Adjoint Formulations for Topology, Shape and Discrete Optimization

Abstract

The use of adjoints for PDE constrained optimization problems has become a common design tool in many areas of applied engineering. The use of adjoint methods for shape optimization has received much attention as it exploits the inherent advantage of computing the Frechet derivative of a single or small number of objectives with minimal computational effort. Topology optimization primarily for problems in structural mechanics has also used adjoints to efficiently compute sensitivities of the weight of the structure. Inverse design to recover the shape that results in a particular scattering pattern have used adjoints to quickly morph topologies. Eulerian network models of transportation (personal and commercial) fleets have also been controlled using adjoints to determine the control authority that maximizes throughput of hubs and transportation corridors. The objective of this paper is to formulate and solve the adjoint problem for a variety of PDE constrained optimization problems. The PDEs we will address include Euler equations, linear elasticity, helmholtz and wave equations. For the Euler equations, building on experience of the second author, we plan to tackle new objective functions that include derivatives of the state variables. For the problems in linear elasticity we compare continuous and discrete formulations for minimum weight structures. For the Hemlhotz problem, we study objective functions that contain a mix of polynomial and derivatives in the state-variable. For the wave propagation problem, we use the linear wave equation and its adjoint formulation to determine time dependent optimal controls that enable the identification of the scattering object. Finally, we look at problems in which the PDEs constraining the optimization problem discontinuously change in time and space.