Helmholtz-Hodge Decompositions in the Nonlocal Framework

D’Elia, Marta; Flores, Cynthia; Li, Xingjie; Radu, Petronela; Yu, Yue

doi:10.1007/s42102-020-00035-w

Cited by 12 publications

(3 citation statements)

References 27 publications

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“…A nonlocal vector calculus has been developed (Gunzburger and Lehoucq 2010, Du et al 2012a, Du et al 2013a, Alali, Gunzburger and Liu 2015, Mengesha and Du 2016, Du 2019, D'Elia, Flores, Li, Radu and Yu 2019a to deal with nonlocal models such as (1.5) in much the same way as the classical vector calculus is used to deal with PDE models such as (1.1). Here, mostly following (Du et al 2013a), we provide a brief introduction into the nonlocal vector calculus, including notions that are used in the rest of the article.…”

Section: A Brief Review Of a Nonlocal Vector Calculusmentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

D'Elia,

Du,

Glusa

et al. 2020

Preprint

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Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to adequately model observed phenomena or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modeling and algorithmic extensions which serve to show the wide applicability of nonlocal modeling.

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Section: A Brief Review Of a Nonlocal Vector Calculusmentioning

confidence: 99%

Numerical methods for nonlocal and fractional models

D'Elia,

Du,

Glusa

et al. 2020

Preprint

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“…While the integral form allows one to capture multiscale behavior and discontinuities, it also poses theoretical and numerical challenges. Theoretical challenges include the lack of a complete nonlocal theory [12,18,20], the nontrivial treatment of nonlocal interfaces [2,8,55,29] and nonlocal boundary conditions [24,26,31,69,71], which must be prescribed in a volumetric region surrounding the domain of interest to guarantee the uniqueness of the solution. Computational challenges are related to the integral form that may require sophisticated quadrature rules, yielding discretized systems whose matrices are dense or even full.…”

Section: Introductionmentioning

confidence: 99%

An optimization-based strategy for peridynamic-FEM coupling and for the prescription of nonlocal boundary conditions

D’Elia¹,

Littlewood²,

Trageser³

et al. 2021

Preprint

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We develop and analyze an optimization-based method for the coupling of a static peridynamic (PD) model and a static classical elasticity model. The approach formulates the coupling as a control problem in which the states are the solutions of the PD and classical equations, the objective is to minimize their mismatch on an overlap of the PD and classical domains, and the controls are virtual volume constraints and boundary conditions applied at the local-nonlocal interface. Our numerical tests performed on three-dimensional geometries illustrate the consistency and accuracy of our method, its numerical convergence, and its applicability to realistic engineering geometries. We demonstrate the coupling strategy as a means to reduce computational expense by confining the nonlocal model to a subdomain of interest, and as a means to transmit local (e.g., traction) boundary conditions applied at a surface to a nonlocal model in the bulk of the domain.

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“…Despite their improved accuracy, the usability of nonlocal equations is hindered by several modeling and computational challenges that are the subject of very active research. Modeling challenges include the lack of a unified and complete nonlocal theory [13,18,20], the nontrivial treatment of nonlocal interfaces [2,10,45,28,54,57] and the non-intuitive prescription of nonlocal boundary conditions [24,53,49,58,30]. Computational challenges are due to the integral nature of nonlocal operators that yields discretization matrices that feature a much larger bandwidth compared to the sparse matrices associated with PDEs.…”

mentioning

confidence: 99%

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

D’Elia¹,

Yu²

2021

Preprint

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We introduce a technique to automatically convert local boundary conditions into nonlocal volume constraints for nonlocal Poisson's and peridynamic models. The proposed strategy is based on the approximation of nonlocal Dirichlet or Neumann data with a local solution obtained by using available boundary, local data. The corresponding nonlocal solution converges quadratically to the local solution as the nonlocal horizon vanishes, making the proposed technique asymptotically compatible.The proposed conversion method does not have any geometry or dimensionality constraints and its computational cost is negligible, compared to the numerical solution of the nonlocal equation. The consistency of the method and its quadratic convergence with respect to the horizon is illustrated by several two-dimensional numerical experiments conducted by meshfree discretization for both the Poisson's problem and the linear peridynamic solid model.

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Helmholtz-Hodge Decompositions in the Nonlocal Framework

Cited by 12 publications

References 27 publications

Numerical methods for nonlocal and fractional models

Numerical methods for nonlocal and fractional models

An optimization-based strategy for peridynamic-FEM coupling and for the prescription of nonlocal boundary conditions

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

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