Computational Mechanics 34 (2004) 67–73 Ó Springer-Verlag 2004 DOI 10.1007/s00466-004-0554-4
Iterative coupling of BEM and FEM for nonlinear dynamic analyses D. Soares Jr, O. von Estorff, W. J. Mansur
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D. Soares Jr, W. J. Mansur Department of Civil Engineering, COPPE – Federal University of Rio de Janeiro, CP 68506, CEP 21945–970, Rio de Janeiro, RJ, Brazil
approach has relative benefits and limitations. The finite element method, for instance, is well suited for inhomogeneous and anisotropic materials as well as for dealing with the nonlinear behavior of a body. For systems with infinite extension and regions of high stress concentration, however, the use of the boundary element method is by far more advantageous. More details are given, e.g., by Hughes [17] or Bathe [2] for the FEM and by Becker [3] or Dominguez [10] in the case of the BEM. In fact, it did not take long until some researchers started to combine the FEM and the BEM in order to profit from their respective advantages by trying to evade their disadvantages. Up to now, quite a few publications concentrate on such coupling approaches. Many details are given, e.g., by Zienkiewicz et al. [25, 26], who were among the first suggesting a ‘‘mariage a` la mode – the best of two worlds’’, by von Estorff and Prabucki [13], von Estorff and Antes [14], Belytschko and Lu [4], Yu et al. [23], or Rizos and Wang [21]. A rather complete overview is provided by Beskos [5–7]. It should be mentioned, that in most cases the BEM has been used to model those parts of the investigated bodies which are of semi-infinite extension, while finite parts were represented with the FEM. Moreover, it has been assumed that the considered systems behave linearly, which means that only elastic material and small displacements are considered. The FEM/BEM coupling in the time domain has also been successfully used to take into account nonlinear effects. Thus Pavlatos and Beskos [20] as well as Yazdchi et al. [24], for instance, modeled an inelastic structure and the surrounding soil part expected to become inelastic by the FEM and the remaining soil assumed to behave linearly by means of the BEM. A similar approach has been used in the doctoral thesis by Adam [1] or in some very recent publications by von Estorff and Firuziaan [15], and Firuziaan and von Estorff [16]. The coupling of an inelastic structure with a fluid domain of semi-infinite extension has been investigated by and Czygan and von Estorff [8]. Note that all coupling approaches mentioned so far, except Rizos and Wang [21], were formulated in a way that, first, a coupled system of equation is established, which afterwards has to be solved using a standard direct solution scheme. Doing so, three major problems need to be taken care of:
The financial support by CAPES (Fundac¸a˜o Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior) through scholarship No. BEX0788/02-3 (first author) is gratefully acknowledged.
1. Due to the coupling of the FE system matrices with fully populated BE influence matrices, the coupled system of equations is not banded and sparsely populated anymore. This means that for its solution the optimized
Abstract The present work deals with the iterative coupling of boundary element and finite element methods. First, the domain of the original problem is subdivided into two subdomains, which are separately modeled by the FEM and BEM. Thus the special features and advantages of the two methodologies can be taken into account. Then, prescribing arbitrary transient boundary conditions, a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved. In the case of local nonlinearities within the finite element subdomain, it is straightforward to perform the iterative coupling together with the iterations needed to solve the nonlinear system. The procedure turns out to be very efficient. Moreover, a special formulation allows taking into account different durations of the time steps in each subdomain. Keywords Iterative coupling, Different time steps, Nonlinear analysis, Dynamic response, BEM, FEM
1 Introduction The numerical simulation of arbitrarily shaped continuous bodies subjected to harmonic or transient loads remains, despite much effort and progress over the last decades, a challenging area of research. In most cases, discrete techniques, such as the finite element method (FEM) and the boundary element method (BEM) have been employed and continuously further developed with respect to accuracy and efficiency. Both methodologies can be formulated in the time domain or in the frequency domain, and each
Received: 23 September 2003 / Accepted: 29 January 2004 Published online: 18 March 2004
D. Soares Jr, O. von Estorff (&) Mechanics and Ocean Engineering, TU Hamburg-Harburg, Eissendorfer Str. 42, D-21073 Hamburg, Germany e-mail:
[email protected]
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solvers usually used in the FEM cannot be employed anymore, which leads to rather expensive calculations with respect to computer time. 2. If standard transient coupling approaches are employed, the duration of the time step has to be the same for each subdomain. This sometimes leads to numerical difficulties, in particular, if strongly different materials (with different wave speeds) need to be coupled. 3. In the case of taking into account some nonlinearity within the FE sub-region, the rather big coupled system of equations needs to be solved in each step of the iteration process, i.e. a few times within each time step. This is very computer time consuming.
Taking into account a linear behavior of the considered body, Eq. (1) can be written as
€i þ bi =q ¼ 0 ðc2d c2s Þuj;ji þ c2s ui;jj u
ð4Þ
where cd is the dilatational wave velocity and cs is the shear wave velocity. Furthermore, the following boundary and initial conditions have to be defined: (i) along the boundary C ¼ C1 [ C2 :
i ðX; tÞ for t > 0 on C1 ui ðX; tÞ ¼ u ð5aÞ pi ðX; tÞ ¼ rij ðX; tÞnj ðXÞ ¼ pi ðX; tÞ for t > 0 on C2 ð5bÞ In view of the first aspect, Elleithy et al. [11] got the idea of (ii) at the initial time t ¼ 0 on the boundary (C) and inside coupling the FEM and the BEM iteratively. However, they the domain (X): were interested only in the solution of elastostatic problems. Later, Elleithy and Tanaka [12] applied the iterative ui ðX; 0Þ ¼ u i0 ðXÞ ð6aÞ FE/BE coupling procedure to solve the Laplace equation. _ui ðX; 0Þ ¼ mi0 ðXÞ ð6bÞ In the present paper, an iterative coupling scheme for the investigation of continuous bodies subjected to tran- where the prescribed values are indicated by over bars and sient loads is presented. It turns out to be very efficient pi denotes the traction vector along the boundary with the with respect not only to the first but also to the second and outward normal nj . third aspect mentioned above. The FEM and the BEM parts are based on usual formulations, as suggested, e.g., 3 by Crisfield [9], Jacob and Ebecken [18], Bathe [2] and Time domain elastodynamic BEM analysis Mansur [19], Dominguez [10], Soares Jr et al. [22], In order to implement the numerical scheme for a tworespectively. dimensional time domain BEM analysis to solve the linear The derived coupling scheme is applied to several problem described above, it is necessary to consider a set numerical examples to illustrate how the BE/FE combiof discrete points Xj , j ¼ 1; 2; . . . ; J on the boundary C, and nation can be performed, how accurate the results are, and also a set of discrete values of time tn , n ¼ 1; 2; . . . ; N. The how nonlinearities in some parts of the body can be taken displacement and traction components uk ðX; tÞ and into account. Boundary elements may be used for either pk ðX; tÞ, where k ¼ 1; 2 is indicating the two transversal finite or infinite parts of the body, whereas all finite degrees of freedom at each point, can be approximated element sub-regions have to be bounded domains. using a set of interpolation functions, such that J X M X j m uk ðX; tÞ ¼ /m ð7Þ 2 u ðtÞgu ðXÞukj j¼1 m¼1 Governing equations The momentum equilibrium equation, considering a unit J X M X j m volume of a continuous body, is given by /m ð8Þ pk ðX; tÞ ¼ p ðtÞgp ðXÞpkj m¼1 j¼1 rij;j q€ ui þ qbi ¼ 0 ð1Þ where the following notations are employed: X is a field where rij is the Cauchy stress, using the usual indicial j j notation for Cartesian axes; ui stands for the displacement point on the boundary C, X 2 C; t is the time; gu and gp are spatial interpolation functions related to uk and pk , respecand bi for the body force distribution. Inferior commas m tively, corresponding to a boundary node Xj ; /m u and /p are and overdots indicate partial space (uj;i ¼ ouj =oxi ) and time interpolation functions related to uk and pk , respectime (u_ i ¼ oui =ot) derivatives, respectively. q stands for tively, corresponding to a discrete time tm ; finally, the nodal the mass density. and nodal traction components are defined by The constitutive law can be written, incrementally, as displacement m um kj ¼ uk ðXj ; tm Þ and pkj ¼ pk ðXj ; tm Þ, respectively. 0 Using the above notation, the discretized time domain drij ¼ Dijkl ðdekl dekl Þ ð2Þ BEM equation corresponding to a discrete source point Sl where Dijkl is a tangential matrix defined by suitable state ðl ¼ 1; 2; . . . ; JÞ can be written as (initial condition convariables and the direction of the increment. The incretributions are not taken into account) mental strain ðdeij Þ components are defined in the usual J 2 2 X n X way from the displacement as X 1 X n nm m c ðS Þu þ Hijkl ukj ik l kl 2pqc 1 s k¼1 m¼1 J¼1 k¼1 ðdui;j þ duj;i Þ deij ¼ ð3Þ 2 J 2 X n X 1 X 0 ¼ Gnm pm ð9Þ and eij refers to strains caused by external actions such as 2pqcs k¼1 m¼1 j¼1 ijkl kj temperature changes, creep, etc.
nm where i ¼ 1; 2. Furthermore, Hijkl and Gnm ijkl are the elements of the so-called influence matrices. For the nm complete definitions of Hijkl and Gnm ijkl , as well as for more details about implementing the numerical scheme for two-dimensional time domain BEM analyses, it is referred to Dominguez [10] and Mansur [19], for instance. nm Taking into account that Hijkl and Gnm ijkl only depend on the difference ðn mÞ but not on the values of n and m, separately, one should notice that at time step n only the matrices corresponding to the maximum difference (n m) have to be computed, since all the other matrices are known from previous steps already. Adopting matrix notation and considering Hn and Gn as the influence matrices computed at the current time step n, Eq. (9) can be rewritten as
ðC þ H1 ÞUn ¼ G1 Pn þ
n1 X Gnmþ1 Pm Hnmþ1 Um :
ð10Þ
m¼1
After introducing the boundary conditions in this equation, the following expression is obtained
Axn ¼ Byn þ Qn
ð11Þ
where, as usual in time domain BEM approaches, the entries of xn in Eq. (11) are unknown displacements or traction components at the discrete time tn , while the entries of vector yn are the according known nodal values of displacements and tractions. Qn is the vector related to the convolution process of the BEM. It represents the complete history up to tn1 . The solution of Eq. (11) yields the dynamic response of the BEM subdomain at time tn .
Raphson algorithm can be found, for instance, in Jacob and Ebecken [18]. Equations (13) and (14) enable the computation of the FEM response at time tn .
5 Iterative coupling of BEM and FEM In order to better explain the iterative coupling procedure adopted, the notation IF=B VtðkÞ is going to be used. According to this, the variable V is on the FEM/BEM interface ðI VÞ and it is approached either by the FEM or the BEM subdomain (F V or B V) at time t ðVt Þ and at iterative step ðkÞ ðVðkÞ Þ. The algorithm shown in Table 1 summarizes the iterative coupling procedures being used. It is important to note that the iterative procedure described in Table 1 can be done together with the FEM nonlinear iterations, described in Sect. 4. The algorithm shown in Table 1 solves the FE- and the BE-subsystem separately, which means that different solving procedures can be applied to solve the FEM and the BEM system of equations. Thus the symmetry and sparsity of the FEM matrices can easily be taken into account, which results in a more efficient methodology. By solving the FEM and the BEM apart, one also has a better conditioned system of equations, which is important with respect to the accuracy and efficiency of the analysis. It should be also pointed out that when a Newton–Raphson procedure is considered, which includes a renewal of the effective stiffness matrix at different iterative steps, the BE system of equations is not affected by this renewal. In this way a considerable amount of calculations is avoided. In order to take into account different time discretizations for the FE and the BE subdomains, extrapolations and interpolations of the variables related to each numerical approach can be considered. Fig. 1 and items 2.4.3 and 2.4.5 from Table 1 indicate how this time interpolation–extrapolation procedure should be done, according to the BE formulation adopted (Eqs. (7) and (8)). The parameter b in Table 1, is an interpolation parameter and can be easily evaluated by
4 Nonlinear dynamic FEM analysis In the present formulation, nonlinearities occurring in the FE sub-region can be taken into account. Using, for instance, Newton–Raphson iterative procedures, the governing equilibrium equations describing a nonlinear dynamic problem are given by b ¼ ðB t F tÞ=B Dt : ð15Þ € nðkþ1Þ þ CU_ nðkþ1Þ þ KT DUðkþ1Þ ¼ Fn RnðkÞ MU ð12Þ The parameter a, on the other hand, should be chosen n n appropriately, in order to ensure and/or speed up ð13Þ Uðkþ1Þ ¼ UðkÞ þ DUðkþ1Þ convergence (for more details about the choice of a see where M; C, and KT are mass, damping and nonlinear Elleithy et al. [11]). stiffness matrices, respectively. The nonlinear residual vector is represented by Fn RnðkÞ and DUðkþ1Þ is the var- 6 iation of the incremental displacements, calculated at each Numerical applications €n iterative step. Unðkþ1Þ , U_ nðkþ1Þ and U In this section, three classical but representative models, ðkþ1Þ are the displacement, velocity and acceleration vectors, respectively, at namely two rectangular finite bodies and a semi-infinite time tn and iterative step ðk þ 1Þ. space are investigated. The major objective of these studies Taking into account the Newmark method and Eq. (12) is to show the applicability and accuracy of the new couand (13), the following effective system of equations can be pling approach. Therefore, the results obtained with the achieved, which has to be solved at each iterative step of iterative FEM/BEM coupled formulation are compared – the Newton–Raphson procedure whenever possible – with analytical solutions or with ðkþ1Þ ¼ B ðkÞ ADU ð14Þ uncoupled FEM and BEM procedures. For all the examples in this section, the parameter a (see is the effective nonlinear stiffness matrix Table 1) was adopted equal to 0.5. In the iterative FEM/ In Eq. (14), A ðkÞ is the effective residual vector. Further details on BEM coupling approach, a residual and displacement and B tolerance of 103 was chosen throughout the analyses. an optimized implementation of the Newmark/Newton–
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Table 1. Algorithm for the iterative coupling of BEM/FEM 1. Initial Calculations: 1.1. The Problem domain is subdivided into two sub-domains that are well behaved and solvable, each one being modeled by the FEM and the BEM. 1.2. Time-steps for each sub-domain are selected: F Dt and B Dt Dt 1.3. Initial prescribed values are chosen to the FEM node forces at the interface surface, for example, IF FFð0Þ ¼ 0. 1.4. Standard initial calculations related to the FEM and BEM are considered, for instance, the calculation of the matrices A, B, etc.
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2. Time-step loop: 2.1. Initial time attribution: B t ¼ B Dt and F t ¼ 0. 2.2. Beginning of the evaluations at each time-step: F t ¼ F t þ F Dt 2.3. BEM pre-iterative processing: if F t > B t then 2.3.1. B t ¼ B t þ B Dt 2.3.2. Evaluation of the BEM previous time influence vector QB t 2.4. Iterative loop: Ft 2.4.1. Solve the FE problem. Obtain the displacements at the interface: IF UðkþaÞ . t Ft Ft 2.4.2. Adoption of a relaxation parameter a in order to ensure and/or speed up convergence: IF Uðkþ1Þ ¼ aIF UðkþaÞ þ ð1 aÞIF UFðkÞ . n I Ft I Bt 2.4.3. Time extrapolation of F Uðkþ1Þ in order to obtain U . Since the interpolation function / ðtÞ (equation 7) is usually u B ðkþ1Þ t Bt considered as being linear, one has: IB Uðkþ1Þ ¼ IF UFðkþ1Þ bIB UB tB Dt =ð1 bÞ. t 2.4.4. Solve the BE problem. Obtain the tractions at the interface: IB PBðkþ1Þ . I Bt I Ft 2.4.5. Time interpolation of B Pðkþ1Þ in order to obtain F Pðkþ1Þ . Since the interpolation function /np ðtÞ (equation 8) is usually t Ft considered as being piecewise constant, one has: IF Pðkþ1Þ ¼ IB PBðkþ1Þ . t
2.4.6. From the tractions at the interface, obtain the FEM nodal forces: IF FFðkþ1Þ . 2.4.7. Check for convergence (usual nonlinear convergence check procedures, namely residual, displacement and/or energy convergence checks, can be maintained). Go back to 2.4.1 if convergence has not been achieved. 2.5. Actualization (and impression) of results related to the FEM. 2.6. If F t þ F Dt > B t then: actualization (and impression) of results related to the BEM. 2.7. Go back to 2.2 until the final time-step is reached. 3. End of calculation.
Fig. 1. Time interpolation-extrapolation procedures related to different time step choices: (a) time extrapolaFt tion of IF Uðkþ1Þ in order to obtain I Bt B Uðkþ1Þ ; (b) time interpolation of I Bt I Ft B Pðkþ1Þ in order to obtain F Pðkþ1Þ
Rectangular finite body (1D rod) The first example is that of a rectangular body behaving like a one-dimensional rod. It is fixed at one end and subjected to a Heaviside type forcing function of unitary value (Mansur [19]). A sketch of the model is shown in Fig. 2a. The material properties are: Poisson’s ratio ¼ 0:0; Young’s modulus ¼ 100 N/m2 , and mass
density ¼ 1:5 Ns2 /m4 . The geometry is defined by a ¼ 2 m and b ¼ 1 m. As depicted in Fig. 2b, in the coupled mesh 32 linear boundary elements of equal length and 128 quadrilateral finite elements are employed. The time step for the BEM approach is 0.01s. Different time steps for the FEM approach, namely F Dt ¼ 1:00B Dt; F Dt ¼ 0:50B Dt and F Dt ¼ 0:25B Dt, are selected.
Fig. 2. Rectangular domain (one dimensional rod): (a) geometry and boundary conditions; (b) coupled BEM/FEM mesh
Figure 3 shows the transient displacements at points A and B (see Fig. 2a) for the coupled analysis. The results are compared with the analytical solution (solid line). In addition, the tractions at point B are plotted in Fig. 4. As it can be observed, the use of different time steps in each subdomain is very important, since the FE analyses usually does not give proper results for time steps that are best suited for the BE analyses. Using a time step for the FE subregion of F Dt ¼ 0:25B Dt for instance, yields results that
Fig. 3. Horizontal displacements at points A ð0; b=2Þ and B ða=2; b=2Þ considering different BEM–FEM time step relations
Fig. 4. Horizontal tractions at point B ða=s; b=2Þ considering different BEM–FEM time step relations
are much closer to the analytical solution than in the case of bigger time steps F Dt. Rectangular finite body (cantilever beam) The second example consists of a clamped beamlike body subjected to a suddenly applied uniform load (Heaviside step function) of amplitude one (Fig. 5a). It is aimed to show the applicability of the new scheme to nonlinearities assumed in the FE sub-region. Therefore, a perfectly plastic material obeying the von Mises yield criterion is assumed. The material properties are: Poisson’s ratio ¼ 0:0; Young’s modulus ¼ 100 N/m2 ; mass density ¼ 1:5 Ns2 /m4 ; and uniaxial yield stress ¼ 0:10 N/m2 . The geometry is defined by a ¼ 2 m and b ¼ 1 m. Again, in the coupled mesh 32 linear BE of equal length were used, while 64 quadrilateral FE were employed, as depicted in Fig. 5b. The time step for the BEM approach is 0.015s. For the FEM approach a time step of 0.005s is assumed. Figure 6 depicts the displacements at point A (see Fig. 5a) for the BEM/FEM coupled analysis, as well as for the corresponding uncoupled FEM and BEM analyses. Linear and nonlinear results are plotted. The results obtained by means of the different methodologies show excellent agreement. The time evolution of the plastic zone, obtained by the nonlinear analysis with the coupled BEM/ FEM procedure, is shown in Fig. 7. Semi-infinite space (halfspace) In order to show the applicability of the new formulation in the case of unbounded domains, an elastoplastic halfspace under a continuous stress distribution along its surface (von Estorff and Firuziaan [15]) is considered. The system and its discretization are shown in Fig. 8. Again, in the FE-subdomain a perfectly plastic material obeying the Mohr–Coulomb yield criterion is assumed. The material properties are: Poisson’s ratio ¼ 0:25; Young modulus ¼ 1:77 1010 N/m2 ; mass density ¼ 3:15 104 Ns2 /m4 ; cohesion ¼ 1:25 107 N/m2 ; and internal friction angle ¼ 10 . The geometry is defined by a ¼ 152:4 m and b ¼ 304:8 m. As given in Fig. 8b, in the coupled mesh 90 linear BE of equal length and 300 quadrilateral FE are used. The dimensions of the FE subregion are: c ¼ 762 m and d ¼ 571:5 m in horizontal and vertical direction, respectively. Since a semi-infinite media is investigated, the results related to the uncoupled FEM were obtained by using a mesh, which is big enough to ensure, that the reflections occurring at the edges of the discretization do not affect the results during the time period being considered.
Fig. 5. Rectangular domain (cantilever beam): (a) geometry and boundary conditions; (b) coupled BEM/FEM mesh
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Fig. 6. Vertical displacements at point A ða; b=2Þ considering linear and non-linear analyses
Fig. 9. Vertical displacements at points A and B considering linear and non-linear analyses
– the subdivision of a given domain problem into a number of subdomains, – the modeling of each subdomain either by finite elements or by boundary elements, – the iterative coupling of the FE and BE system matrices, taking into account the current transient boundary conditions (e.g., displacements or loads). Fig. 7. Plastic zone evolution within the FEM mesh
The time step for the BEM approach is 0.05s. For the FEM part it is set to 0.0125s. Figure 9 shows the displacement results obtained at the points A and B (see Fig. 8a). As before, the BEM/FEM coupled analysis and the corresponding uncoupled FEM and BEM analyses are evaluated assuming linear and nonlinear behaviour (see Fig. 9). Once more, the results show excellent agreement among each other. The time evolution of the plastic zone, obtained by the nonlinear analysis with the coupled BEM/ FEM procedure, is shown in Fig. 10.
7 Conclusion In order to increase the efficiency in the case of a FEM/ BEM coupled analysis directly in the time domain, an iterative coupling approach has been developed. It is based on three major steps, namely
The major advantage of such a procedure can be seen in the fact, that the FE and BE system matrices can be solved separately using optimized solution algorithms for each subdomain. Consequently, the systems of equations to be solved are much smaller than the standard coupled matrices. In addition, the iterative coupling offers two advantages: It is straightforward to use different time steps in each subdomain and, moreover, to take into account nonlinearities (within the FE subdomain) in the same iteration loop that is needed for the coupling. The new methodology has been applied to three numerical examples, namely a finite rectangular domain behaving like a finite rod clamped at its one end, a finite rectangular domain loaded like a cantilever beam, and an semi-infinite space (halfspace) with a vertical load at its surface. Assuming linear and also nonlinear material laws, it could be shown that the new algorithm yields excellent results when compared with analytical solutions or other numerical results. Using the advantages of finite and boundary elements in an iterative coupling scheme, the new formulation is well suited to handle more complex semi-infinite systems
Fig. 8. Halfspace: (a) geometry and boundary conditions; (b) coupled BEM/FEM mesh
Fig. 10. Displacement and plastic zone evolution within the FEM mesh
including local nonlinearities. It is clearly superior to formulations where a direct solution scheme is used.
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