Title: An Ultra-high-speed Reproducing Kernel Particle Method Show Chinese title

Title: 一种超高速再生核粒子方法

Abstract: In this work, the fast-convolving reproducing kernel particle method (FC-RKPM) is introduced. This method is hundreds to millions of times faster than the traditional RKPM for 3D meshfree simulations. In this approach, the meshfree discretizations with RK approximation are expressed in terms of convolution sums. Fast Fourier transform (FFT) is then used to efficiently compute the convolutions. Certain modifications to the domain and shape functions are considered to maintain generality for complex geometries and arbitrary boundary conditions. The new method does not need to identify, store, and loop over the neighbors which is one of the bottleneck of the traditional meshfree methods. As a result, the run-times and memory allocations are independent of the number of neighbors and the shape function support size. As a model problem, the method is laid out for a Galerkin weak form of the Poisson problem with the RK approximation, and is verified in 1D, 2D, and 3D. Tables with run-times and allocated memory are presented to compare the performance of FC-RKPM with the traditional method in 3D. The performance is studied for various node numbers, support size, and approximation degree. All the implementation details and the roadmap for software development are also provided. Application of the new method to nonlinear and explicit problems are briefly discussed as well. Show Chinese abstract

Abstract: 在本工作中，引入了快速卷积再生核粒子方法（FC-RKPM）。该方法在三维无网格模拟中比传统RKPM快数百到数百万倍。在此方法中，基于RK近似的无网格离散化用卷积和来表示。然后使用快速傅里叶变换（FFT）来高效计算卷积。考虑对域和形状函数进行某些修改，以保持复杂几何和任意边界条件的通用性。新方法不需要识别、存储和循环遍历邻居，这是传统无网格方法中的瓶颈之一。因此，运行时间和内存分配与邻居数量和形状函数支撑尺寸无关。作为模型问题，该方法用于具有RK近似的泊松问题的Galerkin弱形式，并在1D、2D和3D中进行了验证。给出了运行时间和分配内存的表格，以比较FC-RKPM与传统方法在3D中的性能。研究了不同节点数、支撑尺寸和近似阶数的性能。还提供了所有实现细节和软件开发路线图。也简要讨论了该方法在非线性和显式问题中的应用。