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

Abstract

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 .

Optimizations of a fast multipole symmetric Galerkin boundary element method code

“…The initial FM-SGBEM code and its performance are well detailed in [24,25]. This Fortran code inherits a number of innovative algorithms from the BE community such as (i) the singular integration schemes by Andrä and Schnack [2,8], (ii) the index of severity [17], (iii) the nested Flexible GMRES which makes use of the nearinteraction matrix [4]; and (iv) the extension of the BIEs to multizone configurations [10].…”

“…The goal here is to speed up the existing code by avoiding big changes. A simple observation of Trinh's [24] results (see Table 2) shows that the solving phase is time-consuming. To reduce this duration, it is necessary to reduce the number of iterations or the duration of one iteration.…”

Analysis of two cohesive zone models for crack propagation in notched beams using the BEM

“…Trinh et al employed the fast multipole symmetric Galerkin BEM to solve multizone and multicrack problems in Ref. [14], and Benedetti and Aliabadi employed the BEM in multiscale modeling for polycrystalline materials to analyze material degradation and fracture in Ref. [15].…”

Evaluation of the FMBEM efficiency in the analysis of porous structures

“…The BEM was successfully applied in the analysis of 2D and 3D nonhomogeneous materials containing pores, by Hu et al (2000), Liu (2009), Ptaszny and Fedeliński (2007), Fedeliński et al (2014), Ptaszny et al (2014), Ptaszny and Fedeliński (2011a), Rejwer et al (2014) and Ptaszny (2015), cracks, by Yoshida et al (2001), Liu (2009), Fedeliński et al (2014), Rejwer et al (2014) and Trinh et al (2015) and composite materials, by Kamiński (1999), Liu et al (2005), Chen and Liu (2005), Lei et al (2006), Liu (2009), Ptaszny and Fedeliński (2011b), Fedeliński et al (2014) and Huang et al (2015). However, as far as we know, there is a lack of comprehensive comparison between the higher-order approximation BEM and FEM in 3D large-scale analysis in the literature.…”