State space Newton’s method for topology optimization

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. st 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. t 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.