Solution to Boundary Shape Identification Problems in Elliptic Boundary Value Problems Using Shape Derivatives

This paper concerns the problem identifying geometrical boundary shapes of domains in which elliptic boundary value problems are defined. Such identification problems can be formulated as minimization problems of squared error integrals between the actual solutions of the elliptic boundary value problems and their reference data with respect to perturbation of the uncertain boundary. Mathematicians have presented fundamental theories concerning the shape derivatives of functionals with respect to domain perturbation and the gradient method in Hilbert space. Based on these theories, this paper presents a concrete solution to geometrical domain identification problems. It briefly describes the derivation of the shape gradient functions for two types of shape identification problems with respect to a boundary value on a subboundary and with respect to the gradient in the subdomain and defines the gradient method in Hilbert space. Using the shape gradient functions thus derived and the concept of the gradient method in Hilbert space, a concrete solution to geometrical boundary identification problems is presented. This solution coincides with the traction method proposed previously by the author’s research group.