A semi-Lagrangian constitutive correspondence framework for peridynamics

Behzadinasab, Masoud; Foster, Janet

doi:10.1016/j.jmps.2019.103862

Cited by 82 publications

(28 citation statements)

References 46 publications

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“…Future work would probe the application of the bond-associated models in solving dynamic problems involving crack growth. The presented approach may be readily implemented in other formulations such as the semi-Lagrangian peridynamics [3]. While our analysis shows robustness of the bond-associated model with respect to the number of neighbors, further study could examine the role of horizon and influence function in the developed PD formulations.…”

Section: Discussionmentioning

confidence: 93%

“…To address the issue, stabilization techniques have been proposed, such as (1) addition of penalty terms to deviations from homogeneous deformations [20,31], which involves artificial (non-physical) parameters, and (2) splitting of the integration domain into multiple sub-regions [13], which is unsettling as a continuum theory. A third group of evolving methods [3,8,9] are called bond-associative models, which are continuum-based approaches and do not involve any numerical stabilization. The original correspondence model includes assigning the same kinematic variable (F) to all the bonds connected to a node, which leads to instability issues.…”

Section: Introductionmentioning

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: Discussionmentioning

confidence: 93%

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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“…Future efforts will utilize the developed framework to solve dynamic problems involving material failure. We will apply the presented methodology to semi-Lagrangian PD [4], which is well-suited to simulate very large deformation problems with fragmentation [2]. This framework will be used to study fluid-structure interaction phenomena, where the surrounding fluid imposes stress boundary conditions on the solid structure that are naturally handled in the proposed framework.…”

Section: Discussionmentioning

confidence: 99%

“…We showed that the higher-order corrections cannot eliminate the wellknown issue of numerical instability in correspondence PD theory, which manifest itself as zero-energy mode oscillations in simulations that involve inhomogeneous deformations [3,8,17,24,27], and may lead to large prediction errors in practical applications [16]. To provide stability for the higher-order framework, we developed a continuum-based, bondassociated model, inspired by [4,9,10], which takes into account non-uniform deformations and maintains stability without additional stabilization mechanisms employed. It was shown that the higher-order, bond-associated model can obtain second-order accuracy for smooth problems and first-order accuracy for non-smooth ones (e.g., involving solution-derivative discontinuity).…”

Section: Introductionmentioning

confidence: 99%

A Unified, Stable, and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part II: Wave Propagation and Enforcement of Stress Boundary Conditions

Behzadinasab

Foster

Bazilevs

2020

J Peridyn Nonlocal Model

Self Cite

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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 statebased correspondence PD framework, and (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. Keywords Peridynamics • Meshfree methods • Natural boundary conditions • Non-local derivatives • Reproducing kernel • GMLS • Bond-associative modeling • Higher order • Wave dispersion

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“…7. Ductile fracture: As of the time of writing this review, not many material models and simulation for ductile fracture were available [506][507][508][509]. Note that ductile tearing is challenging for any numerical method, due to the choice of an appropriate ductile failure model.…”

Section: Development Of Adaptive Pd Models and Methodsmentioning

confidence: 99%

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

et al. 2022

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Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized.

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A semi-Lagrangian constitutive correspondence framework for peridynamics

Cited by 82 publications

References 46 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 Unified, Stable, and Accurate Meshfree Framework for Peridynamic Correspondence Modeling—Part II: Wave Propagation and Enforcement of Stress Boundary Conditions

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

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