Mixed FEM of higher-order for time-dependent contact problems

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Abstract

In this paper mixed finite element methods of higher-order for time-dependent contact problems are discussed. The mixed methods are based on resolving the contact conditions by the introduction of Lagrange multipliers. Dynamic Signorini problems with and without friction are considered involving thermomechanical and rolling contact. Rothe’s method is used to provide a suitable time and space discretization. To discretize in time, a stabilized Newmark method is applied as an adequate time stepping scheme. The space discretization relies on finite elements of higher-order. In each time step the resulting problems are solved by Uzawa‘s method or, alternatively, by methods of quadratic programming via a suitable formulation in terms of the Lagrange multipliers. Numerical results are presented towards an application in production engineering. The results illustrate the performance of the presented techniques for a variety of problem formulations.

Introduction

Dynamic contact, including frictional and thermal effects, appears in many engineering processes and has an essential effect on the behavior of machines, tools, workpieces, etc. For instance, the main effects on the dynamic behavior of drilling machines typically result from the contact of the tool and the workpiece in a small contact zone. One of the most decisive factors to control dynamic phenomena in drilling processes is, therefore, the determination of appropriate quantities as contact forces or contact zones. Thus, an essential part of simulation tools for such processes consists in the application of appropriate numerical schemes for contact.

Modeling contact problems involves systems of partial differential equations with inequality conditions describing several aspects of contact as geometrical constraints, friction or thermal effects. In literature, a huge number of numerical schemes is given dealing with the specific phenomena of contact. We refer to the monographs [35], [50] and the survey articles [13], [34] for an overview. Numerical schemes for dynamic problems are usually based on a combination of time and spatial discretization approaches. A usual proceeding is to use Rothe’s method in which the time variable and then the spatial variables are discretized. A well-established approach for the time discretization of hyperbolic problems with finite differences is the Newmark method [39]. The Newmark method can also be applied to discretize dynamic contact problems which, however, requires the use of some special parameters, cf. [3], [7]. It is an easily realizable approach for unilateral contact problems, where the geometrical constraints are ensured in each time step. Finite elements or other Galerkin-type methods are applied for the spatial discretization.

Important issues arising in numerical schemes for dynamic contact problems are, for instance, resolving contact in time preserving energy and momentum [1], [36], stabilizitation to avoid numerical oscillations [10], [19], [27], [28], [29], [33], [37], [41], discretizations with adaptivity [4], [5] and the efficient implementation. Widely used discretization approaches for contact problems are described in [3], [8], [46], [51]. They rely, for instance, on special contact elements with Lagrange multipliers or on penalty methods to capture the geometrical contact conditions.

Using the Newmark approach, one obtains a sequence of partial differential equations of which the solutions are discretized in time. In the framework of linear elasticity, this sequence, also known as the semi-discrete problem, admits static contact problems in each time step. Consequently, techniques for the static case can be applied directly. In literature many approaches for static contact problems are described, which can, in principle, be used to combine them with the Newmark scheme. Again, we refer to the monographs [35], [50]. Solution schemes for static contact problems are still an important subject of current research. We refer to the recent works [11], [24], [25], [32], [49]. Evidently, the use of them in dynamic contact problems opens a wide range of application.

A well-established approach to solve static contact problems is given by the application of mixed methods where the geometrical contact conditions and the frictional conditions are captured by Lagrange multipliers. It is widely studied and enhanced by Haslinger et al. [20], [22], [23] for many applications in frictional contact problems. In particular, efficient domain decomposition techniques are applied in the context of the FETI approach, cf. [12]. The discretization is based on a mixed variational formulation derived from a discretized saddle point formulation. The main advantage of this approach is that the Lagrange multipliers can be interpreted as normal and tangential contact forces. Moreover, the constraints for the Lagrange multipliers are sign conditions and box constraints which are simpler than the original contact conditions. The unique existence of a discrete saddle point is usually verified via an inf-sup condition associated to the discretization spaces. In the case of low-order finite elements, the key to guarantee the inf-sup condition is to use a discretization of the Lagrange multipliers on boundary meshes with a larger mesh size than that of the primal variable, cf. [21]. But, the application of higher-order finite elements is possible as well, which may avoid the use of different mesh sizes by using different polynomial degrees. We refer to [44] for more details, in particular, with respect to the discrete inf-sup condition and solution schemes by some Schur complement techniques. Further benefits of higher-order discretizations are, for instance, the reduction of locking effects and, using hp-adaptivity, high convergence rates or even exponential convergence rates, cf. [42].

In this work, we combine the stabilized Newmark scheme proposed in [10], [33] and the mixed method with a higher-order discretization to obtain a numerical scheme for dynamic contact problems and consider several physical attributes such as damping, friction, thermoelastic coupling and rolling contact. A framework is proposed which enables to include all these attributes in a general setting. The stabilization of the Newmark scheme is based on an additional L2 projection on the admissible set specified by the contact conditions, which can easily be realized for higher-order discretization in space. Stabilization techniques for the Newmark scheme are also proposed in [19], [29], which are, however, more complex to apply in this context, since the efficient construction of the redistributed mass matrix for higher-order basis functions is an open problem.

The physical interpretation of the Lagrange multipliers as contact forces exhibits several advantages. For instance, in thermoelasticity the modeling of the heat induction generated by the frictional contact can directly be realized, using the Lagrange mulitplier associated to the frictional condition. It represents the tangential forces which are proportional to the heat induction.

One of the aims of this work is to show the applicability of the proposed approach with the help of several benchmark examples of dynamic contact. We focus on the stability of the Lagrange multipliers in space and time as well as the conservation of energy. Since the time stepping scheme is slightly dissipative due to the stabilization step, we especially study the dependence of the loss of energy w.r.t. the discretization parameters. Finally, we consider an NC-shape grinding process of free formed surfaces with a toroid grinding wheel as a realistic example of a dynamic contact problems in 3D, which includes all discussed effects.

The article is organized as follows: In Section 2, we introduce some notations and the general mixed formulation of contact problems. Moreover, we propose a discretization of higher-order with finite elements and introduce some solution schemes to solve the resulting systems. In the remaining part of this paper, our aim is to capture contact, friction, damping and thermoelasticity using this general mixed formulation. We show that dynamic contact problems with all these different attributes have, in principle, the same structure in the setting of the mixed method. The first example is a dynamic contact problem of Signorini-type with damping and Tresca friction, which can easily be extended to Coulomb friction. It is discussed in Section 3. We introduce the time discretization using the stabilized Newmark approach and formulate the resulting contact problems in each time step in the sense of the mixed method. We present numerical experiments and show the applicability of the theoretical findings. Section 4 focuses on dynamic contact problems in thermoelasticity where heat generated by friction is fully coupled with linear elastic deformation. Again, we use the stabilized Newmark approach to discretize the contact problem and the Crank–Nicholson scheme for the heat propagation. As in the previous section, we use the mixed method in each time step. We examine numerical experiments again, but now with a thermoelastic coupling. The NC-Shape grinding process is considered in Section 5. To model this process, we have to take friction, thermoelastic coupling and rotational effects into account. In particular, the rotational effects are included in the discretization scheme by an arbitrary Lagrangian Eulerian (ALE) ansatz. Even though this problem is highly complex and includes very different physical phenomena, it is, nevertheless, possible to bring it into the proposed framework of mixed methods. We conclude the article with a discussion of the results and an outlook to future works.

Section snippets

The general mixed method

In this section we present a general mixed method for problems with geometrical and frictional contact. The method is general in the sense that a general bilinear form a and a general linear form on some Sobolev spaces are introduced. Whenever a certain (sub-) problem has the specific form of an energy minimization problem or a variational inequality of second kind describing geometrical contact and/or friction, this general mixed method can be applied to obtain the subsequently proposed

Dynamic Signorini problems with friction

As the first contact problem, we arrange frictional contact problems with Rayleigh damping into the general framework introduced in the last section. In contrast to static problems, the frictional constraint is defined with respect to the velocity and not to the displacements.

Dynamic thermomechanical contact problems

A portion of the energy, dissipated by frictional effects, generates heat, which is induced into the bodies in contact. In this section, we introduce a model for this physical process and extend the discretization techniques presented in the previous sections.

An example from production engineering

To show the applicability of the proposed discretization schemes, we discuss a realistic process in production engineering: The NC-shape grinding process of free formed surfaces with a toroid grinding wheel. A subproblem in the simulation of the grinding process is to simulate dynamic thermomechanical contact with Rayleigh damping. A detailed survey of the engineering process and its simulation is given in [48]. Here, we extend the model by frictional and thermal effects. The simulation of such

Conclusions and outlook

In this paper, we have presented a discretization scheme for dynamic contact problems including damping, frictional, thermal, as well as rotational effects using higher-order finite element methods in space. The mixed scheme of higher order in space known for static contact problems is combined with the stabilized Newmark method in time, which has only been studied in conjunction with low order finite elements, leading to an appropriate discretization of higher order for dynamic contact

Acknowledgments

This research work was supported by the Deutsche Forschungsgemeinschaft (DFG) within the Priority Program 1480, Modelling, Simulation and Compensation of Thermal Effects for Complex Machining Processes, in the Grants BL 256/11-2 and SCHR 1244/2-2.

References (51)

  • P. Wriggers et al.

    Finite element formulation of large deformation impact-contact problems with friction

    Comput. Struct.

    (1990)
  • I. Babuska et al.

    Efficient preconditioning for the p-version finite element method in two dimensions

    SIAM J. Numer. Anal.

    (1991)
  • K.J. Bathe et al.

    A solution method for static and dynamic analysis of three-dimensional contact problems with friction

    Comput. Struct.

    (1986)
  • H. Blum et al.

    Space adaptive finite element methods for dynamic obstacle problems

    ETNA, Electron. Trans. Numer. Anal.

    (2008)
  • H. Blum et al.

    Space adaptive finite element methods for dynamic Signorini problems

    Comput. Mech.

    (2009)
  • D.E. Carlson

    Linear thermoelasticity

  • A. Czekanski et al.

    Optimal time integration parameters for elastodynamic contact problems

    Commun. Numer. Math. Eng.

    (2001)
  • A. Czekanski et al.

    Analysis of dynamic frictional contact problems using variational inequalities

    Finite Elem. Anal. Des.

    (2001)
  • L.F. Demkowicz

    Computing with hp-adaptive finite elements

    (2007)
  • P. Deuflhard et al.

    A contact-stabilized Newmark method for dynamical contact problems

    Int. J. Numer. Methods Eng.

    (2008)
  • T. Dickopf et al.

    Efficient simulation of multi-body contact problems on complex geometries: a flexible decomposition approach using constrained minimization

    Int. J. Numer. Methods Eng.

    (2009)
  • D. Doyen et al.

    Time-integration schemes for the finite element dynamic Signorini problem

    SIAM J. Sci. Comput.

    (2011)
  • H. Ehlich et al.

    Schwankungen von Polynomen zwischen Gitterpunkten

    Math. Z.

    (1964)
  • L.C. Evans

    Partial Differential Equations

    (1998)
  • P.E. Gill et al.

    SNOPT: an SQP algorithm for large-scale constrained optimization

    SIAM J. Optim.

    (2002)
  • Cited by (0)

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