Higher Order Far-Field Boundary Conditions for Crystalline Defects
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Lattice defects in crystalline materials create long-range elastic fields that can be modeled on the atomistic scale. The low rank structures of defect configurations are revealed by a rigorous far-field expansion of the long-range elastic fields, and thus the defect equilibrium can be expressed as a sum of continuum correctors and discrete multipole terms that are essentially computable. In this paper, we develop a novel family of numerical schemes that exploit the multipole expansions to accelerate the simulation of crystalline defects. In particular, the relatively slow convergence rate of the standard cell approximations for defect equilibration could be significantly developed. To enclose the simulation in a finite domain, a theoretically justified approximation of multipole tensors is therefore introduced, which leads to a novel moment iteration as well as the higher order boundary conditions. Moreover, we consider a continuous version of multipole expansions to acquire efficiency in practical implementation. Several prototypical numerical examples of point defects are presented to test the convergence for both geometry error and energy error. The numerical results show that our proposed numerical scheme can achieve the accelerated convergence rates in terms of computational cell size with the higher order boundary conditions.
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