Fast multipole method applied to elastostatic BEM–FEM coupling

https://doi.org/10.1016/j.compstruc.2004.09.007Get rights and content

Abstract

BEM–FEM coupling is desirable for three-dimensional problems involving specific features such as (i) large or unbounded media with linear constitutive properties, (ii) cracks, (iii) critical parts of complex geometry requiring accurate stress analyses. However, for cases with a BEM discretization involving a large number NBEM of degrees of freedom, setting up the BEM contribution to the coupled problem using conventional techniques is an expensive O(NBEM2) task. Moreover, the fully-populated BEM block entails a O(NBEM2) storage requirement and a O(NBEM3) contribution to the solution time via usual direct solvers. To overcome these pitfalls, the BEM contribution is formulated using the fast multipole method (FMM) and the coupled equations are solved by means of an iterative GMRES solver. Both the storage requirements and the solution times are found to be close to O(NBEM). A preconditioner based on the sparse approximate inverse of the BEM block is shown to improve the convergence of the GMRES solver. Numerical examples involving NBEM = O(105  106) unknowns, run on a PC computer, are presented; they include the Eshelby inclusion (as a validation example), a many-inclusion configuration, and a dam structure.

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 O(NBEM2), where NBEM denotes the number of degrees of freedom (DOFs) supported by the BEM mesh, not to mention the O(NBEM3) 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:σij=Cijkuk,(inΩB)σij=CijkFuk,(inΩF)

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 x0:1r=n=0+m=-nnRn,m(yˆ)n=0+m=-nn(-1)nSn+n,m+m(r0)¯Rn,m(xˆ)having put yˆ=y-y0, xˆ=x-x0 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

References (25)

  • A. Frangi et al.

    3D fracture analysis by symmetric Galerkin BEM

    Comput Mech

    (2002)
  • S. Ganguly et al.

    Symmetric coupling of multi-zone curved Galerkin boundary elements with finite elements in elasticity

    Int J Numer Meth Eng

    (2000)
  • Cited by (23)

    • A new insight into the analysis of plane elasticity with coupling of numerical manifold and boundary element methods

      2021, Engineering Analysis with Boundary Elements
      Citation Excerpt :

      Amiri et al. [25,26] and Sadrnejad et al. [27] introduced hybrid numerical models, including a combination of the finite element and finite volume techniques, for multiphase fluid flow in ductile porous surroundings and recommended the use of hybrid numerical techniques. The combination of finite element and boundary element methods have been presented in the area of elastostatics and fluid-structure interaction by Zhang and Zhang [28] and Estorff and Antes [29], in the elastoplastic domain by Soares [30], Boumaiza an Aour [31], and Pavlatos et al. [32], and in the area of elastostatics by Elliethy et al. [33], Dong [34], Lin et al. [35], and Margonari and Bonnet [36]. A combination of the boundary element method and the manifold method for solving 2D potential problems has been presented by Tan and Jiao [37].

    • A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems

      2018, Computers and Structures
      Citation Excerpt :

      The non-symmetric dense matrices appearing in the solution of the traditional BEM restrict its application to small-scale problems. To break through this bottleneck, the fast multipole method (FMM) [12,13] was introduced to improve the efficiency and reduce the memory requirement of the method [5,14–18]. However, the BEM still encounters a time-consuming issue of a large amount of numerical integrations arising from the discretization of boundary integral equations for large-scale problems [19].

    • The fast multi-pole indirect BEM for solving high-frequency seismic wave scattering by three-dimensional superficial irregularities

      2018, Engineering Analysis with Boundary Elements
      Citation Excerpt :

      Among them, the fast multi-pole method can significantly reduce computational complexity and memory requirements and has become dominant in recent years [41]. It has been effectively applied in large-scale electromagnetic wave scattering [42], solid mechanics [43,44], fluid dynamics [45] and acoustics [46,47]. More relevant research papers can be found in the literature [48,49].

    • Finite element-boundary element coupling algorithms for transient elastodynamics

      2015, Engineering Analysis with Boundary Elements
      Citation Excerpt :

      The latter allow for independent mesh sizes for each subdomain. FE–BE coupling algorithms for elastostatics are discussed, among others, by Elleithy et al. [11] and Margonari et al. [12], while Rüberg et al. [13] present an algorithm for time domain elastodynamics using non-conforming interfaces where the coupling conditions are incorporated in a weak sense by means of Lagrange multipliers. This paper aims to present suitable FE–BE coupling procedures for the solution of dynamic SSI problems in the time domain and to compare their computational performance.

    • A 3D FEM/BEM code for ground-structure interaction: Implementation strategy including the multi-traction problem

      2015, Engineering Analysis with Boundary Elements
      Citation Excerpt :

      The use of an iterative solver appears therefore to be mandatory. The GMRES solver with SPAI-left pre-conditioning proposed in [18] has been implemented with the KSPGMRES free solver [19] but it seems that convergence becomes slow for complex problems. The case of Fig. 7 (block in infinite soil) behaves much better than the pile problem of Fig. 20.

    View all citing articles on Scopus
    1

    This author was supported in the framework of a research project funded by MIUR (Cofin 2002).

    View full text