TY - JOUR

T1 - A novel mixed finite element for finite anisotropic elasticity; the SKA-element Simplified Kinematics for Anisotropy

AU - Schröder, Jörg

AU - Viebahn, Nils

AU - Balzani, Daniel

AU - Wriggers, Peter

N1 - 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.

PY - 2016/7/15

Y1 - 2016/7/15

N2 - 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.

AB - 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.

KW - Anisotropic hyperelasticity

KW - Lagrange-multiplier

KW - Mixed finite elements

KW - SKA-element

UR - http://www.scopus.com/inward/record.url?scp=84980494077&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2016.06.029

DO - 10.1016/j.cma.2016.06.029

M3 - Article

AN - SCOPUS:84980494077

VL - 310

SP - 475

EP - 494

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

ER -