Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

Aksoylu, Burak; Beyer, Horst R.; Celiker, Fatih

doi:10.1080/01630563.2017.1320674

Cited by 21 publications

(43 citation statements)

References 29 publications

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“…1.1a-1.1d and 1.2a-1.2d fall into the class of initial boundary value problems. We have studied the above nonlocal wave equations with homogeneous local BC in prior work [1,2,7]. We extended the treatment to inhomogeneous BC in a preliminary study [8].…”

Section: Introductionmentioning

confidence: 99%

“…We studied various aspects of local BC in nonlocal problems [1-7, 9, 11]. Building on [11], we generalized the results in R to a bounded domain [1,2], a critical feature for all practical applications. In [2], we laid the theoretical foundations and in [1], we applied the foundations to numerically solve wave propagation problems using local BC.…”

Section: Introductionmentioning

confidence: 99%

“…Building on [11], we generalized the results in R to a bounded domain [1,2], a critical feature for all practical applications. In [2], we laid the theoretical foundations and in [1], we applied the foundations to numerically solve wave propagation problems using local BC. In [4], we constructed the first 1D operators that agree with the original bond-based peridynamic operator in the bulk of the domain and simultaneously enforce local Neumann or Dirichlet BC which we denote by M N and M D , respectively.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], we constructed the first 1D operators that agree with the original bond-based peridynamic operator in the bulk of the domain and simultaneously enforce local Neumann or Dirichlet BC which we denote by M N and M D , respectively. We carried out numerical experiments by utilizing M N and M D as governing operators in [1]. We extended the operators to higher dimensions in [7].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Nonlocal Problems with Inhomogeneous Local Boundary Conditions

Aksoylu

Gazonas

2020

J Peridyn Nonlocal Model

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This work aims to provide a comprehensive treatment on how to enforce inhomogeneous local boundary conditions (BC) in nonlocal problems in 1D. In prior work, we have presented novel governing operators with homogeneous BC. Here, we extend the construction to inhomogeneous BC. The construction of the operators is inspired by peridynamics. The operators agree with the original peridynamic operator in the bulk of the domain and simultaneously enforce local Dirichlet or Neumann BC. We explain methodically how to construct forcing functions to enforce local BC and their relationship to initial values. We present exact solutions with both homogeneous and inhomogeneous BC and utilize the resulting error to verify numerical experiments. We explain the critical role of the Hilbert-Schmidt property in enforcing local BC rigorously. For the Neumann BC, we prescribe an interpolation strategy to find the appropriate value of the forcing function from its derivative. We also present numerical experiments with unknown solution and report the computed displacement and strain fields.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Nonlocal Problems with Inhomogeneous Local Boundary Conditions

Aksoylu

Gazonas

2020

J Peridyn Nonlocal Model

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“…In addition, the ability to enforce local BC makes these operators attractive because our approach will avoid altogether the surface effects seen in peridynamics. In the work of Aksoylu et al, the theoretical foundations were laid, and in the work of Aksoylu et al, the foundations were applied to prominent BC such as Dirichlet and Neumann, and the numerical implementation of the corresponding wave propagation was presented. The novel operators were extended to higher dimensions .…”

Section: Introductionmentioning

confidence: 99%

On smoothers for multigrid of the second kind

Aksoylu

Kaya

2019

Numerical Linear Algebra App

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Summary We study smoothers for the multigrid method of the second kind arising from Fredholm integral equations. Our model problems use nonlocal governing operators that enforce local boundary conditions. For discretization, we utilize the Nyström method with the trapezoidal rule. We find the eigenvalues of matrices associated to periodic, antiperiodic, and Dirichlet problems in terms of the nonlocality parameter and mesh size. Knowing explicitly the spectrum of the matrices enables us to analyze the behavior of smoothers. Although spectral analyses exist for finding effective smoothers for 1D elliptic model problems, to the best of our knowledge, a guiding spectral analysis is not available for smoothers of a multigrid of the second kind. We fill this gap in the literature. The Picard iteration has been the default smoother for a multigrid of the second kind. Jacobi‐like methods have not been considered as viable options. We propose two strategies. The first one focuses on the most oscillatory mode and aims to damp it effectively. For this choice, we show that weighted‐Jacobi relaxation is equivalent to the Picard iteration. The second strategy focuses on the set of oscillatory modes and aims to damp them as quickly as possible, simultaneously. Although the Picard iteration is an effective smoother for model nonlocal problems under consideration, we show that it is possible to find better than ones using the second strategy. We also shed some light on internal mechanism of the Picard iteration and provide an example where the Picard iteration cannot be used as a smoother.

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A novel and effective way to impose boundary conditions and to mitigate the surface effect in state‐based Peridynamics

Scabbia

Zaccariotto

Galvanetto

2021

Numerical Meth Engineering

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Peridynamics is a nonlocal continuum theory capable of modeling effectively crack initiation and propagation in solid bodies. However, the nonlocal nature of this theory is the cause of two main problems near the boundary of the body: an undesired stiffness fluctuation, the so-called surface effect, and the difficulty of defining a rational method to properly impose the boundary conditions. The surface effect is analyzed analytically and numerically in the present paper in a state-based peridynamic model. The authors propose a modified fictitious node method based on an extrapolation with a truncated Taylor series expansion. Furthermore, a rational procedure to impose the boundary conditions is defined with the aid of the fictitious nodes. In particular, Neumann boundary conditions are implemented via the peridynamic concept of force flux. The accuracy of the proposed method is assessed by means of several numerical examples for a state-based peridynamic model: with respect to the peridynamic model adopting no corrections, the results are significantly improved even if low values of the truncation order for the Taylor expansion are chosen.

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Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems

Cited by 21 publications

References 29 publications

On Nonlocal Problems with Inhomogeneous Local Boundary Conditions

On Nonlocal Problems with Inhomogeneous Local Boundary Conditions

On smoothers for multigrid of the second kind

A novel and effective way to impose boundary conditions and to mitigate the surface effect in state‐based Peridynamics

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