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Abstract This paper is devoted to numerical analysis of history-dependent variational– hemivariational inequalities arising in contact problems for viscoelastic material. We introduce both temporally semi-discrete approximation and fully discrete approximation for the problem, where the temporal integration is approximated by a trapezoidal rule and the spatial variable is approximated by the finite element method. We analyze the discrete schemes and derive error bounds. The results are applied for the numerical solution of a quasistatic contact problem. For the linear finite element method, we prove that the error estimation for the numerical solution is of optimal order under appropriate solution regularity assumptions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the existence, the stability and the numerical approximation of a multidimensional dynamic contact problem modeling the evolution of displacement and temperature in a viscoelastic body that may come into contact with a deformable foundation. The viscoelastic body is assumed to behave according to Kelvin–Voigt constitutive law with added thermal effects under the Green–Lindsay theory. We prove that the presence of viscoelastic terms in the equations provides additional regularity and then an existence and uniqueness result is obtained using the Faedo–Galerkin method. An energy decay property is also shown under the assumption of radial symmetry. Then, a numerical approximation based on the finite element method is proposed. A stability result is proved from which the decay of the discrete energy is deduced. A priori error estimates are shown from which the linear convergence is derived under suitable additional regularity conditions. Finally, some numerical experiments are described to support our results. [ABSTRACT FROM AUTHOR]

This paper mainly analyzes the general elliptic variational–hemivariational inequalities with or without constraints by using the virtual element method. The approximations can be internal or external and a C e ́ a's type inequality is derived for a priori error estimates. Then, we apply the results to a variational–hemivariational inequality arising in frictional contact problems, and the optimal order error estimate is obtained for the linear virtual element solution under appropriate solution regularity assumptions. Finally, numerical simulation results are reported to show the performance of the proposed method, in particular, numerical convergence orders are in good agreement with the theoretical predictions. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this work is to propose new contact elements of higher order of differentiability for analysing two-dimensional frictionless contact problems. Several methods were proposed in the literature to solve the problem caused by the lack of continuity resulting from the discretization. Among them are Bézier interpolation, Hermitian interpolation and splines. One of the difficulties in using Hermitian interpolation is to verify the partition of unity. Therefore, new elements that satisfy the and continuity at the interface are presented in this paper. These new contact elements are based on Hermitian polynomials for ensuring a higher order of continuity. The advantage is that this approach can be easily developed not only for linear elements but also for quadratic elements with higher order of differentiability. The performance of different surface representations is assessed through a comparison with a surface discretization. Some numerical examples are used for assessing the accuracy and the convergence behaviour. [Copyright &y& Elsevier]

Abstract: Efficient numerical methods for describing a garment’s mechanical behavior during wear have been identified as the key technology for garment simulation. This paper presents a finite-element mechanical contact model based on Mindlin–Reissner shell theory for a three-dimensional human body and garment. In this model, the human body and the garment are meshed as basic contact cells, these contact cells between the human body and the garment are defined as the contact pair to describe the contact relationship, and the mathematical formulation of the finite-element model is defined to describe the strain–stress performance of the three-dimensional human body and garment system. By using the solution given by the computer code and the programs specifically developed, the calculations of the mechanics in the basic cells of the human body and the garment have been able to be carried out. The simulation results show that the model of rationality, a good simulation results and simulation efficiency. [Copyright &y& Elsevier]

Abstract: In this paper we revisit a quasi-static contact problem of a thermoviscoelastic beam between two rigid obstacles which was recently studied in . The variational problem leads to a coupled system, composed of an elliptic variational inequality for the vertical displacement and a linear variational equation for the temperature field. Then, its numerical resolution is considered, based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. Error estimates are proved from which, under adequate regularity conditions, the linear convergence is derived. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution. [Copyright &y& Elsevier]

Abstract: This paper analyzes a discrete form of 3D contact problems with local orthotropic Coulomb friction and coefficients of friction which may depend on the solution itself. The analysis is based on the fixed-point reformulation of the original problem. Conditions guaranteeing the existence and uniqueness of discrete solutions are established. Finally, numerical results of a model example are presented. [Copyright &y& Elsevier]

Abstract: We present in this paper an improved non-smooth Discrete Element Method (DEM) in 3D based on the Non-Smooth Contact Dynamics (NSCD) method. We consider a three-dimensional collection of rigid particles (spheres) during the motion of which contacts can occur or break. The dry friction is modeled by Coulomb’s law which is typically non-associated. The non-associativity of the constitutive law poses numerical challenges. By adopting the use of the bi-potential concept in the framework of the NSCD DEM, a faster and more robust time stepping algorithm with only one predictor-corrector step where the contact and the friction are coupled can be devised. This contrasts with the classical method where contact and friction are treated separately leading to a time stepping algorithm that involves two predictor-corrector steps. The algorithm has been introduced in a 3D version of the NSCD DEM software MULTICOR. Numerical applications will show the robustness of the algorithm and the possibilities of the MULTICOR software for solving three-dimensional problems. [Copyright &y& Elsevier]

The NSCD method has shown its efficiency in the simulation of granular media. Since the number of particles and contact increases, the shape of the discrete elements becomes more complicated and the simulated problems becomes more complex, the numerical tools need to be improved in order to preserve reasonable elapsed CPU time. In this paper we present a parallelization approach of the NSCD algorithm and we investigate its influence on the numerical behaviour of the method. We illustrate the efficiency on an example made of hard disks: a free surface compaction. [Copyright &y& Elsevier]

Dynamic granular mechanics systems are problems with variable connectivity between the particles. This property makes it difficult for rigourous analysis since it implies reformulation of the problem each time the connectivity changes. In this paper an improved procedure for selecting contact particles is considered for large granular mechanics applications (≃104 particles). The method for the selecting uses a connectivity table. The main claimed advantage of the method is simplicity since it uncouples the system of particle in subsystems. It is shown, for example, that for N particles system, the classic method of selecting requires (N(N+1))/2 operations compared to N for the new approach. [Copyright &y& Elsevier]

This paper presents recent methods to improve numerical simulation of contact problems by smoothing. The main idea is to combine contact surfaces regularization with an automatic adjustment of both penalty parameter and load step. The underlying goal is to provide handle situations frequently met in an industrial context. [Copyright &y& Elsevier]

In this paper, high-order prediction–correction algorithms are presented to solve unilateral contact problems. These algorithms are built up using the asymptotic numerical method which associates a perturbation technique with a numerical one as the finite element method. [Copyright &y& Elsevier]