Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems

authored by: Bing Bing Xu, Xiao Wei Gao
Abstract: In this article, a novel weak-form zonal Petrov–Galerkin free element method is proposed for two- and three-dimensional linear mechanical problems. By absorbing the advantages of finite block method and strong-form finite element method, the block mapping technique is used in the free element method. Combining the characteristics of the meshless local Petrov–Galerkin method, the local Petrov–Galerkin formulation based on the zonal free element method is formed at last. Besides, the local integral domain selected in the local collocation element is circular or spherical to simplify programming. The transformation of the local integral domain between the physical and normalized spaces is given for two- and three-dimensional problems. The comparison of accuracy and convergence between the new proposed Petrov–Galerkin method and the conventional methods is carried out. Some challenging examples including fracture mechanics problems and a complex 3D problem are given to validate the convergence and accuracy of the proposed method.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Dalian University of Technology
Type: Article
Journal: International Journal for Numerical Methods in Engineering
Volume: 124
Pages: 5047-5068
No. of pages: 22
ISSN: 0029-5981
Publication date: 10.10.2023
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Numerical Analysis, Engineering(all), Applied Mathematics
Electronic version(s): https://doi.org/10.1002/nme.7337 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{6a0fde23df5c41b4a8ebd2915ff135bb,
title = "Petrov–Galerkin zonal free element method for 2D and 3D mechanical problems",
abstract = "In this article, a novel weak-form zonal Petrov–Galerkin free element method is proposed for two- and three-dimensional linear mechanical problems. By absorbing the advantages of finite block method and strong-form finite element method, the block mapping technique is used in the free element method. Combining the characteristics of the meshless local Petrov–Galerkin method, the local Petrov–Galerkin formulation based on the zonal free element method is formed at last. Besides, the local integral domain selected in the local collocation element is circular or spherical to simplify programming. The transformation of the local integral domain between the physical and normalized spaces is given for two- and three-dimensional problems. The comparison of accuracy and convergence between the new proposed Petrov–Galerkin method and the conventional methods is carried out. Some challenging examples including fracture mechanics problems and a complex 3D problem are given to validate the convergence and accuracy of the proposed method.",
keywords = "free element method, mesh-free method, weak-form Petrov–Galerkin method",
author = "Xu, {Bing Bing} and Gao, {Xiao Wei}",
note = "Funding Information: The first author of the article thanks the Humboldt Foundation for its support. ",
year = "2023",
month = oct,
day = "10",
doi = "10.1002/nme.7337",
language = "English",
volume = "124",
pages = "5047--5068",
journal = "International Journal for Numerical Methods in Engineering",
issn = "0029-5981",
publisher = "John Wiley and Sons Ltd",
number = "22",
}