SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A1935-A1949, January 2020.
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.

中文翻译：

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将本地边界条件转换为非本地体积约束的物理一致，灵活且高效的策略

SIAM科学计算杂志，第42卷，第4期，第A1935-A1949页，2020年1月。
非本地模型为广泛的科学和工程应用提供了出色的仿真保真度。但是，一些建模和数值挑战阻碍了非局部模型的广泛部署。其中，我们关注非局部边界条件或体积约束的非平凡处方，必须在构成非局部方程的区域周围的一层上提供非局部边界条件。挑战来自这样一个事实，即通常以力或压力数据的形式在表面（而不是体积）上提供数据。在本文中，我们介绍了一种高效，灵活，用于将表面（局部）数据自动转换为体积数据的物理一致技术，该技术对域的几何形状或非局部解的规律性没有任何限制，并且不依赖于任何离散化。我们证明了我们的公式是正确的，随着非局部性的消失，非局部解的极限是对应于可用表面数据的局部解。对于强能量和$ L ^ 2 $收敛，证明了二次收敛速率。我们用一维数值测试说明了该理论，其结果为现实模拟提供了基础。是对应于可用表面数据的局部解。对于强能量和$ L ^ 2 $收敛，证明了二次收敛速率。我们用一维数值测试说明了该理论，其结果为现实模拟提供了基础。是对应于可用表面数据的局部解。对于强能量和$ L ^ 2 $收敛，证明了二次收敛速率。我们用一维数值测试说明了该理论，其结果为现实模拟提供了基础。