Analysis of a Sinclair-type domain decomposition solver for atomistic/continuum coupling

TL;DR

This paper extends the Sinclair-type domain decomposition method for atomistic/continuum coupling to bounded domains, introduces a boundary element correction, analyzes convergence, and demonstrates improved stability and efficiency through numerical experiments.

Contribution

The authors generalize the Sinclair method to bounded domains using a boundary element approach and provide a detailed convergence analysis with stability results.

Findings

The method is unconditionally stable in one-dimensional cases.

The boundary element correction effectively enforces far-field conditions.

Numerical examples validate improved convergence and stability.

Abstract

The "flexible boundary condition" method, introduced by Sinclair and coworkers in the 1970s, remains among the most popular methods for simulating isolated two-dimensional crystalline defects, embedded in an effectively infinite atomistic domain. In essence, the method can be characterized as a domain decomposition method which iterates between a local anharmonic and a global harmonic problem, where the latter is solved by means of the lattice Green function of the ideal crystal. This local/global splitting gives rise to tremendously improved convergence rates over related alternating Schwarz methods. In a previous publication (Hodapp et al., 2019, Comput. Methods in Appl. Mech. Eng. 348), we have shown that this method also applies to large-scale three-dimensional problems, possibly involving hundreds of thousands of atoms, using fast summation techniques exploiting the low-rank nature of the asymptotic lattice Green function. Here, we generalize the Sinclair method to bounded domains and develop an implementation using a discrete boundary element method to correct the infinite solution with respect to a prescribed far-field condition, thus preserving the advantage of the original method of not requiring a global spatial discretization. Moreover, we present a detailed convergence analysis and show for a one-dimensional problem that the method is unconditionally stable under physically motivated assumptions. To further improve the convergence behavior, we develop an acceleration technique based on a relaxation of the transmission conditions between the two subproblems. Numerical examples for linear and nonlinear problems are presented to validate the proposed methodology.