A second order level-set algorithm and the topological state derivatve

Abstract

In this talk we discuss first and second order methods to solve topology optimisation problems. In the first part we review the concept of the first/second topological derivative using the so-called first and second order topological state derivative associated with a partial differential equation. Usually the topological derivative of a cost functional is defined by a circular singular perturbation, however, also other types of topological perturbations can be defined. The topological state derivative and consequently the topological derivative depends on the shape/design variable and the shape of the singular perturbation chosen. For example, in dimension two the perturbation can be a ball or a circle, in dimension $n\ge 3$ a $n$-ball or a closed hypersurface. We show how the second order topological derivative can be used to define a second order level-set algorithm following the popular level-set algorithm of Amstutz-Andr\"a. In this algorithm the shape is described by a scalar level-set function $\psi$, the interior of the design is defined by negative values of $\psi$ and the outside by positive values of $\psi$. The evolution of $\psi$, which is topically only driven by the first order topological derivative, is proposed to evolve by an update using second order information. Finally, we present numerical experiments for the Newton-type method for some model problems and compare it to the usual gradient-type level-set algorithm.