一种物理上一致的、灵活的、有效的将局部边界条件转换为非局部体积约束的策略
A physically-consistent, flexible and efficient strategy to convert local boundary conditions into nonlocal volume constraints
摘要：
非局部模型为广泛的科学和工程应用提供了卓越的仿真保真度。然而，非局部模型的广泛部署受到几个建模和数值挑战的阻碍。其中，我们专注于非平凡处方的非局部边界条件，或体积约束，必须提供一个层周围的域的非局部方程构成。挑战来自于这样一个事实，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，我们将介绍一种高效，灵活和物理上一致的技术，用于将表面（局部）数据自动转换为体积数据，该体积数据对域的几何形状和非局部解的规律性没有任何约束，并且不与任何离散化相关联。我们表明，我们的配方是适定性和非局部解的极限，作为非局部性消失，是当地的解决方案，对应于现有的表面数据。在强能量和L-2收敛条件下证明了算法的平方收敛速率。我们用一维数值试验说明了理论，其结果为现实的模拟提供了基础。
英文摘要：
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 and 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 ground work for realistic simulations.
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