[Paper Review] A physically-consistent, flexible and efficient strategy to convert local boundary conditions into nonlocal volume constraints

Marta D’Elia, Xiaochuan Tian+1 more|arXiv (Cornell University)|Jun 10, 2019

Numerical methods in engineering25 references4 citations

TL;DR

This paper presents a physically consistent, flexible, and computationally efficient method to convert surface-based local boundary conditions into volumetric nonlocal volume constraints for nonlocal models. The approach ensures second-order convergence in both energy and $L^2$ norms as nonlocality vanishes, without geometric, dimensional, or regularity constraints, and requires only a single nonlocal solve per problem instance.

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 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.

Motivation & Objective

• To address the challenge of prescribing nonlocal volume constraints when only surface data (e.g., force or pressure) are available in nonlocal modeling.
• To develop a method that is physically consistent, computationally efficient, and applicable to arbitrary geometries and dimensions.
• To ensure convergence of nonlocal solutions to the corresponding local PDE solutions as the horizon parameter vanishes.
• To achieve second-order convergence in energy and $L^2$ norms without requiring high regularity of the nonlocal solution.
• To provide a framework that is independent of spatial discretization and can be implemented with standard PDE and nonlocal solvers as black boxes.

Proposed method

• The method introduces a volumetric extension of surface boundary data using a modified body force distribution within the nonlocal horizon layer.
• It formulates the nonlocal problem with volume constraints derived from surface data via a physically motivated transformation that preserves the asymptotic limit to the local PDE.
• The approach uses a variational formulation to ensure well-posedness and stability, with the volume constraint derived from a consistent lifting of surface data into the domain.
• The method is implemented using a Galerkin finite element discretization, with the nonlocal operator defined via a kernel function with bounded support.
• It enables the use of standard nonlocal and PDE solvers as black boxes, avoiding complex coupling or optimization procedures.
• The formulation is shown to be well-posed and to converge to the local solution in the limit of vanishing nonlocality.

Experimental results

Research questions

• RQ1 Can surface-based boundary data be converted into physically consistent nonlocal volume constraints without geometric or dimensional restrictions?
• RQ2 Does the proposed method achieve second-order convergence in energy and $L^2$ norms as the nonlocal horizon shrinks to zero?
• RQ3 Is the nonlocal solution obtained via this method consistent with the corresponding local PDE solution in the limit of vanishing nonlocality?
• RQ4 Can the method be applied to non-smooth or irregular domains without requiring high regularity of the nonlocal solution?
• RQ5 Does the method maintain computational efficiency while ensuring accuracy and physical consistency?

Key findings

• The proposed method achieves second-order convergence in the energy norm and $L^2$ norm as the nonlocal horizon parameter $\varepsilon$ tends to zero, under the assumption that the local solution is in $C^4$.
• Numerical results show quadratic convergence rates for both the Neumann and Dirichlet approaches, with observed convergence orders close to 2.0 in energy and $L^2$ norms.
• The Dirichlet-based approach yields lower errors than the Neumann-based approach across all tested $\varepsilon$ and $h$ values, indicating superior accuracy.
• The method is robust across different discretization levels $h$ and horizon sizes $\varepsilon$, with convergence maintained even when $h$ and $\varepsilon$ are simultaneously refined as $h = \varepsilon^2$ or $h = \varepsilon/4$.
• The formulation is well-posed and guarantees that the nonlocal solution converges to the local PDE solution in the limit $\varepsilon \to 0$, ensuring asymptotic consistency.
• The computational cost is equivalent to a single nonlocal simulation, making the method highly efficient compared to optimization-based alternatives.