A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem

authored by: Timon Rabczuk, Huilong Ren, Xiaoying Zhuang
Abstract: A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations. Common differential operators as well as the variational forms are defined within the context of nonlocal operators. The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity, which is necessary for the eigenvalue analysis such as the waveguide problem. The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields. The governing equations are converted into nonlocal integral form. An hourglass energy functional is introduced for the elimination of zero-energy modes. Finally, the proposed method is validated by testing three classical benchmark problems.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Ton Duc Thang University
Bauhaus-Universität Weimar
Tongji University
Type: Article
Journal: Computers, Materials and Continua
Volume: 59
Pages: 31-55
No. of pages: 25
ISSN: 1546-2218
Publication date: 2019
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Biomaterials, Modelling and Simulation, Mechanics of Materials, Computer Science Applications, Electrical and Electronic Engineering
Electronic version(s): https://doi.org/10.32604/cmc.2019.04567 (Access: Open)
https://doi.org/10.15488/4766 (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{fe94e779642f4a509328b5183e399176,
title = "A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem",
abstract = "A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations. Common differential operators as well as the variational forms are defined within the context of nonlocal operators. The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity, which is necessary for the eigenvalue analysis such as the waveguide problem. The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields. The governing equations are converted into nonlocal integral form. An hourglass energy functional is introduced for the elimination of zero-energy modes. Finally, the proposed method is validated by testing three classical benchmark problems.",
keywords = "Hourglass mode, Nonlocal operator method, Nonlocal operators, Variational principle",
author = "Timon Rabczuk and Huilong Ren and Xiaoying Zhuang",
note = "Funding information: The authors acknowledge the support of the German Research Foundation (DFG).",
year = "2019",
doi = "10.32604/cmc.2019.04567",
language = "English",
volume = "59",
pages = "31--55",
journal = "Computers, Materials and Continua",
issn = "1546-2218",
publisher = "Tech Science Press",
number = "1",
}