A higher order nonlocal operator method for solving partial differential equations

authored by: Huilong Ren, Xiaoying Zhuang, Timon Rabczuk
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.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Bauhaus-Universität Weimar
Tongji University
Ton Duc Thang University
Type: Article
Journal: Computer Methods in Applied Mechanics and Engineering
Volume: 367
No. of pages: 27
ISSN: 0045-7825
Publication date: 01.08.2020
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Mechanics of Materials, Mechanical Engineering, General Physics and Astronomy, Computer Science Applications
Electronic version(s): https://doi.org/10.48550/arXiv.1905.02809 (Access: Open)
https://doi.org/10.1016/j.cma.2020.113132 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@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",
}