近场动力学最小二乘和有限元耦合方法研究

准确高效地对损伤和断裂问题进行建模是计算力学中的关键研究课题之一。将近场动力学最小二乘在处理含裂纹等非连续问题上的优势和有限元计算效率高及便于施加边界条件的优势结合，提出了近场动力学最小二乘和有限元耦合方法。将裂纹及其可能扩展区域划分为近场动力学区域，边界及其他区域划分为有限元区域，并将其中的结点类型分为近场动力学结点和有限元结点。有限元结点仅与同单元中的其他结点产生作用，近场动力学结点则与其族内的所有结点产生作用。将以上的单元刚度矩阵和质量矩阵进行组装得到整体刚度矩阵和整体质量矩阵。本文的耦合方法数值实现简单有效，相对于键基和常规态基近场动力学，该耦合方法包含了应力和应变的概念，同时不受零能模式的影响。一维和二维静态和动态问题的研究，验证了本文的耦合方法的有效性和准确性。

Study on the coupling method of peridynamic least square minimization and finite element method

Accurately and efficiently modeling the damage and fracture problem is one of the key research topics in computational mechanics.A coupling method is proposed in this paper that combines the advantages of peridynamic least squares minimization (PDLSM) and finite element method (FEM) to deal with discontinuous problems to impose the boundary conditions easily.To this end,the crack and its possible propagation region are divided into PD region.The boundary and other regions are divided into FEM region as well as the node types.FEM nodes only interact with other nodes in the same element,while PD nodes interact with all nodes in its family.The global stiffness matrix and mass matrix can then be obtained.The coupling method includes the concepts of stress and strain,and has no zero-energy modes,compared with traditional PD methods.Several numerical examples are used to validate the accuracy and efficiency of the proposed coupling method.