Coupled Peridynamics Least Square Minimization with Finite Element Method in 3D and Implicit Solutions by Message Passing Interface

In this work, we present a new framework to couple peridynamics least squares minimization with finite element method (PDLSM-FEM) for three-dimensional (3D) problems based on the weighted residual method (WRM). In PDLSM-FEM, the problem domain is divided into a PD domain and a FEM domain, and both domains are discretized into uniform or non-uniform elements. Within this framework, the coupling between the PD domain and the FEM domain is straightforward, and the formulation can be converted conveniently between PDLSM, FEM, and PDLSM-FEM. To directly and implicitly solve for displacements, we present an algorithm to assemble the global stiffness matrix in a compressed sparse row (CSR) format by the message passing interface (MPI) technique. The system equations are solved by the parallel direct sparse solver (PDSS) of the Intel Math Kernel Library (MKL). Three static examples are performed and the results show the proposed PDLSM-FEM can substantially reduce computational cost in terms of model efficiency and memory usage. Comparison between PDLSM-FEM results and FEM results using ANSYS demonstrates the good accuracy of the proposed PDLSM-FEM.