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sn hs ehd,teue a oeesl mlmn exis implement easily more alternativ may tensor, user strain the methods, the these mod to Using mo respect tangent tangent with analytical tensor for that stress solutions given However, analytical [2]. plastici intractable of (viscoelasticity, effects implementation inelastic and when as instabiliti s such stiffness moduli of While tangent detection enable component, computation. and simulated convergence the require of moduli nature e physical tangent finite implicit consistent an and within models constitutive implement To differentiation, numerical moduli, elasticity WORDS: KEY . . . Received n.J ue.Mt.Engng Meth. Numer. J. ENGINEERIN Int. IN METHODS NUMERICAL FOR JOURNAL INTERNATIONAL ∗ ulse niei ie nieLbay(www.onlinelibra Library Online Wiley in online Published ro eiaino ifiutaayia ouin.Copyri solutions. constitutive analytical difficult advanced of implementing derivation prior of means numeric effective that concluded and is It e ac tec effort. convergence computational more the obtain greater be of methods to approximated Application appr numerically found moduli. the the are analytical floating approximation of the higher of of order Using evaluation orders the solution. all as for analytical moduli that the found towards is tend It approximati precision. accur higher-order quadruple The using i precision. numerically floating-point schemes investigated increased difference implement finite to approximation e higher-order lat the computational for approximation, and numerical so precision analytical accuracy known floating-point their with accur models and constitutive for hyperelastic developed methods are approximation moduli numerical real-domain Two orsodnet:SehnJh only mi:stephen. Email: Connolly. John Stephen to: Correspondence a edt ifrne ewe hsvrinadteVersion 10.1002/nme.6176 the doi: and as version article this between differences and to pagination lead typesetting, may copyediting, the undergon and through publication been for accepted been has article This ihrodradhge otn-on rcso numerica precision floating-point higher and Higher-order eateto ehncl&ArsaeEgneig Univers Engineering, Aerospace & Mechanical of Department prxmtoso nt taneatct moduli elasticity strain finite of approximations tpe onCnol* oadMceze egnGorash Yevgen Mackenzie, Donald Connolly*, John Stephen Accepted elemen finite nonlinear hyperelasticity, precision, point Article 2019; 00 :1–19 .INTRODUCTION 1. hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This SUMMARY ywlycm.DI 10.1002/nme.6176 DOI: ry.wiley.com). ght [email protected] lapoiaino lsiiymdl sapowerful a is moduli elasticity of approximation al e en oe nti otx.Agnrlformula general A context. this in novel being ter uin.Temtosaehge-re n higher and higher-order are methods The lutions. n nsadr obepeiinadincreased and precision double standard in ons

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(unperturbed) tangent stationary The and tensor. perturbed stress th symmetric in the perturbed are of components stress the wherein technique rgamn eurd hs ehd r eeal escom appro obtain less [2]. to Miehe generally methods differentiation are deri numerical methods the applies b these to required required, related form evaluations programming derivat elementary the the of 7], in 6, accumulation 5, usage [4, methods numerical differentiation automatic developed, for or expressions expres analytical used, these the give is to software symbolically differentiated mathematics algebraic method numerical [3], and automatic symbolic, into categorised e ns ihntefiieeeetmto sn nytestre the only using method model. element finite the within ones, new 2 nti ae,tonmrclapoiainmtosfraccu for methods approximation numerical two paper, this In for allow numbers complex utilising methods Approximation h owr ifrnefruai rtodrapproximati first-order a is formula difference forward The o ohLgaga n ueinfiiesri configuration strain finite Eulerian and Lagrangian both For Accepted Article t applied previously methods differentiation Alternative hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This in.Floigti,tepormcodifies program the this, Following sions. v sotie ueial yalgorithmic by numerically obtained is ive mlxapoiainmtosalwfor allow second methods and approximation first omplex for [18], al et Tanaka by y, 1] hsmto a lobe applied been also has method This [16]. n iae agn oui spooe by proposed as moduli, tangent ximated r netgtdi ealt uniythe quantify to detail in investigated are y ehdi eododrapproximation second-order a is method ro[9.Tetomtosaeexpected are methods two The error[19]. t tantnetmdl,i rpsd In proposed. is moduli, tangent strain ite ntera-oanol.Temethods The only. real-domain the in n mlmnaini oedfcl than difficult more is implementation , uhacag sntpsil nolder in possible not is change a Such fhge-re prxmto methods approximation higher-order of i nqedrcin n orientations and directions unique six e dmteaia ucin oalwfor allow to functions mathematical ed ofiieflaigpitpeiini these in precision point floating finite to .I yblcdfeetainmethod differentiation symbolic a In s. ytewl-nw owr difference forward well-known the by aie ehp u oteadditional the to due Perhaps vative. eiin smnind higher-order mentioned, As recision. e acyitga,ipeetdby implemented integral, Cauchy ted srltosiso ie constitutive given a of relationships ss etsoftware. ment uiu oteegt-re.From[14], eighth-order. the to up duli brcierudoferr ia and Kiran error. round-off ubtractive euet tls ihrfloating-point higher utilise to cedure fapoiainicessbtthe but increases approximation of r rubto ed oad eo the zero, towards tends erturbation e agn ouii eedn on dependent is moduli tangent ted the of connection by found quently nbthge-re approximations higher-order but on bantnetmdl a be can moduli tangent obtain o itn oreo ro:subtractive error: of source flicting ,Mee[]apidaperturbation a applied [2] Miehe s, uhta h teseutosare equations stress the that such nt lmn otaeAbaqus. software element finite e agn oui u hi optimal their but moduli; tangent o.Amr rvln approach prevalent more A mon. 5.I 1,1,1,1] tis it 15], 14, 13, [11, In 15]. , feetainmto sshyper- uses method ifferentiation cierudoferr Complex error. round-off active npolm yPe-outet Prez-Foguet by problems in mle etrainmagnitudes perturbation smaller h nt lmn ovr In solver. element finite the y mrcldfeetainmethod differentiation umerical atn h xrsin othe to expressions the dapting aecmuaino elasticity of computation rate . o l ueia prxmto ehd.TeFrrnprogr Fortran [20]. dataset The a in methods. available approximation openly numerical all for computationa investigate solutions to employed analytical is analys Abaqus/Standard, element the Finite precision. to floating-point program compared increased these are In moduli purpose. this elasticity for created Lagr the programs both Fortran in assessed hyperel is two Accuracy for solutions. analy analytical compared known and with validated compared are methods if proposed investigated be only can solutio method analytical when is previo method approximation as numerical Although, efficiency. computational and accuracy ycmaio oteaayia ouin ocuin r p are Conclusions solution. analytical c the and to behaviour comparison convergence by the m where i method, derived are element methods analytically finite approximation the to 4 section compared const In configurations. and hyperelastic two evaluated for are A 3 discussed. moduli section in are presented error is attainable methods minimum the pre and ar floating-point magnitude precision, higher and floating-point approximations higher higher-order enable to procedure the ob opeetr,sc htbt rnainadround-of and moduli. truncation elasticity of both approximations that numerical accurate such complementary, be to atsa oriae n aevcosaeoitd h fixe The omitted. ela are finite vectors of base and relationships coordinates kinematic Cartesian fundamental the Here, o rvt.Tetopitdfraingain tensor gradient deformation two-point configuration. 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AcceptedX Article 2 A nd ∂ h oa ouertoi ie by given is ratio volume local The . ∂ C S il-ichf stress Piola-Kirchhoff C Ω sthe is ee nytefis-re owr prxmto sshown. is approximation forward first-order the only Here, . n fabd r ohdfie ntredmninlra coordina real three-dimensional in defined both are body a of th odrapoiain nLgaga n ueinconfigura Eulerian and Lagrangian in approximations -order 4 χ th f where , odrmtra lsiiytensor elasticity material -order ′ ( x ∆ = ) X S ∈ ∆ n h tanincrement strain the and Ω f χ hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This S 0 ( :Ω 2 = x smpe oiscret rsail qiaetpoint equivalent spatial, or current, its to mapped is S 0 + → nd ∂ stkna ucino h ih Cauchy-Green right the of function a as taken is ∂ ǫ C ǫ S ) il-ichf stress Piola-Kirchhoff R − 3 : uhthat such , f 2 1 ( ∆ x so tnadbnhakpolm using problem, benchmark standard a of is ) ninadElra ofiuain,using configurations, Eulerian and angian fcec,wt ota user-subroutines Fortran with efficiency, l C F saeitatbe h fetvns fthe of effectiveness the intractable, are ns ie by given s iinhv nteotmlperturbation optimal the on have cision seMee2 n h eeecstherein references the and Miehe[2] (see + eetdi eto 5. section in resented usqetagmnsaeoitdhere omitted are arguments Subsequent peetdadivsiae ihnthe within investigated and mplemented ealdi eto .Teefc that effect The 2. section in detailed e sysae,tepiayavnaeo a of advantage primary the stated, usly J ttv oes prxmt elasticity Approximate models. itutive a hnb endas defined be then may ,hge-re prxmtoso the of approximations higher-order s, sn tnaddul rcso and precision double standard using O si osiuiemdl ihknown with models constitutive astic e ( det = m n uruie eeoe are developed subroutines and ams e,rsetvl,gvnby given respectively, res, muainlefiinyaeassessed are efficiency omputational tct r endwt epc to respect with defined are sticity roscnb eue oprovide to reduced be can errors f h-re nt tantnosare tensors strain finite chy-Green ( ueia netgto ftetwo the of investigation numerical dl nbt aeiladspatial and material both in oduli eeec configuration reference d ia ouin.Frti esn the reason, this For solutions. tical ae ntrso hi numerical their of terms in gated ∆ ǫ x ) C = C hsipisthat implies This . lsiiytno[1,telinear the tensor[21], elasticity sdfie as defined is χ F ( ) X where , S t , ihrsett h right the to respect with ) and , > J t ∈ R 0 F h general The C stetime the is u othe to due = in and tions C a also may espace te Ω ∂χ/∂ = 0 F and (2) (1) T X F 3 . ouiaewl-nw 2] rosaeoitdadol h fi the only and omitted are Proofs given. fr [22]. this well-known from are and validation corrections moduli for and Abaqus/Standard conversions of Although, use method. later Abaqus/Standar the by to required due form is the is which stress, Cauchy prto,tesaileuvln f()i ie as given is (1) of equivalent spatial the operation, d eoiytensor velocity ,3 ,5 ;i DAadB=1 ,3 )teapoiae material approximated the 4) 3, 2, 1, = B and A 2D form convenient in the 6; 5, 4, 3, 2, acyGentensor Cauchy-Green tensor o optto fteapoiaesaileatct tens elasticity spatial approximate the of computation For tensor elasticity Spatial 2.2. h ple etrain eoe as denoted perturbation, applied The ybl dpigti ocmuetemtra lsiiymo elasticity material the compute to this Adapting symbol. etre eomto rdet r acltdusing calculated are gradients deformation perturbed tan,adsbeunl h etre tess h pert the stresses, by perturbed the subsequently pertu applied and the strains, of orientation and direction the indicate 1,2,3,1,1,2) epciey sn h indices the Using respectively. 23), 13, 12, wit 33, employed 22, is (11, notation Voigt probl conditions. the axisymmetric if four or or approximations, six only using computed where 4 esr()hsmnrsmere uhthat: such symmetries s and minor stress the has both of (3) symmetry the tensor Noting (3). using moduli lob ueial prxmtd oee,ulk h mate the unlike However, approximated. numerically be also acyGentnosadmyohrieb enda nargum an as defined be gradient otherwise may and tensors Cauchy-Green Here, h lry-ae rLetm eiaie fteKrhofs Kirchhoff the of derivative, time Lie or Oldroyd-rate, the and (4) in gradient deformation the of indices the Including = xadn h nrmn ftesri tensor strain the of increment the Expanding h 4 The h etre tessaecluae,alwn calculati allowing calculated, are stresses perturbed The 1 2 S ˆ ǫ d L IJ th sasalprubto magnitude, perturbation small a is F ˆ as − odrtno eoe h pta lsiiytno,which tensor, elasticity spatial the denotes tensor -order ( KL ersnsteprubdsrs esr hc r ahafun a each are which tensors stress perturbed the represents L L T ) v ∆ h indices The . L ( τ hc ntr sdfie ntrso h aeo h deformatio the of rate the of terms in defined is turn in which , = Acceptedτ Article : = ) − 1 2 τ b d FF ˙ r sd h pta lsiiytno sdrvda h Jau the as derived is tensor elasticity spatial The used. are h aeo eomto a t sa ento ntrso t of terms in definition usual its has deformation of rate The . ∆ ∆ − F ˆ FF 1 ( F IJ ˆ KL − ( − KL r h opnnso h 2 the of components the are 1 ) C FF ˙ ) IJ iJ − C = ( − A KL = τ ∆ 1 ( 2 ǫ F B ˆ F ∆ ) F T ˆ ( ) ≈ KL iJ ≈ F ( hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This FF KL h eiaie a elnaie[1 n a therefore can and linearized[21] be may derivatives The . − 1 + ǫ O ) 1 ǫ T − h ) h = E ( sdfie as defined is , S 2 1 ǫ S ˆ ∆ ǫ ˆ K ) IJ F C A T C F stetucto ro and error truncation the is ⊗ IJKL + : = iK F ˆ = F ˆ E ∆ ( ( − B L KL F ˆ T ) F 1 2 + = A δ btos ocmuetefradperturbed forward the compute To rbations. ( re eomto rdet r calculated are gradients deformation urbed T r h ichf testensor stress Kirchhoff the or, KL ) LJ h ∆ no h prxmtdmtra elasticity material approximated the of on − F C = nd tress ∆ mi eue oasm Dpaestrain plane 2D assume to reduced is em − − ) uii h ompooe yMiehe[2] by proposed form the in duli F nie 1 ,3 ,5 )representing 6) 5, 4, 3, 2, (1, indices h JIKL + S mwr oatraiesailtangent spatial alternative to amework iltno,goerccnrbtosare contributions geometric tensor, rial T il-ichf testnosand tensors stress Piola-Kirchhoff IJ netgtosuigtefiieelement finite the using investigations + .Temtvto o hsframework this for motivation The d. FF A S E F ∆ i ri entosue,teelasticity the used, definitions train sn h rnce delta Kronecker the using IJ L s-re owr prxmto is approximation forward rst-order L iL and − v F ⊗ n fteprubddeformation perturbed the of ent i = 1 − ( T T τ lsiiytno 3 sgvnin given is (3) tensor elasticity E F C B + δ sfrhrdfie ntrsof terms in defined further is ) KJ K n h aeo deformation of rate the and , IJLK to fteprubdright perturbed the of ction = olwdb push-forward a by followed ∆ KL FF tmyteeoebe therefore may It . O rdet sfollows as gradient, n − i DAadB=1, = B and A 3D (in stebig the is 1 T anrt fthe of mann-rate i τ n h left the and O espatial he Landau δ ij the , KL (7) (4) (8) (6) (3) (5) . endi em fteOdodrt fteKrhofsrs i stress Kirchhoff the of Oldroyd-rate ma the the of in terms as in interpretations defined same the with notation, Voigt In hnicuigteidcsadteKoekrdlai 1) t (11), in delta Kronecker by the calculated and indices the including Then hnb usiuino 9 h emti otiuin can contributions geometric form compact the the in (9) stress of Cauchy substitution by Then n xrse nVitntto ti ie as given is it notation Voigt in expressed And qiaett h aeilipeetto,tesailper spatial the implementation, material the to Equivalent as defined ap therefore forward is first-order The objectivity. preserve to present netgto stertucto ro xoetal dec methods: exponentially backward error truncation diff their investig as central investigation are Furthermore, methods accuracy[24]. difference numerical higher central only possible, f of approximat are approximations higher-order higher-order While for moduli. derived elasticity are equations General approximations numerical Higher-order 2.3. re prxmto n oi eerdt stesecond-orde the th as example, to referred For is approximation. so of and approximation order order their by to referred etre eomto gradients deformation perturbed tesadteJuanrt fteCuh tesaerelated are stress Cauchy as the here of Jaumann-rate denoted the stress, and Cauchy stress the of Jaumann-rate the of where oee,tesailprubddfraingainsaec are gradients deformation perturbed spatial the However, nrdcdfrcneine si[1,t inf the signify to in[21], as convenience, for introduced oecmeca nt lmn ovr eur h pta e spatial the require solvers element finite commercial Some τ ˆ ij Accepted Article a which tensors, stress Kirchhoff perturbed the represents O ij ǫ ( a kl 2 ( n ) b ≈ ) ABQ ≈ oprdwith compared 1 F ǫ ( ˆ 1 ǫ ∆ ABQ A h ( F ˆ ijkl kl h τ ˆ ABQ ( F ˆ τ ⊙ ˆ ij kl ) ( a kl ) ij B = iJ , F ˆ ) ( a F ) ˆ 1 kl ( F ijkl ( ˆ = J ( 1 = b kl sthe is ) b ( ) hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This ≈ ) kl ≈ ) h F 2 ǫ ijkl ) = iJ Jǫ − ( Jǫ − = 1 1 e 2 O 1 k 2 + τ ( + h τ F h nd ( a ⊗ τ ˆ ij τ ( ˆ A 2 i ǫ + ǫ odriett esradtesymbol the and tensor identity -order ij a i 1 e n ik − ( l ∆ ) − 4 δ ⊙ F F ˆ B o lrt,cnrldfeec cee are schemes difference central clarity, For . F ˆ ik th ( ( F ( ˆ 1 ( + τ jl odrpouto two of product -order b F kl 1 esstoodcmae ihfradand forward with compared twofold reases ( ) ⊙ kl + rxmto ftesaileatct tensor elasticity spatial the of proximation lJ + ) e ubddfraingain sgvnby given is gradient deformation turbed ⊙ radadbcwr ifrnemethods difference backward and orward rnemtosaemr ovnetfor convenient more are methods erence l ) ABQ luae using alculated τ − tdi eal ste r eeal of generally are they as detail, in ated τ 2]b h following the by [23] eiltno,tesaileatct tensor elasticity spatial the tensor, terial A + ⊗ τ eta ifrneapproximation. difference central r − eprubddfraingainsare gradients deformation perturbed he ipie to simplified s oso ohtemtra n spatial and material the both of ions + il ⊙ τ rtcnrldfeec sasecond- a is difference central first e + δ e ogv h amn-aeo the of Jaumann-rate the give to cel e atct esrt edfie nterms in defined be to tensor lasticity τ a B il k h lry-aeo h Kirchhoff the of Oldroyd-rate The . τ 1 ij i τ F F jk ) ⊙ i ) kJ ijkl ⊙ ) 1 ) 1 ) i ) a ij ( eafnto ftespatial the of function a re b ( ) kl ) 2 nd odrtensors: -order ⊙ a been has (13) (12) (15) (16) (10) (17) (14) (11) (9) 5 . rvos n3 the 3D In previous. 1)t give elasti to approximate (18) difference central for formula general satiebcwr etre on,wihi o-eofrce for argument non-zero the is Then which greater. point, and perturbed order backward twice a is on ttecnr on,wihe ofcet r prescri are coefficients weighted of point, used difference centre are the perturbations at backward found and forward appro app difference equidistant higher-order central of the the In this, points. For perturbed backward error. truncation the reduce nVitntto,where notation, Voigt In 6 h notation The ...Hge-re prxmtoso aeilelasticit material of approximations Higher-order 2.3.2. ...Hge-re prxmtoso aeilelasticit material of approximations Higher-order II Table 2.3.1. f in given coefficients are The twelfth-order, them. to up obtain study, to this in used used be may Fornberg[26] of ntesailcngrto,teprubddfraingra deformation perturbed equ the corresponding configuration, spatial the the give In to equivalently modified then are etre ons h coefficients the points, perturbed tess h tessaeafnto fteprubddefor perturbed (4) the of of modification ad a function using of a first calculation calculated are require stresses moduli The elasticity stresses. material the of Here, o xdprubto antd,icesn h re ft of order the increasing magnitude, perturbation fixed a For aulcluaino h eurdcefiinscnb tedio be can coefficients required the of calculation Manual m eest h oiino h prxmto si twr oli to were it if as approximation the of position the to refers z}|{ n ( m th · ) Accepted Article where -order, C sitoue oietf h position, the identify to introduced is C IJ 6 n A ( KL ( ( B 4 A f ) n ) ′ ≈ and ( ≈ o D etre 2 perturbed 2D) for x     = )    n 1 ǫ B 1 ǫ sa vnnme,fo[5,i ie as given is from[25], number, even an is     c z

r speiul end hsi endas defined is this defined, previously as are    m c z 2 F ˆ m 2 1 m ǫ r acltdb h equation the by calculated are F ( ˆ m }| KL m }| = 2 n ( = m = X ( kl n m even − X x even − ) ) = n n m m hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This X n { { even even − even ( = ) = = n ( − even n 2 F F ( 1) − ( nd c c x + + m m ( m m il-ichf tessaete acltd The calculated. then are stresses Piola-Kirchhoff c + ) +1 m m ) m S ˆ S mǫ ˆ ! ) A IJ f z ∆ ∆ n 2 m n ) 2 moduli y ( F z iymdl sfudb obnn 3 and (3) combining by found is moduli city ˆ x F F ˆ ˆ + scluae.Frfradadbackward and forward For calculated. is fteapoiain with approximation, the of , F insaecluae using calculated are dients iain ple ee neulnumber equal an here, applied ximations ( ˆ ! }| m ( ( m moduli y B nApni A. Appendix in oiain eur oefradand forward more require roximations KL kl e.Tegnrlfruafracentral a for formula general The bed. ( iinlprubd2 perturbed ditional m 2 }| m aingains hc r therefore are which gradients, mation ) KL ) oesr htteapoiainis approximation the that ensure To . ) tosfrtesaileatct tensor. elasticity spatial the for ations ! sads nagrtmsmlrt that to similar algorithm an so and us ) A ! { ta ifrneshmso fourth- of schemes difference ntral ) +     { eapoiainwl,i theory, in will, approximation he rtecnrldfeec schemes difference central the or    h ihrodrapproximations higher-order The O        + nalna rd i.e. grid, linear a on e ihrodrapproximations Higher-order ǫ + 2n O O ǫ 2 ǫ n 2 n nd Piola-Kirchhoff m endas defined m = (18) (21) (20) (22) (23) (19) − 2 . esri em fteJuanrt fteCuh tesas stress Cauchy the of Jaumann-rate the of terms in tensor n nVitntto sdefined is notation Voigt in And n nVitntto ti ie by given is it notation Voigt in And hl ihrodrapoiain nraeteexponenti the increase O approximations higher-order While precision floating-point Higher 2.4. h eea qainfrtesaileatct esri te in tensor by elasticity given spatial is the stress for equation general The prahb ih[]cnldsta o rtodrapprox first-order a for co that tensor perturbation concludes optimal all which Miehe[2] in by minimisation sca approach obtaine a be a require can even would perturbation for context, perturbati optimal the vary optimal of to exact approximation known the An are determining these variable, scalar As mi exactly errors. which round-off that and is magnitude perturbation optimal The magnitude perturbation Optimal 2.5. lowe in result moduli. elasticity to the expected of approximations is This magnitudes. numb investigaperturbation the or twice enables implemented thes precision t been quadruple values previously equating approximation, not precision by has higher method precision the The double truncates to computat This returned After counterparts. interest. are of values values the calculated wh other precision, double any in and equivalently defined variables also The are precision. solver quadruple incr in defined may In initially used. negligible are is is precision quadruple error compilers, Fortran round-off current floating-poi which higher for with magnitudes However, 27]. errors[9, round-off si 1) h emti otiuin aclt ietege the give to cancel contributions geometric the (16), in As ǫ 2 n o increasing for ij a ( kl ( b ) ) ≈ ≈   

Accepted Article   1 ǫ 1 ǫ ij ǫ       a opt ( kl ( 2 2 b m n ) ) m a eapoiae as approximated be may h eeto hsi elkont ersrce yteconfl the by restricted be to well-known is this of benefit the , = ≈ ≈ n = n X X even − even −       n n even even Jǫ Jǫ 1 1       ( ( c c 2 2 m m m m ˆ ) ˆ ) = = n n ǫ τ X X τ even even − − opt ij a hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This n n z z even even ≈ F ˆ F ˆ }| }| m ( m ( kl p kl ( ( ) c c ) macheps m m { {       ˆ ) ˆ ) τ τ − − a ij z z ( ( F 1 ˆ 1 F ˆ ota,alary,vralsadconstants and variables arrays, all Fortran, }| m m fteOdodrt fteKirchhoff the of Oldroyd-rate the of rms nmgiuei eeal o possible. not generally is magnitude on ( n rasrqie ytefiieelement finite the by required arrays and }| ⊙ m ⊙ ( b ro infiatfiue loigsmaller allowing figures significant of er figures. significant 16 approximately o kl tpeiin h ag fperturbation of range the precision, nt c r h tesadeatct tensors, elasticity and stress the are ich ) o fteqarpepeiinarrays, precision quadruple the of ion τ ldcyo h rnainerr by error, truncation the of decay al { τ ) iie h mato ohtruncation both of impact the nimises accurate more and error truncation r yotmsto 1] hc,i this in which, [10], optimisation by d ea oml o h pta elasticity spatial the for formula neral pnnsaecniee.Asimpler A considered. are mponents    { + ae u oisaalblt within availability its to Due ease. e nti otx.Fralodr of orders all For context. this in ted +    mto ftefis eiaie the derivative, first the of imation rast hi obeprecision double their to arrays e    τ τ    + ⊙ a ucino nindependent an of function lar ⊙ + O 1 1 O ) ) a ij ǫ ( 2 b ( ǫ kl n ) 2    n )    + + O O ǫ 2 ǫ n 2 n icting (25) (24) (26) (27) (28) 7 . em foeo eea tanmaue.Cnetn l hype function all energy Connecting strain d measures. Helmholtz strain is a several or that Further of behaviours one respectively. macroscopic of mechanisms, terms to molecular fit internal mathematical the phenomenologi a be defines may it equation constitutive hyperelastic A equations hyperelasticity Constitutive 3.1. investi numerical here methods. allow approximation considered numerical These is solutions. hyperelasticity analytical only known However, damage[32, 35]. includ plasticity[31], [34, materials 30], k Most 29, method with configurations. viscoelasticity[28, comparison precision spatial through and quadruple accuracy material numerical is proposed their methods the of approximation terms with numerical and higher-order precision of use The is (29) equation of validity section. the doub to and in magnitude equal approximations perturbation approximately higher-order is into error investigation an whose solutions obtain will argument yeeatcmtras h eomto ssltit isoc into split is deformation the materials, hyperelastic pia etrainmgiuei acltdas calculated is magnitude perturbation optimal rcso eue h ro yasurdpwr ie htdo that Given power. squared 10 a by error the reduces precision nqarpepeiin( precision quadruple in snwpeitdas predicted now is n httecntnsaeafnto of function a a are errors constants round-off the and that and truncation when occurs error minimum h s fqarpepeiinwl oe h pia pertur optimal ( the precision lower double will in approximation precision forward quadruple conflicti first-order of to use due increases the exponentially size perturbation twelfth-o the to up val of complete approximations and a second derivatives not and order is is approximations and second-order 18] studies to 14, up previous 13, investigated of 11, plots studies[1, error previous the with to true hold to seen fPe-oute l1] ipegnrleuto o high for lo equation as general in here simple proposed 18 graphically is a 14, approximations al[11], true 13, et 11, hold Prez-Foguet studies[1, to of later seen from magnitude be perturbation can (28) approximation antd a eapoiae y(29). by approximated be may magnitude h superscripts The hnices h oa ro by error total the increase then 8 h qaindaso h bevto httettlerrco error total the that observation the on Error draws equation The − h rpsdeuto 2)peit htfricesnl h increasingly for that predicts (29) equation proposed The pertu optimal the of approximation an permits equation This h ahn epsilon, machine The 16 infiatfiue,ti rdcsta rtodrapproxi first-order a that predicts this figures, significant ≈ ( γǫ · ) ersnsaysri esr() u otedfeec fs of difference the to Due measure(s). strain any represents n where ,

Accepted Articlen and ǫ opt γ macheps d sacntn,utli necpsterudoferr.The errors. round-off the intercepts it until constant, a is macheps ≈ r h reso h prxmto n ftedrvtv,re derivative, the of and approximation the of orders the are 10 − .NMRCLINVESTIGATION NUMERICAL 3. Error 16 ntrso h rnainerror truncation the of terms In . 10 = stesals computable smallest the is ǫ macheps opt ≈ − hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This 32 ( Ψ ωǫ ≈ , − ( · n ) macheps d 1 = where , xss endi em fui oueweethe where volume unit of terms in defined exists, ǫ opt uhta 2)i aifid h pia perturbation optimal the satisfied, is (28) that such and ≈ 10 ω 3 racmiaino hs behaviours these of combination a or 33] horic d macheps ) − n sacntn.Teeoe yasmn the assuming by Therefore, constant. a is 1) = grudoferr tas rdcsthat predicts also It error. round-off ng + 1 eadqarpepeiin h optimal the precision, quadruple and le dri sesdi eto 3.3. section in assessed is rder 8 odrdrvtvs h aiiyfrfirst- for validity The derivatives. -order oe h ento a egvnin given be may definition the more, d ainadvldto fteproposed the of validation and gation as such inelasticity, of aspect some e ihtesm prxmto scheme approximation same the With . ge-re prxmtosteoptimal the approximations igher-order .Bsdo 2)adteobservations the and (28) on Based ]. bepeiinnmesaesoe with stored are numbers precision uble icse nfrhrdti ntenext the in detail further in discussed nareetwt 2) Comparisons (28). with agreement in al rmcaitclydrvd i.e. derived, mechanistically or cally rodrdrvtvsadhigher-order and derivatives er-order dto steeapoiain only approximations these as idation F o ipecntttv oeswith models constitutive simple for eeul si rzFge tal[11], et Prez-Foguet in as equal, re h nt rcso ii.Following limit. precision finite the eatcmtrasi h assumption the is materials relastic h pia etrainmagnitude perturbation optimal the ainmgiue o xml,a example, For magnitude. bation onaayia ouin o both for solutions analytical nown n volumetric and rsod otetucto error: truncation the to rresponds κ ainuigqarpeprecision quadruple using mation bto antd,wihcnbe can which magnitude, rbation h ehd r sesdin assessed are methods The . netgtdi tnaddouble standard in investigated -o lt frltv ro vs error relative of plots g-log for fie yeutosdescribing equations by efined 10 = O + 1 eradbl eaiusin behaviours bulk and hear ( − ǫ n 16 > κ ) , h s fquadruple of use the , n 1 = 1 J 1 ohl re The true. hold to / on-f errors round-off 3 and I components: d spectively. 1) = (29) the . h v eomto rdet a ersetvl describe and dilation respectively as given without be are shear may compression, uniaxial gradients pr tension, deformation homog these a five moduli, represent The elasticity gradients and deformation gradien these tensor deformation Physically, stress The all the gradients. for calculate calculated deformation invest is unique is error five moduli relative elasticity average The approximate solutions. the of accuracy The moduli elasticity vol approximate of the Investigation where 3.2. B, Appendix in given completeness. g for are functions included energy C These strain the invariant. for of simplified Green Jaumann-rate further and is stress Cauchy here the used of respectively. terms stress i in Kirchhoff defined defined the initially of are Oldroyd-rate Rothert[41], the and and tenso Kaliske elasticity by material defined equivalent as the and tensor stress ucino h aiu rtinvariant first maximum the of function a l3]adHsanadSenan4] r epcieydefi respectively are an Steinmann[40], stress and material Hossain and The al[39] configurations. spatial and material ftelf rrgtCuh-re tantensor, strain Cauchy-Green right or defined left is the models these of of energy to isochoric the The disregarded. hybri to contributions incompressible volumetric an the incompressibility, As contribution. energy isochoric hyp for Hartmann[38] by discussed he as Stability increments, material. strain the and of limit extensibility smoot finite increa a the stiffness the and where limit, stable stability numeric defined unconditionally in clearly is differences model their to neo-Hookean due The chosen are These models. throughout. eopsdit qiaetcomponents. equivalent into decomposed F towards antds h netgtosueprubtosof perturbations use investigations The magnitudes. nryfnto ssmlrydculdit incompressibl into decoupled similarly components is function energy au fteotmlprubto antd.Frdul pre double For magnitude. perturbation 10 optimal perturbati the suitable of of range value the into insight good a provide ucin o h e-oka n etmodels, Gent and neo-Hookean the for functions F h eaieerri acltdfrec prxmto met approximation each for calculated is error relative The 1 = = naayia mlmnainrqie lsdfr soluti closed-form requires implementation analytical An h e-oka n etsri nryfntosaedefined are functions energy strain Gent and neo-Hookean The throughout: used are models constitutive hyperelastic Two nteeequations, these In 0   onto down 1 0 0 0 1 0 0 0 1 J 1 I / 3 1 I m   h parameters The . F ; W 10 F = 2 − = ( · 18

Accepted  Article J ) 1 4 0 and 0 0 / r sd o udul rcso,ternei xeddfro extended is range the precision, quadruple For used. are . 0 0 5 3 C F √ hc implies which , 1 1 4 U and . 5 ( J √ C ) µ 0 1 4 . 1 5 , uhthat such r aaeesrltdt h ha tfns and stiffness shear the to related parameters are   W µ ; and G F 3 = = W J C   − m nH hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This 0 = 0 0 0 . µJ I 0 0 2 = Ψ r e o05(P) . Ma n 25()respectively (-) 22.5 and (MPa) 1.0 (MPa), 0.5 to set are 1 2 J = m m √ − 1 0 hc ensa smttclmtwhen limit asymptotic an defines which , C 2 . ln W 2 / I W 1 3 1 C √ nH + = I 1 0 ¯ 0 1 1 1 and . 2 U − × e smttclya h tantnstowards tends strain the as asymptotically ses tr erfr otepstv entns fstress of definiteness positive the to refers re −   and a nry tesadeatct ouiare moduli elasticity and stress energy, tal .Tesailsrs n lsiiytensors, elasticity and stress spatial The r. 10 I Vitntto)tno opnnsusing components tensor notation) (Voigt ¯ 3 nmgiue n ofida approximate an find to and magnitudes on ; tesadeatct esr r also are tensors elasticity and Stress . b C 1 hyaete epcieyaatdt be to adapted respectively then are They ha ihdlto.I arxfr,they form, matrix In dilation. with shear gasaeaalbei h dataset[20]. the in available are ograms F J nosydfre lmn fmaterial. of element deformed eneously x lsiiytnos si timn et Steinmann in as tensors, elasticity d = W 4 − m omlto sltrue oenforce to used later is formulation d vnol ntrso h rtCauchy- first the of terms in only iven e ntrso h 2 the of terms in ned gtdb oprsnt h analytical the to comparison by igated scoi n iainlvolumetric dilational and isochoric e = rlsi aaee estimation. parameter erelastic em fteKrhofsrs tensor stress Kirchhoff the of terms n and ntrso h rtpicplinvariant principal first the of terms in n o h lsiiymdl nboth in moduli elasticity the for ons = iin etrainmgiue from magnitudes perturbation cision, saeiptt ota rga to program Fortran a to input are ts lsaiiyadfnto smoothness. function and stability al G   3 J sflos neomd uniaxial undeformed, follows: as d h e-oka[6 n Gent[37] and neo-Hookean[36] the o sn ag fperturbation of range a using hod − tr 0 0 0 0 1 0 1 1 3 epciey r endby defined are respectively, ucin h etmdlhsa has model Gent The function. h uh tes h implementation The stress. auchy 2 3 / b × 3 mti otiuin r also are contributions umetric b . h scoi tanenergy strain isochoric The . 10 5 where ,   x ; where , F 5 nyi em fan of terms in only = J   = nd 1 x 0 0 0 0 0 . J m 0 1 det m Piola-Kirchhoff sa nee,to integer, an is 10 F . = 550 9535 h strain- The . . 0 0 2 I to 1 I m 1 . 10 9535 − tends . . 2 2 − (31) (30) 3 (32) 36   is 9 . . oui eoe () ob nlddi h oprsn h e equation following The the by comparison. notation Voigt in the tru error in the relative of The included result calculation. a is be moduli analytical to precision t quadruple A(8), for allows denoted also moduli, This respectively. configurations spatial App. euti nuprubddfraingain o eldoma real for gradient deformation unperturbed an in result sfle h eaieerri acltdb oprsno t of comparison by calculated is error relative The false. is ugsigta h prxmto ehd r o affected not are methods and approximation neo-Hookean the the that between difference suggesting little p is double for there shown show is moduli elasticity material moduli analytical elasticity Material 3.2.1. followi number the and (16)), approximation. is precision brackets u quadruple in and scheme (8) number approximation the of respectively), fol type differences the the central indicate In r letters magnitude. two vs perturbation first magnitude optimal perturbation approximate of upthe graphs approximations log-log for co using precision are plotted quadruple moduli and spatial double and using material approximated precision, etrainmgiue mle hnteerne r much are ranges these than smaller magnitudes Perturbation 10 oidpnetyivsiaeteefc fhge-re ap higher-order of effect the investigate independently To Relative Error Relative Error 10 10 10 10 10 10 10 10 oteqarpepeiinaayia lsiiymoduli, elasticity analytical precision quadruple the to , 10 10 - 3 2 - 2 4 - 1 6 - 1 6 - 1 2 - 8 - 8 - 4 10 10 0 0 - 3 6 - 1 8 (c) neo-Hookean, quadruple precision (a) neo-Hookean, double precision 10 10 - 3 0 - 1 5 Perturbation Magnitude Perturbation Magnitude 10 10

Accepted appr moduli elasticity material of error Relative 1. Figure Article E - 2 4 - 1 2 R 10 = 10 - 1 8 - 9   a,b X 10 10 6 =1 - 1 2 - 6 E R 10 10 C scluae n vrgdfrec ne fteeatct t elasticity the of index each for averaged and calculated is h euto h oprsnbtenteapoiae and approximated the between comparison the of result The - 6 - 3 A 16 10 10 ab 0 0 − hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This C

Relative Error Relative Error App. 10 10 10 10 10 10 10 10 10 10 - 1 6 - 1 2 - 3 2 - 2 4 - 1 6 - 8 - 4 - 8 10 10 0 0 ab - 3 6 - 1 8 2 10 10   (d) Gent, quadruple precision - 3 0 - 1 5 1 2 (b) Gent, double precision Perturbation Magnitude Perturbation Magnitude cto n onigerr hogotits throughout errors rounding and ncation stepeiinue dul rcso is precision (double used precision the is / eiini iue1 n b hs plots These 1b. and 1a Figure in recision edul rcso nltclelasticity analytical precision double he   10 10 gtehpe niae h re fthe of order the indicates hyphen the ng rxmtosadhge floating-point higher and proximations a,b X prdfrbt osiuiemodels constitutive both for mpared e F n Daetefradand forward the are CD and (FD sed - 2 4 - 1 2 oignmrclivsiain,the investigations, reveal numerical also lowing plots These error. elative etmdl o l approximations, all for models Gent 6 eapoiaemoduli, approximate he otetefhodr h eut are results The twelfth-order. the to C ysaiiydfeecso function or differences stability by =1 napoiain,i.e. approximations, in A 10 10 oe than lower 16 - 1 8 - 9 C rrbtentedul and double the between rror and A 10 oximations. 16 10 - 1 2 - 6 ab A 16 10 10 2 o h aeiland material the for ,   - 3 - 6 macheps 2 1 10 10 0 0 C A(8) CD(16)-12 CD(16)-10 CD(16)-8 CD(16)-6 CD(16)-4 CD(16)-2 FD(16)-1 A(8) CD(8)-12 CD(8)-10 CD(8)-8 CD(8)-6 CD(8)-4 CD(8)-2 FD(8)-1 + 1 n thus and App. > ǫ ensor (33) and 1 . etrainmgiuedcess l reso approxima of orders approximation. r first-order all e Once decreases, returns. relative diminishing magnitude lower with perturbation but models, approximation constitutive of order both For smoothness. o ihrodrapoiain ic h pia perturb optimal the modu since elasticity i approximations material higher-order increases the for of diminishing those to show similar are eighth-order results lo the achieves than generally invalid approximation greater by of affected t order less to increased are similar the approximations are shown spatial results are the These results but models. The constitutive stress. Gent Cauchy and the of Jaumann-rate the moduli elasticity Spatial 3.2.2. o to expected is perturbation optimal perturbation, co invalid whose Gent inval an approximations to in and due result neo-Hookean first-order therefore than both is greater error in The present smoothness. is or stability it this as precision, t quadruple Also, to and errors. double equal both gradient in logarithmic present is expected this the follow not does ic hs r o oti rnain oee,gvntheir given However, truncation. in lost negligible. not are these since o-eotrs tews,vle ls ozr a tl ha still may anal zero precision to double close values the otherwise, to terms; identical non-zero a generally precision quadruple are these moduli precision, double solut to analytical precision truncated double the than accuracy higher ihqarpepeiin hw nFgr c&1,teapprox the 1d, & 1c Figure in shown precision, quadruple With tlre etrainmagnitudes, perturbation larger At Relative Error Relative Error 10 10 10 10 10 10 10 10 10 10 - 3 2 - 2 4 - 1 6 - 1 6 - 1 2 - 8 - 8 - 4 10 10 0 0 - 3 6 - 1 8 (c) neo-Hookean, quadruple precision (a) neo-Hookean, double precision 10 10 - 3 0 - 1 5 Perturbation Magnitude Perturbation Magnitude

Accepted10 Article10 iue2 eaieerro pta lsiiymdl appro moduli elasticity spatial of error Relative 2. Figure - 2 4 - 1 2 10 10 - 1 8 - 9 10 10 - 1 2 - 6 h netgto fteapoiae pta lsiiym elasticity spatial approximated the of investigation The 10 10 - 6 - 3 ≥ 10 10 10 0 0 hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This − 2 ti on htterltv ro fteapproximations the of error relative the that found is it ,

Relative Error Relative Error 10 10 10 10 10 10 10 10 10 10 - 1 6 - 1 2 - 3 2 - 2 4 - 1 6 - 8 - 4 - 8 10 10 0 0 - 3 6 - 1 8 10 10 (d) Gent, quadruple precision hc etit h cuayo higher-order of accuracy the restricts which - 3 0 - 1 5 (b) Gent, double precision Perturbation Magnitude Perturbation Magnitude i huhtererrcniust decrease to continues error their though li, o o l reso prxmto.When approximation. of orders all for ion to cusa oe vld perturbation (valid) lower at occurs ation dtucto ro 1] Approximations [10]. error truncation id ro sntdet rnaino rounding or truncation to due not is error poiain ftemtra elasticity material the of pproximations etrainerr ndul precision, double In error. perturbation e ro.Hwvr prxmtosof approximations However, error. wer 10 10 incneg otesm ro sthe as error same the to converge tion crwti hsrange. this within ccur antd,teeerr r physically are errors these magnitude, oeo h aeileatct moduli elasticity material the of hose cuay h udul precision quadruple The accuracy. n udoferr r infiat sthe as significant, are errors ound-off eodro h prxmto.Since approximation. the of order he - 2 4 - 1 2 earsda au faon 10 around of value residual a ve tclsltos hsi refrall for true is This solutions. ytical rri bandwt nincreased an with obtained is rror nFgr o ohneo-Hookean both for 2 Figure in siuiemdl,i sntdeto due not is it models, nstitutive 10 10 - 1 8 - 9 mtdeatct ouireach moduli elasticity imated 10 10 ximations. - 1 2 - 6 10 10 - 3 - 6 10 10 0 0 A(8) CD(16)-12 CD(16)-10 CD(16)-8 CD(16)-6 CD(16)-4 CD(16)-2 FD(16)-1 A(8) CD(8)-12 CD(8)-10 CD(8)-8 CD(8)-6 CD(8)-4 CD(8)-2 FD(8)-1 dl uses oduli − 11 16 . hycniu oflcut u edtwrsa nietypro 2.5. indirectly section fu of F show an discussion results related towards 3. These the error. tend Figure the but and in magnitude fluctuate perturbation highlighted to as continue fluctuate, t they approaching errors when round-off However, dec approximation. the steadily the errors of of truncation order order the the each that for reveals magnitude points perturbation more the approxi of difference steps central linear eighth-order and fourth- spati using neo-Hookean The error. minimum the around magnitudes antdsaenti gemn o obepeiinhigher precision approxi double of for error. agreement orders perturbation lower in m not for ne nearest are reasonable spatial the magnitudes generally to measured are i taken (29) the are used of to values approximations measured compared where of are magnitudes, approximations orders the all I and calcu are precision magnitudes quadruple perturbation optimal approximate The magnitude perturbation Optimal a 3.3. precision double the than moduli approximation. elasticity use accurate the moduli, more elasticity material the with As magnitudes. 12 iue3 eaieerro orh n ihhodrapprox eighth-order and fourth- of error Relative 3. Figure udul:measured Quadruple: udul:calculated Quadruple: obe measured Double: obe calculated Double: h aueo rnainadrudoferr r xmndf examined are errors round-off and truncation of nature The Accepted fro magnitudes, perturbation optimal Approximate I. Table Article 1 1 1 Relative error 1 . . . . D1C- D4C- D8C-0CD-12 CD-10 CD-8 CD-6 CD-4 CD-2 FD-1 0 0 10 10 10 10 10 10 10 10 10 10 10 10 10 0 0 E E E E - 3 2 - 3 1 - 3 0 - 2 9 - 2 8 - 2 7 - 2 6 - 2 5 - 2 4 - 2 3 - 2 2 - 2 1 - 2 0 − − − − 10 17 16 8 8 - 8 1 2 1 4 1 . . . . 0 2 10 CD(16)-4 CD(16)-8 0 6 E E Perturbation Magnitude E E hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This - 7 1 − − − − 11 11 5 6 modulus. 10 1 4 1 6 - 6 . . . . 0 1 0 3 1 E E E E 4 − − − − 10 7 7 3 4 - 5 ain nqarpepeiinwt 450 with precision quadruple in mations mtoso e-oka pta elasticity spatial neo-Hookean of imations fhge otn-on rcso obtains precision floating-point higher of es talgrtmcgain qa to equal gradient logarithmic a at rease qain(9 n iue2 n 2c. and 2a Figure and (29) equation m te gemn iheuto 2)and (29) equation with agreement rther 1 2 1 5 . . antd,soni iue3 Using 3. Figure in shown magnitude, . . ae sn 2)frbt obeand double both for (29) using lated ain u h pia perturbation optimal the but mation; odrapoiain u oinvalid to due approximations -order 0 7 0 2 leatct ouiaeapproximated are moduli elasticity al 10 ayia ouinfralodr of orders all for solution nalytical E E E E loigteotmlperturbation, optimal the ollowing eotmlprubto magnitude, perturbation optimal he otoa eainhpbtenthe between relationship portional giue h prxmt values approximate The agnitude. h rvosscin nTable In section. previous the n - 4 − − − − -oka pia perturbation optimal o-Hookean 5 5 3 3 rhruigmr perturbation more using urther 1 8 10 1 2 1 1 . . . . 0 8 0 7 - 3 E E E E − − − − 4 4 2 2 1 1 1 3 . . . . 0 2 0 5 E E E E − − − − 4 3 2 2 1 3 1 5 . . . . 0 5 0 9 E E E E − − − − 3 3 2 2 . h eomdcnorpo hwn h ha tesi h 1-2 the in stress shear the showing plot contour deformed magni The perturbation optimal with methods approximation All magnitudes perturbation optimal with Convergence 4.2. th with models, constitutive Gent and neo-Hookean both for mod N the discretised methods, usi is approximation body 3D all The in For constant. is incompressibility. simulated is face is thickness one model its The that where such face. membrane opposite tapered mesh the a of at level of applied third consists the It for 4a. dimensions assessme Figure the with shown in is used model commonly The is problem membrane Cook’s The membrane Cook’s 4.1. dataset[21] the in available are here modul used elasticity subroutines UMAT analytical equivalent the using approximat obtained the of efficiency computational and magnitudes convergence perturbation optimal approximate measured th The solving by investigated are method, precision b quadruple wi approximations, must Higher-order arrays. trunca these pre-defined then the codes, are to which precision arrays, precision quadruple quadruple For declared problems. 2D respectively for DDSDDE(6,6) and d indices STRESS(6) with arrays o notation, terms pre-defined Voigt in in computed tensor are elasticity tensors spatial These and d stress the Cauchy provides the interface of the subr iteration, with each interface at UMAT elasticity, its Abaqus/Stand through efficiency. materials computational implemented and eleme finite convergence in of methods approximation the of performance The o context the in investigated is This section. methods. following these of impor costs another Though moduli. precisio elasticity prec quadruple precise double obtaining first-order the a beyond Therefore, accuracy truncation. in to improvement ele any finite hence However, and moduli. analytical approxima precision methods, quadruple these of both Utilising complementary. accompli been otherwise have t to approximation. known of not first-order is computation a approximations even elastici the for approximated solutions in precision analytical the errors quadruple from the round-off errors in solutions, to rounding novel the approximations, Due by numerical development. resu influenced precision which be quadruple perturbation, to of larger likely a also for is requirement diminish with the hi but to increased due and generally approxima is of moduli higher-order order elasticity e increasing approximated With that of configurations. accuracy spatial show the improve investigations both approximations numerical The findings investigation numerical of Discussion 3.4. Accepted Article p floating-point higher and higher-order the that shown is It .FNT LMN INVESTIGATION ELEMENT FINITE 4. hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This fraingain n eurscalculation requires and gradient eformation prxmto sastsatr en of means satisfactory a is approximation n o ee eeso ehrfieet The refinement. mesh of levels seven for i elkonCo’ ebaeproblem. membrane Cook’s well-known e atfco steascae computational associated the is factor tant hsadr obepeiinadtenovel the and precision double standard th sn yrdCDHeeet oenforce to elements C3D8H hybrid using these than accurate more are moduli ty li iuae ihasern odo 15 of load shearing a with simulated is el them equating by precision double to ted real-domain with accuracy of level This n eun.Tedmnsigrtrsare returns diminishing The returns. ing h amn-aeo h acystress. Cauchy the of Jaumann-rate the f etslesueol obeprecision double only use solvers ment in h cuayo h approximated the of accuracy the tion, ue eefudt tanconvergence. attain to found were tudes o ehd r oprdt solution a to compared are methods ion fie si h rvosscin ythe by section, previous the in as efined . e lsiiymdl edtwrsthe towards tend moduli elasticity ted rmTbeIaeue hogot The throughout. used are I Table from t nivldprubto rosand errors perturbation invalid in lts r sue,wihalw o user- for allows which used, is ard hscnet stemr significant more the is context, this uie rte nFrrn o finite For Fortran. in written outines hdi hscontext. this in shed aaeesdfie nscin3.1. section in defined parameters e terms in investigated is simulation nt atct oui ntemtra and material the in moduli, lasticity eiinapoiainmtosare methods approximation recision gpaesri onayconditions boundary strain plane ng rSRS()adDDSDDE(4,4) and STRESS(4) or , h nt lmn ehdi the in method element finite the f ln ssoni iue4 n 4c and 4b Figure in shown is plane acltduigindependently using calculated e ul xdadasern odis load shearing a and fixed fully in elements 2048 with refinement so ii sicneunildue inconsequential is limit ision diinlcefiins h use The coefficients. additional edul rcso analytical precision double he to nt lmn procedures. element finite of nt hrflaigpitprecision floating-point gher 13 . iue4 a oksmmrn,dmnin nm;()neo-Hoo (b) mm; in dimensions membrane, Cook’s (a) 4. 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Absolute Max. Force Residual (N) iue5 aiu oc eiul uigcnegnefr(a for convergence during residuals force Maximum 5. Figure sn ee eeso ehrfieet h opttoa effi computational the refinement, mesh of levels seven Using 10 10 10 10 10 10 -10 -8 -6 -4 -2 0 1

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AcceptedIteration Number Article 3 4 5 hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This tes(MPa). stress CD(16)-2 FD(16)-1 A(16) CD(8)-12 CD(8)-10 CD(8)-8 CD(8)-6 CD(8)-4 CD(8)-2 FD(8)-1 A(8) CD(16)-12 CD(16)-10 CD(16)-8 CD(16)-6 CD(16)-4 e sA8 n (6 epciey Though respectively. A(16) and A(8) as ted

Absolute Max. Force Residual (N) 10 10 10 10 10 10 e-oka n b etsimulations. Gent (b) and neo-Hookean ) n prxmt ehd’sletimes solve methods’ approximate and l -10 -8 -6 -4 -2 h aerslsaeotie o all for obtained are results same the , l smr opttoal efﬁcient computationally more is uli convergence quadratic methods, ferent 0 nebhvormyb bevdby observed be may behaviour ence en ha tes(P) c et shear Gent, (c) (MPa); stress shear kean, l.Gvnta h tesi computed is stress the that Given ely. ie o h lblmti iterations matrix global the for uired ,2 5 8 aesonta with that shown have 18] 15, 2, 1, h tesadtnetcomponents tangent and stress the f xmmfrersdasaeplotted are residuals force aximum lqarpepeiinmto,is method, precision quadruple el e2 htapoiainmethods approximation that he[2] numerical the While guaranteed. 1 lsltostpclyrequiring typically solutions al antd hwquadratic show magnitude n b.Bt obeadquadruple and double Both (b). 2

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Table in given and (19) equation hswr a upre nfl ya niern n Physica and Engineering an 18 by studentship - full EP/N509760/1 number: in Grant grant. supported (EPSRC) was work This implementation. pr here of proposed method approximation precision quadruple 16 h eomto gradient deformation the ( h nvriyo tahld Pr”dt rhv thttp:/ at archive data “Pure” Strathclyde 48b8-8778-fb4a2aacc5fd. re of this University during created the subroutines and programs Fortran The h ofcet o eta ifrneapoiain pt up approximations difference central for coefficients The h nltclsrs n lsiiytnosaedfie for defined invariant are first the tensors of elasticity terms in and defined stress analytical The a .AAYIA LSIIYMDL O IS NAIN HYPERE INVARIANT FIRST FOR MODULI ELASTICITY ANALYTICAL B. , b , c - 12 8 6 2 n 0- 10 4 & even al I egtdcoefficients Weighted II. Table d srqie,wihaedfie sfollows as defined are which required, is ) 5544 6- 4- 2- 6 5 4 3 2 1 -1 -2 -3 -4 -5 -6 .FNT IFRNEEUTOSADCOEFFICIENTS AND EQUATIONS DIFFERENCE FINITE A. Accepted1 Article - 1260 385 1 1 J o ohmtra n pta entos sripto fou of input user definitions, spatial and material both For . a 2 = 504 280 56 1 1 5 ∂W ∂ ¯ I 1 ACKNOWLEDGMENTS - c I ; - - m 105 1 63 84 60 AASTATEMENTDATA 1 5 4 5 b fteCuh-re tantnosadtedtriatof determinant the and tensors strain Cauchy-Green the of o eta ifrneapoiain pt 12 to up approximations difference central for 4 = hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This APPENDIX 15 56 21 20 12 ∂ 5 3 1 5 1 ∂ 2 ¯ I W 1 2 − - - - - - ; 6 5 3 2 4 7 6 5 4 3 1 2 c m = 1 7 6 6 5 5 4 4 3 3 2 2 ∂U ∂J vdsa cuaeadefciemeans effective and accurate an ovides erhaemd pnyaalbefrom available openly made are search ; h 12 the o - - - - /dx.doi.org/10.15129/b202f1b6-4d39- - 56 21 20 12 15 5 1 5 3 1 n yeeatccntttv model constitutive hyperelastic any d 11648. = ∂ 105 ∂J 63 84 60 5 5 1 4 2 cecsRsac Council Research Sciences l th U 2 odraecluae using calculated are -order - - - 504 280 56 5 1 1 1260 385 1 1 th LASTICITY -order. coefficients r 5544 1 (34) . h aeileatct esri endas defined is tensor elasticity material The ntesailcngrto,teKrhofsrs sdefined is stress Kirchhoff the configuration, spatial the In tensors spatial Analytical B.2. osiuiemdl endi eto .,teecoefficien these 3.1, section in defined models constitutive h amn-aeo h acysrs r eurd h Cauc the required, are stress by Cauchy stress the of Jaumann-rate the ntemtra ofiuainte2 the configuration material the In tensors material Analytical B.1. of terms in defined models constitutive hyperelastic all h pta lsiiytno endi em fteOldroyd the of terms in defined as tensor elasticity spatial The o nt lmn oe hr h acysrs esradsp and tensor stress Cauchy the where codes element finite For h mlmnain r hrfr eea n eur only require and general therefore are implementations The C iso σ = = 2 + Gent neo-Hookean 3 al I.Cefiinsfrcluaino nltclstres analytical of calculation for Coefficients III. Table a J bJ J − − iso 1 Accepted− Article 3 2 τ 4 3 S ogive to = I = ( C 1 1 2 + 3 vol a S ⊗ C iso b − 1 = τ I 1 ) 1 + b ¯ = vol ⊙ J − ( 1 1 − ⊗ S 2 ( C τ 3 1 c vol C ⊙ I d c b a = µ ¯ b J iso 1 ¯ nd − I + m 1 − 1 1 1 J 3 = il-ichf tesi endas: defined is stress Piola-Kirchhoff ) − dJ + ( − − c a C τ 1 3 ) + C − vol I hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This J J 1 2 C 1 dJ 1 C m µ = − ⊗ − = = ⊗ − 2 1 ( ) C 1 / 1 1 1 iso a 3 b ¯ ⊗ − ⊗ iso 1 ⊗ 2 0 I − ¯ C ⊗ J 1 + 1 b + ¯ 1 1 b ¯ m + − − 1 − 3 − vol 1 1 + C + ) b ¯ 2 ⊗ − 1 3 1 3 vol ⊗ − I I I b ¯ 1 C 2 1 1 1 saegvni al III. table in given are ts 1 2 cJ as C ⊗ ⊗ 1 − cJ k and 2 D rt fteKrhofsrs sdefined is stress Kirchhoff the of -rate − 1 0 1 C ( 1 h ento fteecefiinsfor coefficients these of definition the 1 + + 1 ysrs srltdt h Kirchhoff the to related is stress hy − J ( C n lsiiymoduli. elasticity and s J ⊙ + J 1 1 3 c + − − o h e-oka n Gent and neo-Hookean the For . ta lsiiytno ntrsof terms in tensor elasticity atial I ( − 1 + 1 9 1 J 1 9 1 ) J 1 + I ⊙ 1 ( )2 1) c I 1 1 ) 1 2 C 3 1 2 ⊗ J I − ( C C 1 1 1 1 k 2 − − ) 0 ⊗ C 1 1 − 1 + 1 ⊗ D ) 1 C ⊗ 1 − J 1 C 2 1 − 1 (42) (35) (36) (38) (37) (40) (41) (39) 17 . 18 endi 1)t give to (15) in defined 5 i ,SnW opttoa fcec fnmrclappro numerical of efficiency Computational W. Sun H, Liu 15. 8 aaaM aaaaT mt ,Fjkw ,BlaiD Schr D, Balzani M, Fujikawa R, Omote T, Sasagawa M, Tanaka f 18. Numbers Hyper-Dual of Development The J. Methods Alonso of J, Fike Unification and 17. Review JT. Hwang JRRA, Martins 16. 4 ia ,Kadla .Nmrclyapoiae Cauchy approximated Numerically K. Khandelwal R, Kiran 14. 0 ahrR. Mathur 10. 1 ´rzFge ,Rd´ge-ernA uraA Numeri A. Huerta Rodr´ıguez-Ferran A, Pérez-Foguet A, 11. 3 ia ,Kadla .Cmlxse eiaieapproxim derivative step Complex K. Khandelwal R, Kiran 13. Numeri A. Huerta A, Rodr´ıguez-Ferran A, Pérez-Foguet 12. .MeeC ueia optto fagrtmc(consiste algorithmic of computation Numerical C. Miehe 2. numer Robust J. Schröder D, Balzani M, Fujikawa M, Tanaka 1. .YugJ,YoJ aaurmna ,TbrL,PrchoR. Perucchio LA, Taber A, Ramasubramanian J, Yao JM, Young 3. .Krl .Mlilnug n ut-niomn generat multi-environment and Multi-language J. Korelc 4. .Krl .Atmto fpia n estvt nlsso analysis sensitivity and primal of Automation J. Korelc 5. .SnW hio L eeso E ueia Approximation Numerical ME. Levenston EL, Chaikof W, Sun 8. .RteS atanS uoai ifrnito o stres for differentiation Automatic S. Hartmann S, Rothe 6. .Dni E cnblRB. Schnabel JE, Dennis 9. .Wigr P. Wriggers 7. h amn-aeeatct esri eae oteOldroy the to related is tensor elasticity Jaumann-rate The ifrnito ceefrhprlsi aeilmdl b models Engineering material and hyperelastic Mechanics Applied for scheme differentiation inelasticity. Engineering app and its Mechanics and Applied approximation derivative complex-step on based 10.1016/0045-7825(96)01019-5 uruie o imcaia rwhAnalysis. Growth Biomechanical for Subroutines imdclEngineering Biomedical models. mlmnaino brrifre yeeatcmateria hyperelastic fiber-reinforced a of implementation nttt fArnuisadAtoatc;21;Rso,V Reston, 2011; Astronautics; Hori and New Aeronautics the of including Institute Meeting Sciences Aerospace AIAA 49th Models. Computational o:10.1115/1.4002375 doi: nvriyo ea tAsi,http://hdl.handle.net/215 Austin, at Texas of University optr n Structures and Computers Computers 1996. . 09 45:6169 o:10.1007/s00466-009-0395-2 doi: 631–649. 44(5): 2009; 603 o:10.1115/1.2979872 doi: 061003. mlmnain fNnierHprlsi aeilModel Material Hyperelastic Nonlinear of Implementations Mechanics 10.1016/S0045-7825(99)00296-0 opttoa plasticity. computational 00 82:1914 o:10.1002/(SICI)1097-0207(2000 model. doi: MRS-Lade 159–184. the 48(2): 2000; to application an matrices: ABQ iieEeet nAayi n Design and Analysis in Elements Finite 05 58:10–15 o:10.1007/s00419-014-0939-6 doi: 1103–1125. 85(8): 2015; 02 84:3237 o:10.1007/s003660200028 doi: 312–327. 18(4): 2002; naayia praht optn tpszsfrfinite-di for sizes step computing to approach analytical An olna iieEeetMethods Element Finite Nonlinear optrMtosi ple ehnc n Engineering and Mechanics Applied in Methods Computer Accepted Articleiso = aJ + IAJournal AIAA − bJ 06 91) 1118.di 10.1080/10255842.2015.111 doi: 1171–1180. 19(11): 2016; optrMtosi ple ehnc n Engineering and Mechanics Applied in Methods Computer ueia ehd o nosrie piiainadNonl and Optimization Unconstrained for Methods Numerical σ 1 04 4:11.di 10.1016/j.compstruc.2014.04.009 doi: 1–13. 140: 2014; − = 1 1 σ ⊙ b ¯ iso b ⊗ ¯ 05 8:2–5 o:10.1016/j.cma.2014.08.020 doi: 22–45. 283: 2015; 04 6:4440 o:10.1016/j.cma.2013.11.005 doi: 454–470. 269: 2014; ABQ + 03 11) 5229.di 10.2514/1.J052184 doi: 2582–2599. 51(11): 2013; ABQ b ¯ + σ − vol vol b ¯ hsatcei rtce ycprgt l ihsreserved rights All copyright. by protected is article This = 3 1 REFERENCES ⊙ eln edleg pigrBri edleg.2008 . 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Phd . niern with Engineering ) 08 130(6): 2008; 6 Siam 16. . (43) (46) (45) (44) in g . 7 etA.ANwCntttv eainfrRubber. for Relation Constitutive New A AN. Gent 37. 9 timn ,HsanM osr .Hprlsi oesf models Hyperelastic G. Possart M, Hossain P, Steinmann para material 39. of Identifiability RR. Gilbert S, Hartmann 38. 6 ilnR.LreEatcDfrain fIorpcMate Isotropic of Deformations Elastic Large RS. Rivlin 36. 0 osi ,SenanP oehprlsi oesfrrub for models hyperelastic More P. Steinmann M, Hossain 40. 1 aik ,RtetH ntefiieeeetimplementatio element finite the On H. Rothert M, Kaliske 41. 5 agS hse A xeietlcaatrzto n c and characterization Experimental SA. Chester S, Wang 35. 3 ¨keeS ih .Amcomcoapoc orubber-li to approach micro-macro A C. Miehe S, Göktepe 33. 4 ih ,Kc .Sprmoe nt elastic-viscoelast finite Superimposed J. Keck C, Miehe 34. 9 osi ,V K timn .Eprmna td n num and study Experimental P. Steinmann DK, Vu M, Hossain 29. 2 ge W obrhD.Aped-lsi oe o h Mul the for model pseudo-elastic A DG. Roxburgh RW, Ogden 32. 7 sasnE elrHB. Keller E, Isaacson 27. 8 ih ,Gotp .Amcomcoapoc orubber-li to approach micro-macro A S. Göktepe C, Miehe 28. 1 ioJ,Tyo L ossettnetoeaosfrrat for operators tangent Temperatu Consistent S. RL. Zang Taylor J, JC, Wang Simo M, 31. Hossain L, Zhang X, Yao Z, Liao 30. 6 onegB eeaino nt ifrnefrua na on formulas difference finite of Generation B. Fornberg 26. 4 hnI,Ob .Nwfiiedfeec omlsfrnumeri for formulas difference finite New R. Ohba IR, Khan 24. T. Hughes JC, Simo 23. 5 iJ eea xlctdfeec omlsfrnumerica for formulas difference explicit General J. Li 25. 1 ozpe GA. Holzapfel 21. 2 azn P at ,VrlJ okcnuayerri comm in error conjugacy Work J. Vorel M, Gattu Baˇzant ZP, 22. 9 alyD,Bri ,BrenJ.Hg-rcso computa High-precision JM. Borwein R, Barrio DH, Bailey 19. 0 onlyS.Frrnporm n bqssbotns H subroutines: Abaqus and programs Fortran SJ. 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