Weak form of peridynamics for nonlocal essential and natural boundary conditions

Madenci, Erdogan; Dördüncü, Mehmet; Barut, A. O.; Phan, Nam

doi:10.1016/j.cma.2018.03.038

Cited by 90 publications

(33 citation statements)

References 16 publications

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“…Since the dependence of the non-local deformation gradients, i.e., Eqs. (16), (25), and (28), on displacement is linear, the following constitutive model is constructed for consistency with linear elastostatics:…”

Section: Numerical Examplesmentioning

confidence: 99%

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A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods

Behzadinasab

Trask

Bazilevs

2020

J Peridyn Nonlocal Model

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The overarching goal of this work is to develop an accurate, robust, and stable methodology for finite deformation modeling using strong-form peridynamics (PD) and the correspondence modeling framework. We adopt recently developed methods that make use of higher-order corrections to improve the computation of integrals in the correspondence formulation. A unified approach is presented that incorporates the reproducing kernel (RK) and generalized moving least square (GMLS) approximations in PD to obtain non-local gradients of higher-order accuracy. We show, however, that the improved quadrature rule does not suffice to handle instability issues that have proven problematic for the correspondence model-based PD. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability without introducing artificial stabilization parameters. Numerical examples are provided to study the convergence of RK-PD, GMLS-PD, and their bond-associated versions to a local counterpart, as the degree of non-locality (i.e., the horizon) approaches zero. Problems from linear elastostatics are utilized to verify the accuracy and stability of our approach. It is shown that the bond-associative approach improves the robustness of RK-PD and GMLS-PD formulations, which is essential for practical applications. The higher-order, bond-associated model can obtain second-order convergence for smooth problems and first-order convergence for problems involving field discontinuities, such as curvilinear free surfaces. In Part II of this paper we use our unified PD framework to (a) study wave propagation phenomena, which have proven problematic for the state-based correspondence PD framework; (b) propose a new methodology to enforce natural boundary conditions in correspondence PD formulations, which should be particularly appealing to coupled problems. Our results indicate that bond-associative methods accompanied by higher-order gradient corrections provide the key ingredients to obtain the necessary accuracy, stability, and robustness characteristics needed for engineering-scale simulations.

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Section: Numerical Examplesmentioning

confidence: 99%

“…While Galerkin formulations can be developed for peridynamics (see, e.g., [12,25]), the weak-form approach involves a six-dimensional (i.e., double) integral [6,21], entailing considerable geometric complication and computational cost. Strong-form strategies are appealing, therefore, for practical applications.…”

Section: Introductionmentioning

confidence: 99%

A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods

Behzadinasab

Trask

Bazilevs

2020

J Peridyn Nonlocal Model

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“…-Boundary conditions: One of the major challenges pertains to the treatment of boundary conditions when dealing with non-local models [37,50,85,86]. This is an issue that one also finds in molecular dynamics simulations or in smooth particles hydrodynamics.…”

Section: Concluding Remarks and Perspectivesmentioning

confidence: 99%

A Review of Benchmark Experiments for the Validation of Peridynamics Models

Diehl

Prudhomme

Lévesque

2019

J Peridyn Nonlocal Model

106

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Peridynamics (PD), a non-local generalization of classical continuum mechanics (CCM) allowing for discontinuous displacement fields, provides an attractive framework for the modeling and simulation of fracture mechanics applications. However, PD introduces new model parameters, such as the so-called horizon parameter. The length scale of the horizon is a priori unknown and need to be identified. Moreover, the treatment of the boundary conditions is also problematic due to the non-local nature of PD models. It has thus become crucial to calibrate the new PD parameters and assess the model adequacy based on experimental observations. The objective of the present paper is to review and catalog available experimental setups that have been used to date for the calibration and validation of peridynamics. We have identified and analyzed a total of 39 publications that compare PD-based simulation results with experimental data. In particular, we have systematically reported, whenever possible, either the relative error or the R-squared coefficient. The best correlations were obtained in the case of experiments involving aluminum and steel materials. Experiments based on imaging techniques were also considered. However, images provide large amounts of information and their comparison with simulations is in that case far from trivial. A total of six publications have been identified and summarized that introduce numerical techniques for extracting additional attributes from peridynamics simulations in order to facilitate the comparison against image-based data.

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“…In addition, the action is considered over a finite distance, embedding a length scale in the governing equations, resulting in a formulation which is non-local in nature. While the peridynamic equations can be solved under the Galerkin formulation [21,22], this approach requires evaluation of a double (six-dimensional) integral [23], which results in considerable computational expense. Therefore, for practical applications, the strong form version is often employed.…”

Section: Introductionmentioning

confidence: 99%

Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation

2019

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State-based peridynamics is a non-local reformulation of solid mechanics that replaces the force density of the divergence of stress with an integral of the action of force states on bonds local to a given position, which precludes differentiation with the aim to model strong discontinuities effortlessly. A popular implementation is a meshfree formulation where the integral is discretized by quadrature points, which results in a series of unknowns at the points under the strong-form collocation framework. In this work, the meshfree discretization of state-based peridynamics under the correspondence principle is examined and compared to traditional meshfree methods based on the classical local formulation of solid mechanics. It is first shown that the way in which the peridynamic formulation approximates differentiation can be unified with the implicit gradient approximation, and this is termed the reproducing kernel peridynamic approximation. This allows the construction of non-local deformation gradients with arbitrary-order accuracy, as well as non-local approximations of higher-order derivatives. A high-order accurate non-local divergence of stress is then proposed to replace the force density in the original state-based peridynamics, in order to obtain global arbitrary-order accuracy in the numerical solution. These two operators used in conjunction with one another is termed the reproducing kernel peridynamic method. The strong-form collocation version of the method is tested against benchmark solutions to examine and verify the highorder accuracy and convergence properties of the method. The method is shown to exhibit superconvergent behavior in the nodal collocation setting.

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Weak form of peridynamics for nonlocal essential and natural boundary conditions

Cited by 90 publications

References 16 publications

A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods

A Unified, Stable and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part I: Core Methods

A Review of Benchmark Experiments for the Validation of Peridynamics Models

Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation

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