Thermo-mechanical contact problems on non-matching meshes

doi:10.1016/j.cma.2008.11.022

Computer Methods in Applied Mechanics and Engineering

Volume 198, Issues 15–16, 15 March 2009, Pages 1338-1350

https://doi.org/10.1016/j.cma.2008.11.022 Get rights and content

Abstract

Non-matching meshes and domain decomposition techniques based on Lagrange multipliers provide a flexible and efficient discretization technique for variational inequalities with interface constraints. Although mortar methods are well analyzed for variational inequalities, its application to dynamic thermo-mechanical contact problems with friction is still a field of active research. In this work, we extend the mortar approach for dynamic contact problems with Coulomb friction to the thermo-mechanical case. We focus on the discretization and on algorithmic aspects of dynamic effects such as frictional heating and thermal softening at the contact interface. More precisely, we generalize the mortar concept of dual Lagrange multipliers to non-linear Robin-type interface conditions and apply local static condensation to eliminate the heat flux. Numerical examples in the two-dimensional and the three-dimensional setting illustrate the flexibility of the discretization on non-matching meshes.

Introduction

The numerical simulation of frictional contact problems is still a challenging task and plays an important role in many industrial applications. Mortar techniques became a promising discretization method for such type of problems involving non-matching meshes, see, e.g. [9], [21], [3], [5]. Recently, a lot of work has been done to generalize these concepts to dynamic contact problems including friction and large deformations. Due to the sliding of the different bodies, most of the frictional work results in the generation of heat. This observation motivates the extension of the mortar method to thermo-mechanical dynamic contact problems including frictional heating and thermal softening effects at the contact interface.

The mortar method is a hybrid formulation in space. The displacement and the temperature field enter as primal variable, whereas the contact stress and the thermal flux at the contact interface are the dual variables. Mathematically speaking, the dual variables, also denoted as Lagrange multipliers, enforce the interface conditions. Here, we have two types of constraints: the non-penetration condition and the friction law for the mechanical unknowns and the generation of heat as well as the flow condition for the thermal variables. The main focus of this paper is on the treatment of the Robin-type thermal interface condition within the mortar framework. Due to the use of biorthogonal basis functions for the mechanical dual variable, the mechanical interface constraints at the contact zone decouple for each node. Although the mortar approach can be seen as a segment-to-segment formulation, the possible decoupling results in a node-to-segment approach which can be handled more easily from the numerical point of view. In contrast, a straightforward application of the concept of biorthogonality to Robin-type constraints does not decouple the nodes. To benefit from static condensation, we introduce a stable operator which can be interpreted as mass lumping at the interface.

For the formulation of the linear thermo-elastic constitutive equations, we follow [2], [17]. A more general thermo-plasticity formulation is given in [15]. The extension to dynamic thermo-mechanical contact mechanics can be found, e.g. in [10], [13], [12] and is also considered in the textbooks [20], [17]. For the modeling of the contact heat flux, we use the linear model proposed in [13]. More general models can be found in [18]. For the considered formulation with a coefficient of friction depending on the temperature we refer to [13], [12].

We start with the introduction of the constitutive equations and the interface conditions in Section 2. Section 3 presents the time discretization based on the midpoint scheme. The mortar formulation and the space discretization can be found in Section 4, followed by the resulting algebraic formulation in Section 5. Some comments on the applied numerical algorithm to solve the arising non-linear equations are given in Section 6. The last section shows numerical examples both in the two- and three-dimensional setting and illustrates the flexibility of the considered discretization.

Section snippets

Problem formulation for linear thermo-elasticity

We consider two bodies in their reference configuration $Ω^{i} \subset R^{d}, i \in {m, s}$ , with the dimension $d = 2, 3$ . The superscript s stands for the slave body and m for the master body, as it is common in the framework of mortar techniques. We are interested in the displacement field $u^{i} (x, t)$ and the temperature $θ^{i} (x, t)$ for $(x, t) \in Ω^{i} \times (0, T)$ , where $(0, T)$ is the given time interval. The local balance of momentum is given by $ϱ^{i} {\ddot{u}}^{i} - Div (P^{i}) = f^{i} in Ω^{i} \times (0, T) .$ Here we denote by $ϱ^{i}$ the mass density of the body $Ω^{i}$ and by $P^{i}$ the

Discretization in time

Both the hyperbolic mechanical equilibrium condition (2.6a) and the parabolic thermal one (2.6b) are discretized using the midpoint scheme which is of order two. We start with introducing some notation. Firstly, we define $t^{k} ≔ k Δ t, k = 0, \dots, N_{T}$ , where the time step size is given by $Δ t ≔ T / N_{T}$ . Secondly, we denote by $u^{k} \approx u (\cdot, t^{k})$ and $θ^{k} \approx θ (\cdot, t^{k})$ the approximation of the displacement field and the temperature at time $t^{k}$ , respectively. Setting $Δ u^{k} ≔ u^{k + 1} - u^{k}$ , the midpoint rule yields $u^{k + 1 / 2} ≔ \frac{1}{2} (u^{k + 1} + u^{k}) = u^{k} + \frac{Δ u^{k}}{2},$ ${\dot{u}}^{k +}$

Space discretization and mortar formulation

In this section, the space discretized variational formulation of the time discretized thermodynamic contact problem is derived. Introducing Lagrange multipliers $λ_{u}$ and $λ_{θ}$ for the mechanical boundary stress $p_{c}$ and the thermal heat flux $q_{c}^{s}$ on the possible contact boundary, respectively, we end up with a mortar based formulation. Our main focus is on the treatment of the Robin-type thermal interface conditions (2.17a), (2.17b) which leads to a generalized saddle point formulation. A local static

Algebraic representation

To formulate the algebraic representation of the first two equations in (4.28), we introduce the mass matrices $M_{j}, j = 1, 2$ , resulting from the bilinear forms $m_{j} (\cdot, \cdot)$ with respect to the nodal basis. To avoid numerical oscillations in space due to the possible non-zero heat flux at the interface, we use a standard lumping technique and replace the mass matrix $M_{2}$ by a diagonal matrix $M_{2}^{L} [p, p] ≔ \sum_{q} M_{2} [p, q]$ . We remark that using the lumped version of the mass matrix does not deteriorate the numerical

Numerical algorithm

One of the challenges in the numerical simulation of such type of problems is the design of fast and robust solvers which can handle the non-linearities in the mechanical and thermal interface constraints efficiently. For the highly non-linear mechanical condition we use a semi-smooth Newton method; at the same time, we update the non-linearities of the thermal condition in the iteration process using a fixed point approach. The non-linear mechanical contact constraints (5.4), (5.5), are

Numerical examples

In this section, numerical examples for the proposed formulation and algorithm are presented both in the three- and two-dimensional setting. The numerical implementation is based on the finite element toolbox UG [1] and the fast direct solver package PARDISO [14].

Conclusion

In this paper, we apply mortar techniques with a dual Lagrange multiplier on non-matching meshes to thermo-mechanical contact problems. Discretization methods based on Lagrange multipliers are quite attractive in the case of sliding geometries. The new variables on the interface represent the surface forces and the heat flux and enter in a consistent way into the variational problem. Therefore, these formulations pass a patch test for highly non-matching meshes in contrast to simple node to

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Thermo-mechanical contact problems on non-matching meshes

Abstract

Introduction

Section snippets

Problem formulation for linear thermo-elasticity

Discretization in time

Space discretization and mortar formulation

Algebraic representation

Numerical algorithm

Numerical examples

Conclusion

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

C.R. Akad. Sci. Paris

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

UG – a flexible software toolbox for solving partial differential equations

Comput. Vis. Sci.

Linear thermoelasticity