2023/11/22 by Sören Bartels, Bartels, Sören, Hedwig Keller +3
Engineering · #Advanced Numerical Analysis Techniques #Composite Structure Analysis and Optimization #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Topology Optimization in Engineering
paper · pdf · doi:10.48550/arxiv.2311.13386
openalex publication_date 2023/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the optimization of convex domains under a PDE constraint numerical difficulties arise in the approximation of convex domains in ℝ3. Previous research used a restriction to rotationally symmetric domains to reduce shape optimization problems to a two-dimensional setting. In the current research, two approaches for the approximation in ℝ3 are considered. First, a notion of discrete convexity allows for a nearly convex approximation with polyhedral domains. An alternative approach is based on the recent observation that higher order finite elements can approximate convex functions conformally. As a second approach these results are used to approximate optimal convex domains with isoparametric convex domains. The proposed algorithms were tested on shape optimization problems constrained by a Poisson equation and both algorithms achieved similar results.