Primal-dual Newton Methods in Structural Optimization

We consider the numerical solution of optimization problems for systems of partial differential equations with constraints on the state and design variables as they arise in the optimal design of the shape and the topology of continuum mechanical structures. After discretization the resulting nonlinear programming problems are solved by an “all-at-once” approach featuring the numerical solution of the state equations as an integral part of the optimization routine. In particular, we focus on primal-dual Newton methods combined with interior-point techniques for an appropriate handling of the inequality constraints. Special emphasis is given on the efficient solution of the primal-dual system that results from the application of Newton’s method to the Karush–Kuhn–Tucker conditions where we take advantage of the special block structure of the primal-dual Hessian. Applications include structural optimization of microcellular biomorphic ceramics by homogenization modeling, the shape optimization of electrorheological devices, and the topology optimization of high power electromotors.