The bond-based peridynamic system with Dirichlet-type volume constraint

Mengesha, Tadele; Du, Qiang

doi:10.1017/s0308210512001436

Cited by 116 publications

(111 citation statements)

References 27 publications

Supporting

Mentioning

103

Contrasting

Unclassified

Order By: Relevance

“…Thus, our result also recovers the analogue of the compactness result in [10] for standard fractional spaces which can be applicable to other studies of variational problems associated with fractional PDEs [28]. Finally, we refer to [37,38,43] for more discussions on applications and generalizations of the compactness result of [10] to nonlocal problems involving spaces of either scalar-valued or vector-valued functions.…”

Section: A New Compactness Resultsupporting

confidence: 78%

“…which is equivalent to the original norm · 2c S as demonstrated by the Poincaré inequality given later (see also [37,38]). We use (·, ·) S to denote the equivalent inner product induced by the norm · S , that is, …”

Section: Model Equationmentioning

confidence: 80%

“…The natural solution space is the Hilbert space S given by the closure of C ∞ c (Ω ) iñ S. It is a dense subspace of L(Ω ) containing H 1 0 (Ω ) when restrictions on Ω are considered [37,38]. Let · S denote the seminorm in S defined by…”

Section: Model Equationmentioning

confidence: 99%

“…First, we will apply a nonlocal Poincaré inequality just for n = 1 that has already been developed in the literature [22,38], that is, there exists C 1 > 0 such that u 2 L 2 ≤ C 1 u 2 S1 ∀u ∈ S 1 . Now by the definition of the modified kernels, we know that γ n is a sequence of nonnegative kernels that is increasing in n, which means that u S1 ≤ u Sn for every n ≥ 1.…”

Section: Properties Of Energy Spaces and Bilinear Formsmentioning

confidence: 99%

See 3 more Smart Citations

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Tian¹,

Du²

2015

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

We present a nonconforming discontinuous Galerkin finite element scheme for nonlocal variational problems associated with some linear nonlocal diffusion and nonlocal peridynamic operators subject to volumetric constraints. The nonlocal operators under consideration have nonlocal interaction kernels that exhibit singularities at the origin so that the natural energy spaces do not allow conforming discontinuous finite element functions. The key idea in our method is based on the introduction of modified nonlocal interaction kernels near the origin. It works for multidimensional problems defined on bounded domains partitioned with general meshes and approximations using general basis functions. A convergence theory is established for two types of finite dimensional spaces, with the first type including finite element spaces that lead to unconditional convergence if they contain all continuous piecewise linear functions and the other type being the discontinuous piecewise constant function space that is shown to be conditionally convergent. The analysis is based on the recently developed framework of asymptotically compatible schemes for parametrized variational problems. It also relies crucially on a successful extension of a compactness result of Bourgain, Brezis, and Mironescu to function sequences associated with a sequence of kernels that is assumed to be convergent to a more general limiting kernel. Numerical experiments are carried out to offer additional observations on the performance of our proposed method.1. Introduction. This paper serves as a first attempt to develop discontinuous and nonconforming Galerkin finite element discretization of nonlocal diffusion (ND) and nonlocal peridynamic models defined on a bounded spatial domain in any given space dimension. Since first introduced by Silling in [45], peridynamics (PD), which is an integral-type nonlocal continuum theory, has demonstrated effectiveness in modeling material singularities [4,32,47,48]. As pointed out in [21], linear scalar PD operators also share similarities with ND operators, thus making the study of PD relevant to the study of ND models, as well as fractional diffusion models, in various applications [2,11,13,19,22,27,29,30,31,33,36,40]. Along with research efforts on the mathematical analysis of PD/ND models, a variety of numerical methods have been studied and implemented, including finite difference, finite element, quadrature, and particle based methods [1,8,14,24,25,32,35,42,44,46,50,51,52,53,54,56,57]. In terms of Galerkin type approximations, most of the existing analyses have focused on the conforming ones and very few studies have been carried out for nonconforming discontinuous Galerkin (DG) finite element discretizations.

show abstract

Section: A New Compactness Resultsupporting

confidence: 78%

Section: Model Equationmentioning

confidence: 80%

Section: Model Equationmentioning

confidence: 99%

Section: Properties Of Energy Spaces and Bilinear Formsmentioning

confidence: 99%

See 2 more Smart Citations

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Tian¹,

Du²

2015

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, adapting the argument used in the proof of [14, Lemma 2.3], we can actually show that S(R d ; k) is a separable Hilbert space with inner product (·, ·) L 2 + [·, ·] S(R d ;k) . See also similar results in [12,25,26]. We denote the dual space of S(R d ; k) by S * (R d ; k).…”

Section: Introductionmentioning

confidence: 83%

Solvability of nonlocal systems related to peridynamics

Kaßmann¹,

Mengesha²,

Scott³

2019

Communications on Pure &Amp; Applied Analysis

Self Cite

View full text Add to dashboard Cite

In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lamé system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to nonsymmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces.

show abstract

A well‐posed parameter identification for nonlocal diffusion problems

Zhang

Nie

2022

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper concerns the problem of identifying the diffusion parameter in the nonlocal diffusion model. The parameter identification problem is formulated as an optimal control problem, and the objective of the control is a cost functional formulated by the energy functional. By using the nonlocal vector calculus, in a manner analogous to the local partial differential equations counterpart, we proved that the cost functional is a strictly convex functional with a unique global minimizer. Moreover, the existence and uniqueness of parameter identification are further demonstrated. Finally, one-dimensional numerical experiments are given to illustrate our theoretical results and show that continuous and discontinuous parameters can be estimated.

show abstract

The bond-based peridynamic system with Dirichlet-type volume constraint

Cited by 116 publications

References 27 publications

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Solvability of nonlocal systems related to peridynamics

A well‐posed parameter identification for nonlocal diffusion problems

Contact Info

Product

Resources

About