Efficient Acceleration Strategies for Multigrid Preconditioned Conjugate Gradients in Fast 3D Topology Optimization (Journal of Computational Mathematics, 2025)
This paper presents various acceleration techniques tailored for the traditional 3D topology optimization problem. Firstly, the adoption of the finite difference method leads toa sparser stiffness matrix, resulting in more efficient matrix-vector multiplication. Additionally, a fully matrix-free technique is proposed, which only assembles stiffness matricesat the coarsest grid level and does not require complex node numbering. Moreover, an innovative N-cycle multigrid (MG) algorithm is proposed to act as a preconditioner withinconjugate gradient (CG) iterations. Finally, to further enhance the optimization processon high-resolution grids, a progressive strategy is implemented. The numerical results confirm that these acceleration techniques are not only efficient, but also capable of achievinglower compliance and reducing memory consumption. MATLAB codes complementing thearticle can be downloaded from Github.
Keywords: Topology optimization; Linear elasticity; Fully matrix-free; Multigrid preconditioned conjugate gradient; Finite difference method.