Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

Leng, Yu; Tian, Xiaochuan; Trask, Nathaniel A.; Foster, John T.

doi:10.1016/j.cma.2020.113264

Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

Journal Article · Sat Jul 18 00:00:00 UTC 2020 · Computer Methods in Applied Mechanics and Engineering

DOI:https://doi.org/10.1016/j.cma.2020.113264· OSTI ID:1670723

Leng, Yu ^[1]; Tian, Xiaochuan ^[2]; Trask, Nathaniel A. ^[3]; Foster, John T. ^[1]

Univ. of Texas, Austin, TX (United States). Dept. of Petroleum and Geosystems Engineering
Univ. of Texas, Austin, TX (United States). Dept. of Mathematics
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research

Here, we study reproducing kernel (RK) collocation method for peridynamic Navier equation. In the first part, we apply a linear RK approximation to both displacement and dilatation, and then back-substitute dilatation and solve the peridynamic Navier equation in a pure displacement form. The RK collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish. The stability is shown by comparing the collocation scheme with the standard Galerkin scheme using Fourier analysis. In the second part, we apply the RK collocation to the quasi-discrete peridynamic Navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed. The analysis is carried out on a special family of rectilinear Cartesian grids for the RK collocation method with a designated kernel with finite support. We assume the Lamé parameters satisfy

λ \geq μ

to avoid extra assumptions on the nonlocal kernel. Finally, numerical experiments are conducted to validate the theoretical results.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: AC04-94AL85000; NA0003525

OSTI ID:: 1670723

Alternate ID(s):: OSTI ID: 1639112

Report Number(s):: SAND--2020-0004J; 681958

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Vol. 370; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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