International Journal for Numerical Methods in Engineering , Vol. 69, No. 12 , pp. 2441 - 2468 , 2006
S. Wang
Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL 61801, U.S.A.
E. de Sturler
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, U.S.A.
G.H. Paulino
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205 North Mathews Avenue, IL 61801, U.S.A.
Abstract
The computational bottleneck of topology optimization is the solution of a large number of linear systems arising in the finite element analysis. We propose fast iterative solvers for large three-dimensional topology optimization problems to address this problem. Since the linear systems in the sequence of optimization steps change slowly from one step to the next, we can significantly reduce the number of iterations and the runtime of the linear solver by recycling selected search spaces from previous linear systems. In addition, we introduce a MINRES (minimum residual method) version with recycling (and a short-term recurrence) to make recycling more efficient for symmetric problems. Furthermore, we discuss preconditioning to ensure fast convergence. We show that a proper rescaling of the linear systems reduces the huge condition numbers that typically occur in topology optimization to roughly those arising for a problem with constant density. We demonstrate the effectiveness of our solvers by solving a topology optimization problem with more than a million unknowns on a fast PC.
Key words: topology optimization, three-dimensional analysis, iterative methods, Krylov subspace recycling, preconditioning, large-scale computation.