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Advanced Numerical Methods in the Mathematical Sciences

Pavel Bochev, Sandia National Laboratories
Optimization-Based Coupling of Local and Non-Local Continuum Models: A Divide and Conquer Approach for Stable and Physically Consistent Heterogeneous Numerical Models

Abstract

We formulate and analyze an optimization-based method for coupling of local continuum models such as Partial Differential Equation (PDE), with nonlocal continuum descriptions in which interactions can occur at distance, without contact. Examples of the latter include nonlocal continuum mechanics theories such as peridynamics or physics-based nonlocal elasticity, which can model pervasive material failure and fracture and can also result from homogenization of nonlinear damage models.

The purpose of such Local-to-Nonlocal (LtN) couplings is to combine the computational efficiency of PDEs with the accuracy of nonlocal models, which can incorporate strong nonlocal effects due to long-range forces at the mesoscale or microscale. The need for local-nonlocal couplings is especially acute when the size of the computational domain is such that the nonlocal solution becomes prohibitively expensive to compute, yet the nonlocal model is required to accurately resolve small scale features such as crack tips or dislocations that can affect the global material behavior.

The main idea of the presented approach is to couch the coupling of the two models into an optimal control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the local and nonlocal problem domains and the controls are the nonlocal volume constraint and the local boundary condition. We present the method in the context of local-to-nonlocal diffusion coupling. Numerical examples in one-dimension illustrate the theoretical properties of the approach.