I³MS - Wriggers Seminar
Prof. Dr. Peter Wriggers - Contact Modelling, New Formulations and Insights
Institute of Continuum Mechanics, University of Hanover
The numerical simulation of contact problems is nowadays a standard procedure for many engineering applications. The contact constraints are usually formulated using special interpolations like the mortar method- at the interface. . The aim of this lecture is twofold: (a) exploring the strength of isogeometric methods in contact applications and (b) introducing a new scheme that is based on a space filling mesh in which the contacting bodies can move and interact. The latter is based on a non-classical approach where the medium will be formulated as an isotropic/anisotropic material with changing characteristics and directions.
In the first part of the lecture we will study NURBS-based isogeometric analysis of contact problems and compare with standard C0-continuous Lagrange finite elements. A knot-to-surface (KTS) algorithm is developed to treat the contact constraints with NURBS contact surface discretizations. Qualitative studies deliver satisfactory results for various finite deformation frictionless thermoelastic contact problems. Quantitative studies based on the Hertz problem suggest the need for a relaxation of the mechanical contact constraints that appear in the standard KTS approach. The improved mortar-based KTS algorithm delivers robust and accurate results for NURBS discretizations. As a numerical example the finite deformation of a Grosch wheel is depicted in Figure 1. It is easy to see that knot points of the isogeometric mesh do not relate to the contact constraints. In total, we conclude that NURBS-based isogeometric analysis is a viable technology for contact problems and offers potential accuracy as well as convergence improvements over C0-continuous finite elements. For more details see ,  and . In the second part of the lecture a new approach is followed that provides an alternative method to treat contact problems. It avoids the complexity of the classical schemes related to the exact formulation and enforcement of the contact constraints. The contacting bodies will be imbedded in a medium. This medium will have assigned specific material properties that on one hand approximate rigid body movements before contact and on the other hand account for the contact constraints whenever necessary. Thus the constitutive equations of the medium have to change their properties. This is related to the magnitude of the parameters in space with respect to the movement of the bodies. But also a change from isotropic to anisotropic behaviour will be necessary in order to model frictionless contact in a correct way. A typical computation can be found in Figure 2 which shows the mesh of the entire domain and the deformed contact state. Details can be found in .
 I. Temizer, P. Wriggers and T. J. R. Hughes, Contact Treatment in Isogeometric Analysis with NURBS, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 1100–1112  L. de Lorenzis, I. Temizer, P.Wriggers and G. Zavarise, A large deformation frictional contact formulation using NURBS-based isogeometric analysis, International Journal for Numerical Methods in Engineering 87 (2011) 1278–1300.  I. Temizer, P. Wriggers and T. J. R. Hughes, Three-Dimensional Mortar-Based Frictional Contact Treatment in Isogeometric Analysis with NURBS, Computer Methods in Applied Mechanics and Engineering, 209-211 (2012), 115–128.  P. Wriggers, J. Schr¨oder and A. Schwarz, A finite element method for contact - using a third medium, Computational Mechanics, to appear.