# Contact modelling

Completed Research project

Efficient algorithms using dual Lagrange multipliers for three-dimensional dynamic contact problems.

contact stresses

## Overview

• Mortar based contact formulation
• Dual shape functions for the Lagrange multipliers
• Stable time integration schemes for contact problems
• Contact between thin-walled structures using surface oriented shell elements
• Three-dimensional mortar based contact formulations
• Contact search

## Project description

### Mortar based contact formulation

Most of the contact algorithms developed in the past are based on so-called node-to-segment approaches, which enforce the contact constraints at special collocation points (usually the finite element nodes). Although easy to implement, this formulation can only roughly represent the contact pressure and shows, in certain cases, locking due to over constraint. The mortar method, developed as a technique to join dissimilar meshes, fulfils the impenetrability condition in a weak, integral sense and hence avoids too stiff behaviour. In addition, the improved coupling of the contacting interfaces yields a better representation of the contact pressure. Overall, the mortar method is considerably more robust than traditional formulations.

Simulation of contact between two elastic half rings; contact stresses pictured

Based on a classical Lagrange multiplier formulation, the mortar method uses additional unknowns, the Lagrange multipliers, to formulate the contact coupling conditions. Thus, the size of the system of equations increases by the number unknown Lagrange multipliers. If these additional unknowns are formulated in terms of the displacements, using a penalty method, the system of equations keeps its original size, but penetration of the contacting bodies gets unphysical and user-dependent. However, using dual shape functions for the discretization of the Lagrange multipliers instead yields an easy condensation of the additional unknowns. The resultant system matrix is of constant size and solved just for the unknown displacements. Calculation of the contact stresses is done in a post process step.

Simulation of contact between two stacked beams; contact stresses pictured
Contact patch test; contact stresses pictured

### Modified definition of the dual shape functions

Depending on which of the contacting bodies is chosen for the discretisation of the Lagrange multipliers ("slave" body), in certain situations an inconsistency at the boundary of the contact area can appear. This leads to an inconsistent calculation of the gap between the bodies and to an inconsistent transmission of the contact forces. To avoid this problem, a modified definition of the dual shape functions combined with an appropriate weighting procedure has been developed at the institute. So, for example, the contact patch test (see picture on the right) is fulfilled independent of the choice of the "slave" side.

### Stable modelling of dynamic contact problems

Considering dynamic problems including contact, additional challenges due to time discretisation arise. In non-linear structural dynamics, usually an algorithm based on the Newmark equations is used. These algorithms can be adjusted or actually are designed to behave stable for smooth problems. Stability is assured either by controlled loss of energy due to numerical damping or by exact conservation of energy (for non-dissipative problems). However, in the presence of contact conditions these naturally stable algorithms might become unstable because of growth of energy in a release step. In addition, also for stable solutions oscillations in the contact forces as a result of oscillations in the inertia term may appear. As a consequence of this unphysical results and active set instabilities might appear. Goal of the work at the institute is the investigation and further development of time integration schemes, which avoid artificial oscillations and at the same time leave the system energy preferably unaffected.

### Modelling of thin-walled structures under large deformations

This part of the project applies the dual mortar method on contact problems between thin-walled structures under large deformations. Modelling of thin-walled structures is done using surface-oriented shell elements, which has the benefit, that the contact conditions can directly be applied at the surface of the elements und thus no projection is necessary. Discretization in time is done by the "Generalized-α" and the "Generalized-Energy-Momentum-Method" and combined with appropriate strategies to achieve stable modelling of the contact. Matter of the current work is applying the formulation on frictional contact and on threedimensional multibody contact.

Simulation of an elastic tube deformed by a rigid plate

### Three-dimensional mortar based contact formulations

The contact boundary in two-dimensional contact problems consists of piecewise straight lines if the body is discretized with bilinear finite elements. In a three-dimensional description the contact boundary no longer consists of lines, but instead of surfaces. These surfaces are in general curved, even if the body is discretized with eight node trilinear hexahedra elements. For evaluation of the contact part in the finite element formulation, integration over the contact surface is necessary. In order to simplify this integration, the curved surface is normally approximated by piecewise flat surfaces. When using a three-dimensional problem description, the dual shape functions depend on the geometry of the surface element. This is a remarkable difference compared with the two-dimensional formulation.

Three-dimensional ironing problem with rotating indenter

### Contact search

To check which regions of the modeled bodies are in contact the distance between their boundaries is determined. Because this is computationally very expensive, contact search algorithms are used to identify the regions which are close to each other, if there are any. Only for those regions a distance calculation is worthwhile. An efficient way to determine the approximate relative positions of the boundaries is the investigation of bounding volume hierarchies (see movie below). The contact boundaries and their subregions are described with bounding volumes which are stored in a binary tree. Due to the simple description of the bounding volumes, intersections can be determined in a computationally cheap way. The distance calculation, hence, only needs to be done for those regions where the effort is worth it

Process of contact search with bounding volumes

## Project data

Project titel:
Effiziente Algorithmen mit dualen Lagrange-Multiplikatoren für dreidimensionale, dynamische Kontaktprobleme
Funding:
German Research Foundation (DFG), Research Grant BI 722/7-1, GEPRIS project number 168822784
Researcher:
Thomas Cichosz, Christoph Wilking

### Publications

1. Wilking, C., & Bischoff, M. (2017). Alternative integration algorithms for three-dimensional mortar contact. Computational Mechanics, 59, 203–218. https://doi.org/10.1007/s00466-016-1345-4
2. Wilking, C. (2017). Effiziente Integration und verbesserte Kontaktspannungen für duale Mortar-Formulierungen. Doktorarbeit, Bericht Nr. 66, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-9244
3. Tkachuk, A., Wohlmuth, B., & Bischoff, M. (2013). Hybrid-mixed discretization of elasto-dynamic contact problems using consistent singular mass matrices. International Journal for Numerical Methods in Engineering, 94, 473–493. https://doi.org/10.1002/nme.4457
4. Mangold, O., Prohl, R., Tkachuk, A., & Trickov, V. (2012). Reduction of Numerical Sensitivities in Crash Simulations on HPC-Computers (HPC-10). Wolfgang E. Nagel, Dietmar B. Kröner, Michael M. Resch, (Eds.): High Performance Computing in Science and Engineering ’11. Transactions of the High Performance Computing Center Stuttgart. Springer, 631–636. https://doi.org/10.1007/978-3-642-23869-7_46
5. Cichosz, T. (2012). Stabile und konsistente Kontaktmodellierung in Raum und Zeit. Doktorarbeit, Bericht Nr. 58, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-495
6. Tkachuk, A., & Bischoff, M. (2011). Buckling under Contact Constraints as a Source of Scatter in Car Crash Simulations. II International Conference on Computational Contact Mechanics 15-17 June 2011, Hannover, Germany.
7. Cichosz, T., & Bischoff, M. (2011). Consistent treatment of boundaries with mortar contact formulations using dual Lagrange multipliers. Computer Methods in Applied Mechanics and Engineering, 200, 1317–1332. https://doi.org/10.1016/j.cma.2010.11.004
8. Tkachuk, A. (2010). A contact-stabilized Newmark method for coupled dynamical thermo-elastic problem. 3rd International Conference on Nonlinear Dynamic, September 21-24, 2010, Kharkov, Ukraine.
9. Hartmann, S., & Ramm, E. (2008). A mortar based contact formulation for non-linear dynamics using dual Lagrange multipliers. Finite Elements in Analysis and Design, 44, 245–258. https://doi.org/10.1016/j.finel.2007.11.018
10. Hartmann, S., Brunssen, S., Wohlmuth, B., & Ramm, E. (2007). Unilateral non-linear dynamic contact of thin-walled structures using a primal-dual active set strategy. International Journal for Numerical Methods in Engineering, 70, 883–912. https://doi.org/10.1002/nme.1894
11. Hartmann, S. (2007). Kontaktanalyse dünnwandiger Strukturen bei großen Deformationen. Doktorarbeit, Bericht Nr. 49, Institut für Baustatik und Baudynamik, Universität Stuttgart. https://doi.org/10.18419/opus-263