Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

Pham, Anh-Duc; Mouhoubi, Saïda; Bonnet, Marc; Chazallon, Cyrille

doi:10.1016/j.enganabound.2012.07.004

Cited by 18 publications

(11 citation statements)

References 44 publications

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“…where the symmetric matrix K ∈ R N×N sym arises from discretizing the bilinear forms, X ∈ R N collects all unknown DOFs on S p , S c and S u , and F ∈ R N corresponds to the right-hand side of (11). The operators K, K ′ are in practice reformulated in regularized form [36] to avoid actual evaluation of principal value integrals.…”

Section: Solution Strategy For the Sgbem Equationsmentioning

confidence: 99%

“…The hierarchical subdivision aims at producing new pairs of non-adjacent children cells (at a lower level) whenever two parent cells are adjacent, thus reducing the cases where double integrals need to be computed by "traditional" (and costlier) methods. These considerations are at the root of the well-known multi-level fast multipole method, used here and described in more detail (for the present elastostatic SGBEM context) in [36]. It implicitly recasts the SGBEM system (12) in the form…”

Section: Fast Multipole Algorithm In Bemmentioning

confidence: 99%

“…This article is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for multi-zone elasticity problems with cracks, and is intended as a continuation of [36] where FM-SGBEM were investigated for homogeneous elastic solids containing (possibly many) cracks. The subdomain coupling approach, following [21,18], is based on a minimal set of interfacial unknowns (i.e.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Solving multizone and multicrack elastostatic problems: A fast multipole symmetric Galerkin boundary element method approach

Trinh¹,

Mouhoubi²,

Chazallon³

et al. 2015

Engineering Analysis with Boundary Elements

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Symmetric Galerkin boundary element methods (SGBEMs) for three-dimensional elastostatic problems give rise to fully-populated (albeit symmetric) matrix equations, entailing high solution times for large models. This article is concerned with the formulation and implementation of a multi-level fast multipole SGBEM (FM-SGBEM) for multi-zone elasticity problems with cracks. The subdomain coupling approach is based on a minimal set of interfacial unknowns (i.e. one displacement and one traction vector at any interfacial point) that are defined globally for the complete multizone configuration. Then, unknowns for each subdomain are defined in terms of the global unknowns, with appropriate sign conventions for tractions induced by subdomain numbering. This formulation (i) automatically enforces the perfect-bonding transmission conditions between subdomains, and (ii) is globally symmetric. The subsequent FM-SGBEM basically proceeds by assembling contributions from each subregion, which can be computed by means of an existing single-domain FM-SGBEM implementation such as that previously presented by the authors (EABE, 36:1838-1847, 2012). Along the way, the computational performance of the FM-SGBEM is enhanced through (a) suitable storage of the near-field contribution to the SGBEM matrix equation and (b) preconditioning by means of nested GMRES. The formulation is validated on numerical experiments for 3D configurations involving many cracks and inclusions, and of sizes up to N ≈ 10 6 .

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Section: Solution Strategy For the Sgbem Equationsmentioning

confidence: 99%

Section: Fast Multipole Algorithm In Bemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Solving multizone and multicrack elastostatic problems: A fast multipole symmetric Galerkin boundary element method approach

Trinh¹,

Mouhoubi²,

Chazallon³

et al. 2015

Engineering Analysis with Boundary Elements

Self Cite

View full text Add to dashboard Cite

show abstract

“…a node of the tree, represents a part of the simulation box and is used by the algorithm to factorize the interactions between elements. The FMM was later extended for different types of physical simulations and different approximation kernels (Barba & Yokota, 2011;Blanchard et al, 2016;Darve et al, 2013;Darve & Havé, 2004;Frangi et al, 2003;Malhotra & Biros, 2015;Pham et al, 2012;Sabariego et al, 2004;Vazquez Sabariego, 2004).…”

Section: Introductionmentioning

confidence: 99%

TBFMM: A C++ generic and parallel fast multipole method library

Bramas¹

2020

JOSS

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TBFMM, for task-based FMM, is a high-performance package that implements the parallel fast multipole method (FMM) in modern C++17. It implements parallel strategies for multicore architectures, i.e. to run on a single computing node. TBFMM was designed to be easily customized thanks to C++ templates and fine control of the C++ classes' inter-dependencies. Users can implement new FMM kernels, new types of interacting elements or even new parallelization strategies. As such, it can be used as a simulation toolbox for scientists in physics or applied mathematics. It enables users to perform simulations while delegating the data structure, the algorithm and the parallelization to the library. Besides, TBFMM can also provide an interesting use case for the HPC research community regarding parallelization, optimization and scheduling of applications handling irregular data structures.

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“…Other FMM implementations based on multipole expansions have been reported in the literature for modeling elastic but unconnected linear [24,26,27,36,37], circular [28,[38][39][40][41], and penny-shaped [23,28,[39][40][41] fracture problems in absence of fluid flow. However, these works used a direct formulation of the boundary element method, usually called the fast multipole-boundary element method or FMBEM, which are not suitable for modeling fractures problems in unbounded reservoir domains.…”

Section: Introductionmentioning

confidence: 99%

Fast multipole displacement discontinuity method (FM-DDM) for geomechanics reservoir simulations

Verde

Ghassemi

2015

Int. J. Numer. Anal. Meth. Geomech.

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Summary The displacement discontinuity method (DDM) is frequently used in geothermal and petroleum applications for modeling the behavior of fractures in linear‐elastic rocks. The DDM requires O(N2) memory and O(N3) floating point operations (where N is the number of unknowns) to construct the coefficient matrix and solve the linear system of equations by direct methods. Therefore, the conventional implementation of the DDM is not computationally efficient for very large systems of cracks, often limiting its application to small‐scale problems. This work presents an approach for solving large‐scale fracture problems using the fast multipole method (FMM). The approach uses both the DDM and a kernel‐independent version of the FMM along with a preconditioned generalized minimal residual algorithm to accelerate the solution of linear systems of equations using desktop computers. Using the fundamental solutions for constant displacement discontinuity in a two‐dimensional elastic medium, several numerical examples involving fracture networks representing fractured reservoirs are treated. Numerical results show good agreement with analytical solutions and demonstrate the efficiency of the FMM implementation of the DDM for large‐scale simulations. Copyright © 2015 John Wiley & Sons, Ltd.

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Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

Cited by 18 publications

References 44 publications

Solving multizone and multicrack elastostatic problems: A fast multipole symmetric Galerkin boundary element method approach

Solving multizone and multicrack elastostatic problems: A fast multipole symmetric Galerkin boundary element method approach

TBFMM: A C++ generic and parallel fast multipole method library

Fast multipole displacement discontinuity method (FM-DDM) for geomechanics reservoir simulations

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