Efficient Local-to-Nonlocal Coupling Method for Weak Forms

Core Concepts

The authors propose a splice method that couples local and nonlocal diffusion models using weak (variational) formulations, enabling computational efficiency, unintrusiveness, and desirable properties such as patch tests and asymptotic compatibility.

Abstract

The authors present a method to efficiently couple local and nonlocal diffusion models using weak (variational) formulations. The key highlights are:

The proposed splice method inherits desirable properties from related splice and optimization-based coupling schemes, such as patch tests, asymptotic compatibility, and unintrusiveness.
The method enables the use of finite element discretizations for both local and nonlocal problems, providing a flexible analysis framework.
Compared to optimization-based coupling, the splice method significantly reduces computational cost as it does not require an iterative minimization procedure.
The authors prove well-posedness of the coupling scheme and demonstrate its effectiveness through various numerical examples in 1D and 2D.
The method can be extended to vector-valued descriptions such as peridynamics.
The authors consider two families of kernel functions: fractional kernels and integrable kernels with singularity.
The authors discuss the discretization of local and nonlocal problems using continuous piecewise linear finite elements, and the coupling of these discretizations.
The authors interpret the splice method as a special case of an optimization-based coupling approach, which allows them to leverage existing results on well-posedness and patch test properties.
Numerical results demonstrate that the splice method passes patch tests and converges to the local model as the nonlocal horizon vanishes.

Stats

The authors use the following key metrics and figures to support their work:

Fractional kernel function (2) with fractional order s and horizon δ
Integrable kernel function (4) with singularity strength α and horizon δ
Computational domains ΩL, ΩN, ΩN,I, ΩN,e
I and their corresponding meshes TL, TN, TN,I, TN,e
I
Finite element spaces Vh,L, Vh,N and their corresponding degrees of freedom IL, IN, IΓ, IN,I, IN,e
I
Restriction operators RL, RN, RΓ, RN,I

Quotes

"The increased flexibility encapsulated in the kernel allows one to capture effects that classical models using partial differential equations cannot reproduce in general, without the use of multiscale coefficients."
"Commonly used nonlocal descriptions are diffusion operators [2] of the form Lu(x) = ∫(u(y) - u(x))γ(x, y) dy, with kernel γ and x, y ∈ Rd."
"Local-to-Nonlocal (LtN) coupling aims to combine a nonlocal model posed on a sub-region of the computational domain with a local model that is prescribed on the complement."

Key Insights Distilled From

A Splice Method for Local-to-Nonlocal Coupling of Weak Forms

by Shuai Jiang,... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2404.13744.pdf

Deeper Inquiries

How can the proposed splice method be extended to vector-valued nonlocal models, such as peridynamics, and what are the key considerations in the extension

The proposed splice method can be extended to vector-valued nonlocal models, such as peridynamics, by considering each component of the vector separately. In the context of peridynamics, which describes material behavior at the mesoscale, the displacement field is a vector quantity representing the deformation of the material.
To extend the splice method to handle vector-valued nonlocal models, the key considerations include:

Discretization: Each component of the vector field needs to be discretized separately using appropriate finite element spaces. This ensures that the splice method can be applied to each component individually.
Coupling: The coupling between the local and nonlocal components of the vector field should be done in a consistent manner to maintain the integrity of the overall vector field. This may involve modifying the splice matrix to account for vector quantities.
Boundary Conditions: Ensuring that the boundary conditions are appropriately handled for vector-valued fields is crucial. The splice method should be adapted to accommodate vector boundary conditions and interactions at the boundaries of the local and nonlocal domains.
Asymptotic Compatibility: Extending the splice method to vector-valued models requires verifying that the method maintains asymptotic compatibility as the nonlocal effects diminish. This ensures that the method converges to the local model as the interaction horizon approaches zero for each component of the vector field.

By addressing these considerations, the splice method can be effectively extended to handle vector-valued nonlocal models like peridynamics, enabling the coupling of different components of the vector field across local and nonlocal domains.

Can efficient preconditioners be designed for the non-symmetric splice coupling matrix to enable the use of Krylov methods like GMRES or BiCGStab

Efficient preconditioners can be designed for the non-symmetric splice coupling matrix to enable the use of Krylov methods like GMRES or BiCGStab. Preconditioning is essential for accelerating the convergence of iterative solvers when dealing with non-symmetric matrices like the splice coupling matrix.
Key steps in designing efficient preconditioners for the non-symmetric splice coupling matrix include:

Block Preconditioning: Partitioning the splice matrix into blocks corresponding to the local and nonlocal components can help in designing effective block preconditioners. By preconditioning each block separately, the overall convergence of the iterative solver can be improved.
Approximate Inverse Preconditioning: Constructing an approximate inverse of the splice matrix can be an effective preconditioning strategy. This involves approximating the inverse of the matrix using techniques like incomplete factorization or algebraic multigrid.
Domain Decomposition Preconditioners: Utilizing domain decomposition techniques can also be beneficial for preconditioning the non-symmetric splice matrix. By decomposing the problem into subdomains and applying preconditioners locally, the overall convergence can be enhanced.
Adaptive Preconditioning: Adaptive preconditioning techniques, where the preconditioner is adjusted based on the convergence behavior of the iterative solver, can further improve the efficiency of solving the non-symmetric system.

By implementing these strategies, efficient preconditioners can be designed for the non-symmetric splice coupling matrix, enabling the use of Krylov methods like GMRES or BiCGStab for faster and more reliable solutions.

What are the potential applications of the local-to-nonlocal coupling approach beyond diffusion problems, and how would the method need to be adapted to handle different types of physics

The local-to-nonlocal coupling approach has potential applications beyond diffusion problems in various fields of physics and engineering. To adapt the method for different types of physics, the following considerations need to be taken into account:

Mechanical Systems: In the context of mechanical systems, the coupling approach can be applied to problems involving elasticity, plasticity, and fracture mechanics. The method would need to be adapted to handle stress-strain relationships, material properties, and crack propagation in such systems.

Fluid Dynamics: For fluid dynamics applications, the coupling approach can be used to model nonlocal interactions in flows, turbulence, and multiphase flows. Adapting the method to handle Navier-Stokes equations, boundary conditions, and fluid properties is essential in this context.

Electromagnetics: In electromagnetic problems, the coupling approach can be utilized for nonlocal interactions in wave propagation, antenna design, and electromagnetic compatibility studies. Adapting the method to handle Maxwell's equations, boundary conditions, and material properties specific to electromagnetics is crucial.

Multi-Physics Simulations: The method can be extended to handle multi-physics problems where different types of physics govern different parts of the domain. This involves developing coupling schemes that can effectively integrate local and nonlocal models across different physics domains.

By adapting the local-to-nonlocal coupling approach to these diverse applications, researchers and engineers can address a wide range of complex problems in physics and engineering, enabling more accurate and efficient simulations across various disciplines.