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：许多复杂几何固体力学问题传统上采用离散边界方法求解。然而，当问题涉及演化的域边界时，由于需要追踪边界并不断重新划分网格，此类方法可能显得繁琐。本文采用一种鲁棒的光滑边界方法（SBM），通过在更大、更简单的计算域中，将复杂几何隐式表示为光滑指示函数的支撑集。我们给出了由此导出的力学平衡方程，其中非齐次边界条件被源项替代。得到的力学平衡问题是半定的，求解较为困难。为此，本文提出一种高效求解近奇异SBM弹性问题的计算策略。我们采用块结构化自适应网格细化（BSAMR）方法适当分辨演化边界，并结合几何多重网格求解器高效求解力学平衡问题。本文讨论了实现该方法的一些实用数值策略，尤其强调了网格中心场与节点中心场的重要性。我们通过三个代表性算例验证了求解器的精度与性能：a) 空穴周围的塑性应变演化，b) 脆性材料中的裂纹成核与扩展，c) 结构拓扑优化。在每个算例中，即使存在大范围近奇异区域，求解器仍能实现良好的收敛，而任何收敛问题均源于其他复杂性（如应力集中）。我们将该框架作为研究涉及变几何的广泛固体力学问题的通用工具。