Topological asymptotic expansion of shape functionals via adjoint based methods and nonsmooth analysis in structural optimisation

Abstract

Design optimisation deals with the minimisation (maximisation) of a given shape functional, which mapssubsets of R^dto the real numbers, with respect to the design variable. Therein, one distinguishes betweenthe terminology shape optimisation, which is concerned with smooth deformations of a given shape andtopology optimisation, which addresses topological changes. Put into the context of mechanical engineering, we refer to both topics as structural optimisation. Recent advances in the manufacturing process gaverise to a large variety of design possibilities. Consequently, the demand for optimal designs and appropriate optimisation methodologies has increased significantly.We first follow the approach of classical shape optimisation. We apply the well established theory revolving around the shape derivative to a model problem in the framework of linear elasticity with pointwisestress constraints. These constraints are then compactly formulated by the maximum norm, which resultsin a nonsmooth optimisation problem. We employ methods from nonsmooth analysis to derive optimality conditions and draw a connection to the Clarke subgradient. Additionally, we consider three simplegeometries to address the numerical applicability of our methodology.Next, we investigate topological sensitivities in the framework of topology optimisation. We comparethree different adjoint based methods to derive the first and second order topological derivative. We apply these methods to a PDE constrained problem in the framework of linear elasticity and highlight thedifferences in view of applicability and efficiency.Based on our observations, we then employ the averaged adjoint method to compute the complete topological asymptotic expansion for a PDE constrained model problem including the Laplacian and a perturbation of the right hand side. We observe that, depending on the objective functional, the asymptoticanalysis of the adjoint variable can be more involved. In fact, a L^2tracking-type cost functional requiresthe introduction of the fundamental solution of the biharmonic equation.Finally, we utilise the notion of topological state derivatives to investigate numerical schemes in thecontext of topology optimisation. We approximate a state of the art level-set algorithm and introduce asteepest descend scheme in the context of one-shot type methods.