Multi-connected boundary conditions in solid mechanics and surgery theory

authored by: Huilong Ren, Xiaoying Zhuang, Cosmin Anitescu, Timon Rabczuk
Abstract: Boundary conditions are critical to the partial differential equations (PDEs) as they constrain the PDEs ensuring a unique and well defined solution. Based on combinatorial and surgery theory of manifolds, we develop multi-element boundary conditions as the generalization of the traditional boundary conditions in classical mechanics: Dirichlet boundary conditions, Neumann boundary conditions and Robin boundary conditions. The multi-element boundary/domain conditions glue the physical quantities at several points of different boundaries or domains on the fly, where the point-to-point correspondence (point mapping) on several boundaries are established on the common local coordinate system and the interactions are realized through the “wormhole” (i.e. the constraint equations). The study on weak form shows that the general multi-element boundary conditions are inconsistent with the variational principle/weighted residual method. To circumvent this dilemma, a numerical scheme based on augmented Lagrange method and nonlocal operator method (NOM) is proposed to deal with the mechanical problem equipped with general multi-element boundary conditions. Numerical tests show that the structures have completely different deformation modes for different multi-element boundary conditions.
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: 251
ISSN: 0045-7949
Publication date: 15.07.2021
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Civil and Structural Engineering, Modelling and Simulation, General Materials Science, Mechanical Engineering, Computer Science Applications
Electronic version(s): https://doi.org/10.1016/j.compstruc.2021.106504 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{6b1d14ab1b6d4ff0a834f4cec411a085,
title = "Multi-connected boundary conditions in solid mechanics and surgery theory",
abstract = "Boundary conditions are critical to the partial differential equations (PDEs) as they constrain the PDEs ensuring a unique and well defined solution. Based on combinatorial and surgery theory of manifolds, we develop multi-element boundary conditions as the generalization of the traditional boundary conditions in classical mechanics: Dirichlet boundary conditions, Neumann boundary conditions and Robin boundary conditions. The multi-element boundary/domain conditions glue the physical quantities at several points of different boundaries or domains on the fly, where the point-to-point correspondence (point mapping) on several boundaries are established on the common local coordinate system and the interactions are realized through the “wormhole” (i.e. the constraint equations). The study on weak form shows that the general multi-element boundary conditions are inconsistent with the variational principle/weighted residual method. To circumvent this dilemma, a numerical scheme based on augmented Lagrange method and nonlocal operator method (NOM) is proposed to deal with the mechanical problem equipped with general multi-element boundary conditions. Numerical tests show that the structures have completely different deformation modes for different multi-element boundary conditions.",
keywords = "Augmented Lagrange method, Continuum mechanics, Multi-element boundary, Nonlocal operator method, Topological surgery",
author = "Huilong Ren and Xiaoying Zhuang and Cosmin Anitescu and Timon Rabczuk",
note = "Funding Information: The authors acknowledge the supports from RISE-BESTOFRAC. ",
year = "2021",
month = jul,
day = "15",
doi = "10.1016/j.compstruc.2021.106504",
language = "English",
volume = "251",
journal = "Computers and Structures",
issn = "0045-7949",
publisher = "Elsevier Ltd.",
}