  We propose an efficient probabilistic method to solve a deterministic problem
-- we present a randomized optimization approach that drastically reduces the
enormous computational cost of optimizing designs under many load cases for
both continuum and truss topology optimization. Practical structural designs by
topology optimization typically involve many load cases, possibly hundreds or
more. The optimal design minimizes a, possibly weighted, average of the
compliance under each load case (or some other objective). This means that in
each optimization step a large finite element problem must be solved for each
load case, leading to an enormous computational effort. On the contrary, the
proposed randomized optimization method with stochastic sampling requires the
solution of only a few (e.g., 5 or 6) finite element problems (large linear
systems) per optimization step. Based on simulated annealing, we introduce a
damping scheme for the randomized approach. Through numerical examples in two
and three dimensions, we demonstrate that the stochastic algorithm drastically
reduces computational cost to obtain similar final topologies and results
(e.g., compliance) compared with the standard algorithms. The results indicate
that the damping scheme is effective and leads to rapid convergence of the
proposed algorithm.
