Numerical Analysis Seminar

Date: April 18, 2018

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Marta D’Elia, Sandia National Laboratories

Title: An optimization-based coupling strategy for local and nonlocal elasticity problems

Abstract: Nonlocal continuum theories such as peridynamics and nonlocal elasticity can capture strong nonlocal effects due to long-range forces at the mesoscale or microscale. For problems where these effects cannot be neglected, nonlocal models are more accurate than classical Partial Differential Equations (PDEs) that only consider interactions due to contact. However, the improved accuracy of nonlocal models comes at the price of a computational cost that is significantly higher than that of PDEs. The goal of Local-to-Nonlocal (LtN) coupling methods is to combine the computational efficiency of PDEs with the accuracy of nonlocal models. LtN couplings are imperative when the size of the computational domain or the extent of the nonlocal interactions are such that the nonlocal solution becomes prohibitively expensive to compute, yet the nonlocal model is required to accurately resolve small scale features. We propose an optimization-based coupling strategy for the solution of a nonlocal elasticity problem. Our 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. We present the implementation of our coupling strategy using Sandia's agile software components toolkit, which provides the groundwork for the development of engineering analysis tools. We show that our method passes linear and quadratic patch tests and we present numerical convergence studies. Using three-dimensional geometries, we also show that our approach can be successfully applied to challenging, realistic, problems.