Efficient optimization-based quadrature for variational discretization of nonlocal problems

Pasetto, Marco; Shen, Zhaoxiang; D’Elia, Marta; Tian, Xiaochuan; Trask, Nathaniel; Kamensky, David

doi:10.1016/j.cma.2022.115104

Cited by 3 publications

(3 citation statements)

References 47 publications

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“…We set up some of the symbols used throughout the paper and present the relevant lemmas and results. Before we start to discuss the problem (1), we introduce the strong form of a nonlocal volume-constrained Poisson problem [28,[36][37][38][39]:…”

Section: Nonlocal Peridynamic Differential Operator Problemmentioning

confidence: 99%

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Numerical analysis of a nonlocal Volterra integro-differential equation

Zhu,

Dong,

Zheng

et al. 2024

Preprint

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We study the numerical approximation to a nonlocal Volterra integro-differential equation, in which the integral term is the convolution product of a positive-definite kernel and a nonlocal peridynamic differential operator. Compared with the classical differential operators, the nonlocal peridynamic differential operators describe, e.g., discontinuities, and have domonstrated more widespread applications. The equation is discretized in space by the Galerkin finite element method, and we accordingly prove its error estimate. We then discretize the equation in time by the backward Euler method, and a positive quadrature rule is combined to approximate the convolution term. The convergence rate of the fully-discrete finite element scheme is proved, and numerical experiments are carried out to substantiate the theoretical findings.

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Section: Nonlocal Peridynamic Differential Operator Problemmentioning

confidence: 99%

“…Here, u = u(x, t), u t denotes ∂u/∂t, β is a positive-definite kernel function, i.e., β ∈ L 1,loc (0, T ], and the space L 1,loc (0, T ] is a function space with local integrability on the interval (0, T ]. The nonlocal peridynamic differential operator (PDDO) is given by [27,28]…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a nonlocal Volterra integro-differential equation

Zhu,

Dong,

Zheng

et al. 2024

Preprint

View full text Add to dashboard Cite

show abstract

“…In the present work, the Gaussian quadrature method is used, which has been discussed in detail in [31]. For other numerical quadrature methods, refer to [40].…”

Section: Assembling the Element Equationsmentioning

confidence: 99%

ABAQUS and ANSYS Implementations of the Peridynamics-Based Finite Element Method (PeriFEM) for Brittle Fractures

Han¹,

Li²,

Zhang³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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In this study, we propose the first unified implementation strategy for peridynamics in commercial finite element method (FEM) software packages based on their application programming interface using the peridynamics-based finite element method (PeriFEM). Using ANSYS and ABAQUS as examples, we present the numerical results and implementation details of PeriFEM in commercial FEM software. PeriFEM is a reformulation of the traditional FEM for solving peridynamic equations numerically. It is considered that the non-local features of peridynamics yet possesses the same computational framework as the traditional FEM. Therefore, this implementation benefits from the consistent computational frameworks of both PeriFEM and the traditional FEM. An implicit algorithm is used for both ANSYS and ABAQUS; however, different convergence criteria are adopted owing to their unique features. In ANSYS, APDL enables users to conveniently obtain broken-bond information from UPFs; thus, the convergence criterion is chosen as no new broken bond. In ABAQUS, obtaining broken-bond information is not convenient for users; thus, the default convergence criterion is used in ABAQUS. The codes integrated into ANSYS and ABAQUS are both verified through benchmark examples, and the computational convergence and costs are compared. The results show that, for some specific examples, ABAQUS is more efficient, whereas the convergence criterion adopted in ANSYS is more robust. Finally, 3D examples are presented to demonstrate the ability of the proposed approach to deal with complex engineering problems.

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An Efficient Jacobi Spectral Collocation Method with Nonlocal Quadrature Rules for Multi-Dimensional Volume-Constrained Nonlocal Models

Zhao

Nie

2023

Int. J. Comput. Methods

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In this paper, an efficient Jacobi spectral collocation method is developed for multi-dimensional weakly singular volume-constrained nonlocal models including both nonlocal diffusion (ND) models and peridynamic (PD) models. The model equation contains a weakly singular integral operator with the singularity located at the center of the integral domain, and the numerical approximation of it becomes an essential difficulty in solving nonlocal models. To approximate such singular nonlocal integrals in an accurate way, a novel nonlocal quadrature rule is constructed to accurately compute these integrals for the numerical solutions produced by spectral methods. Numerical experiments are given to show that spectral accuracy can be obtained by using the proposed Jacobi spectral collocation methods for several different nonlocal models. Besides, we numerically verify that the numerical solution of our Jacobi spectral method can converge to its correct local limit as the nonlocal interactions vanish.

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Efficient optimization-based quadrature for variational discretization of nonlocal problems

Cited by 3 publications

References 47 publications

Numerical analysis of a nonlocal Volterra integro-differential equation

Numerical analysis of a nonlocal Volterra integro-differential equation

ABAQUS and ANSYS Implementations of the Peridynamics-Based Finite Element Method (PeriFEM) for Brittle Fractures

An Efficient Jacobi Spectral Collocation Method with Nonlocal Quadrature Rules for Multi-Dimensional Volume-Constrained Nonlocal Models

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