Solving dynamic contact problems with local refinement in space and time

Abstract

Frictional dynamic contact problems with complex geometries are a challenging task – from the computational as well as from the analytical point of view – since they generally involve space and time multi-scale aspects.

To be able to reduce the complexity of this kind of contact problem, we employ a non-conforming domain decomposition method in space, consisting of a coarse global mesh not resolving the local structure and an overlapping fine patch for the contact computation. This leads to several benefits: First, we resolve the details of the surface only where it is needed, i.e., in the vicinity of the actual contact zone. Second, the subproblems can be discretized independently of each other which enables us to choose a much finer time scale on the contact zone than on the coarse domain. Here, we propose a set of interface conditions that yield optimal a priori error estimates on the fine-meshed subdomain without any artificial dissipation. Further, we develop an efficient iterative solution scheme for the coupled problem that is robust with respect to jumps in the material parameters. Several complex numerical examples illustrate the performance of the new scheme.

Introduction

The numerical simulation of dynamic contact problems plays an important role in many applications in mechanics, like the forming of sheet metal, crash tests and other examples of structures under impact, fracture dynamics, or tire rolling. Especially the latter application is a challenging task from the point of view of simulation, as the problem usually features a complex three-dimensional geometry, nonlinear elastic materials as well as dynamic effects. In addition, the contact zone is usually quite small compared to the size of the tire but needs to be resolved very accurately to get a good picture of the evolution of the contact pressures during rolling contact. More importantly, many other multiscale contact simulations require local refinement in the vicinity of the contact zone which is a priori unknown.

In order to be able to perform an accurate simulation of a car tire, there is a huge demand for a sound numerical scheme that combines a suitable multi-scale discretization in space and in time with an efficient solution algorithm for the frictional contact conditions. The algorithm has to be robust with respect to jumps in the material parameters as well as to provide an energy-consistent description of the dynamics. The aim of this work is to design such an algorithm by combining several well-established mathematical methods with some new approaches. All of them are described in the following.

The basis of the algorithm is provided by a decomposition of the original structure into several overlapping subdomains which have different mesh sizes. The global d-dimensional structure, d ∈ {2, 3}, is discretized with a relatively coarse mesh that does not resolve the details along the contact boundary, whereas at the contact area, an overlapping patch with a fine triangulation is introduced. A two-dimensional example of such a geometry is sketched in Fig. 1. In order to avoid an expensive volume coupling, the transfer between the subdomains is only performed at the inner (d − 1)-dimensional interface. Here, we employ the variationally consistent mortar method (see, e.g., [1], [2]) with dual Lagrange multipliers [3] to enforce the weak continuity of the traces.

The subject of domain decomposition methods is already well-established in the literature; we refer to [4], [5], [6] and the references therein for an overview of the topic. The construction of domain decomposition schemes which are robust with respect to the mesh size as well as to jumps in the material parameters has been the topic of several papers (e.g., [7], [8], [9]). In this work, we make use of the overlapping decomposition in order to obtain an iterative solution scheme whose convergence rate is bounded independently of the mesh size or the Lamé parameters in the subdomains.

The next important item is the treatment of the contact inequality constraints (see [10], [11], [12], [13], [14] and the references therein for an overview of the topic). These conditions are enforced in a variationally consistent way using dual Lagrange multipliers, allowing for the application of a primal–dual active set strategy [15], [16], [17]. This scheme can be interpreted as a semismooth Newton method [15] applied to a set of nonlinear equations describing the contact conditions [18], [19]. In combination with the iterative subdomain coupling, an inexact Newton scheme is obtained which still shows superlinear local convergence provided that appropriate stopping criteria for the inner iteration are satisfied [20], [21], [22].

The incorporation of inertia effects into the formulation makes the simulation of the nonlinear contact problem even more challenging. Standard time stepping algorithms like the trapezoidal rule generally lose their energy conservation property if applied to nonlinear problems. Possible remedies are presented in, e.g., [23], [24], [25] for nonlinear material laws and in [26], [27] for contact. But even with the energy-consistent formulation of [27], the computed results for the contact stresses show spurious oscillations in time. This is avoided employing a local modification of the mass matrix along the potential contact boundary [28], [29], [30].

The last feature of the algorithm is the possibility to use different time step sizes in the subdomains. Here, the main challenge lies in the construction of the interface constraints. A suboptimal choice can lead to numerical instability or to artificial dissipation at the interface, even if the time integrators in the subdomains are stable and energy-conserving [31], [32], [33]. In [34], [35], a time substepping scheme with linear interpolation of the velocity and the Lagrange multipliers has been proposed which is stable but dissipative at the interface. An improved energy-conserving scheme with linear interpolation of the multipliers has been analyzed in [36], [37]. However, both methods rely on the expensive exact solution of the resulting coupled system. In contrast, we present a time-discrete coupled system that uses different time step sizes in each subdomain, conserves the energy over a coarse time step and does not require the exact solution of the coupled problem.

At this point, one might ask whether the algorithm presented in this work possibly is just a combination of known numerical techniques. As a return, we recall that we aim for formulating an algorithm that is able to solve challenging dynamic contact problems with complex local geometries. This cannot be achieved without building upon the knowledge and experience gained in the above mentioned topics of numerical simulation, i.e., we combine domain decomposition with mortar coupling, contact modeling via semismooth Newton methods and energy-consistent time integration. However, the algorithm presented in this work also contains important new aspects. The main novelty is the consistent use of the overlapping domain decomposition approach for both the discretization and the solver, leading to a two-scale formulation both in space and in time which has been implemented with contact constraints and dynamic terms. This is complemented by an extensive theoretical and numerical investigation of the properties of the resulting method.

We now turn to the structure of the rest of this work. In Sections 2 Problem formulation for the linear setting, 3 Iterative coupling algorithm, 4 Numerical results for the linear setting, we present and investigate the proposed algorithm for a linear dynamic problem. Section 2 introduces the notation and the governing equations as well as the algebraic formulation of the resulting fully coupled system. In Section 3, we describe an efficient iterative solution scheme based on overlapping domain decomposition and analyse its convergence rate. Section 4 numerically confirms the theoretical results by means of several tests.

In Section 5, we extend the domain decomposition approach to nonlinear problems, leading to an inexact semismooth Newton method. A special focus is put on the approximation of the frictional contact conditions. Section 6 contains several numerical examples illustrating the efficiency and the robustness of the resulting iterative scheme.

The case of different time step sizes is analyzed in Section 7, where we present an energy-conserving formulation that can efficiently be solved by the iterative scheme used before. The numerical results in Section 8 illustrate that the scheme can considerably decrease the number of global systems to be solved and still provide a good local accuracy. Section 9 concludes the work with a short summary.