Abstract

In the present paper three algorithms are applied to a finite element model of two thermoelastic bodies in frictional wearing contact. All three algorithms utilize a modification of a Newton method for B-differentiable equations as non-linear equation solver. In the first algorithm the fully-coupled system of thermomechanical equations is solved directly using the modified method, while in the other two algorithms the equation system is decoupled in one mechanical part and another thermal part which are solved using an iterative strategy of Gauss–Seidel type. The two iterative algorithms differ in which order the parts are solved. The numerical performance of the algorithms are investigated for two two-dimensional examples. Based on these numerical results, the behaviour of the model is also discussed. It is found that the iterative approach where the thermal subproblem is solved first is slightly more efficient for both examples. Furthermore, it is shown numerically how the predicted wear gap is influenced by the bulk properties of the contacting bodies, in particular how it is influenced by thermal dilatation.

Highlights

  • The present paper concerns the numerical treatment of the thermoelastic model of contact, friction and wear developed in (Strömberg et al, 1996)

  • One might note that the difference between the direct approach (MBN) and the iterative approaches (GSMT and GSTM) is decreased when the number of time steps is increased while the difference is increased when the number of contact nodes is increased, when is assigned a higher value or when wear is excluded from the model

  • The first approach is a direct use of a modified Newton method for B-differentiable equations on the fully-coupled problem, while in the second approach the mechanical and thermal problems are solved uncoupled from each other using an iterative strategy

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Summary

Introduction

The present paper concerns the numerical treatment of the thermoelastic model of contact, friction and wear developed in (Strömberg et al, 1996). A possible drawback with the direct use of the Newton method is the numerical difficulties reported in (Strömberg, 1998) In that paper, these problems were solved by eliminating the temperature algebraically before the Newton step was performed. A modification of Pang’s Newton method was introduced in (Strömberg, 1997) to solve the augmented Lagrangian formulation of frictional contact (including wear). In such manner the step of solving the search direction is linearized This approach of modifying Pang’s Newton method has proven numerically to be very successful for solving both friction problems as well as wear problems. The performance of the algorithms are discussed as well as the behaviour of the model based on the numerical results

Governing equations
The system of discrete equations
Variational formulation
Space discretization
Time discretization
Projections
System of semismooth equations
Algorithms
Semismoothness
Pang’s Newton method
Direct approach – modified Newton method
Modified directional derivatives
Numerical tests
The two examples
Comparison of the three algorithms
Results for different constitutive settings
Concluding remarks
Full Text
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