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

Peter Gangl, Michael Helmut Gfrerer*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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.
Original languageEnglish
Article number46
JournalApplied Mathematics & Optimization
Volume88
Issue number2
DOIs
Publication statusPublished - 2023

Keywords

  • Design optimization
  • Finite element method
  • Shape derivative
  • Topological derivative
  • Unified sensitivity analysis

ASJC Scopus subject areas

  • Control and Optimization
  • Applied Mathematics

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