A new, optimization-based coupling strategy for nonlocal and local diffusion models

The use of nonlocal models in science and engineering applications has been steadily increasing over the past decade. The ability of nonlocal theories to accurately capture effects that are difficult or impossible to represent by local partial differential equation (PDE) models motivates and drives the interest in a wide range of alternative nonlocal models. For instance, nonlocal continuum theories such as peridynamics or physics-based nonlocal elasticity allow interactions at distance without contact and can accurately resolve small scale features such as crack tips and dislocations. However, the improved accuracy of nonlocal models comes at the price of a significant increase in computational costs compared to, e.g., traditional PDEs. In particular, a complete nonlocal simulation remains computationally untenable for many science and engineering applications. Further, nonlocal models generally require the application of volume constraints, which are more difficult to specify in practice than the boundary conditions of the local theory. As a result, many researches have focused attention on the development of various Local-to-Nonlocal coupling strategies, which aim to combine the accuracy of nonlocal models with the computational efficiency of PDEs. The basic idea is to use the more efficient PDE model everywhere except in those parts of the domain that require the improved accuracy of the nonlocal model.

Sandia researchers have recently developed a new, optimization-based method for the coupling of nonlocal and local diffusion problems. The approach formulates the coupling as a 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 nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. The new coupling approach is supported by rigorous mathematical theory establishing the well-posedness of the optimization formulation. Sandia researchers have implemented and demonstrated the optimization coupling approach using Sandia’s agile software components toolkit; the latter provides the groundwork for the development of engineering analysis tools.

Simulation of a bar containing a discontinuity. The discontinuity extends vertically from the top of the bar halfway through the height of the bar. Dirichlet boundary conditions are applied to the ends of the bar (local domain), circumventing the need to prescribe volume constraints in the nonlocal domain.
Simulation of a bar containing a discontinuity. The discontinuity extends vertically from the top of the bar halfway through the height of the bar. Dirichlet boundary conditions are applied to the ends of the bar (local domain), circumventing the need to prescribe volume constraints in the nonlocal domain.
Contact
Marta D'Elia, mdelia@sandia.gov

April 1, 2016

News story url: https://www.sandia.gov/ccr/news/a-new-optimization-based-coupling-strategy-for-nonlocal-and-local-diffusion-models/