Antoine Laurain

Abstract: The structure theorem of Hadamard–Zolesio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. Actually the domain representation, also known as distributed shape derivative, is more general than the boundary expression as it is well-defined for shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithms. In this paper we describe several advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We identify a tensor representation of the distributed shape derivative, study its properties and show how it allows to recover the boundary expression directly. We use a novel Lagrangian approach, which is applicable to a large class of shape optimization problems, to compute the distributed shape derivative. We also apply the technique to retrieve the distributed shape derivative for electrical impedance tomography. Finally we explain how to adapt the level set method to the distributed shape derivative framework and present numerical results.

TL;DR: Second-order topological expansions in electrical impedance tomography problems with piecewise constant conductivities and interactions between several simultaneous perturbations are considered, aimed at determining the relevance of non-local and interaction terms from a numerical point of view.

TL;DR: In this paper, a level set based shape and topology optimization approach to electrical impedance tomography (EIT) problems with piecewise constant con- ductivities is introduced, which relies on the notion of shape derivatives to update the shape of the domains where the conductivity takes its different values.

TL;DR: In this paper, a shape optimization problem is formulated by introducing a tracking-type cost functional to match a desired rotation pattern, and shape sensitivity analysis is rigorously performed for the nonlinear problem by means of a new shape-Lagrangian formulation adapted to nonlinear problems.

TL;DR: Numerical results confirm that the level set method for shape optimization of the energy functional for the Signorini problem is efficient and gives better results compared with the classical shape optimization techniques.