Adaptive Multigrid Methods for Signorini’s Problem in Linear Elasticity

Contact problems in computational mechanics are of significant importance for a variety of practical applications. Examples are metal forming processes, crash analysis, the design of gear boxes, bearings, car tires or implants in biomechanics. Here, we focus on Signorini’s problem in linear elasticity [10, 21, 22, 29] describing the linearized frictionless contact of an elastic body with a rigid foundation. Even in this simple case, the construction of fast, reliable solvers is far from trivial, due to the intrinsic non–differentiable nonlinearity of the problem. Regularization techniques [7, 11, 13, 14] require careful handling of regularization parameters in order to find a reasonable compromise between efficiency and accuracy. Dual techniques (cf. e.g. [5, 13, 14, 15]) are based on saddle point formulations incorporating the constraints by means of Lagrange multipliers. Active set strategies [2, 16, 18, 20] iteratively provide approximations of the contact set. A linear subproblem with given contact set has to be solved in each iteration step and multigrid methods are typically used for this purpose. An active set strategy with inexact linear solver has been proposed by Dostál [8]. Recently, Schöberl [27, 28] has developed an approximate variant of the projection method (cf. e.g. [13, p. 5]) using a domain decomposition preconditioner and a linear multigrid solver on the interior nodes. Mesh–independent convergence rates are proved, provided that the number of interior nodes is growing with higher order than the number of nodes on the Signorini boundary. Several authors have applied multigrid techniques to scalar obstacle problems directly (see e.g. [4, 6, 12, 16, 19, 23, 26]). Block versions of these methods can be applied in linear elasticity, provided that normal directions are constant along the Signorini boundary. New difficulties arise, as soon as spatially varying normals occur. In this case, the slip conditions at the contact boundary can not be represented on coarse grids. In this paper we use a direct approach as introduced in [23, 26]. Our algorithm does not involve any regularization or dual formulation and should be considered as a descent method rather than an active set strategy. The basic idea is to minimize the energy on suitably selected d–dimensional subspaces, where d = 2, 3 is the dimension of the deformed body. In this way, we obtain nonlinear variants of successive subspace correction methods in the sense of Xu [34]. See e.g. [9, 30] for a similar approach to smooth nonlinear problems. Well–known projected block Gauß–Seidel relaxation is recovered by choosing the d–dimensional subspaces spanned by the fine grid nodal basis functions associated with a fixed node. In order to increase convergence speed by better representation of the low–frequency components of the error, we additionally minimize on subspaces spanned by functions with large support. The suitable selection of these coarse grid spaces is crucial for the efficiency of the resulting method. Our choice is based on sophisticated modifications of the multilevel nodal basis. Straightforward implementation of the resulting algorithm requires additional prolongations in order to check the constraints prescribed on the fine grid. As a consequence, the complexity of one iteration step is O(nJ lognJ) for uniformly refined triangulations and might be even O(nJ ) in the adaptive case. Optimal complexity of the multigrid V–cycle is recovered by approximating fine grid constraints on coarser grids using so–called monotone restrictions. This modification may slow down convergence, as long as the algebraic error is too large. In our numerical experiments we observed that initial iterates as provided by nested iteration are usually accurate enough to provide fast convergence throughout the whole iteration process. Our approach can be extended to more complicated situations like elastic contact or contact with friction. This will be the subject of forthcoming work. The paper is organized as follows. First we give a brief introduction to Signorini’s problem. A general framework for our method including basic convergence results is presented in Section 3. In particular, it turns out that the discrete coincidence set is detected in a finite number of steps, if the given discrete problem is non–degenerate. Then, our nonlinear iteration automatically becomes a linear