Domain expression of the shape derivative and application to electrical impedance tomography
- Laurain, Antoine
- Sturm, Kevin
2010 Mathematics Subject Classification
- 49Q10 35Q93 35R30 35R05
- Shape optimization, distributed shape gradient, electrical impedance tomography, Lagrangian method, level set method
The well-known 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. However a volume representation (distributed shape derivative) is more general than the boundary form and allows to work with 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 algorithm. In this paper we describe the numerous advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We give several examples of numerical applications such as the inverse conductivity problem and the level set method.