A novel mixed finite element for finite anisotropic elasticity; the SKA-element Simplified Kinematics for Anisotropy

authored by: Jörg Schröder, Nils Viebahn, Daniel Balzani, Peter Wriggers
Abstract: A variety of numerical approximation schemes for boundary value problems suffer from so-called locking-phenomena. It is well known that in such cases several finite element formulations exhibit poor convergence rates in the basic variables. A serious locking phenomenon can be observed in the case of anisotropic elasticity, due to high stiffness in preferred directions. The main goal of this paper is to overcome this locking problem in anisotropic hyperelasticity by introducing a novel mixed variational framework. Therefore we split the strain energy into two main parts, an isotropic and an anisotropic part. For the isotropic part we can apply different well-established approximation schemes and for the anisotropic part we apply a constant approximation of the deformation gradient or the right Cauchy–Green tensor. This additional constraint is attached to the strain energy function by a second-order tensorial Lagrange-multiplier, governed by a Simplified Kinematic for the Anisotropic part. As a matter of fact, for the tested boundary value problems the SKA-element based on quadratic ansatz functions for the displacements, performs excellent and behaves more robust than competitive formulations.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): University of Duisburg-Essen
Technische Universität Dresden
Type: Article
Journal: Computer Methods in Applied Mechanics and Engineering
Volume: 310
Pages: 475-494
No. of pages: 20
ISSN: 0045-7825
Publication date: 15.07.2016
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Mechanics of Materials, Mechanical Engineering, General Physics and Astronomy, Computer Science Applications
Electronic version(s): https://doi.org/10.1016/j.cma.2016.06.029 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{eb2e9f26cf564721a1a779e4ed07a58f,
title = "A novel mixed finite element for finite anisotropic elasticity; the SKA-element Simplified Kinematics for Anisotropy",
abstract = "A variety of numerical approximation schemes for boundary value problems suffer from so-called locking-phenomena. It is well known that in such cases several finite element formulations exhibit poor convergence rates in the basic variables. A serious locking phenomenon can be observed in the case of anisotropic elasticity, due to high stiffness in preferred directions. The main goal of this paper is to overcome this locking problem in anisotropic hyperelasticity by introducing a novel mixed variational framework. Therefore we split the strain energy into two main parts, an isotropic and an anisotropic part. For the isotropic part we can apply different well-established approximation schemes and for the anisotropic part we apply a constant approximation of the deformation gradient or the right Cauchy–Green tensor. This additional constraint is attached to the strain energy function by a second-order tensorial Lagrange-multiplier, governed by a Simplified Kinematic for the Anisotropic part. As a matter of fact, for the tested boundary value problems the SKA-element based on quadratic ansatz functions for the displacements, performs excellent and behaves more robust than competitive formulations.",
keywords = "Anisotropic hyperelasticity, Lagrange-multiplier, Mixed finite elements, SKA-element",
author = "J{\"o}rg Schr{\"o}der and Nils Viebahn and Daniel Balzani and Peter Wriggers",
note = "Funding information: The authors gratefully acknowledge Annalisa Buffa and Pablo Antolin Sanchez for fruitful discussions, starting 2015 at the Workshop for Computational Engineering of the MFO. The authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Novel finite elements for anisotropic media at finite strain” under the project “Reliable Simulation Techniques in Solid Mechanics, Development of Non-standard Discretization Methods, Mechanical and Mathematical Analysis” (SCHR 570/23-1) (WR 19/50-1). In addition to that the author Daniel Balzani appreciates funding through the “Institutional Strategy” at TU Dresden, as part of the DFG-Excellence Initiative. For support and discussions regarding the implementation into AceGen the authors would like to acknowledge Joz?e Korelc.",
year = "2016",
month = jul,
day = "15",
doi = "10.1016/j.cma.2016.06.029",
language = "English",
volume = "310",
pages = "475--494",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",
}