On the stability analysis of hyperelastic boundary value problems using three- and two-field mixed finite element formulations

verfasst von: Jörg Schröder, Nils Viebahn, Peter Wriggers, Ferdinando Auricchio, Karl Steeger
Abstract: In this work we investigate different mixed finite element formulations for the detection of critical loads for the possible occurrence of bifurcation and limit points. In detail, three- and two-field formulations for incompressible and quasi-incompressible materials are analyzed. In order to apply various penalty functions for the volume dilatation in displacement/pressure mixed elements we propose a new consistent scheme capturing the non linearities of the penalty constraints. It is shown that for all mixed formulations, which can be reduced to a generalized displacement scheme, a straight forward stability analysis is possible. However, problems based on the classical saddle-point structure require a different analyses based on the change of the signature of the underlying matrix system. The basis of these investigations is the work from Auricchio et al. (Comput Methods Appl Mech Eng 194:1075–1092, 2005, Comput Mech 52:1153–1167, 2013).
Organisationseinheit(en): Institut für Kontinuumsmechanik
Externe Organisation(en): Universität Duisburg-Essen
Università degli Studi di Pavia
Typ: Artikel
Journal: Computational mechanics
Band: 60
Seiten: 479-492
Anzahl der Seiten: 14
ISSN: 0178-7675
Publikationsdatum: 05.05.2017
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Numerische Mechanik, Meerestechnik, Maschinenbau, Theoretische Informatik und Mathematik, Computational Mathematics, Angewandte Mathematik
Elektronische Version(en): https://doi.org/10.1007/s00466-017-1415-2 (Zugang: Geschlossen)
: Details im Forschungsportal „Research@Leibniz University“

BibTeX

@article{e125dfc1698a465ca4c7e66a901b1a55,
title = "On the stability analysis of hyperelastic boundary value problems using three- and two-field mixed finite element formulations",
abstract = "In this work we investigate different mixed finite element formulations for the detection of critical loads for the possible occurrence of bifurcation and limit points. In detail, three- and two-field formulations for incompressible and quasi-incompressible materials are analyzed. In order to apply various penalty functions for the volume dilatation in displacement/pressure mixed elements we propose a new consistent scheme capturing the non linearities of the penalty constraints. It is shown that for all mixed formulations, which can be reduced to a generalized displacement scheme, a straight forward stability analysis is possible. However, problems based on the classical saddle-point structure require a different analyses based on the change of the signature of the underlying matrix system. The basis of these investigations is the work from Auricchio et al. (Comput Methods Appl Mech Eng 194:1075–1092, 2005, Comput Mech 52:1153–1167, 2013).",
keywords = "Condensed formulations, Signature of a matrix, Stability analysis, Three-field and two-field mixed FEM",
author = "J{\"o}rg Schr{\"o}der and Nils Viebahn and Peter Wriggers and Ferdinando Auricchio and Karl Steeger",
note = "Funding information: The authors gratefully acknowledge Gerhard Starke, Fleurianne Bertrand and Rainer Niekamp for helpful discussions. The authors appreciate the 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).",
year = "2017",
month = may,
day = "5",
doi = "10.1007/s00466-017-1415-2",
language = "English",
volume = "60",
pages = "479--492",
journal = "Computational mechanics",
issn = "0178-7675",
publisher = "Springer Verlag",
number = "3",
}