Efficient GPU accelerated topology optimization of composite structures with spatially varying fiber orientations

This paper presents an efficient method for topology optimization of composite structures with spatially varying fiber orientations using GPU acceleration. One of the challenges in solving such problems is that the element stiffness matrices are not the same even for congruent elements, thus all the element stiffness matrices have to be computed on-the-fly or pre-computed and stored in advance, which can significantly increase the computational burden or memory requirement. To address this challenge, this work exploits both the structure of the Cartesian grids and the element stiffness matrices. By extending the idea of template stiffness matrices (TSMs) for 2D problems, each element stiffness matrix in 3D problems can be expressed as a linear combination of 21 TSMs, with the coefficients being derived from the corresponding elasticity matrix. As a result, only the TSMs need to be pre-computed and stored in memory, reducing the computational burden and memory requirement. Combining with the MGPCG method, an efficient linear solver for problems with spatially varying composite materials is developed. Further, the Node-by-Node (NbN) and Element-by-Element (EbE) strategies for sparse matrix–vector multiplication (SpMV) computation and two strategies for handling the elasticity matrices are implemented and compared to determine their effectiveness. Additionally, decoupled sensitivity analysis and design variables updating methods are introduced to further enhance the efficiency of topology optimization. The effectiveness of the proposed method is demonstrated through numerical examples using both consumer-grade and high-end GPU cards. It is shown that topology optimization problems with 33.6 million elements and 100.8 million design variables can be solved with 150 iterations in approximately 2.9 h using a single NVIDIA Tesla V100 GPU.