Nonlocal operator method with numerical integration for gradient solid

authored by: Huilong Ren, Xiaoying Zhuang, Timon Rabczuk
Abstract: The nonlocal operator method (NOM) is initially proposed as a particle-based method, which has difficulties in imposing accurately the boundary conditions of various orders. In this paper, we converted the particle-based NOM into a scheme with approximation property. The new scheme describes partial derivatives of various orders at a point by the nodes in the support and takes advantage of the background mesh for numerical integration. The boundary conditions are enforced via the modified variational principle. The particle-based NOM can be viewed as a special case of NOM with approximation property when nodal integration is used. The scheme based on numerical integration greatly improves the stability of the method. As a consequence, the requirement of the operator energy functional in particle-based NOM is avoided. We demonstrate the capabilities of the proposed method by solving gradient elasticity problems and comparing the numerical results with exact solutions.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Bauhaus-Universität Weimar
Tongji University
Ton Duc Thang University
Type: Article
Journal: Computers and Structures
Volume: 233
ISSN: 0045-7949
Publication date: 06.2020
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Civil and Structural Engineering, Modelling and Simulation, Materials Science(all), Mechanical Engineering, Computer Science Applications
Electronic version(s): https://doi.org/10.1016/j.compstruc.2020.106235 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{877f9e35aefc4174bd178d490feefa74,
title = "Nonlocal operator method with numerical integration for gradient solid",
abstract = "The nonlocal operator method (NOM) is initially proposed as a particle-based method, which has difficulties in imposing accurately the boundary conditions of various orders. In this paper, we converted the particle-based NOM into a scheme with approximation property. The new scheme describes partial derivatives of various orders at a point by the nodes in the support and takes advantage of the background mesh for numerical integration. The boundary conditions are enforced via the modified variational principle. The particle-based NOM can be viewed as a special case of NOM with approximation property when nodal integration is used. The scheme based on numerical integration greatly improves the stability of the method. As a consequence, the requirement of the operator energy functional in particle-based NOM is avoided. We demonstrate the capabilities of the proposed method by solving gradient elasticity problems and comparing the numerical results with exact solutions.",
keywords = "Gradient solid, Modified variational principle, Nonlocal operator method, Numerical integration, Peridynamics",
author = "Huilong Ren and Xiaoying Zhuang and Timon Rabczuk",
note = "Funding Information: The first author acknowledges the support from the COMBAT Program ( Computational Modeling and Design of Lithium-ion Batteries , Grant No. 615132 ). ",
year = "2020",
month = jun,
doi = "10.1016/j.compstruc.2020.106235",
language = "English",
volume = "233",
journal = "Computers and Structures",
issn = "0045-7949",
publisher = "Elsevier Ltd.",
}