A unified approach to shape and topological sensitivity analysis of discretized optimal design problems. (English) Zbl 1517.49026
Summary: 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.
MSC:
| 49Q12 | Sensitivity analysis for optimization problems on manifolds |
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 65K10 | Numerical optimization and variational techniques |
| 35K57 | Reaction-diffusion equations |
Keywords:
shape derivative; topological derivative; unified sensitivity analysis; design optimization; finite element methodSoftware:
CutFEMReferences:
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