Coupling Finite and Boundary Element Methods for Static and Dynamic

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Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458

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Comput. Methods Appl. Mech. Engrg.

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Coupling ﬁnite and boundary element methods for static and dynamic elastic problems with non-conforming interfaces

Thomas Rüberg a, Martin Schanz b,* a Graz University of Technology, Institute for Structural Analysis, Graz, Austria b Graz University of Technology, Institute of Applied Mechanics, Technikerstr. 4, 8010 Graz, Austria article info abstract

Article history: A coupling algorithm is presented, which allows for the flexible use of finite and boundary element meth- Received 1 February 2008 ods as local discretization methods. On the subdomain level, Dirichlet-to-Neumann maps are realized by Received in revised form 4 August 2008 means of each discretization method. Such maps are common for the treatment of static problems and Accepted 26 August 2008 are here transferred to dynamic problems. This is realized based on the similarity of the structure of Available online 5 September 2008 the systems of equations obtained after discretization in space and time. The global set of equations is then established by incorporating the interface conditions in a weighted sense by means of Lagrange Keywords: multipliers. Therefore, the interface continuity condition is relaxed and the interface meshes can be FEM–BEM coupling non-conforming. The field of application are problems from elastostatics and elastodynamics. Linear elastodynamics Ó Non-conforming interfaces 2008 Elsevier B.V. All rights reserved. FETI/BETI

1. Introduction main, this approach is often carried out only once for every time step which gives a staggering scheme. A comprehensive overview The combination of finite and boundary element methods (FEM of such methods is given by von Estorff and Hagen [5]. Although and BEM) for the solution of problems arising in structural the iterative coupling is very attractive from the point of software mechanics is attractive because it allows for an optimal exploita- design, the convergence commonly depends on relaxation param- tion of the respective advantages of the methods [14,32]. Consider, eters which are rather empirical [5]. For this reason, a direct cou- for instance, the problem of soil-structure interaction, where a fi- pling approach is preferred in this work which is independent of nite element method is well-suited for the treatment of the struc- such parameters. Direct FEM–BEM coupling itself can be separated ture and the near-field with its capability of tackling nonlinear into substructuring methods, where the interface conditions are di- phenomena. On the other hand, the finite element mesh has to rectly fulfilled by setting equal the nodal unknowns, and into La- be truncated which spoils the quality of the numerical analysis grange multiplier methods which enforce the interface conditions especially in dynamics where spurious wave reflections would oc- by auxiliary equations. The substructuring concept for FEM–BEM cur. Boundary element methods are appropriate for the represen- coupling is given, for instance, in the books of Beer [3] and Hart- tation of infinite and semi-infinite media and it seems thus mann [13]. The drawback of the classical substructuring is that natural to employ both methods in combination for such problems. the assembly spoils the structure of the system matrices if, for in- The idea of combining these two discretization methods goes stance, a sparse symmetric positive definite finite element stiffness back to Zienkiewicz et al. [32] who pointed out the complementary matrix and fully-populated nonsymmetric boundary element sys- characters of the methods and the benefits of their combined use. A tem matrix are assembled. Moreover, these methods require con- mathematical survey of the coupling of FEM and BEM is given by forming interface meshes, i.e., the nodes of the interface Stephan [29]. One branch of FEM–BEM coupling is the iterative discretizations have to coincide and the interpolation orders have coupling in which the individual subdomains are treated indepen- to be equal on both sides of the interface. The Lagrange multiplier dently by either method based on an initial guess of the interface approach circumvents both of the mentioned drawbacks because unknowns. Then, the newly computed displacements or tractions matrix entries are never mixed from the different subdomains on the interface are synchronized and based on these updated val- and the interface conditions can be posed in a weighted sense such ues another subdomain solve yields enhanced results. In time do- that non-conforming interfaces can be handled. The mathematical analysis of Lagrange multiplier methods can be found in the book * Corresponding author. Tel.: +43 316 873 7600. of Steinbach [27] and this concept has been transferred to acoustic- E-mail address: [email protected] (M. Schanz). structure coupling by Fischer and Gaul [10]. In this work, a

Lagrange multiplier approach is preferred because the possibility with the Lamé parameters k and l and the mass density q. For sim- of combining non-conforming interface discretizations is of great plicity, the following shorthand is used for (1) benefit especially when combining finite and boundary elements. qu€ðx; tÞþðLuÞðx; tÞ¼fðx; tÞ; ð3Þ In [11,12], a similar method is proposed for elastostatic problems. But in that approach a three-field method is used. Between where the dot-notation is used for the temporal derivatives and L is the interfaces of two subdomains an additional reference frame a partial differential operator which is in fact elliptic [28]. with displacement unknowns is introduced. Then, so-called local- The function u is assumed to have a quiescent past and, there- ized Lagrange multipliers between each interface and this frame fore, vanishing initial conditions at t ¼ 0 ensure the coupling conditions. In addition, the Lagrange multipli- uðx; 0þÞ¼u_ ðx; 0þÞ¼0: ð4Þ ers are assumed as pointwise constraints in order to avoid the te- dious integration over shape functions from different surface The boundary trace of the displacement field is denoted by discretizations. But the placement of the reference frame degrees uC ¼ Tru and the corresponding traction field by t ¼ T u, where Tr of freedom is not straightforward and requires additional work. is the trace to the boundary C and T is the traction operator. By pre- In the presented method, the algebraic structure of the final sys- scribing boundary conditions on these quantities, an initial bound- tem of equations is similar to the FETI method (see the survey arti- ary value problem is given cle of Farhat and Roux [9]) and the treatment of floating qu€ðx; tÞþðLuÞðx; tÞ¼f ðx; tÞðx; tÞ2X Âð0; 1Þ subdomains is according to [7]. The application of the FETI method to three dimensional elasticity problems especially for the case of uCðy; tÞ¼gDðy; tÞðy; tÞ2CD Âð0; 1Þ ð5Þ large discontinuities in the material parameters is presented in tðy; tÞ¼gNðy; tÞðy; tÞ2CN Âð0; 1Þ [17]. An extension of the FETI technique to the coupling of finite together with the initial condition (4). This problem is formulated and boundary element methods is given in [18] which is based for the domain X with the boundary C which is subdivided into on the idea of method-independent Dirichlet-to-Neumann maps. the two parts C and C where Dirichlet and Neumann boundary On the other hand, the treatment of non-conforming interface dis- D N conditions are prescribed for the boundary trace u and the cretizations is well-established within the context of the mortar C traction t, respectively. methods (see the book of Wohlmuth [31]), where special shape functions for the Lagrange multiplier fields are used such that 2.2. Variational principle the corresponding unknowns can be eliminated from the final system of equations. The combination of the FETI method with the A possible variational principle for the initial boundary value mortar method has also been carried out in [2] and [25] for the problem (5) is to require the equation [16] solution of elliptic problems. Here, the concept of the FETI coupling € framework for non-conforming interface discretization is followed. hqu; viþatðu; vÞ¼FtðvÞð6Þ It is extended for the employment of boundary element methods to hold for suitably chosen functions v. In this expression, hu; vi is as an alternative discretization method and, moreover, carried over the L -scalar product of the displacement field and an admissible to the treatment of dynamic problems in time domain. 2 test function v. The bilinear form a ðu; vÞ is introduced which is de- The finite element method used in this work is the standard ap- t fined as proach for linear elasticity which can be found in the book of Hughes [16]. The employed boundary element method is a colloca- atðu; vÞ¼ rðuÞ : eðvÞdx; ð7Þ tion approach for static and dynamic problems. See the book of ZX Schanz [24] for a collocation boundary element method for elasto- where rðuÞ is the stress tensor due to the displacement field u and dynamic problems. Here, the formulation is slightly altered in or- eðvÞ the strain tensor due to the test function v. Moreover, F ðvÞ is der to realize Dirichlet-to-Neumann maps as in [26]. In both t the linear form cases, i.e., finite and boundary element discretizations, the method itself is not a major point of this work but their combination. More- over, the established coupling framework is not restricted to the FtðvÞ¼ f Á vdx þ gN Á vC ds; ð8Þ X CN chosen finite and boundary element formulations. Each of the local Z Z discretization schemes is easily replaced as long as a Dirichlet-to- where vC is the boundary trace of the test field v. Note that both the Neumann map can be formulated. bilinear and the linear form carry the subscript t which indicates that they are time dependent. 2. Linear elastodynamics 2.3. Boundary integral equation 2.1. Basic equations Alternatively, the solution u to the initial boundary value problem Within the framework of linear elasticity, the dynamics of an (5) can be expressed by the boundary integral representation [24] elastic solid are governed by the Lamé–Navier equations [1] t o2 u x t UÃ x y t t y ds d uðx; tÞ 2 2 fðx; tÞ ð ; Þ¼ ð À ; À sÞ ð ; sÞ y s 2 À c1rðr Á uðx; tÞÞ þ c2r Âðr Â uðx; tÞÞ ¼ ð1Þ 0 C ot q Z Zt T Ã > in the presence of the body forces fðx; tÞ.In(1), the vector field À ð yU ðx À y; t À sÞÞ uCðy; sÞdsy ds: ð9Þ 0 C uðx; tÞ describes the displacement of a material point at point x Z Z and time t. Both the three-dimensional and the plane strain cases In this equation, UÃðx À y; t À sÞ is the fundamental solution of the are governed by this equation and, therefore, one has x 2 Rd with Lamé–Navier Eq. (1), see, e.g., [1]. Note that in expression (9) the d ¼ 2ord ¼ 3 in the following. Moreover, the speeds of the com- volume force term f is assumed to vanish and the material proper- pression and shear wave, c1 and c2, are used which are defined by ties are constant throughout the whole domain X. Moreover, the integration and the differentiation involved in the application of k þ 2l l c and c 2 the traction operator T to the fundamental solution are with re- 1 ¼ q 2 ¼ q ð Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffi spect to the variable y which is indicated by the corresponding sub- T. Rüberg, M. Schanz / Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458 451

I scripts. The application of the trace Trx to the integral representation (9) yields the boundary integral equation in operator notation uhðx; tÞ¼ uiðxÞuiðtÞð17Þ i¼1 [4] X with the shape functions ui and the time dependent approximation Vt Ã t ¼ CuC þ Kt Ã uC; ð10Þ coefficients ui. Using the test functions v from the space spanned by where the asterisk denotes temporal convolution, i.e., the shape functions ui and inserting the approximation (17) into t g Ã h ¼ 0 gðt À sÞhðsÞds. The introduced operators are the single the variational formulation (6) yields the system of coupled ordin- layer operator Vt , the double layer operator Kt, and the integral free ary differential equations R term C which are defined as follows for x, y 2 C and 0 < t < 1 Mu€ðtÞþAuðtÞ¼fðtÞ: ð18Þ t Ã ðVt Ã tÞðx;tÞ¼ U ðx Ày;t ÀsÞtðy;sÞdsy ds; The matrices used in this expression are the mass matrix M, the 0 C stiffness matrix A and the force vector f. Moreover, u is the assembly Z Z t Ã T of the time dependent coefficients of the approximation (17) and u€ ðKt ÃuCÞðx;tÞ¼lim ½ðT yU ðx Ày;t ÀsÞÞ uCðy;sÞdsy ds; ð11Þ e!0 Z0 ZCe its second time derivative. In fact, one has Ã T ðCuCÞðx;tÞ¼uCðx;tÞþlim ðT yU ðx Ày;0ÞÞ uCðx;tÞdsy: M½i; j¼hqu ; u i; e!0 j i Zce A½i; j¼atðuj; uiÞ; ð19Þ In these expressions, Ce and c are integration regions defined e f by ½i¼FtðuiÞ;

o with the bilinear form at of (7) and the linear form Ft of (8). The use Ce ¼ C n BeðxÞ and ce ¼ X \ BeðxÞ; ð12Þ of a classical time integration scheme such as the Newmark method where the ball of radius e with center x is denoted by BeðxÞ and its [22] gives the series of systems of algebraic systems of equations on surface by oB ðxÞ. e the equidistant time grid 0 ¼ t0 < t1 ¼ Dt < ÁÁÁ< tn ¼ nDt. In any case, this series of equations can be abbreviated by 2.4. Static case Aun ¼ fn þ hn ð20Þ In the limiting case of a static model, the previously presented with the dynamic stiffness matrix A, the coefficients of the approxi- equations are simplified. First of all, the boundary value problem e mation un at time point tn, the force vector at that time point, and a is now given by history term hn depending on previouslye computed coefficients of ðLuÞðxÞ¼fðxÞ x 2 X; the approximation of the displacement field and possibly of its first and second time derivatives. Note that due to the assumptions of a uCðyÞ¼gDðyÞ y 2 CD; ð13Þ linear material behavior and the equidistant time grid the left hand tðyÞ¼g ðyÞ y 2 C N N side matrix A is not altered throughout the computation. In the static with the d-dimensional displacement field u which is now only case, the system is slightly different, since there is no history term dependent on the position x. The variational expression (6) reduces and the dynamice stiffness boils down to the classical stiffness matrix to requiring that as defined in (19). The resulting system of equation has then the form Au ¼ f ð21Þ aðu; vÞ¼FðvÞð14Þ with the stiffness matrix A resulting from the static bilinear form a holds for all admissible test functions v. Note that the bilinear form and the force vector f due to the linear form F, both defined simi- aðu; vÞ and the linear form FðvÞ do not depend on time anymore and, larly to the dynamic versions in (7) and (8). therefore, the subscript t has been omitted. But their definitions are totally equivalent to (7) and (8). Finally, the static boundary integral 3.2. Boundary element method equation reads

Vt ¼ CuC þ KuC ð15Þ Among the numerous boundary element formulations (see, e.g., with the single and double layer operators V and K, respectively, [15]) a special collocation method is chosen in this approach. To which are now deﬁned as begin with, the ﬁelds of the boundary unknowns uC and t are approximated by the trials [28]

V Ã I ð tÞðxÞ¼ U ðx À yÞtðyÞdsy u y t y u t C C;hð ; Þ¼ uið Þ ið Þ; Z ð16Þ i¼1 Ã T X ð22Þ ðKuCÞðxÞ¼lim ðT yU ðx À yÞÞ uCðyÞdsy: J e!0 Ce Z thðy; tÞ¼ wjðyÞtjðtÞ j¼1 Of course, in these expressions the fundamental solution UÃðx À yÞ X of the operator L is involved which is independent of time. with the boundary shape functions ui and wj. Now, the ui are the UÃðx À yÞ is commonly referred to as the Kelvin solution and can coefficients of the boundary displacement field uC, not to be con- be found in [19]. The integral free term C in (15) is the same as in fused with the approximation in (17). The shape functions for the the definition in (11). unknown uC are piecewise linear continuous functions, whereas the unknown t is approximated by piecewise linear discontinuous Ã 3. Approximation methods functions. Locating K distinct collocation points xk on the boundary C yields the system of convolution equations 3.1. Finite element method ðV Ã tÞðtÞ¼CuðtÞþðK Ã uÞðtÞð23Þ In order to obtain a finite element scheme, the unknown func- with the time dependent matrices resulting from the single and tion u is approximated by means of the trial [16] double layer operators defined in (11) 452 T. Rüberg, M. Schanz / Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458

Ã Ã with the d Â d-submatrices of the discretized single and double V½k; jðt À sÞ¼ U ðx À y; t À sÞw ðyÞdsy; k j layer operators ZC ð24Þ T Ã Ã T K½k; iðt À sÞ¼lim ð yU ðxk À y; t À sÞÞ uiðyÞdsy Ã Ã !0 V½k; j¼ U ðx ; yÞw ðyÞdsy e Ce k j Z ZC ð32Þ and the integral free term T Ã Ã T K½k; i¼C½k; iþlim ð yU ðxk; yÞÞ uiðyÞdsy !0 e Ce T Ã Ã T Ã Z C½k; i¼lim ð yU ðxk À y; 0ÞÞ uiðxkÞdsy: ð25Þ e!0 ande the integral-free term C similar to (25). Obviously, the force vec- Zce tors fD and fN are defined in the same manner as in the dynamic con- In both previous expressions, the integration regions C and are e ce sideration and the mass matrix B is exactly the same as before. used as defined in (12). Since UÃ is a d Â d-matrix, the entries given in Eqs. (24) and (25) are consequently d Â d-submatrices of the cor- 4. Coupled solution algorithm responding system matrices. Note, that the integral free term C does not depend on time which is accomplished by taking the third and 4.1. Partitioned problem formulation fourth argument of the fundamental solution UÃðx À y; t À sÞ equal as in (25). The remaining convolutions can be discretized by either The considered mixed initial boundary value problem of linear using the time domain fundamental solution and carrying out the elastodynamics (5) and the corresponding static case of the mixed integrations analytically as done, for instance, by Mansur [21] or boundary value problem (13) are now formulated for a spatial par- by using the convolution quadrature method of Lubich [20]. The lat- titioning of the computational domain X. For sake of simplicity, at ter has been transferred to time domain boundary element methods first only the static case of (13) is considered and the extension to for various materials by Schanz [24] and, in this approach, only the dynamic problems is given afterwards. Laplace domain fundamental solution is required by means of The domain X is subdivided into N subdomains XðrÞ, i.e., which quadrature weights for the convolution integrals are gener- s ated. Independent of this choice, the time-discretized version of Ns ðrÞ Eq. (23) has the form X ¼ X ; ð33Þ r¼1 n n [ ðrÞ VmtnÀm ¼ Cun þ KmunÀm ð26Þ each of which has a boundary C which is decomposed into a m¼1 m¼1 Dirichlet, a Neumann, and an interface part X X for the time step tn ¼ nDt. The Dirichlet boundary conditions are incorporated by requiring that the approximation (22) ðrÞ ðrÞ ðrÞ ðrpÞ C ¼ CD [ CN [ C : ð34Þ directly fulfills those conditions posed on u . By means of the 0 ðrÞ 1 C p[2J abbreviation @ A In this expression, JðrÞ is the set of indices of subdomains which ðrÞ K0 ¼ C þ Ko and Km ¼ Km; 0 < m 6 n; ð27Þ share an interface with the subdomain X . Note that not every subdomain has its share of the Dirichlet and the Neumann boundaries, the Eq. (26) is converted to ðrÞ ðrÞ e e CD and CN, respectively. In that case, one of the parts CD or CN or n n N both vanish in (34). The mixed boundary value problem for the rth VmtnÀm À KN;muN;nÀm ¼ KD;mgD;nÀm: ð28Þ subdomain of the partitioning (33) now reads m¼1 m¼1 m¼1 X X X LðrÞ ðrÞ Here, the subscriptse D and N refere to columns of K associated with ð uÞðxÞ¼fðxÞ x 2 X ; ðrÞ ðrÞ the given Dirichlet and unknown Neumann data, respectively. The ðTr uÞðyÞ¼gDðyÞ y 2 CD ; ð35Þ e approximation of the given Dirichlet datum gDðtÞ at time point tn T ðrÞ ðrÞ ð uÞðyÞ¼gNðyÞ y 2 CN : is denoted by gD;n. Similarly, the given Neumann datum gNðtÞ at time point tn is approximated by gN;n and is included in a weighted form Depending on the geometric constellation of the considered subdomain XðrÞ, the Dirichlet or the Neumann boundary condition is not Btn ¼ BgN;n ð29Þ applicable if the subdomain does not have any share of the corresponding part of the boundary C of the original problem. In this using the mass matrix B½i; j¼hw ; u i. Note that the vector g is j i N;n boundary value problem, the operators L, Tr, and T carry the sub- padded with zeros for coefficients belonging to the Dirichlet bound- domain superscripts in order to emphasize the fact that they can ary C where this datum is unknown. This corresponds to an exten- D be different for each subdomain. For instance, the material param- sion of the Neumann datum g with zero to the Dirichlet boundary. N eters k, l, and q could vary from subdomain to subdomain. For The series of systems of equations then reads notational purposes, let uðrÞ denote the restriction of the unknown ðrÞ n u to the subdomain X and by Cs the skeleton of the partitioning, V0 À KN;0 tn fD;n tnÀm ¼ À Vm À KN;m ð30Þ which is B un fN;n unÀ !m¼1 m e X Ns ÀÁ ðrÞ e Cs ¼ C n C: ð36Þ with the abbreviations fD;n ¼ KD;ngD;n and fN;n ¼ BgN;n. Using as many r¼1 ! collocation points as trial functions for the unknown traction field t, [ the matrix V0 becomes quadratic.e Note that the use of piecewise Problem (35) is not yet completely stated. Therefore, suitable inter- constant shape functions for the approximation of the traction field face conditions have to be formulated in order to embed this local in (22) would result in numerical instabilities in the solution of sys- problem into the global constellation. These conditions are com- tem (30), see [28]. monly the continuity of the displacement field and the equilibrium The static case contains the reduced system of equations of the interface tractions

ðrÞ ðpÞ V À K t fD uC ðyÞ¼uC ðyÞ; ð37aÞ N ¼ ð31Þ u f tðrÞ y tðpÞ y 0 37b B ! N ð Þþ ð Þ¼ ð Þ e T. Rüberg, M. Schanz / Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458 453 for all points y on the interface CðrpÞ and for all such interfaces. Although in dynamics a statement similar to (38) is not avail- Equipping the local mixed boundary value problem (35) with one able, the above procedure can be repeated exactly in the same of the interface conditions (37) yields a well-posed problem. Note manner for the systems (20) and (30). In these cases, the ﬁnite ele- that if a subdomain does not have enough share of the global ment realization of the Dirichlet-to-Neumann map at time point

Dirichlet boundary CD and only the traction interface condition tn ¼ nDt is (37b) is used, its local boundary value problem will not have a un- ~ SfeuC;n ¼ gfe;n ð42Þ ique solution. The displacement field can then be altered by the rigid body motions which are possible due to the lack of prescribed where the following abbreviations have been used e ~ ~ ~ À1 ~ displacements. Such a subdomain is referred to as floating, see also Sfe ¼ ACC À ACIðAIIÞ AIC; ð43Þ [7] for more details on floating subdomains. À1 g~fe n ¼ fC;n þ hC;n À ACIðAIIÞ ðfI;n þ hI;nÞ: The above considerations are the same for the dynamic case of e ; In case of a dynamic boundary element method, these algebraic the initial boundary value problem of linear elastodynamics (5). e e Therefore, the partitioning and the formulation of local problems manipulations are applied to system (30). This gives the equation ~ is not repeated for this case. Moreover, the same interface condi- SbeuC;n ¼ gbe;n ð44Þ tions (37) have to hold now for all times t 2ð0; 1Þ. Nevertheless, with the components floating subdomains do not appear in dynamics due to the inertia e À1 terms in (1). Sbe ¼ BV K0N 0 ð45Þ À1 g~be ¼ fN;n À BV ðfD;n À ‘nÞ: 4.2. Dirichlet-to-Neumann maps e e For simplicity, the part of the right hand side of system (30) due to

Before the introduction of the coupling strategy, the notion of the convolution has been abbreviated by ‘n, i.e.,

Dirichlet-to-Neumann maps is presented which enables a coupling n formulation independent of the chosen discretization method. ‘n ¼ ðVmtnÀm À KN;muC;nÀmÞ: ð46Þ Reconsider the boundary integral Eq. (15) which will now be m¼1 X solved for the traction t Eqs. (42) and (44)e thus allow for Dirichlet-to-Neumann maps at À1 t ¼ V ðC þ KÞ uC: ð38Þ each time step using either a ﬁnite or a boundary element discret- S ization, respectively.

Confer|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}[28] for the invertibility of the single layer operator V. The 4.3. FETI framework newly introduced operator S is the so-called Steklov–Poincaré operator [27] and it maps the boundary displacement uC onto the trac- By means of the previously presented Dirichlet-to-Neumann tion t. Eq. (38) is one possible representation of this operator in the maps, the local boundary value problem (35) can be represented continuous setting. It remains to realize this mapping by means of a by the equivalent boundary-based formulation [27] discretization method. SðrÞ ðrÞ ðrÞ ðrÞ Reordering the finite element stiffness matrix of (21) according ð uC ÞðyÞ¼gC ðyÞ y 2 C ; to degrees of freedom associated with the interior of the domain X ðrÞ ðrÞ uC ðyÞ¼gDðyÞ y 2 CD ; ð47Þ or the boundary C yields the system of equation ðrÞ ðrÞ t ðyÞ¼gNðyÞ y 2 CN : AII AIC uI fI ¼ : ð39Þ ðrÞ This statement for all subdomains X ,1< r < Ns, together with the ACI ACC uC fC interface conditions (37) represents the global mixed boundary va- The subscripts I and C refer to the interior and the boundary de- lue problem (13). Now, a global variational principle is formulated grees of freedom, respectively. The discrete mapping of boundary by incorporating the displacement continuity condition (37a) in a displacements to boundary nodal forces is now formally established weighted form [27] by SðrÞ ðrÞ ðrÞ ðrpÞ ðrÞ ðpÞ ðrÞ ðrÞ À1 À1 ð uC ÞÁvC ds þ k ÁðvC À vC Þds ¼ gC Á vC ds ðrÞ ðrpÞ ðrÞ ðACC À ACIðAIIÞ AICÞ uC ¼ fC À ACIðAIIÞ fI ð40Þ C ðrÞ C C Z pX2J Z Z Sfe gfe ðrpÞ ðrÞ ðpÞ l ÁðuC À uC Þds ¼ 0: ð48Þ ðrpÞ with|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} the newly introduced|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} abbreviations S for the finite element r C fe p2Jð Þ Z S X realization of the operator and gfe for the boundary forces. ðrÞ In order to obtain a boundary element realization of this map, These two expressions have to hold for subdomains X . For the simply the first equation of the system (31) is eliminated weighted formulation of the interface conditions a Lagrange multiplier field k has been introduced on the skeleton Cs of the partition- À1 À1 rp ðBV KNÞ uC ¼ fN À BV fD : ð41Þ ing, where kðrpÞ denotes its restriction to the interface Cð Þ. ðrÞ g ðrpÞ Sbe be Moreover, the functions vC and l are the test functions which e correspond to the fields of the local boundary displacement uðrÞ |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}Now, S represents|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} the mapping of the boundary displacements u C be C and the Lagrange multiplier kðrpÞ, respectively. to the boundary forces g . Note that in this context the subscript C be Using the framework of the FETI method of Refs. [8,9], the dis- has been added to the unknown coefficients u even in the case of a cretized version of the variational form (48) is boundary element method in order to emphasize the structural ð1Þ ð1Þ ð1Þ ð1ÞT equivalence between (40) and (41). S C uC gC In summary, (40) and (41) give possible realizations of the con- T ð2Þ ð1Þ Sð2Þ Cð2Þ u g sidered Dirichlet-to-Neumann map by means of a finite element or 0 10 C 1 0 C 1 . . . . . a boundary element discretization. Whereas the former finite ele- B . . CB . C ¼ B . C: ð49Þ B T CB C B C ment realization is symmetric, the boundary element realization is B ðNsÞ ðNsÞ CB ðNsÞ C B ðNsÞ C B S C CB u C B g C nonsymmetric. Possible symmetric boundary element discretiza- B CB C C B C C B Cð1Þ Cð2Þ ÁÁÁ CðNsÞ CB C B C S B CB k C B 0 C tions of the operator can be found in Refs. [26,27]. @ A@ A @ A 454 T. Rüberg, M. Schanz / Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458

In this system, the matrices SðrÞ represent the local discretizations of future research. Especially, the lack of symmetry due to the chosen the operator SðrÞ and can be either a finite element or a boundary boundary element formulation and, therefore, of the matrix F element realization according to (40) or (41), respectively. The would require a lot more effort in the design of such a solver. There- ðrÞ boundary force vectors gC result from the same considerations. fore, direct solution routines are used for both the static and dy- The matrices CðrÞ are the discretizations of the terms corresponding namic problems considered here. to the interface conditions of the variational expression (48) and will be henceforth referred to as connectivity matrices. Finally, 4.4. Connectivity matrices the vector k gathers the coefficients of suitable approximation of the Lagrange multiplier field k. This approximation will be dis- In the original FETI algorithm [8], the Lagrange multipliers are cussed in more detail below. In dynamic problems, exactly the same used as node-wise constraints. Therefore, the connectivity matri- system of equations results at each time step. Then the left and right ces are just such that C½j; i2f0; 1; À1g. This simple and efficient hand sides have to be replaced by the finite element matrices of approach implies conforming interface discretizations, i.e., the (43) or the boundary element matrices of (45). nodes of adjacent subdomains have to spatially coincide at their In order to solve system (49), at first it is assumed that the common interface and, moreover, the polynomial orders of the lo- ðrÞ matrices S are invertible. Then, the local boundary displacements cal discretizations have to be equal. In such a case, the interface are given by condition (37a) is fulfilled exactly at every point of the interface. À1 T Here, this requirement shall be relaxed and non-conforming inter- uðrÞ ¼ SðrÞ ðgðrÞ À CðrÞ kÞ: ð50Þ C C faces are allowed. Therefore, the introduced Lagrange multiplier Inserting this expression into the last line of the system (49), yields field is approximated by the equation for the Lagrange multiplier coefficients Nk

Ns Ns khðyÞ¼ wjðyÞkj ð56Þ r r À1 r T r r À1 ðrÞ Cð ÞSð Þ Cð Þ k ¼ Cð ÞSð Þ g : ð51Þ j¼1 C X r¼1 r¼1 X X with the shape functions wj and the approximation coefficients kj. F d By means of this approximation, the connectivity matrices become |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}In case of dynamic problems,|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} the regularity of expression (50) is ðrÞ ðrÞ ðrÞ guaranteed. But in the static case, the matrix S has no unique in- C ½j; i¼Æ ui ðyÞwjðyÞds: ð57Þ verse if the subdomain XðrÞ is floating. Then a so-called generalized Z inverse and the null-space of the matrix are needed in order to give In this expression,S uðrÞ refers to a boundary element function for the the local boundary displacements ðrÞ approximation of uC or to the boundary trace of a finite element þ T function for the approximation of uðrÞ. The integration is carried uðrÞ ¼ SðrÞ ðgðrÞ À CðrÞ kÞþNðrÞaðrÞ: ð52Þ C C out over all interfaces of the considered subdomain XðrÞ as indicated The generalized inverse is denoted by the superscript + and NðrÞ is by the union in the integration limit. The sign is adjusted such that the null-space of SðrÞ, i.e., SðrÞNðrÞ ¼ 0. Moreover, the vector aðrÞ col- at adjacent sides of the interface it becomes opposite. For instance, lects the amplitudes of the rigid body motions of the subdomain. one can assume the convention that if p < r the sign in (57) is posi-

Here, the procedure described in [7] is adopted which yields a ro- tive and it is negative for r < p. Note that if the shape functions wj bust scheme for the computation of both the generalized inverse are taken to be Dirac delta distributions, i.e., wjðyÞ¼dðy À xjÞ with and the null-space of a rank-deficient matrix. Expression (52) intro- xj being the jth interface node, then (57) reduces to the classical duces another field of unknowns a and, therefore, more equations FETI connectivity matrices. are required. The local solvability condition reads [9] In order to establish a non-conforming coupling, let at any ðrpÞ ðrÞ ðpÞ r T r T interface C for r < p the subdomain X be the slave and X Nð Þ ðgð Þ À CðrÞ kÞ¼0; ð53Þ R C be the master as in the mortar method [31]. This indicates that ðrÞ ðrÞ ðrÞT ðrÞ with the right null-space NR of S , i.e., NR S ¼ 0. Due to the sym- the approximation of the Lagrange multiplier field (56) is defined ðrÞ metry of the finite element realization Sfe , the right and left null- with respect to the interface mesh inherited from the slave side. ðrÞ ðrÞ spaces, NR and N , coincide. In case of the nonsymmetric boundary With this arbitrary convention, the interface matrices obtain the ðrÞ ðrpÞ element realization Sbe , this symmetry is assumed here based on contributions at the interface C the fact that both matrices represent the same physical ðrÞ : ðrÞ ðrÞ ðrÞ characteristics. X C ½i; j¼À ui ðyÞwj ðyÞds ðrpÞ Inserting expression (52) into the last line of system (49) and ZC ð58Þ ðpÞ : ðpÞ ðpÞ ðrÞ assembling the solvability conditions (53) for all subdomains final- X C ½k; j¼þ uk ðyÞwj ðyÞds: ðrpÞ ly gives the system of equations ZC F ÀG k d The first of these contributions is again of the standard mass-matrix ¼ : ð54Þ type but in the second contribution CðpÞ½k; j the difficulty occurs that ÀGT a ÀeT the L2-product of two shape functions is performed which are de- In addition to the quantities F and d due to expression (51), there fined on different meshes. are the matrices For sake of simplicity, the shape functions wjðyÞ are assumed piecewise constant. Then, the second and crucial expression in ð1Þ ð1Þ ... ðNsÞ ðNsÞ G ¼ðC N ; ; C N Þ (58) becomes T T ð55Þ ð1Þ ð1Þ ... ðNsÞ ðNsÞ e ¼ðN gC ; ; N gC Þ: ðpÞ ðpÞ C ½k; j¼þ uk ðyÞds ð59Þ sðrÞ The original FETI method due to [8] has been tailored for finite ele- Z j ðrÞ ðrÞ ment discretizations of static problems and is equipped with pro- with the element sj associated with the shape function wj of the jected conjugate gradient solver for an optimally parallelized interface discretization of subdomain XðrÞ. The interfaces CðrpÞ are solution procedure. Such concepts have been transferred to dy- assumed to be flat such that for any discretization of the adjacent namic problems in [6]. Nevertheless, the development of fast itera- subdomains the computational representations of the geometry tive solution procedures is here not the principal aim and left as cannot overlap or form voids. The first step in the computation of T. Rüberg, M. Schanz / Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458 455

ðpÞ ðrÞ the value of c ½k; j is to determine the overlap of the element sj of representation of the cantilever beam. The beam is fixed at x1 ¼ 0, ðrÞ ðpÞ subdomain X with the support of the shape function uk . Hence, which is the given Dirichlet datum, i.e., gD ¼ 0. The opposite side at one has to find the intersection x1 ¼ 10 m is subjected to a vertical uniformly distributed load with 2 the non-zero component g ¼À1N=m e3. Every other part of the sðrpÞ ¼ sðrÞ \ suppðuðpÞÞ: ð60Þ N kj j k surface C is traction free. The material is assumed to be steel with- This task is trivial in case of a two-dimensional analysis, where only out any lateral contraction having the material parameters k ¼ 0 11 2 the intersection of one-dimensional intervals has to be computed. and l ¼ 1:055 Â 10 N=m . Contrary to this, in a three-dimensional analysis the geometric over- A two- and a three-dimensional analysis are carried out where ðrÞ ... lap of two-dimensional surface elements has to be determined. the domain X is subdivided into four subdomains X , r ¼ 1; ; 4. Therefore, the elements of the master domain XðpÞ are transformed Each of these subdomains is of equal size and has the dimensions to the reference space of the slave domain XðpÞ and then the tech- 5mÂ 1mÂ 0:5 m and the constellation is shown in Fig. 1 together niques of polygon clipping are used as described in [30]. Once the re- with the numbering of the subdomains. ð1Þ ð4Þ ðrpÞ In the numerical analysis, subdomains X and X are always gion skj is computed a quadrature rule is applied to the integral of (59). Since the computed overlap is a convex polygon of rather arbi- treated by the same discretization method with the same mesh ð2Þ ð3Þ trary shape, it will thus be subdivided into triangles on each of which size and so are subdomains X and X . The three combinations a quadrature rule is carried out. In this context, another difficulty ap- of coupling boundary with boundary elements (BEM–BEM), ðpÞ boundary with finite elements (BEM–FEM), and finite with finite pears because the shape function uk is defined with respect to the reference elements of subdomain XðpÞ and, therefore, its evaluation elements (FEM–FEM) are considered. Each of these cases is treated at quadrature points expressed in the reference space of the slave with a coarse, a middle, and a fine discretization. The mesh widths subdomain XðrÞ is not straightforward. Suitable coordinate transfor- are given in Table 1 and an example of these discretizations is mations between these space are required as pointed out in [23]. shown in Fig. 2 where the case BEM–FEM for the middle mesh is Once the connectivity matrices are computed according to the displayed. above described procedure, the FETI solution process can be car- In order to judge the numerical outcome of the different analy- ried out as in the original algorithm of [8]. The only difference is ses, the results of a finite element solution with the whole cantile- that in the original FETI algorithm only extraction procedures are ver beam as one domain is used as a reference solution uref . This required due to the simple structure of the matrices CðrÞ, whereas finite element solution is obtained with h ¼ 1=40 m for the two- the procedure described here requires floating point arithmetic dimensional and h ¼ 1=20 m for the three-dimensional analysis, for the matrix–matrix products in (51) and (55). On the other side, respectively. An error measure is defined by considering the value so-called cross points, that are points at which more than two sub- of domains meet, do not pose any problem in this approach. The La- ju3ðx1; 0; 0ÞÀuref;3ðx1; 0; 0Þj grange multiplier field is associated with elements and not with eðx1Þ¼ ; ð61Þ juref;3ðx1; 0; 0Þj nodes and, therefore, the multipliers cannot be redundant. Note that the case of conforming interface discretizations is fully in- where u3 denotes the vertical displacement of the numerical analy- cluded in this approach and would yield equal expressions for sis of the centerline at x2 ¼ x3 ¼ 0 and uref;3 is the vertical compo- the contributions in (58). nent of the reference solution uref . In Fig. 3, the outcome of the numerical analysis with the various 5. Numerical examples combinations described above is plotted in terms of the error measure defined in (61). Obviously, the results of the two-dimensional The approximations for the following examples are all of the BEM–BEM coupling are worse than the other combinations. The following type. For the finite element discretizations bilinear quad- BEM–FEM combination performs better and the best results are gi- rilaterals or trilinear hexahedra elements are used in two or three ven by the FEM–FEM coupling. On the other hand, in the three- dimensions, respectively. The boundary element analysis is based dimensional analysis the order of the quality of the results is re- on piecewise linear shape functions which are continuous for the versed. Here, the BEM–BEM coupling yields better results than the other combinations and the FEM–FEM coupling has the worst boundary displacements uC and discontinuous for the tractions t. In two dimensions simple line elements are used and in three performance. Concentrating on the results closer to the loaded dimensions triangular elements. The mesh sizes h refer either to end of the beam, one can see that in all coupling combinations the length of the sides of the quadrilaterals or hexahedra, the the results improve with finer discretizations. A closer look at the length of the line elements or the catheti of the triangles. For the outcome of the three-dimensional analysis in Fig. 3 reveals small time discretization of the dynamic finite element method the New- discontinuities in the curves at x1 ¼ 5 m. This effect can be explained by the weak statement of the displacement continuity mark algorithm with parameters b ¼ 0:25 and c ¼ 0:5 is taken. This choice corresponds to an unconditionally stable scheme with second order accuracy and without numerical dissipation [16]. Table 1 Different discretizations of the cantilever problem 5.1. Static analysis of a cantilever beam Mesh hð1Þ ¼ hð4Þ hð2Þ ¼ hð3Þ

The ﬁrst test case to consider is the numerical analysis of a can- Coarse 1=6m 1=4m Middle 1=6m 1=8m tilever beam by means of the presented coupling approach. There- Fine 1=10 m 1=8m fore, a domain of dimensions 10 m Â 1mÂ 1 m is considered as a

Fig. 1. Model of cantilever beam with four subdomains. 456 T. Rüberg, M. Schanz / Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458

Fig. 2. Discretization of the three-dimensional model of the cantilever beam with ﬁnite and boundary elements.

a ﬁxed left end at x ¼ 0 and the longitudinal step load F HðtÞ is ap- 0.2 1 0 coarse BEM-BEM plied at the right end at x1 ¼ ‘. Here, HðtÞ is the Heaviside function, middle BEM-FEM fine FEM-FEM i.e., HðtÞ¼0 for t 6 0 and HðtÞ¼1 for t > 0. The analytical solution 0.15 of the problem is given, for instance, in [24]. Here, again the mate- [-] ε rial steel is used with the parameters k ¼ 0, l ¼ 1:055 Â 1011 N=m2, and q ¼ 7850 kg=m3. The other parame- 0.1 ters are the length of the rod with the value ‘ ¼ 3 m and the mag-

nitude of the load F0 ¼ 1N.

error measure 0.05 The domain of dimensions 3 m Â 1mÂ 1 m is subdivided into three unit cubes. The first and the third of which are discretized by the boundary element method with a mesh width of 0 1 3 0246 810hð Þ ¼ hð Þ ¼ 0:25 m. The middle cube is discretized by finite ele- coordinate x [m] ð2Þ ð2Þ 1 ments with h ¼ 0:5 m for a coarse and h ¼ 0:2 m for a fine discretization. The constellation with the fine discretization is shown in Fig. 5. 0.03 coarse BEM-BEM The choice of the time step size has been determined according middle BEM-FEM to stability conditions of the convolution quadrature method of the fine FEM-FEM dynamic boundary element method. Such considerations can be [-]

ε 0.02 found in detail in [24] and lead to the observation that the size of the time step is bounded from below. As in other time discreti-

zation methods, the ratio b ¼ðc1DtÞ=h is fundamental for stability 0.01 analyses. According to [24] this value is here ﬁxed to approxi-

error measure mately b ¼ 0:2 which corresponds to a time step size of Dt ¼ 10À6 s.

In the numerical analysis, the longitudinal displacements u1 0 0246 810along the middle axis are considered at the loaded face and the x coordinate 1 [m] interfaces. Moreover, the traction component t1 at the ﬁxed end is regarded. Therefore, the points A, B, and C refer to the coordi-

nates x1 ¼ 3m, x1 ¼ 2 m, and x1 ¼ 1 m, respectively along this

Fig. 3. Results of the coupled analysis of the cantilever beam — error measure middle axis, i.e., at x2 ¼ x3 ¼ 0:5 m. These coordinates correspond according to (61) along the coordinate x1 for the various considered combinations in to the coordinate system in Fig. 4 if placed along the centerline two and three dimensions. of the cuboid domain. The outcome is plotted in Fig. 6 together with the analytical solution. The numerical solution for the dis- condition (37a) and, obviously, it does not deteriorate the results placements reproduces well the zigzag curve of the analytical solu- away from the interface. tion. Especially, the finer discretization yields results which are hardly distinguishable from the analytical solution. The traction 5.2. Unit step load on a rod solution of the coarse discretization deviates significantly from the analytical solution. Nevertheless considering the fact that a As a dynamic test case, an elastic rod which is fixed at one end discontinuous function is approximated, the results for the fine and subject to a unit step load at the other end is considered. The discretization are reasonably good despite the overshoots at each problem is depicted in Fig. 4, where a rod of length ‘ is shown with jump.

Fig. 4. Rod with a unit step force. Displacement results are considered for the points A, B, and C. T. Rüberg, M. Schanz / Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458 457

Fig. 7. Discretization of a foundation on an elastic halfspace.

Fig. 5. Boundary element and ﬁnite element discretization of the dynamic loaded rod. 3e-09

2.5e-09 [m]

3 2e-09 3e-11 u A 2.5e-11 analytical 1.5e-09 coarse foundation (top)

[m] fine B 1e-09 foundation (bottom)

1 2e-11 u

displacement soil static solution 1.5e-11 5e-10

1e-11 0 0 0.01 0.02 0.03 0.04 0.05 0.06 t displacement C time [s] 5e-12 Fig. 8. Vertical displacements of the foundation and the soil against time. 0 0 0.002 0.004 0.006 0.008 0.01 time t [s] The constellation of the discretizations of the foundation and the halfspace is shown in Fig. 7. In the dynamic analysis, 400 steps of size Dt ¼ 1:5 Â 10À4 s are computed. In Fig. 8, the vertical displacements at three different positions are plotted, at the mid- 2 points of the top and bottom surfaces of the foundation and at ] 2 another point on the surface of the soil. With respect to the coordinate system in Fig. 7, these points have the coordinates

[N/m analytical 1 t 1 coarse ð1m;1m; À 1mÞ, ð1m;1m;0Þ, and ð4:5m;1m;0Þ, respectively. fine For these three positions the corresponding static result is given

traction by the horizontal lines. Clearly, the static solution is reached after 0 approximately half the computed time which indicates that the waves have been absorbed due to the infinite geometry of the elas- 0 0.002 0.004 0.006 0.008 0.01 tic halfspace. Considering the point on the surface of the soil, one can see that the pressure wave is arriving after approximately time t [s] 0:01 s, followed by the shear wave and the Rayleigh surface waves (cf. [1] for these wave types). No reflections of these waves occur Fig. 6. Coupled boundary and finite element solution for the three-dimensional rod due to the truncated surface mesh. – displacements at points A, B, and C and traction at the fixed end against time. 6. Conclusion

5.3. Foundation on an elastic halfspace A framework for the coupling of ﬁnite and boundary element discretizations of dynamic and static problems has been estab- Finally, the static and dynamic analyses of an individual footing lished which allows for non-conforming interface discretizations. on an elastic halfspace is considered. The foundation is assumed to Basically, the concepts of hybrid domain decomposition methods be of cuboid shape and made of concrete with parameters [27] have been transferred to the treatment of dynamic problems, k ¼ 9:72 Â 109 N=m2, l ¼ 1:46 Â 1010 N=m2, and q ¼ 2400 kg=m3. where the key point is the realization of Dirichlet-to-Neumann It is represented by the subdomain Xð1Þ which is a cube of dimen- maps by the chosen discretization methods in time domain. In fact, sions 1 m Â 1mÂ 1 m and will be discretized by 216 trilinear ﬁ- the presented algorithm falls in the category of FETI/BETI methods nite elements of size hð1Þ ¼ 1=6 m. The top surface of the (see [18]) but with the extension to dynamic problems and non- foundation is subject to a uniform vertical load of magnitude conforming interface discretizations. The performance in terms of 1:0N=m2 which varies as a unit step in time. the quality of the results is good both for the analysis of static 3 The halfspace fx 2 R : x3 > 0g is numerically represented by and dynamic problems. Nevertheless, a further improvement can the surface patch Cð2Þ of dimensions 5 m Â 2 m which is discretized be expected if a symmetric Galerkin boundary element formula- by 320 boundary elements of size hð2Þ ¼ 1=4 m. The material of this tion is used in order to obtain symmetric positive system matrices halfspace is soil with k ¼ l ¼ 1:36 Â 108 N=m2 and q ¼ throughout each step and, therefore, the use of preconditioned pro- 1884 kg=m3. jected conjugate gradient solvers becomes feasible. 458 T. Rüberg, M. Schanz / Comput. Methods Appl. Mech. Engrg. 198 (2008) 449–458

Acknowledgement [15] G. Hsiao, W. Wendland, Boundary element methods: Foundation and error analysis, in: E. Stein, R. de Borst, T. Hughes (Eds.), Encyclopedia of Computational Mechanics, vol. 1, John Wiley & Sons, Ltd., 2004, pp. 339–373 The second author gratefully acknowledges the financial sup- (Chapter 12). port by the Austrian Science Fund (FWF) under grant P18481-N13. [16] T. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, New York, 2000. [17] A. Klawonn, O. Rheinbach, Robust FETI–DP methods for heterogeneous three dimensional elasticity problems, Comput. Methods Appl. Mech. Engrg. 196 References (2007) 1400–1414. [18] U. Langer, O. Steinbach, Coupled finite and boundary element domain [1] J. Achenbach, Wave Propagation in Elastic Solids, North-Holland, 2005. decomposition methods, in: M. Schanz, O. Steinbach (Eds.), Boundary Element [2] H. Bavestrello, P. Avery, C. Farhat, Incorporation of linear multipoint Analysis – Mathematical Aspects and Applications, vol. 29 of Lecture Notes in constraints in domain-decomposition-based iterative solvers – Part II: Applied and Computational Mechanics, Springer, 2007, pp. 61–95. Blending FETI–DP and mortar methods and assembling floating [19] A. Love, Treatise on the Mathematical Theory of Elasticity, Dover Publications, substructures, Comput. Methods Appl. Mech. Engrg. 196 (2007) 1347–1368. 1944. [3] G. Beer, Programming the Boundary Element Method: An Introduction for [20] C. Lubich, Convolution quadrature and discretized operational calculus I&II, Engineers, Wiley, 2001. Numer. Math. 52 (1988) 129–145 & 413–425. [4] M. Costabel, Time-dependent problems with the boundary integral equation [21] W. Mansur, A Time-stepping Technique to Solve Wave Propagation Problems method, in: Encyclopedia of Computational Mechanics, John Wiley & Sons, using the Boundary Element Method, Ph.D. thesis, University of Southampton, Ltd., 2004, pp. 703–721. 1983. [5] O. Estorff, C. Hagen, Iterative coupling of FEM and BEM in 3D transient [22] N. Newmark, A method of computation for structural dynamics, ASCE J. Engrg. elastodynamics, Eng. Anal. Boundary Elements 29 (2005) 775–787. Mech. Div. 85 (1959) 67–93. [6] C. Farhat, L. Crivelli, A transient FETI methodology for large-scale parallel [23] T. Rüberg, Non-conforming FEM/BEM Coupling in Time Domain, vol. 3 of implicit computations in structural mechanics, Int. J. Numer. Methods Engrg. Computation in Engineering and Science, Verlag der Technischen Universität 37 (1994) 1945–1975. Graz, 2008. [7] C. Farhat, M. Géradin, On the general solution by a direct method of a large- [24] M. Schanz, Wave propagation in Viscoelastic and Poroelastic Continua – A scale singular system of linear equations: application to the analysis of floating boundary element approach, Springer, 2001. structures, Int. J. Numer. Methods Engrg. 41 (1998) 675–696. [25] D. Stefanica, Parallel FETI algorithms for mortars, Applied Numerical [8] C. Farhat, F.-X. Roux, A method of finite element tearing and interconnecting Mathematics 54 (2005) 266–279. and its parallel solution algorithm, Int. J. Numer. Methods Engrg. 32 (1991) [26] O. Steinbach, Mixed approximations for boundary elements, SIAM J. Numer. 1205–1227. Anal. 38 (2000) 401–413. [9] C. Farhat, F.-X. Roux, Implicit parallel processing in structural mechanics, [27] O. Steinbach, Stability Estimates for Hybrid Domain Decomposition Methods, Comput. Mech. Adv. 2 (1994) 1–124. Springer, 2003. [10] M. Fischer, L. Gaul, Fast BEM–FEM Mortar coupling for acoustic-structure [28] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value interaction, Int. J. Numer. Methods Engrg. 62 (2005) 1677–1690. Problems, Springer, 2008. [11] J. González, K. Park, C. Felippa, FEM and BEM coupling in elastostatics using [29] E. Stephan, Coupling of boundary element methods and finite element methods, localized Lagrange multipliers, Int. J. Numer. Methods Engrg. 69 (2007) 2058– in: E. Stein, R. de Borst, T. Hughes (Eds.), Encyclopedia of Computational 2074. Mechanics, vol. 1, John Wiley & Sons, Ltd., 2004, pp. 375–412 (Chapter 13). [12] J. González, K. Park, C. Felippa, R. Abascal, A formulation based on localized [30] B. Vatti, A generic solution to polygon clipping, Commun. ACM 35 (1992) 56– Lagrange multipliers for BEM–FEM coupling in contact problems, Comput. 63. Methods Appl. Mech. Engrg. 197 (2008) 623–640. [31] B. Wohlmuth, Discretization Methods and Iterative Solvers Based on Domain [13] F. Hartmann, Introduction to Boundary Elements. Theory and Applications, Decomposition, Springer, 2001. Springer, 1989. [32] O. Zienkiewicz, D. Kelly, P. Bettess, The coupling of the finite element method [14] G. Hsiao, The coupling of boundary and finite element methods, Zeitschrift für and boundary solution procedures, Int. J. Numer. Methods Engrg. 11 (1977) angewandte Mathematik und Mechanik 70 (1990) T493–503. 355–375.