A higher order nonlocal operator method for solving partial differential equations

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Autoren

  • Huilong Ren
  • Xiaoying Zhuang
  • Timon Rabczuk

Organisationseinheiten

Externe Organisationen

  • Bauhaus-Universität Weimar
  • Tongji University
  • Ton Duc Thang University
Forschungs-netzwerk anzeigen

Details

OriginalspracheEnglisch
Aufsatznummer113132
Seitenumfang27
FachzeitschriftComputer Methods in Applied Mechanics and Engineering
Jahrgang367
Frühes Online-Datum27 Mai 2020
PublikationsstatusVeröffentlicht - 1 Aug. 2020

Abstract

A higher order nonlocal operator method for the solution of boundary value problems is developed. The proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method (Ren et al., 2020) which only ensures first-order convergence. Furthermore, it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions. Only the functionals based on the nonlocal operators (termed as operator functional) are needed to obtain the final discrete system of equations, which significantly facilitates the implementation. Several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the Poisson equation in 2–5 dimensional space, Kirchhoff and von Kármán plate problems, incompressible elastic materials as well as phase field modeling of fracture.

ASJC Scopus Sachgebiete

Zitieren

A higher order nonlocal operator method for solving partial differential equations. / Ren, Huilong; Zhuang, Xiaoying; Rabczuk, Timon.
in: Computer Methods in Applied Mechanics and Engineering, Jahrgang 367, 113132, 01.08.2020.

Publikation: Beitrag in FachzeitschriftArtikelForschungPeer-Review

Ren H, Zhuang X, Rabczuk T. A higher order nonlocal operator method for solving partial differential equations. Computer Methods in Applied Mechanics and Engineering. 2020 Aug 1;367:113132. Epub 2020 Mai 27. doi: 10.48550/arXiv.1905.02809, 10.1016/j.cma.2020.113132
Download
@article{40a97b02afe742ec9c679c4690343962,
title = "A higher order nonlocal operator method for solving partial differential equations",
abstract = "A higher order nonlocal operator method for the solution of boundary value problems is developed. The proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method (Ren et al., 2020) which only ensures first-order convergence. Furthermore, it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions. Only the functionals based on the nonlocal operators (termed as operator functional) are needed to obtain the final discrete system of equations, which significantly facilitates the implementation. Several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the Poisson equation in 2–5 dimensional space, Kirchhoff and von K{\'a}rm{\'a}n plate problems, incompressible elastic materials as well as phase field modeling of fracture.",
keywords = "Higher order nonlocal operators, Operator energy functional, PDEs, Strong form",
author = "Huilong Ren and Xiaoying Zhuang and Timon Rabczuk",
note = "Funding Information: The supports from National Basic Research Program of China (973 Program: 2011CB013800 ) and NSFC ( 51474157 ), the Ministry of Science and Technology of China (Grant No. SLDRCE14-B-28 , SLDRCE14-B-31 ) are acknowledged. ",
year = "2020",
month = aug,
day = "1",
doi = "10.48550/arXiv.1905.02809",
language = "English",
volume = "367",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",

}

Download

TY - JOUR

T1 - A higher order nonlocal operator method for solving partial differential equations

AU - Ren, Huilong

AU - Zhuang, Xiaoying

AU - Rabczuk, Timon

N1 - Funding Information: The supports from National Basic Research Program of China (973 Program: 2011CB013800 ) and NSFC ( 51474157 ), the Ministry of Science and Technology of China (Grant No. SLDRCE14-B-28 , SLDRCE14-B-31 ) are acknowledged.

PY - 2020/8/1

Y1 - 2020/8/1

N2 - A higher order nonlocal operator method for the solution of boundary value problems is developed. The proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method (Ren et al., 2020) which only ensures first-order convergence. Furthermore, it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions. Only the functionals based on the nonlocal operators (termed as operator functional) are needed to obtain the final discrete system of equations, which significantly facilitates the implementation. Several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the Poisson equation in 2–5 dimensional space, Kirchhoff and von Kármán plate problems, incompressible elastic materials as well as phase field modeling of fracture.

AB - A higher order nonlocal operator method for the solution of boundary value problems is developed. The proposed higher order nonlocal operator brings several advantages as compared to the original nonlocal operator method (Ren et al., 2020) which only ensures first-order convergence. Furthermore, it can be applied to directly and efficiently obtain all partial derivatives of higher orders simultaneously without the need of using shape functions. Only the functionals based on the nonlocal operators (termed as operator functional) are needed to obtain the final discrete system of equations, which significantly facilitates the implementation. Several numerical examples are presented to show the effectiveness and accuracy of the proposed higher order nonlocal operator method including the solution of the Poisson equation in 2–5 dimensional space, Kirchhoff and von Kármán plate problems, incompressible elastic materials as well as phase field modeling of fracture.

KW - Higher order nonlocal operators

KW - Operator energy functional

KW - PDEs

KW - Strong form

UR - http://www.scopus.com/inward/record.url?scp=85085257298&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1905.02809

DO - 10.48550/arXiv.1905.02809

M3 - Article

VL - 367

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 113132

ER -