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

Journal Article · · Computer Methods in Applied Mechanics and Engineering
 [1];  [2];  [2];  [1]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
  2. Lehigh Univ., Bethlehem, PA (United States). Dept. of Mathematics
+ Show Author Affiliations

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.

Research Organization:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Lehigh Univ., Bethlehem, PA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); SNL Laboratory Directed Research and Development (LDRD) Program; National Science Foundation (NSF)
Grant/Contract Number:
NA0003525; SC0009247; DMS 1620434
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, Vol. 343; ISSN 0045-7825
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

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Cited By (2)

Peridynamic Modeling of Frictional Contact journal April 2019
Asymptotically compatible meshfree discretization of state-based peridynamics for linearly elastic composite materials preprint January 2019

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Related Subjects

42 ENGINEERING
nonlocal
peridynamics
meshfree
asymptotic compatibility