Stabilized low-order finite elements for frictional contact with the extended finite element method

Abstract

Contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity, in which the normal contact pressure exhibits spurious oscillation. This oscillation does not go away with mesh refinement, and in some cases it even gets worse as the mesh is refined. Using a Lagrange multipliers formulation we trace this problem to non-satisfaction of the LBB condition associated with equal-order interpolation of slip and normal component of traction. In this paper, we employ a stabilized finite element formulation based on the polynomial pressure projection (PPP) technique, which was used successfully for Stokes equation and for coupled solid-deformation–fluid-diffusion using low-order mixed finite elements. For the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method. We use low-order linear triangular elements (tetrahedral elements for 3D) for both slip and normal pressure degrees of freedom, and show the efficacy of the stabilized formulation on a variety of plane strain, plane stress, and three-dimensional problems.

Introduction

There exists a large body of literature addressing the computational aspects of contact problems in nonlinear solid mechanics using the finite element (FE) method (see [25], [48], [51] and references therein). A challenging aspect of the problem is the enforcement of the contact condition, whether it be in the context of classical nonlinear contact mechanics in which element sides are aligned to the contact faces [26], [36], [37], [39], [40], [42], [49], [50], or in the framework of the assumed enhanced strain or extended FE method in which contact faces are allowed to pass through and cut the interior of finite elements [2], [3], [8], [11], [12], [13], [17], [18], [19], [20], [23], [24], [27], [28], [29], [31], [34], [35], [38]. The contact condition inhibits interpenetration of the contact faces, as well as requires that the contact pressure be strictly nonnegative. Mathematically, these constraints are represented by classical Karush–Kuhn–Tucker (KKT) conditions in nonlinear programming, which is a generalization of the method of Lagrange multipliers to inequality constraints. Frictional contact adds complexity to the problem in that a second layer of KKT conditions is necessary to describe stick-slip conditions for the case when the frictional faces are in contact mode [8], [19], [24], [27], [28], [29].

The FE method provides a natural tool for simulating frictionless and frictional contact problems. If the contact faces are well defined prior to the beginning of the simulation, then one can simply employ the standard nonlinear contact mechanics approach by aligning element sides with the contact surfaces [25], [26], [36], [37], [39], [40], [42], [48], [49], [50], [51]. However, if the contact faces are not a priori given and are expected to evolve in an unknown fashion during the course of the simulation, then an extended FE method would be more appropriate [3], [4], [19], [29], [33]. The latter approach is generally more robust since it permits the use of instability models [5], [9], [10], [30], [41], [46] to propagate a discontinuity in any direction and at any point in the solution. In either case, exact satisfaction of the contact constraint may be achieved with a formulation based on the Lagrange multipliers method. Approximate satisfaction of the contact constraint also may be imposed by the penalty method particularly for the more complex problem of frictional sliding.

Irrespective of whether one employs the Lagrange multipliers or penalty method, it is generally recognized that certain combinations of discrete interpolation spaces for solid-displacement and normal contact pressure exhibit numerical instability in the form of spurious oscillation in the normal contact pressure. Typically, these oscillations are more pronounced with the Lagrange multipliers method, where contact constraints are imposed exactly, than with penalty method, where contact constraints are imposed only approximately. Oscillation is somewhat reduced by reducing the values of the penalty parameter, but at the expense of accuracy in the form of significant interpenetration of contact faces. Furthermore, the oscillation does not go away with mesh refinement, and in some cases it even gets worse as the mesh is refined.

Some numerical strategies have been proposed in the literature to address the problem of contact pressure oscillation. Existing stabilized methods include Nitsche's method [38], bubble stabilization [17], [18], [34], and reduced Lagrange methods [2], [24], [31]. Bubble stabilization technique introduces additional unknowns, although they can be statically condensed within the element level. It has been shown in [43] that bubble stabilization method is closely related to Nitsche's approach. Unfortunately, the performance of bubble stabilization methods in frictional contact problem has not yet been reported. Mortar method has also been used [24] to address the over-constrained contact problem by reducing the integration points on the interface; however, this approach relies on a heuristic argument for discretizing the interface. In [2], [31], a stabilized Lagrange space is designed to satisfy the Ladyzhenskaya–Babuska–Brezzi (LBB) condition [1], [14], the basic idea being to reduce the number of Lagrange multipliers by certain rules. However, constructing such a stable Lagrange space is quite complicated, and its FE implementation is not trivial particularly in 3D.

We identify the source of the contact pressure oscillation from failure of the discrete subspaces to satisfy the LBB stability condition similar in spirit to the Stokes problem [21], [22], [44]. Specifically, certain combinations of discrete subspaces for slip and contact pressure degrees of freedom, particularly those arising from low-order FE interpolations, result in unstable behavior in the form of contact pressure oscillation. Recently, Bochev et al. [6], [7], [16] quantified the deficiency of some of these low-order mixed finite elements, and proposed a stabilized method aimed at addressing this deficiency. The idea is embodied in so-called polynomial pressure projection (PPP) stabilization, which they used successfully for the Stokes problem. More recently, White and Borja [47] used a similar approach for coupled solid-displacement/fluid-diffusion problem. An analysis of similar pressure projection methods along with a unifying framework for their analysis has also been proposed by Burman [15].

In this paper, we utilize the same PPP technique for stabilizing the frictional contact problem using equal low-order (triangular) interpolations for slip and contact pressure degrees of freedom. Formulation is done with the Lagrange multipliers method for frictionless contact, and with the penalty method for frictional contact. We are not aware of any work in the literature dealing with the implementation of the PPP technique within the framework of the penalty method, and thus, apart from the novel use of this particular technique for the contact problem, we also demonstrate how this technique may be combined with the penalty approach itself. An advantage of the PPP stabilization approach is that the additional stabilizing terms can be assembled locally on each element using standard shape function information, so they introduce minimal additional computational work. Furthermore, the technique is highly suitable for low-order interpolation of displacement and contact pressure fields. To accommodate an evolving slip surface geometry, we implement the stabilized technique in the framework of the extended FE method.