Fast deflated conjugate gradient method with proper orthogonal decomposition for topology optimization

Abstract

. A fast linear solver for topology optimization using a deflation technique with Proper Orthogonal Decomposition (POD) is discussed. The topology optimization method based on evolutionary algorithms requires huge computational cost. In this reason, a deflated Preconditioned Conjugate Gradient (PCG) method is introduced so as to reduce the cost of finite element analysis. The deflation technique decomposes the solution into fast and slowly converging components. The slow components can be solved by direct methods with low computational cost due to small dimensions. Therefore, the deflated PCC method can improve the convergence of PCG. However, the deflated PCG requires to find the slow components. In this study, a POD method with snapshots is introduced. In the optimization process, solution vectors corresponding to parents are used for the snapshots. Orthogonal vectors for the deflation are constructed from the snapshots. Numerical results show that the present method can reduce the computational cost.