Shape derivative techniques in optimal design
Petar Kunštek · 2020
Optimal design theory, also known as shape optimization is quite indispensable in many fields like aeronautics, architecture, medicine, computer science. Applications vary from classical, as construction of an aircraft wing, to more recent as in inverse problems of electrical impedance tomography (non-invasive method of medical scanning), picture segmentation or in 3D printing. From the engineering point of view the main aspect of design process is improving a current design. In such optimal design problems the shape sensitivity analysis plays a central role in finding a solution and creation of numerical methods. In this thesis we consider optimal design problems for stationary diffusion equation, seeking for an arrangement of two isotropic materials, with prescribed amounts, which maximizes a given functional. The optimality of a distribution is measured by an objective function, which is usually an integral functional depending on the distribution of materials and the state function, obtained as a solution of the associated boundary value problem for the corresponding partial differential equation. Commonly, optimal design problems do not have solutions (if they exist, such solutions are usually called classical). Therefore, one can consider a proper relaxation of the original problem by the homogenization method which consists of using generalized composite materials. By enlarging the admissible set of the relaxed problem we can consider an artificial optimal design problem which can be rewritten as a saddle point problem. We further show that it is equivalent to a simpler relaxation problem given only in terms of the local proportion of the original materials for which necessary and sufficient conditions of optimality are obtained. Since every classical solution of the considered artificial optimal design problem is also a (classical) solution of the original problem it can be used to construct a family of classical solutions. The aim of the first chapter of the thesis is to present some classes of optimal design problems on an annulus with classical solutions. The first class is a single state equation problem with a constant right-hand side and homogeneous Dirichlet boundary condition. By analysing the optimality conditions, we are able to show that there exists a unique (classical) solution. We prove that, depending on the amounts of given materials, only two optimal configurations in both two- and three-dimensional case are possible. The second class of problems deals with a two-state optimal design problem. In the second chapter shape derivative results for the considered problem are presented. Assuming that the interface between phases is regular, for the optimal design problem the first and the second order shape derivative are calculated using different techniques e.g. the chain rule approach and the averaged adjoint approach. The presented results are later used in construction of numerical methods. Shape derivatives can be written in a form of domain integral or as an integral over the interface. The domain expression or distributed shape derivative seems more appropriate for numerical implementation since boundary representations include jumps of a discontinuous functions over the interface. The third chapter is devoted to numerical methods for the optimal design problem presented in the first section. Descent methods based on distributed first and second order shape derivatives are implemented and tested. We observe a stable convergence of both descent methods with a novel Newton-like method converging in half as many steps.