Variationally consistent discretization schemes and numerical algorithms for contact problems

Abstract

We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-ordera prioriconvergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-baseda posteriorierror estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

Isogeometric dual mortar methods for computational contact mechanics

“…Since NURBS are naturally associated with higher order approximations, the extension of the dual mortar method to second order Lagrange finite elements in [26,27] including optimal a priori error estimates is also worth mentioning. A comprehensive review on dual mortar methods for contact mechanics can be found in [28,29]. In the context of domain decomposition in IGA, optimality and stability of standard mortar methods have only very recently been investigated in [30][31][32][33], where also the construction of dual B-spline basis functions has been outlined theoretically [33].…”

A Higher Order Phase-Field Approach to Fracture for Finite-Deformation Contact Problems

“…The Karush-Kuhn-Tucker conditions are comprised of the impenetrability condition (34), the condition which only allows compressive tractions (35) and the complementarity condition (36). The Karush-Kuhn-Tucker conditions are shown in Figure 5 for both the penalty method and Lagrange multipliers.…”

“…In particular the inequalities given in (34)- (36) are implemented using the max-operator (see [20] for more informations) as follows Therein λ N and Φ N denote the Lagrange multiplier and corresponding constraint, respectively. Note λ N := t N denotes the exact contact traction and Φ N := g N the gap function.…”

“…like the node-to-surface method, the variational consistent Mortar method has been wellestablished (see e.g. [31,15,30,36]). In a recent publication (see [7]) we applied the Mortar method for thermo-mechanical frictional contact problems.…”

Discretization and numerical realization of contact problems for elastic‐perfectly plastic bodies. PART II – numerical realization, limit analysis

“…The respective necessary and sufficient optimality conditions lead to a system of equations in R n involving Lipschitz continuous terms. Such problems are frequently solved by semi-smooth Newton methods, see [6,17,27,29] for a general theory and [7,8,19,21,28,30,36] for applications in plasticity. In this paper two methods are proposed.…”