Robust, strong form mechanics on an adaptive structured grid: efficiently solving variable-geometry near-singular problems with diffuse interfaces

TL;DR

This paper introduces a robust smooth boundary method combined with adaptive mesh refinement and multigrid solvers to efficiently address complex, evolving geometry problems in solid mechanics, demonstrating high accuracy and convergence.

Contribution

The work presents a novel computational framework integrating SBM, BSAMR, and geometric multigrid for solving near-singular elasticity problems with variable geometry and diffuse interfaces.

Findings

High accuracy and convergence in plastic strain, crack propagation, and topology optimization examples.

Effective handling of large near-singular areas with minimal convergence issues.

Versatile tool for complex solid mechanics problems with evolving boundaries.

Abstract

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.