrease with response no stiffness. compressive (left) in (middle) increase deformation: state); with volumetric response critical dilative constraint the (right) at with is soils (soil of change response volume Cyclic Constitutive 109.5: Figure 3104 of 535 page: . . . AND PROPAGATION ENERGY SEISMIC 109.2. Notes ESSI τ [kPa] -20 -15 -10 10 15 20 -0.02 -5 0 5

Acceleration [m/s2] -0.015 10 -8 -6 -4 -2 0 2 4 6 8 0 -0.01 z =H/2 5 -0.005 γ [-] 10 no dilatancyvs 0 15 0.005 Time [s] 20 0.01 25 surf -nodil τ

surf -dil [kPa] -10 10 15 30 -0.02 -5 0 5 -0.015 35 -0.01 40 -0.005 z =H/2 γ [-]

Amplitude [m/s] 0 0.005 0 1 2 3 4 5 6 0 0.01 2 0.015 4 Frequency [Hz] no dilatancyvs τ [kPa] -50 -40 -30 -20 -10 50 10 20 30 40 -0.035 0 6 eso:2.Ags,22,15:04 2021, August, 26. version: -0.03 8 -0.025 surf -nodil -0.02 10 surf -dil -0.015 z =H/2 γ 12 base [-] -0.01 -0.005 14 0 0.005 0.01 Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et and Wolf in recently on more and discussion ( 1985) thorough Wolf more in A available is presented. (2002). SSII Song is dynamic ones of aspects important specific most and the methods over overview an following withstand the to design the optimizing by period. money return save certain to prevent a help only with also not earthquake but therefore structure an can a system of soil-foundation-structure damage or a parameters collapse of demand the engineering model engineering the numerical the to obtain good design to A order based (EDP’s). in performance needed of are concept models sophisticated the more introduce community to try companies behavior insurance the and on prediction any in makes This and deformations. SSI-system plastic the undergo of difficult. can stiffness very foundation overall the the surrounding modifies soil further the turn as well as a structure acting as forces the occurring inertial large resonance to of lead possibility can This the SSI-system. is the there of structure. cases frequencies a given natural most on a the in by of the 3104 true caused shift of of a loads is 536 of flexibility peak this page: result reduces the if therefore Including Even and system motion. a 2005 ). ground of Preisig, stiffness mainly (Jeremić and overall be the structures to reduces SSII of foundation dynamic behavior regards . . the loading . INTER earthquake to STRUCTURE to SOIL beneficial subject EARTQUAKE structures for 109.3. practice design Interaction Current Structure Soil Eartquake 109.3 Notes ESSI • • ait fmtoso ieetlvl fcmlxt r urnl en sdb nier.In engineers. by used being currently are complexity of levels different of methods of variety A authorities As projects. infrastructure large of design the in important becomes also SSII Dynamic frequency natural it’s in oscillating structure the by caused forces inertial large these of result a As ietmtostetteSIsse sawoe h ueia oe noprtstespatial the incorporates model numerical The whole. a as SSI-system the treat methods Direct radiation the and methods structure incorporateDirect the to with has interaction model The its structure. soil. soil, the case the the from stiffer away to this through For applied In motion structure. be the the to rock. of of has presence or propagation motion the ground soil by the stiff modified soil very get on on doesn’t structures ground structures the flexible of for displacement only the reasonable proportional nodes. is forces the procedure earthquake to This effective applied structure, be can the acceleration of base applying base of the the instead to Alternatively, to building. directly the motion of base ground the the to directly applied is motion ground The SSII No eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 537 page: . . . INTER STRUCTURE SOIL EARTQUAKE 109.3. Notes ESSI • • • o h ietmto ieetlvl fspitcto r possible: are sophistication of levels different method direct the For nt lmn oe slkl omk hmmr oua ihteegneigcommunity. engineering the with popular more them into make full methods to in likely these used is implement rarely are to codes they trying element piles work finite on Current loading analysis. lateral interaction estimating for soil-structure used dynamic simplified widely are under curves types p-y soil if and different Even references in several groups pile lists and (2002) piles cases. pile Naggar single loading on El of loading study and lateral parametric evaluating Mostafa a for provides approach problems. p-y dynamic static the to apply foundations to made been have Attempts rods conical methods of (2003). p-y sequence Preisig a and with Wolf soil and (elastic) (2002) the Song replacing and by Wolf accuracy good foundations with embedded and modeled in subsoil be analysis layered an can of require configurations therefore complex and Relatively pots dash domain. and frequency springs code. dependent frequency element use finite methods used relatively Other is commonly it a as fitting in engineers curve integrated structural a be among connecting by popular to by very determined easy account is been approach into have This taken parameters ( 1994). whose be Wolf masses procedure pots can springs, dash subsoil as and the such masses elements of springs, mechanical configurations several simple Different by for pots. accounted dash is and soil the of behavior The approach stiffness Foundation that fact the is used. however be cannot method method the substructure of the drawback systems nonlinear analysis biggest For an The much assumed. for is is needed method. linearity method substructure time direct the the the equations As of than number faster considerably. increasing an problem with the interface. linearly the of increase on doesn’t size nodes the the the and reduces at separately contributions method analyzed the This be adding can by substructures obtained subdi- Both be generally part. can is soil displacement a SSI-system total and The part structure superposition. a of into principle vided the to refer methods Substructure uplift). (sliding, two those between methods behavior interface Substructure nonlinear the one all also in incorporating and out of soil carried capable the is is structure, system it the as entire of generality the most of provides analysis method The This soil. step. the and structure the of discretization eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI ) h anfaue fteDMaete analyzed then are DRM the of features main addressed. The issues (2003a)). modeling al. practical found. appropriate et is Yoshimura and it (2003a); where al. domain of et size Bielak the or by to domain, soil– compared large a small in represent fairly crack can is a general that be in inhomogeneity can feature of it type local bounded or other The be tunnel...), some can not). dam, feature or building, local closed (bridge, the be system where can foundation–structure problems (that any for surface used continuous be a can by formulated as and methods large of simplified obtain. set the to from easier are difference accelerations local a and constitutes The displacements that which interest. fault object for of made the field, feature man at free or couples local domain, geologic force surrounding any the surface be replace continuous can to a developed feature idea on was main method acting the The counterpart with their structure. mind, with the in and structure, motions soil the ground local and The earthquake only soil with encompassing 2003a)). rock, domain, al., fault, smaller encompassing et much Yoshimura domain, a 2003a; computational to large , al. the method et reduce That (Bielak to accuracy. (DRM) aims high Method with model DRM Reduction models can element Domain and finite mechanics the into rational called input on is waves based seismic as is surface well that as and those method waves body are of a body both them exist Most of There of components mechanics. three waves. Most rational all surface on model present model. properly based always cannot element not methods are finite used such into widely as still motions and currently seismic approaches, input intuitive to simple used on based is methods Model of Element number Finite into A Input Motions Seismic Details Modeling Interaction 109.4.1 Structure Soil Earthquake 109.4 eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 538 page: . . . INTE STRUCTURE SOIL EARTHQUAKE 109.4. Notes ESSI • nwa olw h R sdvlpdi oehtdffrn a hni a oei rgnlpapers original in done was it than way different somewhat a in developed is DRM the follow, what In global–local of variant a essentially is It problems. of range wider much a to applicable is DRM The eore n oeigeotrqie o naayi ti h nymto htrmisvldfor valid remains geometries. loading that complex computational different method and problems, the only contact/interface cases of nonlinearities, the material spite is involving In it problems analysis of soil. kinds an the all for for required also the effort for but modeling only methods and not mentioned resources obtained above be the can forces in and as Displacements structure approach. force’ be ’brute can the interaction as soil-foundation-structure regarded dynamic of modeling three-dimensional nonlinear Full 3d Full eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University domain local scale small large al. relatively Jeremić the et reduce this and somehow time computations to the is scale of developments large presented Most of of feature. goal complexities main local the the the all Besides, of involves feature. effects it the as include will impossible which is problem scale large the solve domain the of size the reduce forces to surface dynamic try by the can bounded that we source model example the smaller For from much loading feature. a the local some transfer to the somehow of of to outside vicinity need domain we immediate the this, the neglect do to almost to order can In and the surrounding, boundaries. to close immediate relatively distance its and the feature and local feature the local of the of size the feature. to local relative this is part influences source. domain machine which forcing large a loading dynamic term dynamic in the of feature sense source local the this small from In the away Similarly, centimeters interest. many of be feature can local the from away kilometers load of source the with system. domain soil-structure physical Large 109.7: Figure 3104 of 539 page: known a is disturbance of source by The field excited behavior. dynamically force . dynamic . a . for INTE of analyzed STRUCTURE history SOIL be EARTHQUAKE time to is 109.4. domain physical Development large (DRM) A Method Reduction Domain The 109.4.1.1 Notes ESSI R Formulation DRM twudb eeca o oaayetecmlt ytm sw r nyitrse ntebehavior the in interested only are we as system, complete the analyze to not beneficial be would It many be can hypocenter earthquake example for large, quite be can analyzed be to system The nodrt prpitl rpgt yai forces dynamic propagate appropriately to order In P P e ( e P t ( ) t e ) P ). 109.7 Figure (see (t) r prpitl rpgtdt h uhsalrmdlboundaries model smaller much the to propagated appropriately are e ( t Seismic source ) htsuc flaigi a wyfo oa etr hc is which feature local a from away far is loading of source That . Ω + Γ ndigs ems ensure must we so doing In 109.7. Figure in shown as u Ω b P e ( u t ) e Local feature n h oa etr i hscs a case this (in feature local the and u i Large scaledomain eso:2.Ags,22,15:04 2021, August, 26. version: Γ P e ( t ) n culyhsto has actually one Γ . Jeremićet al., Real-ESSI r fwitnfrec oan(neir onayadetro of exterior and boundary (interior, domain each for written if or, Γ boundary inside domain domain hsi tl eycmlxpolm u tlatteiflec flclfauei eprrl ae out. taken temporarily is feature local of named influence the is the to least forces at dynamic but problem, the complex propagate very consistently a to still easier is much This is it that so boundary model the simplify the to propagate is consistently to as response dynamic the simulate and it analyze forces simplified to dynamic is possible model the is feature, it local the that of Instead so system. soil–foundation–building a case this (in feature eeice l nvriyo aiona Davis California, of University al. Jeremić et load of source the with domain physical large Simplified 109.8: Figure idea The we system. foundation–building is, 3104 material. of a That and 540 without geometry analyzed. page: model, simpler be simplified much to a a easer shows with much 109.8 crack) is Figure tunnel, that example, building. domain For (bridge, simpler feature a local with . the . feature . replace INTE local STRUCTURE a SOIL EARTHQUAKE replace feature. will local 109.4. we the of behavior details in analyze to able be to as Notes ESSI are h qain fmtosfrtecmlt ytmcnb rte as written be can system complete the for motions of equations The ti ovnett aedffrn at fdmi.Freape h oanisd h boundary the inside domain the example, For domain. of parts different name to convenient is It nodrt rpgt ossetytednmcforces dynamic the consistently propagate to order In h u e M Ω , Ω u + Γ b 0 h ointa ti uhese opoaaetoednmcfre so orerelative. course of is forces dynamic those propagate to easier much is it that notion The . n i ssiltesm si h rgnlmdl hl h hne ipicto,i oeo the on done is simplification, change, the while model, original the in as same the still is h eto h ag cl oan usd boundary outside domain, scale large the of rest The . and P u ¨ e u ( o i t epciey nteoiia domain. original the on respectively, , ) + . h Γ K h ipaeetfilsfretro,budr n neiro h boundary the of interior and boundary exterior, for fields displacement The . P e n i (t) Seismic source u o = n P Ω e + o u 0 Ω b 0 P e ( t ) u Γ Simplified largescaledomain 0 ewl aeasmlfiaini that in simplification a make will we e eaaey h qain banthe obtain equations the separately, ) Γ u ste named then is , 0 i P eso:2.Ags,22,15:04 2021, August, 26. version: Γ e ( t ) and without Ω + h outside The . h local the (109.44) Γ Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et are motion of equations two resulting The equilibrium. and ( boundary 3104 of 541 page: . . . INTE STRUCTURE SOIL EARTHQUAKE 109.4. form: following Notes ESSI and i matrices the equations, these In o airt analyze. to easier now by denoted interior are the forces hand, action–reaction other the On source. force dynamic the are as as therein problem material (denoted the original domain and the region of exterior those the words, to other identical In simplified. significantly is domain interior dis- boundary contain equations both since automatically placements maintained is displacements of Compatibility tl h ae h asadsins arcsadtendlforce nodal the and matrices stiffness (109.47 ). and and (109.46) mass the domain same) exterior the the still to change no was there Since , e     and , h rvoseuto a esprtdpoie htw anantecmaiiiyo displacements of compatibility the maintain we that provided separated be can equation previous The h qain fmto in motion of equations The and respectively) exterior, and boundary (interior, field displacement the model, simplified this For the Thus, domain. interior the from removed is feature local the problem, the simplify to order In cnb sdt bantednmcforce dynamic the obtain to used be can 109.48 equation Previous M M       P 0 e ii bi Ω Ω M M M M M M = b b bb eb bb eb bi ii Ω Ω Ω+ Ω+ Ω+ Ω+ M r eeecn oe nete h neir( interior the either in nodes referencing are u ,wietesuperscripts the while ), M b eb Ω+ bb Ω M M o boundary (on M M M M M M + u bb ib ¨ Ω Ω eb Ω+ ib b 0 be ee be ee Ω Ω+ Ω+ Ω+ Ω+ M   + Ω bb Ω+    0 M     109.8). figure in seen (as removed is features localized the simplified, is ), ee       Ω+ u u ¨ ¨ b M M i u u u u ¨ u ¨ ¨ ¨ ¨    0 ee be Γ Ω+ Ω+ e 0 e e 0 b 0 b ,wieteeulbimi anandtruhato–ecinforces action–reaction through maintained is equilibrium the while ), + +       Ω     K M +   + +        eb Ω+ o h uiir rbe a o ewitnas: written be now can problem auxiliary the for K K     Ω and bi ii u u u u Ω Ω ¨ ¨ u ¨ and K K K K e b 0 b i i 0 , bb eb + bb eb Ω+ Ω+ Ω+ Ω+ K        K K u Ω K b 0 bb ib Ω Ω + ems n tffesmtie epciey h subscripts the respectively; matrices stiffness and mass re , + ee Ω+ K K K K u       eeec oan owihmtie belong. matrices which to domains reference e 0 be ee be ee Ω+ Ω+ u Ω+ Ω+    K K and e 0 0 Ω ii bi Ω Ω u u +     b i P       mtra,goer n h yai oreare source dynamic the and geometry (material, K    b 0 i h niesmlfiddomain simplified entire The . bb retro ( exterior or ) Ω u u u u = K e b e 0 b 0 K + eb    Ω+    ib    Ω K bb P = Ω+ = 0 b          K K P − − , P P P e 0 be ee Ω+ Ω+ e P P e e e oano ntercommon their on or domain ) r h aea nEquations in as same the are b : b 0 in    eso:2.Ags,22,15:04 2021, August, 26. version:     Ω           , u u u in e b i Ω        + = Ω        0 P and 0 0 e (109.46) (109.45) (109.47) (109.48) (109.49)        Ω + P b is . Jeremićet al., Real-ESSI hc,atrmvn h refil motions field free the moving after which, eeice l nvriyo aiona Davis California, of University al. Jeremić et of principle the of application an not and variables 3104 of of change 542 field, a page: displacement just residual is The this that superposition. note to important is It . . . INTE STRUCTURE SOIL EARTHQUAKE 109.4. field residual the and model) simplified Notes ESSI as ffcieforce, effective     h oa displacement, total The oeobtains: one 109.45 Equation in 109.50 Equation substituting By h ih adsd a o ewritten be now can side hand right the 109.52, Equation previous in 109.49 Equation substituting By i yaial ossetrpaeetfre h ocalled so the force, replacement consistent dynamically a is 109.53 equation of side hand right The M M 0     u     bi ii Ω Ω e M M M M = M 0 0 ii bi ii bi Ω Ω Ω Ω u bb Ω e 0 M M + + M M eb P Ω+ ib Ω w M bb bb Ω Ω eff M M e M M bb + + Ω+ eb eb Ω+ Ω+ ib ib Ω Ω M M o h yai oreforces source dynamic the for bb bb Ω+ Ω+ M M 0 ee be Ω+ Ω+ u M M M M e a eepesda h u ftefe field free the of sum the as expressed be can ,     0 0 ee be be ee Ω+ Ω+ Ω+ Ω+                u ¨ e 0               u ¨ + u ¨ ¨ w b i w w e u u u u ¨ ¨ w ¨ ¨ ¨ ¨ u b b i i e e cmn rmtelclfaue sfollowing: as feature) local the from (coming e e 0                      otergthn ie becomes side, hand right the to w + + + e             smaue eaiet h eeec refield free reference the to relative measured is K K K K K K 0 0 P 0 ii bi ii bi Ω Ω Ω Ω bi ii Ω Ω e nohrwrs h yai force dynamic the words, other In . K K K bb bb Ω Ω        bb Ω K K K K K + + K + eb eb Ω+ Ω+ − eb Ω+ ib ib Ω Ω ib K K Ω K M        − bb bb Ω+ Ω+ bb Ω+ ee Ω+ M − M be Ω+ M u ¨ K K K K eb e 0 K K Ω+ be Ω+ u ¨ − 0 0 ee be ee be Ω+ Ω+ Ω+ Ω+ 0 ee be Ω+ Ω+ eso:2.Ags,22,15:04 2021, August, 26. version: e 0 u ¨ 0 K u ¨ − u b 0 e 0         e 0 ee + Ω+ 0     K − fo h background, the (from                      K be Ω+ u K e 0 eb Ω+ w w u u u u be u Ω+ u + b b i i e e e 0 e 0 u u + P u u               b 0 b i e 0 e        w = =               e P 0 0        e (109.51) (109.50) (109.52) (109.53) P =        u e 0 e was . Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et on and inside accelerations) and (displacements unknowns boundary full the comprises (109.53) Equation in boundary scribed not are outgoing. that waves nodes residual the Only all are nodes. nodes those external of the 18 while be to face, will belong that there that on nodes, nodes nodes are 9 nodes boundaruy are bricks, there node brick 27 and 20-17 20, a for between brick. elements node of layer 8 Single DRM: 109.9: Figure boundaries between and on lie that 109.9. nodes Figure the in at shown problem (auxiliary) forces simplified effective the determine from to needed accelerations) domain and in (displacements wave-field elements finite of layer single 3104 of 543 page: . . . INTE STRUCTURE SOIL EARTHQUAKE 109.4. force effective the by replaced consistently Notes ESSI ae ffiieeeet (outside elements finite of layer P for used Elements of Layer Single Discussion DRM eff so onayadetra oe o eto fa2 oebik laent that note Please brick. node 27 a of section a for nodes external and boundary show 109.10 Figure nov nytesub-matrices the only involve P eff = Γ        ( u P P P i e b i eff eff eff and P e (t)        u b = epciey.O h te ad h ouinfrtedmi usd single outside domain the for solution the hand, other the On respectively).        Γ − Ω e M sotie o h eiulukon(ipaeetadaccelerations) and (displacement unknown residual the for obtained is ) + M eb nte neetn bevto sta h ouint rbe de- problem to solution the that is observation interesting Another Ω+ M be P Ω+ be eff u ¨ u ¨ b 0 , u e 0 0 Ω b . + K 0 u Ω − 0 e P be + K hw htteeetv oa forces nodal effective the that shows (109.54) Equation The K eff , eb Ω+ daetto adjacent be M Ω+ : u u eb u 0 Local feature e u 0 i b 0 , e 0 K        Γ Γ eb Γ + hs arcsvns vrweeecp the except everywhere vanish matrices These . Γ and e Γ h infiac fti sta h only the that is this of significance The . Γ e sue ocreate to used is b Γ b eso:2.Ags,22,15:04 2021, August, 26. version: Γ ufc,s o 7node 27 for so surface, P P eff eff o eto of section a for , sta obtained that is Γ e e and Γ e (109.54) Γ e as , Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et boundary the from distance some media at surrounding Ω damping) the and and feature fixity local (including of supports behavior appropriate in is interest boundary the within if (all that means turn in (109.53) This Equation equation feature. the to solution boundary the that the means outside effectively That 542). page on comments (see field, between Brick. elements node of 27 layer and Single DRM: 109.10: Figure 3104 of 544 page: . . . INTE STRUCTURE SOIL EARTHQUAKE 109.4. Notes ESSI oasm httedrvtoswl tl evldwt n yeo aeil(iero olna,elastic nonlinear, or (linear material of type any with valid the possible be of therefor still is use It will no derivations structures. was and the there solids that elastic clearly superposition. linear assume and for of to valid variables, principle only of on is relying change which not superposition, the of is describing principle DRM only the was (inside as 109.50 necessary bowl Equation not plastic The is the elastic linear inside is material inside of material type the on made domain Inside + hsi infiatfranme freasons: of number a for significant is This . • • w h eiulukonfil a emntrdadaaye o nomto bu h dynamic the is about field wave information residual field for the free analyzed Since reference and feature. monitored local the be of can characteristics field unknown residual force source the dynamic boundary the outside to elements distance the of layers then few smaller a magnitudes just encompass to size in Γ reduced be can models large tmigfo h oa feature. local the from stemming e e sgicn euto o,syerhuk rbesweetesz falclfauei resof orders is feature local a of size the where problems earthquake say for, reduction (significant ny hsrsda nnw edi esrdrltv oterfrnefe edo unknowns of field free reference the to relative measured is field unknown residual This only. Ω a einelastic. be can P Γ e Γ (t) n a elc h eairo h ulmdl(outside model full the of behavior the neglect can one ) e u ilol oti diinlwvsfil eutn rmtepeec falocal a of presence the from resulting field waves additional contain only will 0 e h ouinfor solution the , Ω + u n etito was restriction no 109.4.1.1 section in derivations the all In 0 Ω b u 0 e w e u a l h hrceitc fteadtoa aefield wave additional the of characteristics the all has u 0 Local feature e 0 i Γ Γ + Γ Γ e and Γ Γ e e .Ta s h supinta the that assumption the is, That ). sue ocreate to used is P w e b e Γ b b eso:2.Ags,22,15:04 2021, August, 26. version: erhuk hypocenter). (earthquake smaue eaiet the to relative measured is Γ e P in e e e eff Ω + o 0 20-17 20, for , Γ ) e n provide and noregion into e e e Γ e Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et account: into taken be to need layer DRM the regarding conditions of number A Layer. modeled. DRM the outside just inside elements elements finite waves. etc...) of Love, Properties (analytically (Rayleigh, properly surface are and 3104 waves P) of seismic SH, 545 (SV, (real) page: appropriate body all including source, modeled, modeled. seismic ) the are of waves effects the seismic replaces realistic of . . types . INTE All STRUCTURE SOIL EARTHQUAKE 109.4. systems. soil–foundation–structure inside inelastic) or Notes ESSI • • • ha aevelocity. wave shear velocity wave The size) (element spacing grid such elements interpolation quadratic for while elements interpolation linear for exceed size) (element spacing modeling grid maximum maximum if velocity, example For (Bathe 1991). length Mlejnek, wave is and per frequency Argyris needed 1987; are Hughes, bricks) 1976 ; node node , Wilson (27 (8 and elements elements interpolation finite interpolation quadratic linear dependent) 2 10 (mesh or artificial that bricks) no means is That there frequencies. same that certain the so above follow stiffness) filtering to chosen need on layer) (depending DRM size the element of field. for (thickness wave rule elements consistent finite layer have as DRM same to the order be of to in Dimensions need analysis properties field the material free to elastic for similar used used the same, properties of is material All layer, elastic layer. DRM DRM the the for inside surface, elastic- used elastic for yield material linear or inside strain the zero properties if at elastic material beneficial material, elastic is nonlinear plastic for it example layer. elastic-plastic, for DRM properties, be the material similar can inside of very layer portion material or DRM the same, the as have properties inside material to material of need Although part layer elastic DRM the the as within properties elements material finite the elastic. linear for be models to Material need layer DRM the within elements finite The ∆ ∆ h h QE LE f Γ ≤ ≤ max e ihti nmn,teDMbcmsavr oeflmto o nlssof analysis for method powerful very a becomes DRM the mind, in this With . Γ 10 λ 2 λ 20 = ufcs hr ffcieDMforces DRM effective where surfaces, v = = stelws aevlct hti fitrs ntesmlto,uulythe usually simulation, the in interest of is that velocity wave lowest the is 2 10 z n aelnt sgvnas given is length wave and Hz, f v max f v max = = 2 × 10 0Hz 20 v × v 0Hz 20 = = 0Hz 40 ic h ffcieforcing effective the Since v 0 Hz 200 h R aes igelyro finite of layer single a layers, DRM The v P λ eff min r ple,nest ecarefully be to needs applied, are = v/f eso:2.Ags,22,15:04 2021, August, 26. version: max where , ∆ h QE P ∆ eff h slmtdto limited is LE v consistently stewave the is hudnot should Jeremićet al., Real-ESSI ih raepolm.Sml ieritroainatal ih wl)ntstsywv propagation wave satisfy not (will) might actually interpolation linear Simple problems. create might of step time a with qain fmto.Freape ffe edmtosaedvlpduigato SAE rETor EDT or (SHAKE, tool a step using to time needs developed using used are is &c.) motions that field fk, wave free or seismic if SW4, field example, free For a motion. words, other of In equations consistent. be to need DRM. model for Motions Input on Note A eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 546 page: . . . INTE STRUCTURE SOIL EARTHQUAKE 109.4. Layer. well. DRM as the considerations of special outside need elements finite the of Properties Notes ESSI • • • • • ihrhv enue nodrt apoton ae,t oe aito damping. or radiation model 50%, to then waves, 30%, outgoing then damp 20%, to order of in damping used equivalent been larger of have much Values higher larger, progressively damping. be from viscous to damping is physical radiation layers called than additional so those the the in waves, waves, Damping additional outgoing oscilations. any structural out out damp damp to to added are order layers ” in damping layer layer, the first DRM field, this to wave outside adjacent differential elements outside, finite is on that placed elements, be of should Rayleigh, Caughey, damping, Viscous develop to used was that field wave nt lmnsjs usd R ae,adaecnetdt R nt lmns hs reaction, of These nodes change elements. and affects, finite nodes, DRM will element to forces, finite connected are DRM viscous and with layer, shared DRM are outside just that elements then elements finite of damping, nodes viscous on large placed have is layer that DRM the to outside/adjacent P are that elements fact finite the by If explained is requirement, this for reason The inside that material damping. as viscous properties NO similar/same with of and elastic, layer, linear outside DRM be layers, to needs two well. layer DRM just quite outside work layer than layers First more additional have 5 or to 4 recommended example is For it layer. DRM of modeled, of layer is damping one damping If than radiation more If provided. then provided. modeled, be be to be to needs to needs DRM is outside layer damping, elements DRM radiation finite called outside so elements damping, the finite viscous waves, of outgoing of layer addition one with least elastic At linear be to needs etc. layer Rayleigh, Caughey, DRM the outside Material eff P ocswl epouigptnilysgicn ecinfre rmlrevsosdamping viscous large from forces reaction significant potentially producing be will forces eff sapidt nt lmnswti R layer. DRM within elements finite to applied is 543, page on 109.54 equation see force, ∆ t 0 = . 001 ,sml neplto 1 diinlsesfrec fteoiia steps) original the of each for steps additional (10 interpolation simple s, ∆ w t e diinlviscous Additional 542. page on 109.50 equation see waves, ” 0 = esi oin fe ed htaeue o nu noaDRM a into input for used are that field) (free motions Seismic P . P 01 eff eff n hnyudcd htyuwn ornyu analysis your run to want you that decide you then and s ocsi a htwl o ecnitn ihseismic with consistent be not will that way a in forces . iieeeet usd R layer DRM outside elements Finite eso:2.Ags,22,15:04 2021, August, 26. version: fully satisfy Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 2210. free page for on 502.2.3 model section ESSI in the described test as and methods, attention problems pay top. interpolation on to Spatial 3104 structure(s) has of the model. still 547 add ESSI one page: then in however only used and acute, nodes conditions less DRM bit actual simulation. a the ESSI actually to in are close used very be will not it . is . as . good nodes INTE step very STRUCTURE time a SOIL is same EARTHQUAKE It the with model. 109.4. motions the field into motions free frequency generate high to additional, idea introduce will used if and equations Notes ESSI nu oin o h R r ae nFe il oin,ta a edvlpdb ubrof number a by developed be can that motions, Field Free on based are DRM the for motions Input model field free of location if is that done, is interpolation spacial if occur might problem Similar eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI I olbrto ihD.Gazo i n r unFn,adD.HnYang) Han Dr. and Feng, Yuan Dr. and Jie Guanzhou Dr. with collaboration (In (1998-2000-2005-2015-2016-2017-2018-2019-2021-) Computational Mechanics in Computing Parallel 110 Chapter 548 Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et details: forces more external In and internal tolerance. between some equilibrium within until Then, satisfied on matrix. iterated is stiffness updated are element the equations and nonlinear use forces level, then nodal global functions (internal) stiffness on new element constitutive integrate Finite to state, functions. tensors stiffness stress element and finite updated stresses the the to computes them space, delivers point and variable Gauss tensor each internal at and driver stress constitutive in implementation, element iterates finite based of displacement presence the standard is a computations In element finite elasto–plastic of feature distinct Geomechanics The in Computations Element Finite wide 110.2.3.1 a for and decompositions static Walshaw excellent (1999), compute simulations. al. to scientific able et (e.g. of are Monien range that algorithms developed (1998b,a), partitioning been Kumar graph have and multilevel (1999)) Karypis Cross of by (1995), number obtained Leland a usually will and Recently, is boundary Hendrickson decomposition subdomain algorithm. a the partitioning Such of graph size a overhead. balanced the is communications minimizing elements inter-processor while mesh the the of computation, (i.e., minimize number load-balanced processors the a other that in to Ensuring assigned result minimized. elements will is to boundary) adjacent subdomain equal are the roughly that is of processor elements size each mesh to assigned of elements number mesh the of number and the that such computed be to needs Iterations Increments, Stages, Requirements Problem Structures and Solids 110.2.3 Elastic-Plastic for Computing Parallel (FPGAs) 110.2.2 Arrays Gate Programmable Fast (GPGPUs) Units 110.2.1.4 Processiong Graphical Purpose General Computations 110.2.1.3 (SMP) Parallel Memory Shared Computations 3104 110.2.1.2 of (SMP) 549 Parallel FPGA Memory page: GPGPUs, Distributed SMPs, DMPs, 110.2.1.1 on Computing Performance High 110.2.1 HIGHLIGHTS AND Introduction SUMMARY CHAPTER 110.1. 110.2 Highlights and Summary Chapter 110.1 Notes ESSI o aycasso cetfi iuain,a nta sai)dcmoiino nt lmn mesh element finite a of decomposition (static) initial an simulations, scientific of classes many For eso:2.Ags,22,15:04 2021, August, 26. version: w trto levels iteration two . Jeremićet al., Real-ESSI ngeomechanics. in eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 550 page: INTRODUCTION 110.2. Notes ESSI hsilsrtsatofl hleg ihcmuainlla aacn o lsi–lsi simulations elastic–plastic for balancing load computational with challenge two–fold a illustrates This trtosadol rce ihgoa ytmieain fe that. after iterations system level constitutive global complete with slowest to proceed the elements) for only finite wait and elasto–plastic to iterations of having number computa- program large less in with far results one spend (the This thus domain increments. and level stress state constitutive computing elastic in in in time time elements of tional have amount will large domains spend other and The elements might iterations. elasto–plastic domain in the one result com- of example might all every For approach assigned to problems. This be elasto–plastic elements for boundaries. of imbalance subdomain number load of computational same size serious the the domain roughly minimizes splits and assigns DD) node and topological putational geometry as known initial (also the of method on extent DD based the of words, type other preprocessing In traditional time. level The of over. constitutive ahead are additional, known computations not of actual is of extent domain the system The elastic-plastic before the known time. solving not run is part is of dominant iterations 80% This the about where iterations. consumes clock computations level which wall elastic constitutive equations of with in 70% contrast spent than sharp is more analysis in PI, element is are finite the soils elasto–plastic of for an experience during 2002) the time Yang, From Jeremić and constitutive 1999; demanding. particular, computationally 1998, In very al., range. (Jeremić driver et increase plastic algorithms constitutive will in integration load the equations level Computational iterations, constitutive no of solution. are integration form for there closed significantly range, from stresses elastic zones incremental in plastic computes the still loads, just points apply incrementally Gauss we For as but range, develop. until elastic in space are variable points internal Gauss all and Initially, ( stress satisfied in is iterating condition is consistency driver constitutive the deformation mental computations. Elasto–plastic the points is integration work computational of computational of number of same amount is amount the it The the assume that we follows advance. If it in point. element, known integration per is every for point same Gauss the per is work work of amount the case ( equation form ( stresses in increment computing computations. Elastic nw nadvance in known ∆ σ ij = E . ntecs featccmuain osiuiedie a ipets of task simple a has driver constitutive computations elastic of case the In ijkl ∆ kl nteohrhn,freat–lsi rbes o ie incre- given a for problems, elasto–plastic for hand, other the On ∆ .I sipratt oeta nthis in that note to important is It 1997). Sture, (Jeremić and ) F σ ij 0 = o ie eomto nrmn ( increment deformation given a for ) .Tenme fieain sntkoni advance. in known not is iterations of number The ). eso:2.Ags,22,15:04 2021, August, 26. version: same ∆ o aheeetand element each for kl ,truhaclosed a through ), Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et selection the so and computation, the the of dynamically. select course made computational the to be during preferred difficult must change the extremely to Furthermore, is liable employ. is it algorithm to computation, load-balancing algorithm of change load-balancing zones type computational elastic–plastic this of and In type elastic and cycles. soil loading interaction soft during soil–structure with significantly seismic interacts in structure observed stiff is each where (in phenomena computations level Similar (becomes constitutive later rapidly the elements). on unloads the within load solid while computational point the heavy phases, integration change of has loading zone rest The localized initial the narrow, and during The changing. rapidly occurs again). is zones elastic usually narrow zone in change plastic localize Slow unload of to then tends extent rapid. and deformation the and plastic slow that become may both so solid be loading) 3D can of the increments in zones (during problems, dynamic. elastic static and to of the unpredictable case that both the is is in computation phases example, elastic-plastic load-balancing For the computational concerning between whichever issues iterations minimize key (i.e., of the minimize number of to one objective However, primary a dominates). appli- select cost many to For straightforward is communications. inter-processor it minimal domains, the cation obtaining in is algorithm load-balancing a communications the minimizing is over element processing. preferred each parallel is with during cost associated incurred state redistribution of data amount the the minimizing or high, rapidly, requirements relatively change computational domain the the of which regions in different applications of For communications. inter-processor the imizing 3104 computational requirement of time 551 last each scratch This page: from required. partitioning minimized. static is is new load-balancing whole load with a computational associated computing the simply cost from balance us the to prevents that order elements requiring shares in mesh also data the decomposition balance the while dynamic (i.e., redistributing communications), a decomposition inter-processor initial computing the the of minimize computing required problem and of is that The load-balancing as requirements computation. computational same the the periodic computational of Instead, the course ensure the to computation. sufficient during entire not is the elastic–plastic reason, step this and of pre-processing For elastic a load-balance computation. as of the computed structure of decomposition the course static the is, a during That unpredictably and nature. dynamically in changes INTRODUCTION domains dynamic are 110.2. computations these First, Computation Adaptive 110.2.3.2 Notes ESSI o plctosi hc opttoa odblnigocr eyifeunl,tekyojcieof objective key the infrequently, very occurs load-balancing computational which in applications For min- of objective the with odds at is cost redistribution data the minimizing of objective the Often, eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et coherence Cache SMPs for issues Scalability SMPs and DMPs 110.2.4.1 Hardware Computing Parallel computation. have the processor of 110.2.4 every phases idle that the be critical of may is both it these from Instead, work so other of versa. (and amount after vice computation, equal working and plastic an working), be the still still are during may processors work processors these other enough some so, while not to (and lead and for computation may idle), required so elastic are times Doing 3104 the relative processors of sum. during 552 the this work be up page: on much phase sum based too simply each decomposition having to a that compute sufficient requires to not computation and is phase the it each is, of existence That phases The two balanced. the load performed. that individually between then indicates are step check given computations synchronization this a plastic which the for in lengthy required of mesh necessary, is the is computation of computation plastic the regions plastic For the after the if element. only an check the as within to up computations, point possible follow (Gauss) the it integration computations between is phase finished plastic is synchronization is, computation a That elastic is There computations. two-phase computations. ge- are elastic in these computations that elastic-plastic is load-balancing INTRODUCTION computational omechanics with 110.2. associated challenge second The Computation Multi-phase 110.2.3.3 Notes ESSI • • • • • ml SIComputer ESSI Small clusters) Computer of ESSI (clusters machines parallel owned user small, Infiniband, supercomputers 10,000, parallel 1,000, Large 100, 10, latency) memeory) and main (band-with Networks the with latency and (band-with GPUs Cores, CPUs, Nodes, Compute 2CUcrs(M) 4CUcrs(Intel) cores CPU 24 (AMD), cores CPU 32 system file for GigaBit and MPI, for InfiniBand neetwork, dual and space, disk distributed of (48TB) 24TB RAM, distributed of (1056GB) 288GB cores, CPU (784) 208 eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI n Law Amdahl’s 110.2.5.1 Software Computing Parallel 110.2.5 eeice l nvriyo aiona Davis California, of University al. Jeremić et elastic-plastic) and (elastic computations matrices Element parallel Finite embarrassingly and parallel Real 110.2.5.3 etc. mining data search, Partitioning google Graph Dynamic and Static 110.2.5.2 3104 of 553 page: INTRODUCTION 110.2. Notes ESSI sanme fprle processes parallel of number a is • • • oa iet ns wl lc ie with time) clock (wall finish to time Total B hoeia speedup Theoretical lsi-lsi P ovr(P) solver + (P) elastic-plastic (Seq) solver + (P) elastic-plastic (P) solver + (Seq) elastic-plastic (P)) solver + (P) (elements elastic (Seq)) solver + (P) (elements elastic solver(P)) + (Seq) (elements elastic for iterative) Examples and (non-iterative solvers equation of System T S network on-board 4TB RAM, 64GB stefato fagrtmta sserial is that algorithm of fraction the is ( ( n n = ) = ) T T T (1)( ( (1) n ) B = + T (1)( n 1 (1 B − T + B (1) )) n 1 (1 − B )) = n ( B aallprocesses parallel + n 1 (1 1 − B )) T ( n ) eso:2.Ags,22,15:04 2021, August, 26. version: (110.1) (110.2) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Decomposition Domain Plastic 110.2.5.5 3104 of 554 page: INTRODUCTION 110.2. FEM Elastic-Plastic for Computing Parallel 110.2.5.4 Notes ESSI • • • • • • • • • • • clblt strto,sprier...) superlinear (saturation, Scalability iff beneficial is balancing load Computational gain performance Best 1 balancing) load (perfect time execution Best time) clock wall (not time compute Total CPU) (slowest time compute maximum minimize Goal: CPUs among load Computional overhead adds balancing load computational partitions) balanced create and data costs the communications, redistribution inter-processor the both (minimize problem optimization Multi-objective for balancing load computational dynamic Need: are FEM Parallel Current .,nPU nCP ..., , – – – – – – – – – – T T performances performance network multiple performances node compute multiple models material multiple types, element multiple ESSI...) MOOSE, (example DMPs (multi–generation) performance multiple computers, for (DMP) Undeveloped parallel memory distributed homogeneous for developed Well FEM elastic–plastic for Undeveloped FEM elastic for developed Well regen comm oe eeeainfrnwpriinn,apiain(oe)dependent (model) application partitioning, new for regeneration model aacmuiainla eedn nntokconditions. network on depending load communication data T gain := T T max j := − P T i nel =1 best T sum ElemCompLoad T T T overhead best gain := := sum ≥ T T := sum overhead ( T T max T j nPU, /nCP ) comm := [ i ] = j , + max T T comm 1 = ⇒ regen eso:2.Ags,22,15:04 2021, August, 26. version: ( T .,nPU nCP ..., , j T ) + j j ≡ T regen 1 = T best .,nPU nCP ..., , o each for j = Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University decomposition al. Domain Jeremić et years. recent during world engineering and mathematics in in attention more simulations much transient implicit parallel for efficiency ). 1993 of high Farhat, contributions deliver and FETI (Crivelli can The This mechanics it applied condition. structural that orthogonality solution. is an shown local through scheme been multipliers partial direct has Lagrange each a the method to recover and related to problem then are order local modes nodes. in each these interface subdomains from the parallel all at in to compatibility eliminated concurrently enforce are to modes introduced body are Rigid multipliers Lagrange analysis. method, decomposition domain FETI for method In Interconnecting) and Tearing Element (Finite FETI to proposed techniques condensation-type static applying by reduc- analysis at aim structural which in sixties, subdomains. models the of in developed dimension methods the substructuring ing of method paradigm decomposition underlying Domain the Magoulès2007 ). also and , is (Rixena Schwarz H.A. decomposition, domain of chapter. this organize in and examined program be to will ease that relative one the the to is due and modeling computations element is finite decomposition continuum domain in the Non-overlapping used divides extensively subdomains. method overlapping decomposition slightly domain several Decomposi- overlapping into Domain The domain parallel. and problem non-overlapping. in subdomains or domain into overlapping individual domain be each problem can on tion the performed divide be physically finite will parallel calculations to implement element is finite to idea method underlying effective The and method. popular element most the is approach Decomposition Domain 3104 of Introduction 555 page: Algorithm 110.3.1 Decomposition Domain Plastic 110.3 . . . A DECOMPOSITION DOMAIN PLASTIC 110.3. parallelism grained Fine Meta-programs Template 110.2.5.6 Notes ESSI • • oandcmoiinisl a eoeaatv oi sprle rcsigtcnqe receive techniques processing parallel as topic active a become has itself decomposition Domain (1992) Geradin and Farhat (1991); Farhat 1991a); ( Roux and Farhat condensation, static than Other father the by paper 1870 a in found be can method decomposition domain of idea well-known The to able be should algorithm decomposition non-overlapping good a general, In iiieteitraepolmsz ydlvrn iiu onaycnetvt,wihwl help will which connectivity, overheads. boundary communication minimum the delivering reducing by size problem interface the minimize domain. shaped arbitrarily of mesh irregular handle eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University a as defined well al. Jeremić is et which processors, of ). 1999 set al. , a et onto (Schloegel mesh problem a partitioning mapping graph of problem the to corresponds tion for operation balancing load adaptive on proposes focuses chapter elements. This which finite nonlinear algorithm neglected. be Decomposition resulted to issue Domain important imbalance Plastic too Load the is busy. computations equally level achieve processors constitutive to all order nonlinear keep in from to Namely, frameworks want computing. FETI we parallel or performance, of substructuring parallel issue existing balance better the load of fundamental out the break address to more further is not to if chapter solving, this equation of as novelty costly equally The least expensive. at be can computation level the constitutive during the materials, solver equation on set been has focus all that reasonable decades. totally past is it so part expensive , Cai tional and (Hwang spotlight the of part 2007). shared Szyld, and also for Sarkis preconditioners have 2007; Schwartz-type solving ). 2007 system Widlund , decomposition as and domain promising Li han- parallel is ; 2007 constraints preconditioning (Pavarino, interface decomposition-type researchers subdomain domain many of with by solving shown root iterative the of from merging stem The all BDDC dling. and FETI-type as indeed such can methods involved. balancing are nonlinearities load when computational particularly cases, dynamic some time, in run performance program that parallel show improve examples of number Limited min- 2003 ). energy Dohrmann, constrained and using Mandel boundaries 2003; substructure (Dohrmann, on concepts nodes imization of sets disjoint with associated straints (Farhat problems acoustic and structural 2000). dynamic, , 2001 static, al., arising of et discretization equations (FE) of element systems finite symmetric 3104 the of of from solution 556 iterative page: domain-decomposition-based and scalable, (or BDDC recent (or more FETI-DP even Constraints). the the by and namely Decomposition method . effort, Domain . Interconnecting) . Balancing research A and DECOMPOSITION Tearing much have DOMAIN Element receiving papers PLASTIC Finite Many currently Primal Magoulès 110.3. 2007 ). Dual are and , that (Rixena solvers algorithms parallel two for discussed paradigm natural a as revived was Notes ESSI rmteipeetto on fve,frms-ae cetfi opttos oandecomposi- domain computations, scientific mesh-based for view, of point implementation the From nonlinear highly for that observation the from originated has however, chapter, this in presented Work computa- most the is solving equation the that assumed been has it methods, element finite solid In popular most the methods, decomposition domain on presented been have works many Although (1997). McKenna by presented was balancing load computational dynamic on endeavor early con- An enforcing by method substructuring from formulation its derives hand, other the on BDDC, fast, the for developed 2007) al., et (Bavestrello method FETI generation third the is FETI-DP eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI iecmuainlla aacn ceehst ecniee oke l rcsigunits processing all keep to adap- considered simulation, be multiphase of to kind has this scheme for balancing So load computation. computational elastic work tive pure introduces than Plastification heavier processors. much among is that distribution load load even guarantee not does above tioned 2003 ). combinatorial, , al. geometric, et either (Dongarra Various methods as multilevel classified time. or be of techniques, amount can optimization reasonable combinatorial which a spectral, developed, in been size have interesting approaches the of heuristic graphs and for problem partitioning dynamics optimal structure compute to for 2001). well Bittnar, as and (Krysl used although simpler be chapter, algorithms can certain this make node-graph in can the structure partitioning data element-cut that graph and shown fundamental measuring as been load utilized has computational it been for has interfaces graph consistent facilitate element to other migration, computationally the order most On In the method. represent part. element calculations finite element expensive of the basis simulations, the nonlinearity forms material operation for elemental hand, edge-cut). that the fact e., the to (i. of due subdomains sum method different the to minimizing belong while vertices constraint), incident balance whose the edges as the to of (referred weights weight the vertex of amount equal an of vertices the split to ( eeice l nvriyo aiona Davis California, of University al. deterministic Jeremić of et class the P, of outside remain conceivably could time”) that NP polynomial of (”non-deterministic subclass NP smallest problems, in the polynomial-time problems are difficult they most that driver sense the the constitutive are in problems two implementation, NP-complete of the element presence time. finite polynomial the based is displacement standard computations a element In finite (elastic-plastic) levels. inelastic iteration of feature distinct The Element Finite Parallel Inelastic applications. specific our 110.3.2 for ParMETIS parameters the algorithmic through optimal implemented extract is to algorithm performed are PDD Studies the of interface. kernel partitioning graph multi-level tive to simulations. research element this multiphase finite in inelastic do introduced for to been balancing adequate load has not dynamic partitioner is achieve graph algorithm multilevel partitioning parallel graph A static Traditional partition/repartitioning. possible. as much as 3104 of 557 page: . . . A DECOMPOSITION DOMAIN PLASTIC 110.3. Notes ESSI V 1 ; h opeiycasN stesto eiinpolm htcnb ovdb o-eemnsi uigmciein machine Turing non-deterministic a by solved be can that problems decision of set the is NP class complexity The nfiieeeetsmltosivlignniermtra epne ttcgahpriinn men- partitioning graph static response, material nonlinear involving simulations element finite In NP-complete be to known is problem partitioning graph The element finite parallel in used naturally is graph element the mechanics, solid computational In graph undirected weighted, a Given follows. as is problem partitioning graph the Formally, nti hpe,teagrtmo lsi oanDcmoiin(D)i rpsd h adap- The proposed. is (PDD) Decomposition Domain Plastic of algorithm the chapter, this In E ) o hc ahvre n dehsa soitdwih,the weight, associated an has edge and vertex each which for V http://en.wikipedia.org/wiki/NP-complete into k ijitsbes(rsboan)sc htec udmi a roughly has subdomain each that such subdomains) (or subsets disjoint 1 hrfr,gnrlyi sntpossible not is it generally Therefore, . k wygahpriinn rbe is problem partitioning graph -way eso:2.Ags,22,15:04 2021, August, 26. version: qal busy equally G = Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et between 3104 equilibrium of 558 until page: on tolerance. iterated some within are satisfied equations element is nonlinear and forces forces level, external nodal and global (internal) internal on new functions integrate Then, element state, to Finite tensors stress matrix. . . stiffness . functions. updated stiffness A and element DECOMPOSITION the stresses finite DOMAIN updated computes PLASTIC the the space, to use variable 110.3. them then internal delivers and and tensor stress stiffness in constitutive iterates point Gauss each at Notes ESSI • • a escmuainltm ncmuigsrs nrmns hsrslsi rga aigt wait to having program in spend results thus in This and time increments. state of stress elastic computing amount in in large elements time have computational spend will less and far domains elements other The elastic-plastic the iterations. approach level of one This constitutive all example For assigned boundaries. problems. be elastic-plastic subdomain might for of imbalance domain load size computational the serious elements in of minimizes result number and might same the node roughly computational assigns and every topological geometry as to initial known the on (also based of method domain Decomposition extent splits Domain DD) the of words, type other preprocessing In traditional time. The of over. ahead are level known computations constitutive not additional, actual is of domain the extent elastic-plastic before The system known time. the not solving run is is of part iterations 80% dominant about the iterations. consumes where level which computations constitutive equations elastic in of with spent contrast wall is sharp of analysis in 70% element is than finite This More elastic-plastic rocks, an demanding. concrete, during computationally soils, time very for clock are algorithms materials integration granular equations level other constitutive and constitutive of foams particular, integration In for range. significantly plastic form increase closed in will from stresses load incremental Computational computes just range, elastic solution. driver in constitutive still the points iterations, Gauss For no are develop. zones there plastic the loads, ( apply incrementally satisfied we is as but condition consistency until space variable internal 0 and constitutive stress the in deformation incremental iterating given is a driver for problems, elastic-plastic for hand, other the of On amount the that follows Computations it Elastic-Plastic element, per point. points integration integration the every is for of work same number computational the same is the work assume computational we of amount If The advance. in known is increment computing of task simple ( a has ( driver stresses constitutive computations in elastic of case the In Computations Elastic ∆ .Tenme fieain sntkoni dac.Iiily l as onsaei lsi range, elastic in are points Gauss all Initially, advance. in known not is iterations of number The ). σ ij = E ijkl ∆ ∆ σ ij kl o ie eomto nrmn ( increment deformation given a for ) ti motn ont hti hscs h muto okprGuspoint Gauss per work of amount the case this in that note to important is It ) same o aheeetadi is it and element each for ∆ nw nadvance in known kl ,truhacoe omequation form closed a through ), eso:2.Ags,22,15:04 2021, August, 26. version: . F = Jeremićet al., Real-ESSI opttosweesi tutr neat ihsf oladeatcadeat-lsi oe change zones elasto-plastic and elastic and soil interaction soft soil-structure with seismic interacts in structure observed stiff is each where phenomena (in computations level Similar (becomes constitutive later rapidly the elements). on unloads the within load solid while computational point the heavy phases, integration change of has loading zone rest The localized initial the narrow, and during The changing. rapidly occurs again). is zones elastic usually narrow zone in change plastic localize Slow unload of to then tends extent rapid. and deformation the and plastic slow that become may both so solid be loading) 3D can of the increments in zones (during problems, dynamic. elastic static and to of the unpredictable case that both the is is in computation phases example, elastic-plastic load-balancing For the computational concerning between whichever issues iterations minimize key (i.e., of the minimize number of to one objective However, primary a dominates). appli- select cost many to For straightforward is communications. inter-processor it minimal domains, the cation obtaining in is algorithm load-balancing a communications the minimizing is over element processing. preferred each parallel is with during cost associated incurred state redistribution of data amount the the minimizing or high, rapidly, requirements relatively change computational domain the the of which regions in different applications of For communications. inter-processor the imizing computational requirement time last each scratch This from required. partitioning minimized. static is is new load-balancing whole load with a computational associated computing the simply cost from balance us the to prevents that order elements requiring shares in mesh also data the decomposition balance the while dynamic (i.e., redistributing communications), a decomposition inter-processor initial computing the the of minimize computing required problem and of is that The load-balancing as requirements computation. computational same the the periodic computational of Instead, the course ensure the to computation. sufficient during entire not elastic-plastic is the reason, step and this of pre-processing For elastic a load-balance computation. of as the structure computed of decomposition the course static is, the a during That unpredictably and nature. dynamically in changes domains dynamic are computations these First, Computation Adaptive detail. 110.3.2.1 more some in below described is challenges two These mechanics. eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 559 page: . . . A DECOMPOSITION DOMAIN PLASTIC 110.3. Notes ESSI o plctosi hc opttoa odblnigocr eyifeunl,tekyojcieof objective key the infrequently, very occurs load-balancing computational which in applications For min- of objective the with odds at is cost redistribution data the minimizing of objective the Often, in simulations inelastic for balancing load computational with challenge two-fold a illustrates This osiuielvlieain n nypoedwt lblsse trtosatrthat. after complete iterations to system elements) global finite with elastic-plastic proceed of only and number large iterations level with constitutive one (the domain slowest the for eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et constraints. balance of number arbitrary can an which satisfy formulations, can partitioning partitioning that graph graph partitioning-repartitionings adaptive adaptive the new compute from need We progresses recent investigated. be on very will execution chapter research efficient this algorithm its In in ensure dynamic computers. to also parallel adequate is performance not which high are calculation, formulations two-phase partitioning a graph Traditional as understood nature. be can computation FE Elastic-plastic computation. Partitioning the Graph of critical Multiconstraint phases is the it of Instead, both 110.3.2.3 from versa. work vice of and amount working), equal still an are have processors computation, processor other still elastic-plastic every while the may that idle during these be work so, may enough (and not these computation and so elastic idle), (and are the the may processors so up during other Doing work sum after much sum. simply working this too to be on having sufficient based processors not decomposition a some is compute to it re- to lead is, computation and phase That the each of for balanced. required phases load times two individually relative check the be this between phase which step each in synchronization that mesh then the quires are the of computations of existence elastic-plastic required regions The lengthy is For necessary, performed. is computation element. computation elastic-plastic an elastic-plastic the the after within if that only point check indicates (Gauss) as to integration computations, possible given the it a between is for finished phase up is synchronization follow computation a computations elastic is elastic-plastic the There is, That computations. computations. elastic two-phase ge- the are in these computations elastic-plastic that load-balancing is computational omechanics with associated challenge second The 3104 of 560 Computation page: Multiphase selection 110.3.2.2 the so and computation, the the of dynamically. select course made computational the to . be during . preferred difficult . must A change the extremely DECOMPOSITION to Furthermore, DOMAIN is liable PLASTIC employ. is it algorithm to computation, 110.3. load-balancing algorithm of load-balancing type computational this of In type cycles. loading during significantly Notes ESSI • etcsbln odffrn udmis(.. h edge-cut). the (i.e., subdomains different to to (referred belong weight vertices vertex of amount equal an the roughly as has subdomain each that such subdomains) egt the weight, graph undirected weighted, a Given Partitioning Graph Static balance k wygahpriinn rbe st pi h etcsof vertices the split to is problem partitioning graph -way osrit,wiemnmzn h u ftewihso h de hs incident whose edges the of weights the of sum the minimizing while constraint), G ( = ,E V, ) o hc ahvre n dehsa associated an has edge and vertex each which for , eso:2.Ags,22,15:04 2021, August, 26. version: V into k ijitsbes(or subsets disjoint Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 561 page: . . . A DECOMPOSITION DOMAIN PLASTIC 110.3. Notes ESSI • h eattoe hudatmtt aac h attoigwt epc ovre egtwhile weight vertex Thus, to respect vertex. the with of partitioning cost the distribution balance reflects to which metric, attempt well, another should as vertex, repartitioner considered the the be by to carried vertex needs work the be vertex If the to the of minimized. of needs be cost that should computational data load the of the represents amount balance weight to the order graph is, in static That processors the of objective. among characteristics additional redistributed and an requirements adds also the but of partitioning most algorithm. This shares partitioning graph partitioning domain. a graph using the Adaptive by inside achieved plastification be unpredictable can com- to balancing the load due load-balance dynamic progresses dynamically analysis to necessary the is as it putations simulations, FE elasto-plastic scale large For Partitioning Graph Adaptive Mliee Schemes Multilevel 3. Techniques Combinatorial 2. Techniques Geometric 1. rRcrieCodnt Bisection), Coordinate Recursive or oua loihsinclude context. algorithms KL/FM multilevel Popular the as in such powerful schemes high- more computing refinement much of can incremental task become reason, scheme the makes Second coarsening which graph, good easier. coarsest partitioning a the quality on First, edges of reasons. number two large a for hide well partition works the paradigm refine initial multilevel to an The used which be on graph. can finest graph, algorithm the coarsest KL/FM to get the back we Then coarsening, performed. of be vertices rounds will selected After bisection together graph. collapsing by input graph and the coarse partitioning, of form initial we coarsening, Firstly, graph refinement. phases: multilevel three of consists paradigm multilevel The refinement. swapping do to input partition initial an needs which algorithm, include, and methods Popular (LND) They slower. Dissection generally vertices. but the edge-cuts of lower coordinates have the to consider tend not the do on they only graph; based the partitioning of a information other compute adjacency each schemes near partitioning are combinatorial is these That not or space. whether in vertices connected highly together group to Attempt with- include, nodes, methods mesh Popular the of information edge-cut. coordinate considering the out on solely based partitioning Compute ehiusand techniques peeCtigapproach. Sphere-Cutting enga-i/iucaMthye K/M attoigrefinement partitioning (KL/FM) Kernighan-Lin/Fiduccia-Mattheyses Multilevel Recursive Recursive Inertial ieto and Bisection Coordinate ieto (RIB), Bisection Multilevel eso:2.Ags,22,15:04 2021, August, 26. version: Nested k Space-Filling -Way iscin(CND Dissection Levelized Partitioning. Nested Curve size Jeremićet al., Real-ESSI fgahcasnn,ads n ni ucetysalgahi band optto fteinitial the of Computation obtained. round is another graph for small graph sufficiently input a the until as on, acts so then form and graph to coarsening, order constructed in graph newly graph of This input the graph. of vertices coarser selected related together a a collapsing phase, by coarsening graph constructed the is three In graphs of of refinement. consists series multilevel paradigm and This partitioning, initial time. produced coarsening, less those graph substantially than phases: requiring quality while better bisection, or recursive comparable multilevel of by are that partitioning produces scheme partitioning O( is edges of number the a present (1998) Kumar Algorithm and Partitioning Kaypis Graph Multilevel Adaptive performance. parallel 110.3.3 the down brings materials of nonlinearity when balance will redistribution load Element achieve to progresses. triggered simulation condition be the as balancing accordingly load structure multi- graph monitors multi-level, element automatically using updates algorithm and developed This been algorithm. has partitioning (PDD) graph Decomposition objective Domain Plastic the chapter, this In Algorithm PDD Adaptive 110.3.2.4 eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 562 page: . . . A DECOMPOSITION DOMAIN PLASTIC 110.3. Notes ESSI • | .Tesfwr irre EI n aMTSare ParMETIS and METIS libraries software research. The mechanics computational 1999). ( in al. used et widely Schloegel have successful problems very multiconstraint/multiobjective solving been for algorithms partitioning the graph of Multilevel each that constraints subdomains. the across to balanced subject is edge-cuts weight, the size vertex minimizes of the that vector replace partitioning weight we a a If with number, edge-cut). single the a (i.e., is vertex objective which the single (i.e., a constraint only single a minimizes only and balances weight) typically partitioning gives graph traditional (2003) see can al. We et Dongarra Partitioning Graph years, Multiconstraint recent during topic active balance. restore very review. to an up-to-date partitioning been the in has changes method incremental This making by repartitioning final the and called so scratch, from graph Diffusion-Based new a compute simply can Scratch-Remap One available. are approaches size. Different vertex to respect with migration vertex minimizing E | log k for ) k hwta h rpsdmultilevel proposed the that show (1998) Kumar and Kaypis partitioning. -way | eattoe,wihepcel nrdcsmr aardsrbto hnnecessary. than redistribution data more introduces expectedly which Repartitioner, E eattoe tep omnmz h ieec ewe h rgnlpartitioning original the between difference the minimize to attempt Repartitioner | ie,O( (i.e., | E | k ) hra h u ieo utlvlrcriebscinschemes bisection recursive multilevel of time run the whereas )); wymliee attoigagrtmwoerntm slna in linear is time run whose algorithm partitioning multilevel -way m hntepolmbcmsta ffinding of that becomes problem the then , eso:2.Ags,22,15:04 2021, August, 26. version: m weights Jeremićet al., Real-ESSI lsi-lsi nt lmn iuain.Aatv rp attoigdffr rmsai rp partition- that graph sense static the from in differs algorithm partitioning ing graph Adaptive simulations. element finite elastic-plastic eeice l nvriyo aiona Davis California, of University taken been not has distribution al. Jeremić old et the because communications unnecessary more much bring to tends size. to vertex to respect respect the with with partitioning Thus, migration the cost. vertex balancing minimizing redistribution at while its represents aims weight reflects weight algorithm vertex size vertex partitioning the the graph while research, adaptive its element, this of also finite of application but each purpose vertex, of the a minimized. load for of be computational implementation weight should the the our load does In the only considered. balance not be cost, to to redistribution order this in measure processors to the order among In redistributed be to needs the 3104 of 563 page: provide to research this in used is which paradigm. partitioning. (1998d) multilevel static Kumar the and initial Karypis illustrates graph) METIS 110.1 original in Figure available (i.e., is (2003). nest . Finally, . algorithm the . al. This A fast. to et DECOMPOSITION very coarsest Dongarra DOMAIN is the algorithm PLASTIC so from KL/FM-type graph, and a 110.3. level graphs, using each these on of performed smallest) is hence refinement (and partition coarsest the on performed is bisection Notes ESSI dpiegahrpriinn loih a eue oaheednmcla aacn fmultiphase of balancing load dynamic achieve to used be can algorithm repartitioning graph Adaptive eattoigo rp a eotie ipyb attoiganwgahfo cac,which scratch, a from graph new a partitioning by simply obtained be can graph a of repartitioning A (2003) al. et Karypis Scheme Partitioning Graph Multilevel 110.1: Figure one additional betv a ob agtd hti,teaon fdata of amount the is, That targeted. be to has objective eso:2.Ags,22,15:04 2021, August, 26. version: iehave size Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University PARMETIS al. load. Jeremić et the balance to required costs redistribution iterative data the the in and incurred communications computation inter-processor mesh-based the both element minimize should finite repartitionings nonlinear which of progress the during regularly load-balancing routines adaptive static PARMETIS construct while simulations. calling to decomposition, called by domain are achieved static routines is METIS one-step used research. commonly this for in used partitioning are PARMETIS and METIS Both particular, (2003): In al. et simulations. Karypis numerical functionality large-scale following and the routines Ku- computations provides includes PARMETIS parallel and and for Karypis METIS suited by especially fill-reducing provided METIS are functionality and package that the serial partitioning extends PARMETIS widely-used multilevel However, the the (1998c). in on mar implemented based are are are that elements PARMETIS algorithms interface in of ordering numbers algorithms the The that such spent minimized. decompositions time the mesh involving reduces computing simulations dramatically by PARMETIS numerical computation, communication parallel of in matri- for type sparse this suited In particularly of meshes. is orderings unstructured PARMETIS large fill-reducing (2003). computing al. for et and Karypis ces graphs unstructured repartitioning and ing seamless schemes partitioning provides adaptive ParMETIS and that static fact between Warshaw comparison the 3104 consistent. Jostle the considering this of more makes and 564 research on which ( 2003) page: this 4.0 review al. METIS in comprehensive for et chosen a interface Karypis is gives ParMETIS former (2003) in The al. available et (1998). is Dongarra repartitioning the changes Adaptive minimize . incremental balance. . . making to subject. A restore by DECOMPOSITION attempts to repartitioning DOMAIN one partitioning PLASTIC final the which the 110.3. and in in partitioning popular original more the is between difference Repartitioner Diffusion-based account. into Notes ESSI • • • • • • dpiela-aacn hog oanrpriinn samliojcieotmzto rbe,in problem, optimization multi-objective a is repartitioning domain through load-balancing Adaptive partition- for algorithms of variety a implements that library parallel MPI-based an is PARMETIS osrc h ulgah fmeshes. of graphs dual the Construct factorization. direct sparse for orderings fill-reducing Compute partitioning. existing of quality the Improve simulations. multi-physics and multi-phase for graphs Partition meshes. refined adaptively to correspond that graphs Repartition meshes. and graphs unstructured Partition eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI rcsos u ht(u opatfiaino eti olna lmns hsdsrbto spol load poorly is the distribution this among elements) distributed nonlinear well certain is of decomposition balanced. plastification existing to (due the that that but assumes processors, routine This domain. tational eeice l nvriyo aiona Davis California, of University al. Jeremić et an be to load-balancing and is adaptation mesh more either until terminates. iterations simulation respectively. of is the processors, number domain or the another required unbalanced among for redistributed the continue is then on mesh can based the simulation then partitioning The and new load, thus A the elements re-balance nonlinear imbalance. to certain computed load in of of occurs iterations calculation plasticity amount of elemental which number some after of A introducing parallel, amount distribution. in even same performed a the are with simulation (carrying guaranteed the beginning be can very balance the load processors. at computation different work), elastic on are distributed equally elements is all mesh the As Initially, computers. parallel on simulations based performance high (1999) on al. simulations scientific et adaptive Schloegel of execution computers the parallel illustrating diagram A 110.2: Figure 3104 of 565 page: . . . A DECOMPOSITION DOMAIN PLASTIC 110.3. routine the provides Notes ESSI fw osdrec on feeuiganme fieain ftesmlto,ms adaptation, mesh simulation, the of iterations of number a executing of round each consider we If mesh- adaptive of execution the in involved steps common shows (2000) al. et Schloegel 110.2 Figure ParMETIS Shogle al. et Schloegel by, described be can epoch an of time run the then epoch, V3 dpieeatfrrpriinn h rvosublne compu- unbalanced previous the repartitioning for AdaptiveRepart eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University iterations of number the if sufficient the is control al. tradeoffs Jeremić to et the ability limited of a control only This user the objectives. give these but among costs, redistribution tradeoffs data the or edge-cut the either its affect scratch- adversely to than tends edge-cuts also higher it balance in to result order when in can quality. partitioning especially it a costs, However, perturbing redistribution because data imbalanced. methods low remap to slightly leads only usually or is strategy objective, partitioning This secondary the it. a balance to as as edge-cut so the enough focus minimize to to par- is approach original and second the cost The so-called to redistribution be remapping. data will after the partitioning even minimizing high, new on the be similar can how costs good. to redistribution extremely state-of-the-art data as be a titioning, guarantee to Since tends no edge-cut is resulting costs. there the redistribution since partitioning, those However, data the compute to the to labels minimize used subdomain is to the partitioner order graph remap in intelligently partitioning to be original attempt can the then methods of and of scratch family from edge-cut This partitioning the new objective. minimizing a secondary on a focus as to com- only attempt a redistribution called to data gives is the (2000) approach minimize al. first to The et and Schloegel topic. partitioners. this adaptive on review designing prehensive when taken been primarily have adaptive way, problem. this optimization in Viewed multi-objective the in a both partitioning. moved minimize new is be to the partitioning schemes computing to graph partitioning when required adaptive redistribution is for data that critical the is data and It of edge-cut partitioning. amount new total the the realize on to order dependent is time redistribution influences Adaptive balanced data is partitioning. The partitioning new new the the realize well How to order 110.3. t in Equation of moved in function terms be a of to as all required described affects is is repartitioning that time data redistribution of data a amount the as and total described partitioning the is the time of communication edge-cut inter-processor the the of Here, function data. the redistribute to and 3104 partitioning of and 566 simulation, page: the of iteration simulation, the of iteration . . . A DECOMPOSITION DOMAIN PLASTIC 110.3. (2000) Notes ESSI comp hs w ye frpriinralwteue ocmuepriinn htfcso minimizing on focus that partitioning compute to user the allow repartitioner of types two These approaches two general, In problem. dual-objective this handle to how approaches various are There where h ne-rcso omnctostm sdpneto h decto h e partitioning. new the of edge-cut the on dependent is time communications inter-processor The . ( cac-ea eattoe.Teeuesm yeo tt-fteatgahpriinrt compute to partitioner graph state-of-the-art of type some use These repartitioner. scratch-remap t comp iuinbsdrpriinr hs cee tep oprubteoiia attoigjust partitioning original the perturb to attempt schemes These repartitioner. diffusion-based n stenme fieain executed, iterations of number the is + f ( | E cut | )) n + f t ( repart | E t repart cut | + ) stetm opromtecmuiain eurdfrasingle a for required communications the perform to time the is g and ( | V move g ( | V | move ) t comp | ) stetm opromtecmuainfrasingle a for computation the perform to time the is ersn h ie eurdt opt h new the compute to required times the represent eso:2.Ags,22,15:04 2021, August, 26. version: (110.3) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et graph adaptive the of objectives function minimization cost two unified the unify the the to into balancing problem possible with partitioning is associated it redistribution parameter, data this the Using perform communications load. inter-processor to and the processing performing parallel for the during required is incurred times (URA) relative Algorithm the Repartitioning describes parameter Unified This in used parameter key A Algorithm Repartitioning Unified 110.3.3.1 problems. results large the very that better to paper much scalable the and and in fast scheme show inter-processor is also diffusive perform They the to scheme. cost than scratch-remap the repartitioner. results the than than scratch-remap better greater the obtains much scheme than is the redistribution better data communication, and obtains perform repartitioner to scheme diffusive cost the an the the scale, by When similar to obtained of similar similar those are are are than costs that that two better results these results and When repartitioner obtains scratch-remap repartitioner. scheme diffusion-based optimized proposed optimized an the by much costs, are obtained redistribution costs those communication data to inter-processor than when scale that shows in paper greater The of application. costs specific the combined by precise the the minimizing solve directly to attempts By g that simulations problem. scientific optimization of multi-objective load-balancing solid dynamic the the of for rest Algorithm) the and loading rapidly initial (1999). zones during Xenophontos narrow Jeremić occurs and in again) usually elastic localize change (becomes to Slow is rapidly tends zone unloads rapid. deformation plastic and the later may slow of the solid extent both while the 3D be phases, that the can so in change loading) zones of The increments Here, changing. (during unload dynamic. iterations then and and of unpredictable plastic number soil- become both elastic-plastic the the is that concerning is phases issues simulation load-balancing 3104 key earthquake of between the for 567 of required page: one computations example, interaction an As structure computation. the of course the which . in . of . applications A costs DECOMPOSITION combined DOMAIN PLASTIC the of when minimizes value However, 110.3. precisely the low. very (i.e. or high phases very load-balancing either between performs simulation a that Notes ESSI ( | V rsnsaprle dpierpriinn cee(aldthe (called scheme repartitioning adaptive parallel a presents (2000) al. et Schloegel move | E cut | ) h rpsdshm sal ogaeul rdo n betv o h te srequired as other the for objective one tradeoff gracefully to able is scheme proposed the , | + α | V move n | sdffiutt rdc rtoei which in those or predict to difficult is f n ( E snihrvr ihnrvr o,nihrtp fscheme of type neither low, very nor high very neither is cut | ) n and g ( | V move n a hnednmclythroughout dynamically change can | ) nte iavnaeeit for exists disadvantage Another . Unified Relative eso:2.Ags,22,15:04 2021, August, 26. version: n Repartitioning )is ) 110.3 Equation in Cost Unified f atr(RCF). Factor ( E cut Repartitioning Algorithm | (110.4) ) n and Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et be will redistribution data little of requires that variety repartitioning a a for low, set well is work computed. it 1000 If computed. and be 100 will between edge-cut between balancing values is repartitioning/load ITR first that correct default the the suggests By for is ( 2003) situations. (e.g., result al. ascertained The be et cannot Karypis measurement. times time phase), these second case the In by Factor. measurement ITR time first the divide Simply determined be can parameter Factor ITR the perform for to pass recommended required repartition- times to As the the values of on problem. appropriate quality depending optimization (2003), the multi-objective al. describes that a et metric Karypis is single by repartitioning balancing a adaptive with compute though to associated us even redistribution allows ing, data it the such, perform As during to load. incurred time describes the communications the parameter inter-processor This to the compared (2003). performing al. processing for et required parallel Karypis time ParMETIS the in between ITR ratio parameter the single The a (Modi- as scheme. MOSS repartitioning defined partitioning/adaptive a data is the to RCF minimizing facilitate linked to been objectives, model has two analysis library Services) ParMETIS between OpenSees application, tradeoff fied our the In controls edge-cut. or implementation cost URA redistribution the in RCF ParMETIS The in ITR of also Study is and costs 110.3.3.2 redistribution data and shown edge-cut been schemes. the matching has global scheme both than only matching minimize scalable This together to more matched helping partitioning. inherently at original be effective the may very on vertices subdomain be is, same to That the in matching. performed un- are is heavy-edge and they coarsening of partitioning, if phase, variant initial coarsening local graph coarsening, purely the graph a 3104 In of using phases: 568 (2000). three al. page: as et described Schloegel coarsening/refinement be can a which compute 110.1, to Figure attempts Algorithm Repartitioning . . function. . A Unified cost DECOMPOSITION this The DOMAIN minimizing PLASTIC directly while redistribution. 110.3. repartitioning data of amount total where Notes ESSI tedt eitiuinascae ihtels eattoigla aacn phase. balancing repartitioning/load last the with associated and redistribution repartitioning, data last the the 2. since occurred have that communications inter-processor all 1. h nfidRpriinn loih sbsduo h utlvlprdg hti lutae in illustrated is that paradigm multilevel the upon based is Algorithm Repartitioning Unified The α steRltv otFactor, Cost Relative the is 0 . | 001 E cut and | steeg-u ftepriinn,and partitioning, the of edge-cut the is 1000000 fIRi e ih eattoigwt low a with repartitioning a high, set is ITR If . eso:2.Ags,22,15:04 2021, August, 26. version: | V move | sthe is Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et clusters. SMP-based two on out carried been has measurement Performance Computers Parallel 110.4.2 can they fact figures. the repartition load to and/or due the partition studied obtain expose been to to has visualized program. used easily element parallel continuum be been based Only has PDD calculation. the (1997) plasticity of Sture by performance Jeremićimbalance and parallel the scheme study integration to constitutive up Implicit set been have SFSI of models we also is. gain we algorithm Secondly, performance PDD proposed much calculations. the how element scalable finite see how inelastic to show into want 3104 to algorithm we of want PDD 569 Firstly the There page: introducing analysis. investigated. by timing thoroughly have the is can for algorithm PDD focuses proposed major the two of are performance parallel chapter, . . this . ALG In PDD ON STUDIES PERFORMANCE Introduction 110.4. Algorithm PDD 110.4.1 on Studies Performance 110.4 Notes ESSI • • itiue eoyLnxUi lsesaemjrpafrsue nti hpe o pe panalysis. up speed for chapter this in used platforms major are clusters Linux/Unix memory Distributed element finite simulations, element finite SFSI scale large in PDD apply to is objective final our As dacdCmuigRsac CC)a h aionaIsiueo ehooyi Pasadena. in for Technology of Center Institute and California IL, the Argonne, at the in (CACR) at the Laboratory Research National (SDSC) at Computing Argonne Advanced Center (NCSA) Diego, Supercomputer Applications San million Diego Supercomputing California, $53 San of for University the with Center Urbana-Champaign, Foundation National Illinois, Science the of National sites: University the four the to by funding launched in was project TeraGrid The Cluster Linux Intel-Based IA-64 TeraGrid h ewr ecmr sshown is benchmark network The Super- 110.3. Diego Fig San 110.1 . in at Table shown nodes in is P655+ configuration 8-way 176 System of Center. consists computer cluster p655 eServer IBM DataStar The p655 eServer IBM al 1.:LtnyadBnwdhCmaio a fAgs 2004) August of (as Comparison Bandwidth and Latency 110.1: Table Intra-node Inter-node P aece ( Latencies MPI 7.65 3.9 µ sec) adit (MBs) Bandwidth 1379.1 3120.4 eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI adtevria odn sapida .k increments. 5.0kN at Supercomputing applied Diego San nodes). is at The 8-way supercomputer loading (P655+ DataStar vertical Center on the model. out the carried constitutive that been and research simple has 110.3 analysis seemingly this performance Table The for in in even shown shown significant are is is properties It material balancing load plastification. by adaptive by triggered speedup repartitioning show to research constitutive this advanced by through More applied modeled foundation. be is the for can soil assumed laws The is elasticity linear used. and rule) is hardening points kinematic (Gaussian) integration 8 parallel the with study Template3D to element up brick set been 3D has 110.4 performance. Figure in shown as model interaction soil-shallow-foundation A Model Interaction Soil-Foundation 110.4.3 eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 570 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI http://www.sdsc.edu/user DataStar of Configuration System 110.3: Figure h A4cutr nwihtescn ato h efrac td a endone. been has study performance the using of is part and second the Linux of which SuSE configuration on technical running cluster, the is IA64 shows four 110.2 cluster the with Table The network. equipped interconnect are node. cluster nodes Myrinet per The Myricom’s memory teraflops. physical 3.1 Intel of of GHz (GBs) performance 1.5 gigabytes peak dual with a each for nodes, processors, cluster 2 IBM Itanium 256 of consists currently cluster TeraGrid SDSC’s lsopatcmtra oe DukrPae oe ihAmtogFeeiknonlinear Frederick Armstrong with model (Drucker-Prager model material elasto-plastic Template3D oe lhuhtemdlue eesffie h ups of purpose the suffices here used model the although model eso:2.Ags,22,15:04 2021, August, 26. version: (2020) services/datastar/ Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 571 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI ac System Batch Compilers System Operating Disk Interconnect Network Processor Nodes Compute Nodes Access Architecture COMPONENT (tg-login.sdsc.teragrid.org) Cluster IA-64 al 1.:Mtra osat o olFudto neato Model Interaction Soil-Foundation for Constants Material 110.3: Table al 1.:TcnclIfraino A4TrGi lse tSDSC at Cluster TeraGrid IA64 of Information Technical 110.2: Table ieai Hardening Kinematic stoi Hardening Isotropic lsi modulus Elastic modulus Elastic rcinangle Friction oso ratio Poisson ratio Poisson Cohesion otbeBthSse PS ihCtln Scheduler Catalina with (PBS) System Batch Portable ? ? System) 8.0) File SLES (Parallel (SuSE GPFS 2.4-SMP of Linux TB 50 NFS, of TB 1.7 Channel Fiber Ethernet, Gigabit 2000, Myrinet ? ? ? ? ? ? ? ? ? Cluster Linux DESCRIPTION N:Frrn7CC++ C Fortran77 GNU: C++ C Fortran77/90/95 Intel: Tflops 3.1 performance Peak cache L3 MB 6 Integrated GHz 1.5 2, Itanium Intel processors) (524 nodes 262 GB 4 memory: SDRAM ECC dual-processor processors) (8 nodes 2 GB 8 memory: SDRAM ECC quad-processor / olna ( nonlinear A/F Foundation Soil E E 17400 = φ ν h ν 21 = Linear c a 0 = 37 = 0 = 0 = 116 = . GPa . 35 . 2 1˚ kPa . 0 , C r eso:2.Ags,22,15:04 2021, August, 26. version: 80 = . 0 ) Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et repartition. final whole the the of increment, edge-cut minimize factor to ITR tends the ParMETIS the If large, small, cost. very very redistribution be switch data being to a minimize set factor like can is which acts ITR repartitioning It With that do load. kernel. to the tends repartitioning balancing ParMETIS ParMETIS with of associated approaches redistribution algorithmic data the on perform to compared time time processing the parallel between the during ratio to incurred the communications describes inter-processor ParMETIS the in performing ITR for parameter required the 110.3.3.2, Section in described As ITR for Study Numerical 110.4.4 Real Only, (Indication Interaction Soil-Foundation of Section) Model Individual Each Element in Finite Shown Example Model 110.4: Figure 3104 of 572 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI nprle eino D,i eattoigi eesr oaheela aac fe ahload each after balance load achieve to necessary is repartitioning if PDD, of design parallel In AnalysisModel a ob ie fftu e nlsscontainer analysis new a thus off wiped be to has ( 1997) McKenna eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et to tends algorithm URA the then cost. large, redistribution very data be higher to considerably set with is but URA factor cut the ITR edge lowest the small, with if very repartitioning approach hand, give be repartitioning other diffusive to the which factor On in cost, ITR used. final redistribution With is data and minimize partition that algorithms. initial results of present the to types shows tends different 3D 110.12 prototype two Figure for for to analysis figures 110.7 parallel one repartition Figure subsequent The in problems. adopted approaches. interaction be different structure will and of soil performance data performance better Timing the bring study. to investigate parametric tends this to that in collected soil-structure used been Two been have behaves. have 3104 figures algorithm 110.6 of partition/repartition partition Figure 573 the in how page: shown see as research, to models this processors interaction 8 In and URA. 4 the 2, of on application, effectiveness ( out our ITR the for the higher investigate value of much to values . ITR . be extreme . performed adequate ALG two can PDD be an ON cost to determine STUDIES redistribution PERFORMANCE needs to data study order 110.4. preliminary The In only. steps. overhead analysis communication subsequent than reload to defined be can Notes ESSI 0 . 001 and 1 , 000 , 000 r rsrbdadte aallaayi scarried is analysis parallel then and prescribed are ) eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Interaction Soil-Foundation Studying for DOFs) 17,604 Elements, (4,938 Models Problems FE 110.6: Figure Prob- Interaction Soil-Foundation Studying for DOFs) 7,500 lems Elements, (1,968 Models FE 110.5: Figure 3104 of 574 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI 3104 of 575 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI iue107 atto n eatto n2CU IR1-,Ibl o.5) EMdl(1,968 Model FE 5%), Tol. Imbal. (ITR=1e-3, CPUs 2 on DOFs) Repartition 7,500 and Elements, Partition 110.7: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et (1,968 Model FE 5%), Tol. Imbal. (ITR=1e6, CPUs 2 on Repartition DOFs) 7,500 and Elements, Partition 110.8: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI 3104 of 576 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI iue109 atto n eatto n4CU IR1-,Ibl o.5) EMdl(1,968 Model FE 5%), Tol. Imbal. (ITR=1e-3, CPUs 4 on DOFs) Repartition 7,500 and Elements, Partition 110.9: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et (1,968 Model FE 5%), Tol. Imbal. (ITR=1e6, CPUs 4 on DOFs) Repartition 7,500 and Elements, Partition 110.10: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI 3104 of 577 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI iue101:PriinadRpriino Ps(T=e3 ma.Tl %,F oe (1,968 Model FE 5%), Tol. Imbal. (ITR=1e-3, CPUs 7 on DOFs) Repartition 7,500 and Elements, Partition 110.11: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et (1,968 Model FE 5%), Tol. Imbal. (ITR=1e6, CPUs 7 on DOFs) Repartition 7,500 and Elements, Partition 110.12: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI 3104 of 578 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI iue101:PriinadRpriino Ps(T=e3 ma.Tl %,F oe (4,938 Model FE 5%), Tol. Imbal. (ITR=1e-3, CPUs 7 on DOFs) Repartition 17,604 and Elements, Partition 110.13: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et (4,938 Model FE 5%), Tol. Imbal. (ITR=1e6, CPUs 7 on DOFs) Repartition 17,604 and Elements, Partition 110.14: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 5%) Tol. Imbal. DOFs, 7,500 Elements, (1,968 Studies Parametric ITR of Data Timing 110.15: Figure 3104 of 579 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Imbal. DOFs, 7,500 Elements, (1,968 ITR=1e6 over ITR=1e-3 of 5%) Speedup Tol. Relative 110.16: Figure 3104 of 580 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Tol. Imbal. DOFs, 17,604 Elements, (4,938 Studies Parametric ITR of Data 5%) Timing 110.17: Figure 3104 of 581 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI osuyteprle efrac.Tepercsigui,lk edn oe aafo l,hsnot has file, from data model reading like unit, preprocessing The performance. Template3DEP/NewTemplate3Dep) parallel as such the libraries, used study other and to (MOSS PDD in implemented been have routines Timing Analysis Performance Parallel 110.4.5 h td fti hpe,sm ocuin a edrawn. be can conclusions some chapter, this of study the ups st xoetetemr ffiin praht orpriinn o u pcfi aallSFSI parallel specific our for repartitioning do to approach ( efficient approach more scratch-remap either the simulations, the expose to is purpose eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 582 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI Bsdo h iigaayi efre nti hpe,IR1- stebs hieta brings that choice best the is ITR=1e-3 chapter, this in performed analysis timing the on Based 4. inevitably which scheme scratch/remapping the adopts algorithm URA the value, ITR large With 3. partition/repartitioning diffusive for results give to tends algorithm URA the value, ITR 22.1% small to With up is 2. gain performance The (1e6). value larger outperforms (1e-3) ITR of value Smaller 1. so h peu aao aaercsuyo T atr.The factors. ITR on study parametric of data speedup the show 110.17 and 110.16 110.15, Figures outeso h iuieapoc a o asdmc rul norapplication. our in trouble much caused not 110.16 has , 110.15 approach Figures diffusive in the shown of as significant Robustness more processing 110.17. is of gain and number performance of the increase size, the model the With or values. units ITR large over performance better substantially also This scratch/remapping. outperform zones). substantially plastic can of approach formation diffusive of the sense stops why the never part (in repartitioning in and stabilized explains partitioning is initial computation their the of though out even elements all method migrate scratch/remapping totally by to repartitionings tends of Repetitive analysis. be Another same will for repartitionings approach methods. more This diffusive much substructure-type than performed 110.12. on approach scratch/remapping and based the 110.10 method was, observation element 110.8, important finite Figures parallel shown for as meaning preserved great well integrity the is and graph graph quality high original gives approach of this But cost. redistribution data huge introduce as not repartitionings is graph. repetitive algorithm ill-connected that totally diffusive was yield observation the structures to important sense, graph tend very these this One to In scratch/remapping. paid as be calculation. robust must element shown attention as finite careful edge-cut the So high programing very 110.11. when with and graphs 110.9 approach disconnected 110.7, Diffusive even Figures the or diffusive in that the bad high. is fact very drawback very The gives the is typically performance. to approach better research due delivers this thus overall movement in in data possible redistribution application minimizes data our for with performance associated for overhead good is which better. scheme, get to tends speedup the larger, gets model the As processors. 7 for TR IT 1 = e 6 rdffsv prah( approach diffusive or ) eso:2.Ags,22,15:04 2021, August, 26. version: TR IT 1 = e − 3 .Through ). Jeremićet al., Real-ESSI 110.20. 110.19, 110.18, Figure in shown been have PDD by figures partition/repartition DOFs The 4,035 with Model Soil-Foundation 110.4.5.1 chapter. this of end the at discussed be will calculations. of element advantage finite the elastic-plastic investigate nonlinear to in approach results algorithm Decomposition All PDD Domain proposed performance. one-step computational static of with scalability compared the be been show have to will sizes processors different of with number Models various algorithm. on PDD tested adaptive proposed the of effectiveness the indicate element finite chapter. of size this the of as study robustly work application to the able in not increases was 5% model than larger tolerance imbalance load that is algorithm repartitioning Diffusive application. values. our ITR in large scratch/remapping over outperforms performance better brings (0.001) ITR smaller eeice l nvriyo aiona Davis California, of University al. Jeremić et the is other the and factor, ITR the is One tolerance. algorithm. imbalance balancing load load computational adaptive the re- of the performance from partitioner. see parallel can using 3104 by we of element minimized As 583 finite been elasto-plastic has page: research. overhead of partitioning/repartitioning performance this The overall to of the computations. partitioning improves aim graph partitioning basic graph plain the adaptive from below, also switching current sults per- is simply the meaningful which by More In partitioning, gains yet. . graph . partitioning. performance . addressed ALG adaptive consider been graph PDD ON not to by STUDIES has be gain PERFORMANCE problem would algorithmic solving 110.4. spective only equation the reflects research, here this up of phase speed the so timed been Notes ESSI 2. 1. ntefloigscin,tmn aaadpriinrpriinfiue ilb rsne n results and presented be will figures partition/repartition and data timing sections, following the In to clusters Intel IA64 and Power4 IBM DataStar on performed been have studies parametric Detailed conclusion first The illusive. more been has studies imbalance load on tuning hand, that other the previously on stated While as results consistent yield to tend factor ITR on tunings performance The ssae npeiu etoso hscatr hr r opeo e aaeesta control that parameters key of couple a are there chapter, this of sections previous in stated As etrigoeha a esbtnilyhge stefiieeeetmdlsz increases. size model element finite analysis- the subsequent as and higher redistribution data substantially that be fact model, can the element overhead to restarting finite due higher larger set with be speaking, should Basically tolerance the codes. application whole the of performance Tolerance minimizing Load to sections. Imbalance set previous Computational be in will explained focus as algorithm respectively and edge-cut set 1,000,000) or to be redistribution (up can large data value very this or (0.001) interconnections, small network very and to applications different on Depending scheme. ITR stekyprmtrwihdtrie h loihi praho h dpiela balancing load adaptive the of approach algorithmic the determines which parameter key the is steohrkyfco ffciggetyteoverall the greatly affecting factor key other the is eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 584 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI iue101:405DF oe,2CU,IR1-,IblTl5,PDPartition/Repartition PDD 5%, Tol Imbal ITR=1e-3, CPUs, 2 Model, DOFs 4,035 110.18: Figure iue101:405DF oe,4CU,IR1-,IblTl5,PDPartition/Repartition PDD 5%, Tol Imbal ITR=1e-3, CPUs, 4 Model, DOFs 4,035 110.19: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI 3104 of 585 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et iue102:405DF oe,8CU,IR1-,IblTl5,PDPartition/Repartition PDD 5%, Tol Imbal ITR=1e-3, CPUs, 8 Model, DOFs 4,035 110.20: Figure iue102:Tmn aao aallRn n405DF oe,IR1-,IblTl5% Tol Imbal ITR=1e-3, Model, DOFs 4,035 on Runs Parallel of Data Timing 110.21: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Tol Imbal ITR=1e-3, Model, DOFs 4,035 on Runs Parallel of Data 5% Speedup Absolute 110.22: Figure 3104 of 586 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI maac Tolerance Imbalance Factors ITR CPUs of # (DOF) Sizes Model al 1.:Ts ae fPromneStudies Performance of Cases Test 110.4: Table % 0,20% 10%, 5%, 1,000,000 0.001, 64 32, 16, 7, 5, 3, 68,451 32,091, 17,604, 4,035, eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Tol Imbal ITR=1e-3, Model, DOFs 4,035 on DD Static over PDD of 5% Speedup Relative 110.23: Figure 3104 of 587 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 110.30. 110.29 , 110.28, Figure in shown been have PDD 3104 by of 588 figures partition/repartition page: The repartition and data Partition Timing 110.30. balancing 110.24. to load in 110.28 adaptive shown Figure by as from gains elements shown are performance more figures with indicate . . . but to ALG before PDD collected ON described been STUDIES as PERFORMANCE has model 110.4. same DOFs the 17,604 is Elements, This 4,938 with Model Soil-Foundation 110.4.5.2 Notes ESSI iue102:Fnt lmn oe fSi-onainItrcin(,3 lmns 764DOFs) 17,604 Elements, (4,938 Interaction Soil-Foundation of Model Element Finite 110.24: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Imbal ITR=1e-3, Model, DOFs 17,604 Elements, 4,938 on Runs 5% Parallel Tol of Data Timing 110.25: Figure 3104 of 589 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Model, DOFs 17,604 Elements, 4,938 on Runs Parallel of Data 5% Tol Speedup Imbal Absolute ITR=1e-3, 110.26: Figure 3104 of 590 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et ITR=1e-3, Partition/Repartition, PDD CPUs, 2 Model, 5% DOFs Tol 17,604 Imbal Elements, 4,938 110.28: Figure Model, DOFs 17,604 Elements, 4,938 on DD Static over PDD 5% of Tol Imbal Speedup ITR=1e-3, Relative 110.27: Figure 3104 of 591 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI 3104 of 592 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI iue102:498Eeet,1,0 OsMdl Ps D atto/eatto,ITR=1e-3, Partition/Repartition, PDD CPUs, 4 Model, 5% DOFs Tol 17,604 Imbal Elements, 4,938 110.29: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et ITR=1e-3, Partition/Repartition, PDD CPUs, 8 Model, 5% DOFs Tol 17,604 Imbal Elements, 4,938 110.30: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et 3104 of 593 page: 110.34. Figure 110.39. to Figure 110.32 to Figure 110.35 from Figure shown from are . shown . results . are ALG up figures PDD Speed repartition ON STUDIES and 110.31. PERFORMANCE Partition Figure in 110.4. shown DOFs is 32,091 mesh Elements, The 9,297 with Model Soil-Foundation 110.4.5.3 Notes ESSI iue103:Fnt lmn oe fSi-onainItrcin(,9 lmns 201DOFs) 32,091 Elements, (9,297 Interaction Soil-Foundation of Model Element Finite 110.31: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Imbal ITR=1e-3, Model, DOFs 32,091 Elements, 9,297 on Runs 5% Parallel Tol of Data Timing 110.32: Figure 3104 of 594 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Model, DOFs 32,091 Elements, 9,297 on Runs Parallel of Data 5% Tol Speedup Imbal Absolute ITR=1e-3, 110.33: Figure 3104 of 595 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et ITR=1e-3, Partition/Repartition, PDD CPUs, 3 Model, 5% DOFs Tol 32,091 Imbal Elements, 9,297 110.35: Figure Model, DOFs 32,091 Elements, 9,297 on DD Static over PDD 5% of Tol Imbal Speedup ITR=1e-3, Relative 110.34: Figure 3104 of 596 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI 3104 of 597 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI iue103:927Eeet,3,9 OsMdl Ps D atto/eatto,ITR=1e-3, Partition/Repartition, PDD CPUs, 5 Model, 5% DOFs Tol 32,091 Imbal Elements, 9,297 110.36: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et ITR=1e-3, Partition/Repartition, PDD CPUs, 7 Model, 5% DOFs Tol 32,091 Imbal Elements, 9,297 110.37: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI 3104 of 598 page: . . . ALG PDD ON STUDIES PERFORMANCE 110.4. Notes ESSI iue103:927Eeet,3,9 OsMdl 6CU,PDPriinRpriin ITR=1e-3, Partition/Repartition, PDD CPUs, 16 Model, 5% Tol DOFs 32,091 Imbal Elements, 9,297 110.38: Figure eeice l nvriyo aiona Davis California, of University al. Jeremić et ITR=1e-3, Partition/Repartition, PDD CPUs, 32 Model, 5% Tol DOFs 32,091 Imbal Elements, 9,297 110.39: Figure eso:2.Ags,22,15:04 2021, August, 26. version: Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et gains. performance regeneration show model to and done communication been both have handle analysis to Speedup proposed overhead. be will strategy costs. adaptive both globally offset new can operation balancing adaptively load can that we assure which to through overheads strategy extra new a the create monitor and cost regeneration regeneration model and model communication chapter, this needs bigger. in still becomes observed but model As algorithm element performance. partitioning finite best the graph the when the get increases be with to overhead to order inherent has in not addressed model is be element This to finite the regenerated. happen, and communications off data wiped whenever example, For operations. performance implementation. communication the real in the addressed but been algorithm, single not partitioning The has graph the nodes. of processing approach algorithmic among the much differ might patterns communication with associated data overhead the regeneration is model One element units. finite problems. processing is application among one specific load other computational the the and balance cost, to communication wishes one when costs of implementation. PDD naive the of sections operations- previous in balancing had load extra own its PDD the offset that completely implies can otherwise It that overheads. case. related gain DD performance the than bring worse not be not does should PDD of performance as the sharply that dropped algorithm PDD proposed we of hand, 110.34. efficiency 3104 other and the of the 110.33 599 increases, on Figures size page: While in model finite shown problems. the elastic-plastic example as nonlinear on see for observed also been balance can has load up overall Speed improve calculations. can element partitioning graph . . graph . element ALG adaptive that PDD on shown ON been based STUDIES has PERFORMANCE algorithm it sections, previous 110.4. in results analysis performance From Fine-Tuning Algorithm 110.4.6 Notes ESSI hscatrwl rtivsiaeteeeto odblnetlrneo efrac n hna then and performance on tolerance balance load of effect the data investigate both first consider will to chapter hope This we application, our of performance overall the improve to order In repartitioning to overhead problem-dependent extra impose applications certain hand, other the On network the that fact the consider not does algorithm partitioning graph adaptive the Currently levels two consider to have we algorithm. PDD proposed of efficiency overall the we improve problems to the order address In to performed been has fine-tuning algorithm detailed more chapter, this In expects one balancing, load With expected. as work not does PDD of implementation naive the So eso:2.Ags,22,15:04 2021, August, 26. version: TR IT au indicates value Jeremićet al., Real-ESSI eeice l nvriyo aiona Davis California, of University al. Jeremić et Imbal ITR=1e-3, Model, DOFs 32,091 Elements, 9,297 on Runs 20% Parallel Tol of Data Timing 110.40: Figure been has setup Model performance. parallel sections. on previous tolerance in imbalance as of same has effect the DOFs the 32,091 study Elements, to 9,297 with chosen model been foundation Shallow method. decomposition domain static result, a As reduced. performance. be overall can the counts to harm repartition less the 3104 do that of can so 600 cost imbalance and page: regeneration load routine critical model the balancing increase imbalance the to load hopes trigger the one way, can increase this that In to is routine. repartition performance adaptive improve the to of tolerance way . when . natural . overwhelms ALG most cost) PDD regeneration the ON model STUDIES happens, (say, PERFORMANCE overhead repartitioning application-associated the 110.4. that out Tolerance finds Imbalance one If Load on Tuning Fine 110.4.7 Notes ESSI 110.42. and 110.41 110.40, Figures in shown been have results analysis Speedup over speedup showing failed that runs application. previous our of in efficiency effective improving been at not aims has approach and tuning work-around The a as viewed rather should This eso:2.Ags,22,15:04 2021, August, 26. version: