A reproducing kernel enhanced approach for peridynamic solutions

Pasetto, Marco; Leng, Yu; Chen, Jiun‐Shyan; Foster, Janet; Seleson, Pablo

doi:10.1016/j.cma.2018.05.010

Cited by 47 publications

(26 citation statements)

References 54 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[15] showed the equivalence between the PD differential operator [24] and the RK implicit gradient operator [11], which both can be employed to obtain more accurate meshfree gradients, such as a deformation gradient of higher-order accuracy. Reproducing kernel enhanced approaches have been proposed to improve meshfree integration in peridynamic models [15,19,28].…”

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

View full text Add to dashboard Cite

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.

show abstract

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

View full text Add to dashboard Cite

show abstract

“…The effect of numerical integration in the non-local integrals has also been a focus in studies, as it may also have a strong effect on convergence in peridynamics [46][47][48][49][50]. In the first meshfree implementation of peridynamics [24], the peridynamic equation of motion was discretized by nodal integration with the full physical nodal volume as the integration weight, resulting in the so-called full volume (FV) integration.…”

Section: Introductionmentioning

confidence: 99%

“…These techniques exhibit firstorder convergence in displacements [46]. The limitation of first-order convergence was attributed in [50] to the piecewise constant nature of the approximation employed, where the reproducing kernel approximation was introduced in the peridynamic displacement field to increase the convergence rate. However, high-order Gauss integration was employed to achieve the integration accuracy necessary to avoid the aforementioned oscillatory convergence behavior, resulting in an increased computational cost.…”

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In a narrow sense, SPH is a numerical solution, kernel approximation method, for differential terms. The differences between PD and SPH are listed as follows: From the mechanical perspectives, the applied disciplines are initially different in PD and SPH, namely solid mechanics and fluid mechanics. The force between two interaction material points is not always equal in state‐based PD, but is definitely equal in SPH. The material constitution law is included in the particle properties in SPH theory rather than is incorporated in the bond in PD theory. The formulation of the traditional PD employs a total Lagrangian formulation which is based on the material coordinate, while the formulation of the traditional SPH employs an updated Lagrangian formulation which is based on the spatial coordinate. In the same time, the PD theory has many similarities with SPH in physical background and numerical implementation as follows: The same physical model, nonlocal model, is employed in these two theories, which both consider the interaction of a finite influence domain. These two theories both employ a Newton's second law and present a dynamic mechanical method. The material property and tracking perspective are both a Lagrangian view which is fixed on material point in PD and SPH. The discretization equation of PD theory is similar to that of SPH when the meshless technology is used to solve PD theory 23–27 …”

Section: Nonlocal Backgroundmentioning

confidence: 99%

Smoothed peridynamics for the extremely large deformation and cracking problems: Unification of peridynamics and smoothed particle hydrodynamics

Zhou

Yao

Berto

2021

Fatigue Fract Eng Mat Struct

View full text Add to dashboard Cite

Peridynamics (PD) and smoothed particle hydrodynamics (SPH) are definitely attractive theories in amounts of numerical mechanical methods these years, and numerous researchers have studied the similarities of these two methods. In this paper, the smoothed peridynamics (SPD) is proposed to unify these two theories in the meshless view. The SPD employs an update‐Lagrangian (UL) method, which is useful for the extremely large deformation and cracking problem. The SPD governing equation, nonlocal interaction, micro‐bond modules for elastic material are derived. In addition, the choice of the nonlocal kernel and logarithmic stretch is investigated. Finally, numerical experiments are studied to confirm the ability of SPD. The numerical results show that SPD has an excellent performance and application prospect in computational solid mechanics. Since the SPD model formulations are derived for general 3D conditions, it can be straight forwardly extended for large‐scale practical applications across disciplines.

show abstract

A reproducing kernel enhanced approach for peridynamic solutions

Cited by 47 publications

References 54 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

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

Smoothed peridynamics for the extremely large deformation and cracking problems: Unification of peridynamics and smoothed particle hydrodynamics

Contact Info

Product

Resources

About