Peridynamic Galerkin method

an attractive alternative to finite elements

authored by: T. Bode, C. Weißenfels, P. Wriggers
Abstract: This work presents a meshfree particle scheme designed for arbitrary deformations that possess the accuracy and properties of the Finite-Element-Method. The accuracy is maintained even with arbitrary particle distributions. Mesh-based methods mostly fail if requirements on the location of evaluation points are not satisfied. Hence, with this new scheme not only the range of loadings can be increased but also the pre-processing step can be facilitated compared to the FEM. The key to this new meshfree method lies in the fulfillment of essential requirements for spatial discretization schemes. The new approach is based on the correspondence theory of Peridynamics. Some modifications of this framework allows for a consistent and stable formulation. By applying the peridynamic differentiation concept, it is also shown that the equations of the correspondence theory can be derived from the weak form. Likewise, it is demonstrated that special moving least square shape functions possess the Kronecker-δ property. Thus, Dirichlet boundary conditions can be directly applied. The positive performance of this new meshfree method, especially in comparison to the Finite-Element-Method, is shown in the calculation of several test cases. In order to guarantee a fair comparison enhanced finite element formulations are also used. The test cases include the patch test, an eigenmode analysis as well as the investigation of loadings in the context of large deformations.
Organisation(s): Institute of Continuum Mechanics
PhoenixD: Photonics, Optics, and Engineering - Innovation Across Disciplines
External Organisation(s): University of Augsburg
Type: Article
Journal: Computational mechanics
Volume: 70
Pages: 723–743
No. of pages: 21
ISSN: 0178-7675
Publication date: 10.2022
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Ocean Engineering, Mechanical Engineering, Computational Theory and Mathematics, Computational Mathematics, Applied Mathematics
Electronic version(s): https://doi.org/10.1007/s00466-022-02202-w (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{5382a20e1cd24d48a4644b1ffeef1494,
title = "Peridynamic Galerkin method: an attractive alternative to finite elements",
abstract = "This work presents a meshfree particle scheme designed for arbitrary deformations that possess the accuracy and properties of the Finite-Element-Method. The accuracy is maintained even with arbitrary particle distributions. Mesh-based methods mostly fail if requirements on the location of evaluation points are not satisfied. Hence, with this new scheme not only the range of loadings can be increased but also the pre-processing step can be facilitated compared to the FEM. The key to this new meshfree method lies in the fulfillment of essential requirements for spatial discretization schemes. The new approach is based on the correspondence theory of Peridynamics. Some modifications of this framework allows for a consistent and stable formulation. By applying the peridynamic differentiation concept, it is also shown that the equations of the correspondence theory can be derived from the weak form. Likewise, it is demonstrated that special moving least square shape functions possess the Kronecker-δ property. Thus, Dirichlet boundary conditions can be directly applied. The positive performance of this new meshfree method, especially in comparison to the Finite-Element-Method, is shown in the calculation of several test cases. In order to guarantee a fair comparison enhanced finite element formulations are also used. The test cases include the patch test, an eigenmode analysis as well as the investigation of loadings in the context of large deformations.",
keywords = "Correspondence theory, Method of least squares, Peridynamics, Requirements on spacial discretization schemes, Weak form",
author = "T. Bode and C. Wei{\ss}enfels and P. Wriggers",
note = "Funding Information: This project was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany{\textquoteright}s Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453) ",
year = "2022",
month = oct,
doi = "10.1007/s00466-022-02202-w",
language = "English",
volume = "70",
pages = "723–743",
journal = "Computational mechanics",
issn = "0178-7675",
publisher = "Springer Verlag",
number = "4",
}