Velocity Method and Lagrangian Formulation for the Computation of the Shape Hessian

Abstract

The object of this paper is to study the shape Hessian of a shape functional by the velocity (speed) method. It contains a review and an extension of the velocity method and its connections with methods using first- or second-order perturbations of the identity. The key point is that all these methods yield the same shape gradient but different and unequal shape Hessian since each method depends on a choice of “connection.” However, for autonomous velocity fields the velocity method yields a canonical bilinear Hessian. Expressions obtained by other methods can be recovered by adding to that canonical term the shape gradient acting on the acceleration of the velocity field associated with the choice of perturbation of the identity. The second part of the paper is an application of the Lagrangian method with function space embedding to compute the shape gradient and Hessian of a simple cost function associated with the nonhomogeneous Dirichlet problem.