Many solid mechanics problems on complex geometries are conventionally solved using discrete boundary methods. However, such an approach can be cumbersome for problems involving evolving domain boundaries due to the need to track boundaries and constant remeshing. In this work, we employ a robust smooth boundary method (SBM) that represents complex geometry implicitly, in a larger and simpler computational domain, as the support of a smooth indicator function. We present the resulting equations for mechanical equilibrium, in which inhomogeneous boundary conditions are replaced by source terms. The resulting mechanical equilibrium problem is semidefinite, making it difficult to solve. In this work, we present a computational strategy for efficiently solving near-singular SBM elasticity problems. We use the block-structured adaptive mesh refinement (BSAMR) method for resolving evolving boundaries appropriately, coupled with a geometric multigrid solver for an efficient solution of mechanical equilibrium. We discuss some of the practical numerical strategies for implementing this method, notably including the importance of grid versus node-centered fields. We demonstrate the solver's accuracy and performance for three representative examples: a) plastic strain evolution around a void, b) crack nucleation and propagation in brittle materials, and c) structural topology optimization. In each case, we show that very good convergence of the solver is achieved, even with large near-singular areas, and that any convergence issues arise from other complexities, such as stress concentrations. We present this framework as a versatile tool for studying a wide variety of solid mechanics problems involving variable geometry.
翻译:许多复杂几何形状的固体力学问题通常使用离散边界方法来求解。然而,对于涉及到演化域边界的问题,这种方法因需要跟踪边界和不断重新构网而变得繁琐。在本研究中,我们采用强大的平滑边界方法(SBM),将复杂几何形状隐含地表示为平滑指示函数的支持域,在更大、更简单的计算域中。我们在此基础上提出了机械平衡的方程,其中非齐次边界条件被替换成源项。由此产生的机械平衡问题是半定的,使其难以求解。在这项研究中,我们提出了一种计算策略,用于高效地解决近奇异SBM弹性问题。我们采用块状结构自适应网格细化(BSAMR)方法,以适当地解决演化边界,同时采用几何多重网格求解器,以便高效地实现机械平衡的求解。我们讨论了实现此方法的一些实用数值策略,特别是网格与节点中心场之间的重要性。我们针对三个典型案例展示了求解器的精度和性能:a)孔隙周围的塑性应变演化,b)脆性材料中的裂纹形成和扩展,以及c)结构拓扑优化。在每种情况下,我们都展示了求解器的非常好的收敛性,即使是具有大近奇异区域的情况,任何收敛问题也来自于其他复杂性,例如应力集中。我们将这种框架描述为一种用于研究涉及可变形状的各种固体力学问题的通用工具。