A Physically Consistent, Flexible, and Efficient Strategy to Convert Local Boundary Conditions into Nonlocal Volume Constraints
Marta D’Elia, Xiaochuan Tian, Yue Yu
Abstract
Nonlocal models provide exceptional simulation fidelity for a broad spectrum of scientific and engineering applications. However, wider deployment of nonlocal models is hindered by several modeling and numerical challenges. Among those, we focus on the nontrivial prescription of nonlocal boundary conditions, or volume constraints, that must be provided on a layer surrounding the domain where the nonlocal equations are posed. The challenge arises from the fact that, in general, data are provided on surfaces (as opposed to volumes) in the form of force or pressure data. In this paper we introduce an efficient, flexible, and physically consistent technique for an automatic conversion of surface (local) data into volumetric data that does not have any constraints on the geometry of the domain or on the regularity of the nonlocal solution and that is not tied to any discretization. We show that our formulation is well-posed and that the limit of the nonlocal solution, as the nonlocality vanishes, is the local solution corresponding to the available surface data. Quadratic convergence rates are proved for the strong energy and $L^2$ convergence. We illustrate the theory with one-dimensional numerical tests whose results provide the groundwork for realistic simulations.