A Unified Approach to Shape and Topological Sensitivity Analysis of Discretized Optimal Design Problems

作者：Peter Gangl, Michael Helmut Gfrerer · 发表于：Applied Mathematics & Optimization · 年份：2023 · DOI：10.1007/s00245-023-10016-2 · 被引用次数：5 · 研究领域：Topology Optimization in Engineering、Advanced Multi-Objective Optimization Algorithms、Probabilistic and Robust Engineering Design

We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is represented by a piecewise linear and globally continuous level set function on a fixed finite element mesh and relate perturbations of the level set function to perturbations of the shape or topology of the corresponding design. We illustrate the sensitivity analysis for a problem that is constrained by a reaction-diffusion equation and draw connections between our discrete sensitivities and the well-established continuous concepts of shape and topological derivatives. Finally, we verify our sensitivities and illustrate their application in a level-set-based design optimization algorithm where no distinction between shape and topological updates has to be made.