Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains

Seleson, Pablo; Gunzburger, Max; Parks, Michael L.

doi:10.1016/j.cma.2013.05.018

Cited by 104 publications

(69 citation statements)

References 38 publications

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“…Therefore, the model presented in this paper is a generalisation of the models proposed in (Silling and Askari 2005;Seleson et al 2013) for anisotropic materials. Figure 3 shows polar plots of ( ) represented with spherical harmonics up to the second degree, eq.…”

Section: Transversely Isotropic Materials Modelmentioning

confidence: 99%

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A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media

Ghajari

Iannucci

Curtis

2014

Computer Methods in Applied Mechanics and Engineering

172

112

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A new material model for the dynamic fracture analysis of anisotropic materials has been proposed within the framework of the bond-based peridynamic theory. This model enables predicting complex fracture phenomena such as spontaneous crack nucleation and crack branching, curving and arrest, a capability inherited from the bond-based peridynamic theory. An important feature of the model is that the bond properties, i.e. the stiffness constant and critical stretch, are continuous functions of bond orientation in the principal material axes. This facilitates fracture analysis of anisotropic materials with random orientations, such as polycrystalline microstructures. Elastic and fracture behaviour of the model has been verified through simulating uniaxial tension of a composite plate and fracture of a cortical bone compact tension specimen, and making quantitative comparisons to analytical and experimental data. To further demonstrate the capabilities of the proposed model, dynamic fracture of a polycrystalline microstructure (alumina ceramic) has been simulated. The influence of the grain boundary and grain interior fracture energies on the interacting and competing fracture modes of polycrystalline materials, i.e. intergranular and transgranular fracture, has been studied.

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Section: Transversely Isotropic Materials Modelmentioning

confidence: 99%

“…In (Seleson et al 2013), a model for anisotropic materials, in the context of nonlocal diffusion, was presented. Following this reference:…”

Section: Transversely Isotropic Materials Modelmentioning

confidence: 99%

A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media

Ghajari

Iannucci

Curtis

2014

Computer Methods in Applied Mechanics and Engineering

172

112

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“…Other nonlocal models in solid mechanics appear in [13][14][15][16][17]. Closely related models to peridynamics appear in nonlocal heat conduction [18,19] and in nonlocal diffusion [20][21][22].…”

Section: Introductionmentioning

confidence: 97%

“…Studies of different meshing strategies, including comparisons between cubic meshes and centroidal Voronoi tessellations, appear in [29]. Discretizations of peridynamic and related nonlocal diffusion models using weak forms of governing equations appear, on the other hand, in [22,30,31], based on continuous and discontinuous Galerkin finite element methods. Related convergence studies for quadrature methods in peridynamics have been discussed in [32].…”

Section: Introductionmentioning

confidence: 99%

Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations

Seleson

2014

Computer Methods in Applied Mechanics and Engineering

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142

107

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“…Various mathematical and engineering studies have addressed linear peridynamics including [3,4,6,9,14,17,18]. For scalar-valued kernels γ and scalar fields u, the operators in (1) have been studied in the context of nonlocal diffusion, digital image correlation, and nonlocal wave phenomena among other applications, see for example [5,7,16,20]. Several mathematical and numerical studies have focused on nonlocal Laplace operators including [2,10,19].…”

Section: Introductionmentioning

confidence: 99%

Fourier multipliers for nonlocal Laplace operators

Alali

Albin

2019

Applicable Analysis

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Fourier multiplier analysis is developed for peridynamic Laplace operators, which are defined for scalar fields in R n . The focus is on operators L δ,β with compactly supported integral kernels of the form χ B δ (0) (x) 1x β , which are commonly used in peridynamic models [6]. The Fourier multipliers m(ν) of L δ,β are given through an integral representation, which is shown to be well-defined for β < n + 2. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function 2 F 3 in any spatial dimension n and for any β < n + 2. The new general formula for the multipliers is well-defined for any β ∈ R \ {n + 4, n + 6, n + 8, . . .} and is utilized to extend the definition of the peridynamic Laplacians to the case when β ≥ n + 2 (with β = n + 4, n + 6, n + 8, . . .). Some special cases are presented. We show that the Fourier multipliers of L δ,β converge to the Fourier multipliers of ∆ in the limit as β → n + 2. In addition, we show that in the limit as β → n + 2 − , L δ,β u converges to ∆u for sufficiently regular u. Moreover, we identify the limit of the Fourier multipliers of L δ,β as β → −∞ and, furthermore, recognize the limit operator L δ,−∞ as an integral operator with kernel supported in the (n − 1)-sphere.Asymptotic analysis of 2 F 3 is utilized to identify the asymptotic behavior of the Fourier multipliers as ν → ∞. We show that the multipliers are bounded when the peridynamic Laplacian has an integrable kernel (i.e., when β < n), diverge to −∞ at a rate proportional to − log( ν ) when β = n, and diverge to −∞ at a rate proportional to − ν β−n when β > n. The bounds and decay rates are presented explicitly in terms of n, β, and the nonlocality δ.In periodic domains, the Fourier multipliers are the eigenvalues of L δ,β and therefore the presented results provide characterization formulas and asymptotic behavior for the eigenvalues. The asymptotic analysis is applied in the periodic setting to prove a regularity result for the peridynamic Poisson equation.

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Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains

Cited by 104 publications

References 38 publications

A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media

A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media

Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations

Fourier multipliers for nonlocal Laplace operators

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