a dual lagrange method with regularized frictional

11th. World Congress on Computational Mechanics (WCCM XI)
5th. European Conference on Computational Mechanics (ECCM V)
6th. European Conference on Computational Fluid Dynamics (ECFD VI)
July 20 - 25, 2014, Barcelona, Spain
Saskia Sitzmann1 , Kai Willner2 and Barbara I. Wohlmuth3
Central Institute for Scientific Computing, Friedrich Alexander University
Erlangen-Nuremberg, Martensstrasse 5a, 91058 Erlangen, [email protected]
2 Chair of Applied Mechanics, Friedrich-Alexander University Erlangen-Nuremberg,
Egerlandstrasse 5, 91058 Erlangen, [email protected]
Institute for Numerical Mathematics, Technical University Munich, Bolzmannstrasse 3,
85748 Garching, [email protected]
Key words: contact with friction, dual Mortar methods, micro slip, rough surfaces,
constitutive contact equations.
Solving non-linear contact problems within the FEM framework is still a challenging
task from both the mathematical and the engineering point of view. In recent years the
dual mortar method [1] has become of great interest for solving contact problems, since
it enforces the contact conditions in variationally consistent way without increasing the
algebraic system size. But most publications in this area restrict their considerations to
hard contact in normal direction and a perfect Coulomb law in tangential direction [2, 3].
For use in structural dynamics it is important to model the transition from sticking to
slipping in a physical correct way in order to reproduce measured frictional damping.
Taking the surface roughness on the micro scale into account, there are three different
states in the tangential contact on the micro scale [4]: for small shear stresses one gets
a linear elastic stick region, with increasing shear stress, one reaches the region of micro
slip before the whole contact zone is slipping and one reaches macro slip, see figure 1.
Associated mathematical models were introduced by Oden and Pires [5].
The micro slip has significant impact on the dynamic behaviour of the body like in a bolted
connection in the casing of a jet turbine. In order to model this behaviour numerically
within a dual mortar framework, we start with a perturbed Lagrange formulation [6] in
tangential direction and introduce an regularization Operator Gcτ depending on the shear
This project was partly funded by the Federal Republic of Germany, Federal Ministry of Economics
and Technology in cooperation with MTU Aero Engines GmbH under the project LufoIV ”GTF-Turb”
(FKZ: 20T1105)
S. Sitzmann, K. Willner and B.I. Wohlmuth
elastic stick
macro slip
micro slip
Figure 1: A milled steel surface on the micro scale and the resulting tangential contact law taking this
surface roughness into account
stress λτ to model the elastic stick:
Φ(λn , λτ ) := kλτ k − µλn ≤ 0
˙ τ − Gcτ (λ˙ τ ) = γ˙
kλτ k
γ˙ ≥ 0
γ˙ · Φ = 0
A mass lumping technique is used to exploit the full advantages of the duality pairing.
This leads to a regularized saddle point problem, for which a non-linear complementary
function and thus a semi-smooth Newton method can be derived. In order to model the
micro slip the approach has to be adapted.
The derived numerical algorithms were implemented within the free FEM package CalculiX. Selected numerical examples illustrate the robustness and applicability to real life
contact problems of the newly derived dual mortar formulation.
[1] B.I. Wohlmuth. A mortar finite element method using dual spaces for the Lagrange
multiplier. SIAM J. Numer. Anal, 38:989–1012, 1998.
[2] C. Hager, S. H¨
ueber, and B. I. Wohlmuth. A stable energy-conserving approach for
frictional contact problems based on quadrature formulas. International Journal for
Numerical Methods in Engineering, 73(2):205–225, 2008.
[3] M. Gitterle, A. Popp, M.W. Gee, and W.A. Wall. Finite deformation frictional
mortar contact using a semi-smooth Newton method with consistent linearization.
International Journal for Numerical Methods in Engineering, 84:543–571, 2010. DOI.
[4] K.Willner. Constitutive contact laws in structural dynamics. CMES: Computer Modeling in Engineering & Sciences, 48:303–336, 2009.
[5] J.T. Oden and E.B. Pires. Nonlocal and nonlinear friction laws and variational prinziples for contact problems in elasticity. Journal of applied Mechanics, 50(1):67–76,
[6] J. C. Simo, P. Wriggers, and R. L. Taylor. A perturbed Lagrangian formulation for the
finite element solution of contact problems. Computer Methods in Applied Mechanics
and Engineering, 50(2):163 – 180, 1985.