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.

在本手稿中，我们回顾了域分解方法 (DDM) 的性能，该方法作为与有限元求解器集成的黑盒模块实现，用于对具有复杂微结构的材料进行建模。特别是，我们研究了与使用非重叠和重叠 Schwarz 方法相关的准确性和计算成本，以及每种方法所需的调整以避免收敛问题。与流固相互作用等传统应用相比，由于沿与多个材料界面相交的子域边缘的高应力集中，对具有复杂异质结构的材料的机械行为进行 DDM 模拟可能是一项具有挑战性的任务。对于线性弹性问题，这可能会导致沿子域边界的高局部误差，尤其是在子域顶点处，这需要沿这些边缘仔细更新边界条件（节点力和位移）以减轻误差。然而，对于非线性（弹塑性）问题，我们表明这种微观结构特征阻止了非重叠 Schwarz 方法的收敛。解决这种收敛困难的方法是实施重叠 Schwarz 方法，相邻子域之间具有较高的重叠百分比，以实现合理的计算成本。