Abstract:  An efficient multi-grid equation solving method was proposed based on the h-refinement of splines to address the challenges posed by large-scale ITO computation and low efficiency of traditional solving methods. By the proposed method, the weight information obtained through h-refinement interpolation between coarse and fine grids was used to construct the interpolation matrix of the multi-grid method, thereby enhancing the accuracy of mapping information for both coarse and fine grids and improving computational efficiency. Additionally, a comprehensive analysis of the multi-grid solving process was conducted, culminating in the development of an efficient GPU parallel algorithm. Numerical examples illustrate that the proposed method outperforms existing methods, demonstrating speedup ratios of 1.47, 11.12, and 17.02 in comparison to the linear interpolation multi-grid conjugate gradient method algebraic multi-grid conjugate gradient method, and pre-processing conjugate gradient method respectively. Furthermore, the acceleration rate of GPU parallel solution surpasses that of CPU serial solution by 33.86 times, which significantly enhances the efficiency of solving large-scale linear equations.

Key words: isogeometric topology optimization(ITO), system of equations, h-refinement, multi-grid method, GPU parallel computing

摘要： 针对大规模等几何拓扑优化(ITO)计算量巨大、传统求解方法效率低的问题，提出了一种基于样条h细化的高效多重网格方程求解方法。该方法利用h细化插值得到粗细网格之间的权重信息，然后构造多重网格方法的插值矩阵，获得更准确的粗细网格映射信息，从而提高求解速度。此外，对多重网格求解过程进行分析，构建其高效GPU并行算法。数值算例表明，所提出的求解方法与线性插值的多重网格共轭梯度法、代数多重网格共轭梯度法和预处理共轭梯度法相比分别取得了最高1.47、11.12和17.02的加速比。GPU并行求解相对于CPU串行求解的加速比高达33.86，显著提高了大规模线性方程组的求解效率。

关键词: 等几何拓扑优化, 方程组求解, h细化, 多重网格法, GPU并行计算