A fast solution method for three-dimensional many-particle problems of linear elasticity

Abstract

A boundary element method for solving three-dimensional linear elasticity problems that involve a large number of particles embedded in a binder is introduced. The proposed method relies on an iterative solution strategy in which matrix-vector multiplication is performed with the fast multipole method. As a result the method is capable of solving problems with N unknowns using only O(N) memory and O(N) operations. Results are given for problems with hundreds of particles in which N"O(10).

Compositional Development of Parallel Programs

“…There the FMM is used for O(N) matrixvector multiplication. In order to adapt the FMM for applications in fluid and solid mechanics, the classical electrostatics problem must be replaced with a generalized electrostatics problem [17,18]. Such problems involve vector and tensor valued charges, which in essence means that one generalized electrostatics problem is equivalent to several classical electrostatics problems, which share the same geometry.…”

“…Such problems involve vector and tensor valued charges, which in essence means that one generalized electrostatics problem is equivalent to several classical electrostatics problems, which share the same geometry. In particular, FLEMS code [17] relies on the generalized electrostatics problem that is equivalent to 13 classical electrostatics problems.…”

Stress computations on perforated polygonal domains

“…(15) and (16). For corners that are not traction-free, but loaded with a uniform traction, the series l n need to be completed with the exponent l¼1: The integers m 2 and m 4 in Eq.…”

“…The ill-conditioning of this operation increases with the number of discretization points and terms that are included, and with the spacing of the exponents l n and m n of Eqs. (15) and (16). There is, thus, a trade-off between convergence and stability: the more exponents that are included-the more rapid convergence but the lower achievable accuracy in the solution close to corners.…”

Application of a Fast Boundary Element Algorithm for Open Floating Potential Electrostatic Problem

“…Preconditioner based on the block diagonal matrix corresponding to the non-interacting particles for 3D many particle problems is found to be fewer iteration numbers [5]. a preconditioner is well suited for fast multipole BEM to the analysis of 3D crack problems [6].…”