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

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 higher-order gradients. We show, however, that the improved quadrature rule does not suffice to handle correspondence-modeling instability issues. In Part I of this paper, a bond-associative, higher-order core formulation is developed that naturally provides stability. 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 approaches zero. 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 formulations accompanied by higher-order gradient correction provide the key ingredients to obtain the necessary accuracy, stability, and robustness characteristics needed for engineering-scale simulations.

[1]  Marco Pasetto,et al.  A reproducing kernel enhanced approach for peridynamic solutions , 2018, Computer Methods in Applied Mechanics and Engineering.

[2]  T. Belytschko,et al.  An implicit gradient model by a reproducing kernel strain regularization in strain localization problems , 2004 .

[3]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[4]  Philippe H. Geubelle,et al.  Handbook of Peridynamic Modeling , 2017 .

[5]  S. Silling,et al.  Convergence, adaptive refinement, and scaling in 1D peridynamics , 2009 .

[6]  Masoud Behzadinasab,et al.  Peridynamic modeling of large deformation and ductile fracture , 2019 .

[7]  David John Littlewood,et al.  Roadmap for Peridynamic Software Implementation , 2015 .

[8]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[9]  Xiaochuan Tian,et al.  Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation , 2020, ArXiv.

[10]  Ted Belytschko,et al.  A meshfree unification: reproducing kernel peridynamics , 2014, Computational Mechanics.

[11]  Xiaochuan Tian,et al.  Super-convergence of reproducing kernel approximation , 2019, Computer Methods in Applied Mechanics and Engineering.

[12]  S. Silling Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces , 2000 .

[13]  Xiaochuan Tian,et al.  Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion , 2019, SIAM J. Numer. Anal..

[14]  Erdogan Madenci,et al.  Peridynamic differential operator and its applications , 2016 .

[15]  Yuri Bazilevs,et al.  Treatment of near-incompressibility in meshfree and immersed-particle methods , 2020, Computational Particle Mechanics.

[16]  X. Chen,et al.  Continuous and discontinuous finite element methods for a peridynamics model of mechanics , 2011 .

[17]  James W. Foulk,et al.  The third Sandia fracture challenge: predictions of ductile fracture in additively manufactured metal , 2019, International Journal of Fracture.

[18]  Erdogan Madenci,et al.  Weak form of peridynamics for nonlocal essential and natural boundary conditions , 2018, Computer Methods in Applied Mechanics and Engineering.

[19]  S. Silling Stability of peridynamic correspondence material models and their particle discretizations , 2016 .

[20]  John T. Foster,et al.  A semi-Lagrangian constitutive correspondence framework for peridynamics , 2020 .

[21]  S. Silling,et al.  Peridynamic States and Constitutive Modeling , 2007 .

[22]  David John Littlewood,et al.  Simulation of Dynamic Fracture Using Peridynamics, Finite Element Modeling, and Contact , 2010 .

[23]  Timon Rabczuk,et al.  Dual‐horizon peridynamics , 2015, 1506.05146.

[24]  Hailong Chen,et al.  Bond-associated deformation gradients for peridynamic correspondence model , 2018, Mechanics Research Communications.

[25]  Nathaniel Trask,et al.  Asymptotically compatible meshfree discretization of state-based peridynamics for linearly elastic composite materials , 2019, ArXiv.

[26]  Philippe H. Geubelle,et al.  Non-ordinary state-based peridynamic analysis of stationary crack problems , 2014 .

[27]  Kaushik Dayal,et al.  Bond-level deformation gradients and energy averaging in peridynamics , 2018 .

[28]  S. Silling,et al.  A meshfree method based on the peridynamic model of solid mechanics , 2005 .

[29]  Wing Kam Liu,et al.  Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures , 1996 .

[30]  Nathaniel Trask,et al.  An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics , 2018, Computer Methods in Applied Mechanics and Engineering.

[31]  Raul Radovitzky,et al.  An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures , 2014 .

[32]  Marco Pasetto,et al.  Generalized reproducing kernel peridynamics: unification of local and non-local meshfree methods, non-local derivative operations, and an arbitrary-order state-based peridynamic formulation , 2020, Computational Particle Mechanics.

[33]  Debasish Roy,et al.  A modified peridynamics correspondence principle: Removal of zero-energy deformation and other implications , 2019, Computer Methods in Applied Mechanics and Engineering.

[34]  J. Michell,et al.  On the Direct Determination of Stress in an Elastic Solid, with application to the Theory of Plates , 1899 .

[35]  J. Remacle,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[36]  John T. Foster,et al.  On the stability of the generalized, finite deformation correspondence model of peridynamics , 2019, International Journal of Solids and Structures.

[37]  Li,et al.  Moving least-square reproducing kernel methods (I) Methodology and convergence , 1997 .