An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics

Trask, Nathaniel; You, Huaiqian; Yu, Yue; Parks, Michael L.

doi:10.1016/j.cma.2018.08.016

An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics

Journal Article · Wed Sep 05 00:00:00 EDT 2018 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2018.08.016· OSTI ID:1474082

Trask, Nathaniel ^[1]; You, Huaiqian ^[2]; Yu, Yue ^[2]; Parks, Michael L. ^[1]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
Lehigh Univ., Bethlehem, PA (United States). Dept. of Mathematics

In this paper, we present a meshfree quadrature rule for compactly supported nonlocal integro-differential equations (IDEs) with radial kernels. We apply this rule to develop a meshfree discretization of a peridynamic solid mechanics model that requires no background mesh. Existing discretizations of peridynamic models have been shown to exhibit a lack of asymptotic compatibility to the corresponding linearly elastic local solution. By posing the quadrature rule as an equality constrained least squares problem, we obtain asymptotically compatible convergence by introducing polynomial reproduction constraints. Our approach naturally handles traction-free conditions, surface effects, and damage modeling for both static and dynamic problems. We demonstrate high-order convergence to the local theory by comparing to manufactured solutions and to cases with crack singularities for which an analytic solution is available. We verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff–Winkler experiments.

View Accepted Manuscript (DOE)

Research Organization:: Lehigh Univ., Bethlehem, PA (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: National Science Foundation (NSF) (United States); SNL Laboratory Directed Research and Development (LDRD) Program; USDOE; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

Grant/Contract Number:: NA0003525; SC0009247

OSTI ID:: 1474082

Alternate ID(s):: OSTI ID: 1636024

Report Number(s):: SAND--2018-10211J; 667966

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 343; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (21)

Surface corrections for peridynamic models in elasticity and fracture Le, Q. V.; Bobaru, F. Computational Mechanics, Vol. 61, Issue 4 https://doi.org/10.1007/s00466-017-1469-1	journal	August 2017
A meshfree method based on the peridynamic model of solid mechanics Silling, S. A.; Askari, E. Computers & Structures, Vol. 83, Issue 17-18 https://doi.org/10.1016/j.compstruc.2004.11.026	journal	June 2005
Convergence, adaptive refinement, and scaling in 1D peridynamics Bobaru, Florin; Yang, Mijia; Alves, Leonardo Frota International Journal for Numerical Methods in Engineering, Vol. 77, Issue 6 https://doi.org/10.1002/nme.2439	journal	February 2009
Dual-horizon peridynamics: Dual-horizon peridynamics Ren, Huilong; Zhuang, Xiaoying; Cai, Yongchang International Journal for Numerical Methods in Engineering, Vol. 108, Issue 12 https://doi.org/10.1002/nme.5257	journal	July 2016
Mass- and Momentum Conservation of the Least-Squares Spectral Element Method for the Stokes Problem Proot, Michael M. J.; Gerritsma, Marc I. Journal of Scientific Computing, Vol. 27, Issue 1-3 https://doi.org/10.1007/s10915-005-9030-3	journal	December 2005
Reformulation of elasticity theory for discontinuities and long-range forces Silling, S. A. Journal of the Mechanics and Physics of Solids, Vol. 48, Issue 1 https://doi.org/10.1016/S0022-5096(99)00029-0	journal	January 2000
Direct Meshless Local Petrov–Galerkin (DMLPG) method: A generalized MLS approximation Mirzaei, Davoud; Schaback, Robert Applied Numerical Mathematics, Vol. 68 https://doi.org/10.1016/j.apnum.2013.01.002	journal	June 2013
Convergence studies in meshfree peridynamic simulations Seleson, Pablo; Littlewood, David J. Computers & Mathematics with Applications, Vol. 71, Issue 11 https://doi.org/10.1016/j.camwa.2015.12.021	journal	June 2016
A scalable consistent second-order SPH solver for unsteady low Reynolds number flows Trask, Nathaniel; Maxey, Martin; Kim, Kyungjoo Computer Methods in Applied Mechanics and Engineering, Vol. 289 https://doi.org/10.1016/j.cma.2014.12.027	journal	June 2015
Implementing peridynamics within a molecular dynamics code Parks, Michael L.; Lehoucq, Richard B.; Plimpton, Steven J. Computer Physics Communications, Vol. 179, Issue 11 https://doi.org/10.1016/j.cpc.2008.06.011	journal	December 2008
Peridynamics via finite element analysis Macek, Richard W.; Silling, Stewart A. Finite Elements in Analysis and Design, Vol. 43, Issue 15 https://doi.org/10.1016/j.finel.2007.08.012	journal	November 2007
CRKSPH – A Conservative Reproducing Kernel Smoothed Particle Hydrodynamics Scheme Frontiere, Nicholas; Raskin, Cody D.; Owen, J. Michael Journal of Computational Physics, Vol. 332 https://doi.org/10.1016/j.jcp.2016.12.004	journal	March 2017
How to Integrate A Polynomial Over A Sphere Folland, Gerald B. The American Mathematical Monthly, Vol. 108, Issue 5 https://doi.org/10.1080/00029890.2001.11919774	journal	May 2001
On generalized moving least squares and diffuse derivatives Mirzaei, D.; Schaback, R.; Dehghan, M. IMA Journal of Numerical Analysis, Vol. 32, Issue 3 https://doi.org/10.1093/imanum/drr030	journal	September 2011
A New Approach for a Nonlocal, Nonlinear Conservation Law Du, Qiang; Kamm, James R.; Lehoucq, R. B. SIAM Journal on Applied Mathematics, Vol. 72, Issue 1 https://doi.org/10.1137/110833233	journal	January 2012
Asymptotically Compatible Schemes and Applications to Robust Discretization of Nonlocal Models Tian, Xiaochuan; Du, Qiang SIAM Journal on Numerical Analysis, Vol. 52, Issue 4 https://doi.org/10.1137/130942644	journal	January 2014
A High-Order Staggered Meshless Method for Elliptic Problems Trask, Nathaniel; Perego, Mauro; Bochev, Pavel SIAM Journal on Scientific Computing, Vol. 39, Issue 2 https://doi.org/10.1137/16M1055992	journal	January 2017
Theoretical Analysis of the No-Slip Boundary Condition Enforcement in SPH Methods Macia, F.; Antuono, M.; Gonzalez, L. M. Progress of Theoretical Physics, Vol. 125, Issue 6 https://doi.org/10.1143/PTP.125.1091	journal	June 2011
Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity Emmrich, Etienne; Weckner, Olaf Mathematics and Mechanics of Solids, Vol. 12, Issue 4 https://doi.org/10.1177/1081286505059748	journal	November 2005
A position-aware linear solid constitutive model for peridynamics Mitchell, John; Silling, Stewart; Littlewood, David Journal of Mechanics of Materials and Structures, Vol. 10, Issue 5 https://doi.org/10.2140/jomms.2015.10.539	journal	January 2015
On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity Emmrich, Etienne; Weckner, Olaf Communications in Mathematical Sciences, Vol. 5, Issue 4 https://doi.org/10.4310/CMS.2007.v5.n4.a6	journal	January 2007

Cited By (2)

Asymptotically compatible meshfree discretization of state-based peridynamics for linearly elastic composite materials Trask, Nathaniel; Huntington, Benjamin; Littlewood, David arXiv https://doi.org/10.48550/arxiv.1903.00383	preprint	January 2019
Peridynamic Modeling of Frictional Contact Kamensky, David; Behzadinasab, Masoud; Foster, John T. Journal of Peridynamics and Nonlocal Modeling, Vol. 1, Issue 2 https://doi.org/10.1007/s42102-019-00012-y	journal	April 2019

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asymptotic compatibility
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An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics

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