State space Newton’s method for topology optimization

Abstract

We introduce a new algorithm for solving certain classes of topology optimization problems, which enjoys fast local convergence normally achieved by the full space methods while working in a smaller reduced space. The computational complexity of Newton’s direction finding subproblem in the algorithm is comparable with that of finding the steepest descent direction in the traditional first order nested/reduced space algorithms for topology optimization. That is, the space reduction is computationally inexpensive, and more importantly it does not ruin the sparsity of the full-space system of optimality conditions.The fast local convergence of the algorithm allows one to efficiently solve a sequence of optimization problems for varying parameters (numerical continuation). This can be utilized for eliminating the errors introduced by the approximate enforcement of the boundary conditions or 0/1-type constraints on the design field through penalties in many topology optimization approaches.We test the algorithm on the benchmark problems of dissipated power minimization for Stokes flows, and in all cases the algorithm outperforms the traditional first order reduced space/nested approaches by a factor varying from two to almost twenty in terms of the number of iterations while attaining an almost unprecedented accuracy in solving the discretized topology optimization problem. Finally we present a few extensions to the algorithm, one involving computations on adaptively refined meshes and another related to solving topology optimization problems for non-Newtonian fluids.