On the performance of domain decomposition methods for modeling heterogenous materials

摘要

In this manuscript, we review the performance of domain decomposition methods (DDMs), implemented as a black-box module integrated with a finite element solver, for modeling materials with complex microstructures. In particular, we study the accuracy and computational cost associated with using the non-overlapping and overlapping Schwarz methods, together with required adjustments for each method to avoid convergence issues. Compared to conventional applications such as fluid-solid interaction, the DDM simulation of the mechanical behavior of materials with complex heterostructures could be a challenging task due to high stress concentrations along subdomain edges intersecting with multiple material interfaces. For linear elastic problems, this could lead to high local errors along sub-domain boundaries and especially at subdomain vertices, which requires meticulous updating of boundary conditions (nodal forces and displacements) along these edges to alleviate the error. However, for nonlinear (elastoplastic) problems, we show that such microstructural features prohibit the convergence of the non-overlapping Schwarz method. The remedy to such convergence difficulties is to implement the overlapping Schwarz method, with a high overlap percentage between adjacent subdomains to achieve a reasonable computational cost.