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The mortar element method is extremely useful in a variety of settings in computational mechanics, often when the optimal connection or “tying” of dissimilarly meshed domains is desired. This paper describes the benefits of applying such methods to a more complex application: large deformation contact analysis. Although the enhanced accuracy of such contact formulations is to be expected given their sound theoretical grounding, it also turns out that the spatial smoothing provided by mortar contact operators lends considerably more robustness than more traditional, node-to-surface approaches. Issues associated with efficient searching in the surface-to-surface framework are discussed, and some examples are summarized which demonstrate the effectiveness of the approach.

Due to their efficiency and robustness, linear multigrid methods lend themselves to be a starting point for the development of nonlinear iterative strategies for the solution of nonlinear contact problems, see, e.g., [3, 1, 10, 5]. One nonlinear strategy is to reduce the contact problem to a sequence of linear problems and to solve each of these by a linear multigrid method. This approach is often connected to active set strategies or semismooth Newton methods [6]. To avoid solving the linear problems exactly, one can use inexact active set strategies, see [7, 8]. The convergence of this inexact strategy depends on the accuracy the inner problem is solved with, see [7], as well as on algorithmic parameters [8]. A second strategy is to deal directly with the nonlinearity within the multigrid method by using, e.g., nonlinear smoothers and nonlinear interpolation operators, see [10, 9, 1]. Using the convex energy for controlling the iteration process, globally convergent nonlinear multigrid methods can be constructed which allow for solving contact problems with the speed of a linear multigrid method [10]. A third possibility is to employ a saddle point approach [3] and to solve for the primal and dual variables simultaneously using an algebraic multigrid method.

The focus of the contribution is on a detailed discussion of the 2D formulation in contact which can be viewed either as a reduction of the general 3D case to a special cylindrical geometry, or as the contact of 2D bodies bounded by plane curves. In addition, typical frictional characteristics, such as the yield surface and the update of the sliding displacements allow a geometrical interpretation in the chosen coordinate system on the contact surface.

In this work, we will presente a comparison of two formulation for the discretization of elastodynamic contact problems. The first approach consists on a midpoint scheme and a contact condition expressed in terms of velocity. This approach gives an energy conserving scheme. The second one we propose is a new distribution of the solid mass. The problem expressed with the new mass matrix is well posed, energy conserving and has a lipschitz solution. Finally, some numerical results are presented.

For the numerical approximation of nonlinear contact problems, mortar methods provide a powerful and efficient tool. To detect the correct contact zone and to decompose it into the sliding and sticking part for Coulomb friction, we use primal-dual active set strategies. Combining these strategies with optimal multigrid methods, we get an efficient inexact approach. The extension of the mortar approach to thermal contact problems, to the case of large deformations and nearly incompressible materials is shown. Numerical results in 2D and 3D are given to illustrate the flexibility of the algorithm.

The combined finite-discrete element method is a new computational method for the simulation of fracturing and fragmenting solids or particulate media where individual particles are deformable. The method combines the power of the finite element method in capturing deformability of solid particles with the power of the discrete element method to accurately represent interaction between individual particles for systems comprising millions of particles. In addition, the method is suitable for the simulation of extensive fracture and/or fragmentation processes. In recent years the method has captured the attention of researchers in a wide field of potential applications and the interest in the method has grown significantly. In this paper key features of the method are explained with special attention being paid to the processing of contact interaction. Also, the new generation of linear time complexity search algorithms has been presented including the NBS and the MR algorithm. In the computational mechanics context, these algorithms are also relevant for the discrete element method, the finite element method and grid generation.

In this work, a dynamic frictionless viscoelastic contact problem is considered. The contact with the foundation is modelled by a normal compliance contact condition. The mechanical damage of the material, caused by excessive stress or strain, is included into the model through a differential inclusion. The weak formulation leads to a nonlinear system including a parabolic variational inequality for the damage field coupled with a variational equation for the displacement field. The existence of a unique weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and a finite difference to discretize the time derivatives. Error estimates are obtained, from which the linear convergence of the scheme, under suitable regularity conditions, can be derived. Finally, some numerical results on a two-dimensional problem are presented to show the performance of the scheme.

During the last years, the construction of energy conserving time integration methods to solve nonlinear elastodynamic problems has attracted the interest of many researchers ([12, 5, 3] .). Furthermore, many works have been devoted to extend these conservative formulations to frictionless impact; more precisely, Laursen and Chawla [9] and Amero and Petocz [2] have shown the interest of the persistency condition to conserve the energy in the discrete framework. But these contributions concede a contact interpenetration which vanishing as the time step tends towards zero. Recently, this drawback is resolved by Laursen and Love [10] by introducing a discrete jump in velocity and by Hauret [6] by considering a specific penalized enforcement of the contact conditions. In this work, we present an energy-conserving algorithm for hyperelastodynamic contact problems which differs from the approaches mentionned above ([10] and [6]); this approach permits to ensure both the Kuhn- Tucker and persistency conditions at the end of each time step. These two laws are enforced during each time increment by using an extended Newton method. In section 2, we recall some general aspects of nonlinear elastodynamic problems with contact and friction. The section 3 permits also to recall the usual energy conserving frameworks used to solve nonlinear elastodynamic problems. In section 4, we present an energy-conserving algorithm to treat impact problems with an extension to frictional dissipation phenomenon. In the last section 5, representative numerical simulations are presented to assess the performance and also to underscore the conservative or dissipative behaviour of the proposed method.

Sensitivity analysis (SA) is developed for three-dimensional multi-body frictional contact problems. The direct differentiation method (DDM) is applied to obtain response sensitivities with respect to arbitrary design parameters (parameter and shape SA). The FE formulation of contact employs smoothing of the master surface, and the augmented Lagrangian technique is used to enforce the contact and friction conditions. Numerical examples, including application for optimization, illustrate the approach.

Thiswork is concernedwith the finite element modeling of contact/impact problems between rubber materials. The developed algorithm, namely here Bi-First, combines the bi-potential method for solution of contact problems and the first order algorithm for integration of the time-discretized equation of motion. Numerical examples are given in two cases: multi-contact problem between Blatz-Ko hyperelastic bodies and Love-Laursen’s test with a novel hyperelastic model.