Fast multipole method applied to elastostatic BEM–FEM coupling
Introduction
The finite element method (FEM) and the boundary element method (BEM) are important numerical tools for computing the solutions of many engineering problems. FEM is appropriate for very large classes of situations, including e.g. those with heterogeneous or non-linear constitutive properties, or finite deformations. On the other hand, BEM is useful for modelling special situations such as very large or unbounded domains, geometrical singularities (e.g. cracks) or to obtain very accurate results in regions of complicated shape (see e.g. [1], [5], [6]). Coupling the BEM and the FEM allows to exploit their complementary advantages when the geometrical configuration warrants it.
The topic of BEM–FEM coupling has been studied since a long time, and many such coupled formulations have been proposed and analysed [14]. In particular, since the traditional collocation BEM (CBEM) formulations lead to unsymmetric systems of coupled BEM–FEM equations, a number of investigations have been directed towards either forcing the symmetry of the CBEM–FEM equations (like in e.g. [3], [15]), or use a symmetric Galerkin BEM (SGBEM) formulation in order to obtain naturally a symmetric system of BEM–FEM equations (see e.g. [7], [13], [19], [25]). The latter approach is well suited to optimally exploit direct solvers.
As the problem size grows, direct solvers applied to coupled BEM–FEM equations become impractical or infeasible with respect to both computing time and storage, even using specific implementation strategies such as out-of-core procedures, mainly because of the fully-populated nature of the BEM matrices, whose build-up computational cost and storage requirement are both of order , where NBEM denotes the number of degrees of freedom (DOFs) supported by the BEM mesh, not to mention the growth of the solution time. To overcome these pitfalls, one needs to resort to iterative solution algorithms for linear systems, together with an acceleration technique for computing the BEM contribution to the residual of the matrix BEM–FEM equation.
Coupled SGBEM–FEM formulations usually lead to governing matrices that are symmetric but not sign-definite. In such cases, iterative solvers do not take advantage of the symmetry (in contrast with e.g. the conjugate-gradient technique applied to positive definite problems). Hence the final symmetry of the coupled problem is not as important as in connection with direct solvers. Since CBEM is simpler and less costly to set-up, a good case can be made for considering the unsymmetric CBEM–FEM approach.
In this article, a simple CBEM–FEM coupled approach leading to a system of equations solved by means of the generalized minimal residual (GMRES) iterative algorithm [22], [8] is presented. The BEM part of the calculation is accelerated by means of the fast multipole method (FMM), a method originally introduced by Rokhlin [20] and further discussed in e.g. [10] and in the recent review article by Nishimura [18]. When applied to elastostatic BEM, it provides a reduction of both storage requirements and computational cost to O(NBEM). These improvements make BEM a viable tool (either on a stand-alone basis or coupled with FEM) for large problems. In addition, a preconditioning technique known as the SParse Approximate Inverse (SPAI) technique is implemented for improving the convergence (i.e. reducing the number of iterations) of GMRES. The article is organized as follows. In Section 2, CBEM and FEM formulations are outlined and the coupled problem is presented. Then, the FMM treatment of the BEM equations is presented in Section 3. The solution technique, and especially the preconditioning strategy, is discussed in Section 4. Finally, numerical examples are examined in Section 5.
Section snippets
Coupled CBEM–FEM formulation
Consider a solid occupying a three-dimensional region Ω. A coupled BEM–FEM model of the solid (Fig. 1) is defined on the basis of a partition Ω = ΩB ∪ ΩF, where ∂ΩB (the boundary of ΩB) and ΩF respectively support boundary element and finite element discretizations. Let SI = ∂ΩB ∩ ∂ΩF denote the BEM–FEM interface, while SB and SF are the remaining surfaces such that ∂ΩB = SI ∪ SB and ∂ΩF = SI ∪ SF. Both subregions ΩB,ΩF are here endowed with linear elastic properties:
Multipole expansions for 3D elasticity
The FMM used in this article follows closely the treatment presented in [24] for solving the hypersingular CBEM for elastostatic crack problems. The latter is based on the following series expressing the inverse of the distance r = ∣y − x∣ between two points y and x, expanded about two poles y0 and x0:having put , and r0 = y0 − x0, and where the overbar indicates complex conjugation. The (complex-valued) solid harmonics
Preconditioning strategy
As previously mentioned, the GMRES iterative algorithm is used for solving the system (10) of CBEM–FEM equations. The convergence rate of iterative solvers depends strongly on the spectral properties of the coefficient matrix. These spectral properties can be improved by means of suitably chosen linear transformations, i.e. preconditioning. Preconditioning iterative solvers is thus an important practical issue, to which a great deal of attention is devoted in the literature [8], [21].
The linear
Numerical examples
Two numerical examples based on the CBEM–FEM coupling method described above are presented in this section. The first one (Section 5.1), which concerns the response of an infinite medium to a uniform temperature applied over an ellipsoidal region, has an exact solution. The second one (Section 5.2) concerns a real dam structure, in order to demonstrate the present coupling technique on a realistic problem. For both examples, the stopping criterion for the GMRES algorithm was a backward error
Summary
In this paper a coupling technique between the finite element method (FEM) and the collocation boundary element method (CBEM) has been presented. Its main feature is the recourse to the Fast Multipole Method in order to both accelerate the BEM contribution to the overall computation and allow BEM meshes involving large numbers of degrees of freedom. In addition, a preconditioning strategy based on the sparse approximate inverse concept has been implemented. All these issues have been
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This author was supported in the framework of a research project funded by MIUR (Cofin 2002).