Peridynamic (PD) theory is significant and promising in engineering and materials science; however, it imposes challenges owing to the enormous computational cost caused by its nonlocality. Our main contribution, which overcomes the restrictions of the existing fast method, is a general computational framework for the linear bond-based peridynamic models based on the meshfree method, called the matrix-structure-based fast method (MSBFM), which is suitable for the general case, including 2D/3D problems, and static/dynamic issues, as well as problems with general boundary conditions, in particular, problems with crack propagation. Consequently, we provide a general calculation flow chart. The proposed computational framework is practical and easily embedded into the existing computational algorithm. With this framework, the computational cost is reduced from $O(N^2)$ to $O(N\log N)$, and the storage request is reduced from $O(N^2)$ to $O(N)$, where N is the degree of freedom. Finally, the vast reduction of the computational and memory requirement is verified by numerical examples.

翻译：近场动力学（Peridynamic, PD）理论在工程与材料科学中具有重要意义且前景广阔，然而其非局部特性导致计算成本巨大，带来了严峻挑战。本文的主要贡献在于突破现有快速方法的局限性，提出了一种基于无网格方法的线性键基近场动力学模型的通用计算框架，称为基于矩阵结构的快速方法（Matrix-Structure-Based Fast Method, MSBFM）。该方法适用于一般情形，包括二维/三维问题、静态/动态问题以及一般边界条件问题，特别适用于裂纹扩展问题。为此，我们提供了通用的计算流程图。所提出的计算框架实用性强，且易于嵌入现有计算算法。在该框架下，计算复杂度从$O(N^2)$降至$O(N\log N)$，存储需求从$O(N^2)$降至$O(N)$，其中N为自由度。最后，通过数值算例验证了计算与存储需求的大幅降低。