Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D

Yoshida, Kenichi; Nishimura, Naoshi; Kobayashi, Shoichi

doi:10.1002/1097-0207(20010130)50:3<525::aid-nme34>3.0.co;2-4

Cited by 92 publications

(61 citation statements)

References 18 publications

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“…For these reasons many researchers tried to devise algorithms which were hybrids of tree codes and FMM, in order to combine the high accuracy of FMM methods with the simplicity of tree codes. In addition, the extension of the FMM to more general kernels like the modified Laplacian [13], the Stokes [10] , and the Navier [9,26] operators can be quite cumbersome, due to the need to implement efficient translation operators. In this paper, we only review algorithms that could be used to develop kernel independent methods.…”

Section: Synopsis Of the New Methodmentioning

confidence: 99%

A kernel-independent adaptive fast multipole algorithm in two and three dimensions

Ying

Biros

Zorin

2004

Journal of Computational Physics

449

505

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We present a new fast multipole method for particle simulations. The main feature of our algorithm is that it does not require the implementation of multipole expansions of the underlying kernel, and it is based only on kernel evaluations. Instead of using analytic expansions to represent the potential generated by sources inside a box of the hierarchical FMM tree, we use a continuous distribution of an equivalent density on a surface enclosing the box. To find this equivalent density we match its potential to the potential of the original sources at a surface, in the far field, by solving local Dirichlet-type boundary value problems. The far field evaluations are sparsified with singular value decomposition in 2D or fast Fourier transforms in 3D. We have tested the new method on the single and double layer operators for the Laplacian, the modified Laplacian, the Stokes, the modified Stokes, the Navier, and the modified Navier operators in two and three dimensions. Our numerical results indicate that our method compares very well with the best known implementations of the analytic FMM method for both the Laplacian and modified Laplacian kernels. Its advantage is the (relative) simplicity of the implementation and its immediate extension to more general kernels.

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Section: Synopsis Of the New Methodmentioning

confidence: 99%

A kernel-independent adaptive fast multipole algorithm in two and three dimensions

Ying

Biros

Zorin

2004

Journal of Computational Physics

449

505

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“…The boundaries of the model are discretized using boundary elements and the internal domain where local yielding is expected to occur are discretized using internal cells. Then an adaptive quad-tree structure is constructed [18,19,22,24]. The root of the tree, which is at level 0, is a square box containing all the boundaries elements and internal cells.…”

Section: Numerical Implementation Of the Fast Multipole Methodsmentioning

confidence: 99%

“…By the recursive operations on the tree structure which are described in detail in References [18][19][20][21][22][23][24][25], the multipole and local moments of all the boxes are calculated and the matrix-vector product is obtained. For a leaf, the multipole moments are obtained from all the elements in it using multipole expansions.…”

Section: Numerical Implementation Of the Fast Multipole Methodsmentioning

confidence: 99%

“…Usually an expansion of the form in Equation (9) is valid when the locations of the source point x and the field point Y satisfy a certain condition. A typical one is |Y − Y 0 |<|x − Y 0 | [18,19,24].…”

Section: Multipole Expansionmentioning

confidence: 99%

“…The matrix-vector product is obtained by recursive operations on the tree structure without explicitly forming the coefficient matrix. In recent years, the fast multipole BEM and its applications were investigated by many authors, including Peirce and Napier [16] for 2D elastostatics, Popov and Power [17] for 3D elastostatics, Yoshida et al [18,19] for 3D elastostatic crack problems, Liu et al [20] for the modelling of carbon-nanotube composites, Wang and Yao [21,22] for the simulation of composite materials, and Wang and Yao [23] for the analysis of fatigue crack growth. A recent literature review on the fast multipole BEM has been given by Nishimura [24].…”

Section: Introductionmentioning

confidence: 99%

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Fast multipole boundary element analysis of two‐dimensional elastoplastic problems

Wang¹,

Yao²

2006

Commun. Numer. Meth. Engng.

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SUMMARYThis paper presents a fast multipole boundary element method (BEM) for the analysis of two-dimensional elastoplastic problems. An incremental iterative technique based on the initial strain approach is employed to solve the nonlinear equations, and the fast multipole method (FMM) is introduced to achieve higher runtime and memory storage efficiency. Both of the boundary integrals and domain integrals are calculated by recursive operations on a quad-tree structure without explicitly forming the coefficient matrix. Combining multipole expansions with local expansions, computational complexity and memory requirement of the matrix-vector multiplication are both reduced to O(N ), where N is the number of degrees of freedom (DOFs). The accuracy and efficiency of the proposed scheme are demonstrated by several numerical examples.

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Boundary Integral Equation Methods for Elastic and Plastic Problems

Bonnet

2004

Encyclopedia of Computational Mechanics

187

318

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This chapter deals with formulations based on boundary integral equations (BIEs) for elastic and plastic problems. After a brief review of the basic integral identities of solid mechanics and issues associated with the singular character of the fundamental solutions, the collocation and symmetric Galerkin BIE formulations and the associated boundary element methods (BEMs) are presented, in their conventional form where the complete matrix equation is set up using numerical integration and stored. This approach being inadequate for large problem sizes, fast solution techniques are then reviewed, with emphasis on the fast multipole method. Next, BEMs in both collocation and symmetric Galerkin form are described for fracture mechanics and small‐strain elastoplasticity. The treatment of the hypersingular integrals arising in the integral representation of tractions on the crack surface or of strains at potentially plastic interior points is discussed, along with other issues. Shape sensitivity analysis techniques are also presented, based on either the direct differentiation of the primary elastic BIE or an adjoint solution. Finally, the symmetric Galerkin BIE is used to define a symmetric formulation for BEM–FEM coupling.

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Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D

Cited by 92 publications

References 18 publications

A kernel-independent adaptive fast multipole algorithm in two and three dimensions

A kernel-independent adaptive fast multipole algorithm in two and three dimensions

Fast multipole boundary element analysis of two‐dimensional elastoplastic problems

Boundary Integral Equation Methods for Elastic and Plastic Problems

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