A Physically Consistent, Flexible, and Efficient Strategy to Convert Local Boundary Conditions into Nonlocal Volume Constraints
A Physically Consistent, Flexible, and Efficient Strategy to Convert Local Boundary Conditions into Nonlocal Volume Constraints
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DOI：
10.1137/19m1266617
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发表时间：
2019-06
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通讯作者：
M. D'Elia;Xiaochuan Tian;Yue Yu
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文献类型：
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作者：
M. D'Elia;Xiaochuan Tian;Yue Yu
非局部模型为广泛的科学和工程应用提供了卓越的模拟保真度。然而，非局部模型的更广泛部署受到一些建模和数值挑战的阻碍。其中，我们重点关注非局部边界条件或体积约束的非平凡规定，这些规定必须在非局部方程所在域周围的层上提供。挑战源于这样一个事实：通常，数据以力或压力数据的形式在表面（而不是体积）上提供。在本文中，我们介绍了一种高效、灵活且物理一致的技术，用于将表面（局部）数据自动转换为体积数据，该技术对域的几何形状和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。我们表明，我们的公式是适定的，并且随着非局部性消失，非局部解的极限是与可用表面数据相对应的局部解。证明了强能量收敛和L-2收敛的二次收敛速率。我们用一维数值测试来说明该理论，其结果为实际模拟提供了基础。
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.