On the effectiveness of multigrid preconditioned iterative methods for large-scale frequency response topology optimization problems

Large-scale static topology optimization of mechanical structures has been successfully realized for linear problems, including giga-voxel resolution aircraft wings and suspension bridges. Wherein the multigrid preconditioned conjugate gradient method (MG-CG) plays an important role in the repetitive solution of the state equations. However, research on large-scale dynamic topology optimization, e.g., frequency response problems, is still limited. Since the coefficient matrix of the dynamic equation is no longer a positive definite symmetric matrix, yet an indefinite, non-Hermitian and complex matrix, the conjugate gradient method (CG) is no longer applicable and the standard weapon-of-choice, the geometric multigrid preconditioner is no longer guaranteed to work. It is therefore of interest to investigate which iterative methods, if any, posses excellent generality and low computational-cost. In this paper, the effectiveness of several typical preconditioned iterative methods is studied, including conjugate gradient method, biconjugate gradient stabilized method (BICGSTAB), induced dimensionality reduction (IDR), generalized minimum residual method (GMRES). A detailed comparison and analysis of iterative methods' convergence, mesh dependence, and sensitivity to stiffness distribution in dealing with indefinite problems is given first. Then, despite its known disabilities, the geometric multigrid method is applied as a preconditioner for GMRES, BICGSTAB and IDR, i.e., MG-GMRES, MG-BICGSTAB and MG-IDR, to facilitate the efficient solution of large-scale frequency response analysis. In addition, the influence of several smoothers, including damped Jacobian iteration, successive over relaxation, symmetric SOR, and incomplete LU factorization, on the convergence of geometric multigrid iterative methods is also discussed. Numerical examples show that MG-BICGSTAB deals with low-frequency problems well, but for the whole frequency range, MG-GMRES with ILU smoother converges quickly and steadily, even if the model is extremely large. Furthermore, the effectiveness of the proposed procedure is further verified in dynamic topology optimization with up to 2.8 million degrees of freedom using a standard desktop computer.