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

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. The purpose of this work is to present a comprehensive strategy for efficiently solving such problems on an adaptive structured grid, while expositing some of the basic yet important nuances associated with solving near-singular problems in strong form. 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 semidefinite equations for mechanical equilibrium, in which inhomogeneous boundary conditions are replaced by source terms. In this work, we present a computational strategy for efficiently solving near-singular SBM-based solid mechanics problems. We use the block-structured adaptive mesh refinement method, 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.

Appendix A: Convergence data

The linear elastic, strong-form near-singular multigrid solver exhibited nearly universally linear convergence in the example problems presented in this work. Here, we present a more detailed exposition of the solver behavior in time for the fracture cases and the topology optimization example (Fig. 10). We note that in all cases, the results from the initial solve are not included because it is for the initial, non-regularized version of the problem. In all examples, we observe a rapid convergence during the first 3–4 iterations. This rapid convergence lasts until the error is reduced to $10^{-3}$, $10^{-4}$, and $10^{-1}$ for mode I, stress-concentration, and topology optimization, respectively. A sharp reduction follows this in the convergence rate, which is almost always non-decreasing during the remaining solve.

In mode-I fracture and topology optimization, interestingly, the worst convergence occurs during the middle of the simulation; subsequently, convergence improves and approaches a constant rate. Both exhibit a couple of solves in which the convergence rate turned sharply from a higher to a lower value; in both cases, the convergence might be described as “piecewise linear.”

In the stress-concentration fracture case, linear convergence is constantly observed, but the convergence rate decreases steadily as the simulation progresses. We attribute this to the increasing irregularity of the problem, resulting from large chunks of material that have been degraded, and regions of the boundary that are under-regularized. In other words, it appears to be the fracture model, not the solver, that is responsible for the degrading convergence.