Configurational-force-driven adaptive refinement and coarsening in topology optimization

The iterative nature of topology optimization, especially in combination with nonlinear state problems, often requires the solution of thousands of linear equation systems. Furthermore, due to the pixelated design representation, the use of a fine mesh is essential to obtain geometrically well-defined structures and to accurately compute response quantities such as the von Mises stress. Therefore, the computational cost of solving a fine-mesh topology optimization problem quickly adds up. To address this challenge, we consider a multi-level adaptive refinement and coarsening strategy based on configurational forces. Configurational forces based on the Eshelby stress predict configurational changes such as crack propagation or dislocation motion. Due to a relaxation in the calculation of (Eshelby) stresses with respect to the design variables, discrete configurational forces increase not only in highly stressed regions, but also in gray transition regions (design boundaries). For this reason, they constitute an ideal criterion for mesh adaptivity in topology optimization, especially when avoiding stress failure is a priority. By using configurational forces for refinement, we obtain a high-resolution structure where the refined mesh is present along the design boundaries as well as in stress-critical regions. At the same time, multi-level coarsening using the same criterion drastically minimizes the computational effort.