On the prescription of boundary conditions for nonlocal Poisson's and peridynamics models

D’Elia, Marta; Yu, Yue

doi:10.48550/arxiv.2107.04450

Cited by 3 publications

(4 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this work, without loss of generality, we consider the Dirichlet condition B I = I, where I is the identity operator. Other types of conditions, e.g., Neumann [40,42,46,47], Robin [43,48] or periodic [32], are also compatible with our learning algorithm.…”

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 86%

A data-driven peridynamic continuum model for upscaling molecular dynamics

You,

Yu,

Silling

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Nonlocal models, including peridynamics, often use integral operators that embed lengthscales in their definition. However, the integrands in these operators are difficult to define from the data that are typically available for a given physical system, such as laboratory mechanical property tests. In contrast, molecular dynamics (MD) does not require these integrands, but it suffers from computational limitations in the length and time scales it can address. To combine the strengths of both methods and to obtain a coarse-grained, homogenized continuum model that efficiently and accurately captures materials' behavior, we propose a learning framework to extract, from MD data, an optimal Linear Peridynamic Solid (LPS) model as a surrogate for MD displacements. To maximize the accuracy of the learnt model we allow the peridynamic influence function to be partially negative, while preserving the well-posedness of the resulting model. To achieve this, we provide sufficient well-posedness conditions for discretized LPS models with sign-changing influence functions and develop a constrained optimization algorithm that minimizes the equation residual while enforcing such solvability conditions. This framework guarantees that the resulting model is mathematically well-posed, physically consistent, and that it generalizes well to settings that are different from the ones used during training. We illustrate the efficacy of the proposed approach with several numerical tests for single layer graphene. Our two-dimensional tests show the robustness of the proposed algorithm on validation data sets that include thermal noise, different domain shapes and external loadings, and discretizations substantially different from the ones used for training.

show abstract

Section: The Linear Peridynamic Solid (Lps) Modelmentioning

confidence: 86%

A data-driven peridynamic continuum model for upscaling molecular dynamics

You,

Yu,

Silling

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Here, the second condition is the nonlocal counterpart of a Dirichlet boundary condition; differently from the local case, it is prescribed on a volume and it is referred to as a Dirichlet volume constraint. Neumann constraints have also been considered in the literature [19,20]; however, in this work we only consider the Dirichlet case as the prescription of Neumann conditions is not germane to the paper. We note that the last two conditions in (4) couple the equations for u 1 and u 2 and represent jump conditions on the solutions and fluxes; we refer to them as nonlocal interface conditions.…”

Section: Strong Formmentioning

confidence: 99%

“…we use the constant kernel γ 1D,C 1 in Ω 1 , and the fractional kernel γ 1D,F 2 in Ω 2 . Similarly to the previous examples, the source terms and the flux jumps are prescribed so that the analytic solution is again given by (20).…”

Section: Numerical Illustrations and H-convergencementioning

confidence: 99%

An asymptotically compatible coupling formulation for nonlocal interface problems with jumps

Glusa¹,

D’Elia²,

Capodaglio³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce a mathematically rigorous formulation for a nonlocal interface problem with jumps and propose an asymptotically compatible finite element discretization for the weak form of the interface problem. After proving the well-posedness of the weak form, we demonstrate that solutions to the nonlocal interface problem converge to the corresponding local counterpart when the nonlocal data are appropriately prescribed. Several numerical tests in one and two dimensions show the applicability of our technique, its numerical convergence to exact nonlocal solutions, its convergence to the local limit when the horizons vanish, and its robustness with respect to the patch test.

show abstract

“…The latter is generally not given a priori. The strategy adopted in [9] is to first solve for a family of local problems parameterized by δ subject to the local boundary data on the outer rim of the δ-layer, and then to use the computed solutions as the volumetric data for the nonlocal problems. Instead of relying on the information about local solutions beyond the given boundary data, we propose to incorporate suitable nonlocal gradient operators into the nonlocal model to mimic the extrapolation of the boundary data to the volumetric data.…”

mentioning

confidence: 99%

Second order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Lee¹,

Du²

2021

Preprint

View full text Add to dashboard Cite

We present an approach to handle Dirichlet type nonlocal boundary conditions for nonlocal diffusion models with a finite range of nonlocal interactions. Our approach utilizes a linear extrapolation of prescribed boundary data. A novelty is, instead of using local gradients of the boundary data that are not available a priori, we incorporate nonlocal gradient operators into the formulation to generalize the finite differences-based methods which are pervasive in literature; our particular choice of the nonlocal gradient operators is based on the interplay between a constant kernel function and the geometry of nonlocal interaction neighborhoods. Such an approach can be potentially useful to address similar issues in peridynamics, smoothed particle hydrodynamics and other nonlocal models. We first show the well-posedness of the newly formulated nonlocal problems and then analyze their asymptotic convergence to the local limit as the nonlocality parameter shrinks to zero. We justify the second order localization rate, which is the optimal order attainable in the absence of physical boundaries.

show abstract

On the prescription of boundary conditions for nonlocal Poisson's and peridynamics models

Cited by 3 publications

References 52 publications

A data-driven peridynamic continuum model for upscaling molecular dynamics

A data-driven peridynamic continuum model for upscaling molecular dynamics

An asymptotically compatible coupling formulation for nonlocal interface problems with jumps

Second order accurate Dirichlet boundary conditions for linear nonlocal diffusion problems

Contact Info

Product

Resources

About