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Volume 43, Issue 5
Efficient Acceleration Strategies for Multigrid Preconditioned Conjugate Gradients in Fast 3D Topology Optimization

Bingzhen Zhou, Zixian Zhu & Xiaoping Wang

DOI: 10.4208/jcm.2508-m2025-0035

J. Comp. Math., 43 (2025), pp. 1063-1091.

Published online: 2025-09

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  • Abstract

This paper presents various acceleration techniques tailored for the traditional 3D topology optimization problem. Firstly, the adoption of the finite difference method leads to a sparser stiffness matrix, resulting in more efficient matrix-vector multiplication. Additionally, a fully matrix-free technique is proposed, which only assembles stiffness matrices at 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 within conjugate gradient (CG) iterations. Finally, to further enhance the optimization process on high-resolution grids, a progressive strategy is implemented. The numerical results confirm that these acceleration techniques are not only efficient, but also capable of achieving lower compliance and reducing memory consumption. MATLAB codes complementing the article can be downloaded from Github.

  • Keywords

Topology optimization, Linear elasticity, Fully matrix-free, Multigrid preconditioned conjugate gradient, Finite difference method.

  • AMS Subject Headings

49M41, 65M55, 74B05, 74S20, 90C06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-43-1063, author = {Zhou , BingzhenZhu , Zixian and Wang , Xiaoping}, title = {Efficient Acceleration Strategies for Multigrid Preconditioned Conjugate Gradients in Fast 3D Topology Optimization}, journal = {Journal of Computational Mathematics}, year = {2025}, volume = {43}, number = {5}, pages = {1063--1091}, abstract = {

This paper presents various acceleration techniques tailored for the traditional 3D topology optimization problem. Firstly, the adoption of the finite difference method leads to a sparser stiffness matrix, resulting in more efficient matrix-vector multiplication. Additionally, a fully matrix-free technique is proposed, which only assembles stiffness matrices at 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 within conjugate gradient (CG) iterations. Finally, to further enhance the optimization process on high-resolution grids, a progressive strategy is implemented. The numerical results confirm that these acceleration techniques are not only efficient, but also capable of achieving lower compliance and reducing memory consumption. MATLAB codes complementing the article can be downloaded from Github.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2508-m2025-0035}, url = {http://global-sci.org/intro/article_detail/jcm/24471.html} }
TY - JOUR T1 - Efficient Acceleration Strategies for Multigrid Preconditioned Conjugate Gradients in Fast 3D Topology Optimization AU - Zhou , Bingzhen AU - Zhu , Zixian AU - Wang , Xiaoping JO - Journal of Computational Mathematics VL - 5 SP - 1063 EP - 1091 PY - 2025 DA - 2025/09 SN - 43 DO - http://doi.org/10.4208/jcm.2508-m2025-0035 UR - https://global-sci.org/intro/article_detail/jcm/24471.html KW - Topology optimization, Linear elasticity, Fully matrix-free, Multigrid preconditioned conjugate gradient, Finite difference method. AB -

This paper presents various acceleration techniques tailored for the traditional 3D topology optimization problem. Firstly, the adoption of the finite difference method leads to a sparser stiffness matrix, resulting in more efficient matrix-vector multiplication. Additionally, a fully matrix-free technique is proposed, which only assembles stiffness matrices at 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 within conjugate gradient (CG) iterations. Finally, to further enhance the optimization process on high-resolution grids, a progressive strategy is implemented. The numerical results confirm that these acceleration techniques are not only efficient, but also capable of achieving lower compliance and reducing memory consumption. MATLAB codes complementing the article can be downloaded from Github.

Zhou , BingzhenZhu , Zixian and Wang , Xiaoping. (2025). Efficient Acceleration Strategies for Multigrid Preconditioned Conjugate Gradients in Fast 3D Topology Optimization. Journal of Computational Mathematics. 43 (5). 1063-1091. doi:10.4208/jcm.2508-m2025-0035
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