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

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 paper 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 (Pham et al., Eng Anal Bound Elem 2012;36:1838-47 [36]). 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≈10⁶.