University of Birmingham > Talks@bham > Applied Topology Colloquium > A consistent relaxation of optimal design problems for coupling shape and topological derivatives

A consistent relaxation of optimal design problems for coupling shape and topological derivatives

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I will present a procedure for approximating a ‘black and white’ shape and topology optimization problem by a density optimization problem, allowing for the presence of ‘grayscale’ regions. The construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem, and on an interpolation profile, which endows the intermediate density regions with fictitious material properties. In particular, this framework includes the classical SIMP model.

Under mild hypotheses on the smoothing operator and on the interpolation profile, we prove that the features of the approximate density optimization problem (material properties, objective function, etc.) converge to their exact counterparts as the smoothing parameter vanishes. Notably, the Fréchet derivative of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective, depending on whether it is evaluated at the boundary of the domain or in its interior. These results shed new light on the connections between these two different notions of sensitivities for domain functionals and on the construction of consistent interpolation schemes. This also applies to the bi-material case.

The concepts of shape and topological derivatives will be first recalled. Then the approximation procedure will be explained. Related algorithms, including level-set formulations and the incorporation of perimeter penalization, will be finally discussed and illustrated by numerical examples in the linear elasticity context.

This talk is part of the Applied Topology Colloquium series.

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