A partitioned coupling framework for peridynamics and classical theory: Analysis and simulations

Abstract

We develop and analyze a concurrent framework for coupling peridynamics and the corresponding classical elasticity theory, with applications to the numerical simulations of damage problems. In this framework, the peridynamic model and the elastic model are solved separately and coupled with a partitioned approach. In the region where material failure is expected to initiate, we employ the peridynamic theory. In the rest of the problem domain, the material is modeled by the classical elasticity theory. On the peridynamic–classical theory interface, there is a transition region where the two subdomains overlap. The two solvers communicate by exchanging proper boundary conditions at the peridynamic–classical theory interface, which enables a modular software implementation. We analyze different coupling strategies on a 1D simplified problem and obtain expressions for the optimal reduction factor (convergence rate index). The selection of optimal coupling parameters is verified with numerical experiments, where we demonstrate that the optimal Robin coefficient from 1D simplified problem analysis can be extrapolated to more complicated problems, including cases with damage. Both the analysis and the numerical results suggest that the optimal Robin boundary condition on the classical theory side combined with a Dirichlet boundary condition with Aitken relaxation rule on the peridynamic side would be the most robust choice. Comparing with the commonly employed Dirichlet interface conditions, the optimal Robin boundary condition together with Aitken relaxation accelerates the coupling convergence rate by 10 times. With the developed optimal coupling strategy, we also numerically demonstrate the coupling framework’s asymptotic convergence to the local solution and its capability to capture crack initiation and growth in 2D problems.