Numerical solutions in three dimensional elastostatics

Cruse, T. A.

doi:10.1016/0020-7683(69)90071-7

Cited by 491 publications

(179 citation statements)

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“…The integral operator (7) may be recast into alternative, equivalent regularized forms which are free of CPV integrals [3]. Equations (1) and (6) are applicable to either interior or exterior elastodynamic problems.…”

Section: Boundary Integral Equationmentioning

confidence: 99%

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A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain

Chaillat

Bonnet

Semblat

2008

Computer Methods in Applied Mechanics and Engineering

104

144

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To reduce computational complexity and memory requirement for 3-D elastodynamics using the boundary element method (BEM), a multi-level fast multipole BEM (FM-BEM) is proposed. The diagonal form for the expansion of the elastodynamic fundamental solution is used, with a truncation parameter adjusted to the subdivision level, a feature necessary for achieving optimal computational efficiency. Both the single-level and multi-level forms of the elastodynamic FM-BEM are considered, with emphasis on the latter. Crucial implementation issues, including the truncation of the multipole expansion, the optimal number of levels, the direct and inverse extrapolation steps are examined in detail with the backing of numerical experiments. A complexity analysis for both the single-level and multi-level versions is conducted. The correctness and computational performances of the proposed elastodynamic FMM are demonstrated on numerical examples, featuring up to O(10 6 ) DOFs run on a single-processor PC and including the diffraction of an incident P plane wave by a semi-spherical or semi-ellipsoidal canyon, representative of topographic site effects.Keywords: Fast multipole method; Boundary element method; 3-D elastodynamics; Topographic site effects IntroductionThe boundary element method (BEM), pioneered in the sixties [6,41], is a mesh reduction method, subject to restrictive constitutive assumptions but yielding highly accurate solutions. It is in particular well suited to deal with unbounded-domain idealizations commonly used in e.g. acoustics [50], electromagnetics [35,39] or seismology [7,22]. In contrast with domain discretization methods, artificial boundary conditions [18] are not needed for dealing with the radiation conditions, and grid dispersion cumulative effects are absent [24,51].However, in traditional boundary element (BE) implementations, the dimensional advantage with respect to domain discretization methods is offset by the fully-populated nature of the BEM coefficient matrix, with set-up and solution times rapidly increasing with the problem size N . It is thus essential to develop alternative, faster strategies that allow to still exploit the known advantages of BEMs when large N prohibit the use of traditional implementations. Fast BEMs, i.e. BEMs of complexity lower than that of traditional BEMs, appeared around 1985 with an iterative integral-equation [19,20], in the context of many-particle simulations. The FMM then naturally led to fast multipole boundary element methods (FM-BEMs), whose scope and capabilities have rapidly progressed, especially in connection with application in electromagnetics [21,31,32,53], but also in other fields including acoustics [14,36,48] and computational mechanics [30]. Many of these investigations are summarized in a review article by Nishimura [37]. The FMM, as well as other fast BEM approaches [23,27,55,56], intrinsically relies upon an iterative solution approach for the linear system of discretized BEM equations, with solution times typically of order O(N lo...

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Section: Boundary Integral Equationmentioning

confidence: 99%

“…The boundary element method (BEM), pioneered in the sixties [6,41], is a mesh reduction method, subject to restrictive constitutive assumptions but yielding highly accurate solutions. It is in particular well suited to deal with unbounded-domain idealizations commonly used in e.g.…”

Section: Introductionmentioning

confidence: 99%

A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain

Chaillat

Bonnet

Semblat

2008

Computer Methods in Applied Mechanics and Engineering

104

144

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“…Moreover, bandedness may be induced in the resulting matrices which are otherwise fully populated.44 Most conventional subregioning schemes resort to assembling the subregional matrices into one final banded matrix. 48 However in the technique49 which is discussed in this paper, each subregion matrix is inverted and stored immediately after being set up. Hence, it eliminates the need to use a reduction scheme, and less computing time is spent on inversions.…”

Section: Subregioningmentioning

confidence: 99%

The development of the Trefftzian methodology for engineering design analysis

Wearing¹,

Sheikh²

1995

Advances in Engineering Software

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“…A close inspection of the results can show the reference potential level, but in general, it would be better to incorpora te ( 15) as an additional equation in the manner indicated, for instance, by Symm 14 (Fig. 10 (1 )).…”

Section: Examplesmentioning

confidence: 99%