Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation

Mengesha, Tadele; Du, Qiang

doi:10.1007/s10659-013-9456-z

Cited by 130 publications

(141 citation statements)

References 21 publications

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“…This minor change is crucial to this study because we now can apply the result to much more general kernels with less singularity at zero. Such observations on more lenient conditions on the kernel have been noticed in the literature [39,43]. Most notably, we point out that [43] provided a very general argument for the original compactness result that works for d ≥ 2 without the assumption on γ being nonincreasing.…”

Section: A New Compactness Resultsupporting

confidence: 78%

“…Similarly, extensions to nonlocal systems may also be feasible as the general framework of [52] on asymptotically compatible schemes has been applied to systems of nonlocal models previously. Such a work would require further extension of the compactness result proved here, much like those given in [37,39] for systems associated with a sequence of kernels approaching Dirac-Delta measures. Connections of the nonconforming DG methods to their local limits as the nonlocal horizon vanishes is another interesting issue both in theory and in practice.…”

Section: Discussionmentioning

confidence: 83%

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Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Tian¹,

Du²

2015

SIAM J. Numer. Anal.

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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.

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Section: A New Compactness Resultsupporting

confidence: 78%

Section: Discussionmentioning

confidence: 83%

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Tian¹,

Du²

2015

SIAM J. Numer. Anal.

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“…where m and ϑ [·] are given by (20) [2] and Mengesha and Du [3]. Similar analysis could be carried out here to show, among other things, that the quadratic free energy function is non-negative for arbitrary h, but it is outside of the scope of this work.…”

Section: Determination Of the Peridynamic Constantsmentioning

confidence: 59%

On the Determination of a Peridynamic Constant in a Linear Constitutive Model

Aguiar

2015

J Elast

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This work is an extension of previous investigation concerning a free energy function for an isotropic, linearly elastic peridynamic material that depends quadratically on infinitesimal normal and shear strain states. The free energy function contains four peridynamic material constants, from which three constants are related to the classical elasticity coefficients of an isotropic linear elastic material, with one of the three constants being arbitrary. To determine this arbitrary constant, the difference displacement quotient state at a point is decomposed in terms of radial and non-radial components. If the radial component is zero, the quadratic free energy function reduces to an integral expression that multiplies the arbitrary constant. This result together with a correspondence argument is used next to find a general expression for this constant. A simple experiment in mechanics is then used to evaluate this constant in terms of the classical shear modulus and the horizon δ. The correspondence argument can also be used to find a general expression for the fourth peridynamic constant that appears in the quadratic free energy function.

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“…17 The convergence of an equilibrium peridynamic model to the Navier equation in the sense of solution operators is established in the work of Mengesha and Du. 18 Numerical analysis of linear peridynamic models for one-dimensional (1D) bars have been given in the work of Bobaru et al 8 and Weckner and Emmrich. 16 Related approximations of nonlocal diffusion models are discussed in the works of Tian et al, 19 Chen and Gunzburger, 20 and Du et al 21 A stability analysis of the numerical approximation to solutions of linear nonlocal wave equations is given in the work of Guan and Gunzburger.…”

Section: Introductionmentioning

confidence: 99%

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

Jha

Lipton

2018

Numerical Meth Engineering

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Summary We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond‐based peridynamic model with a local elasticity model or a linearized peridynamic model away from the fracture set. The nonlocal model treated here is characterized by a double‐well potential and is a smooth version of the peridynamic model introduced in the work of Silling. The nonlinear peridynamic evolutions are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale ε of nonlocal interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation, the consistency error for the numerical approximation is found to depend on the ratio between mesh size h and ε. More generally, for local Lagrange interpolation of order p≥1, the consistency error is of order hp/ε. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size h is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics.

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Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation

Cited by 130 publications

References 21 publications

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

On the Determination of a Peridynamic Constant in a Linear Constitutive Model

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

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