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In this paper, we consider the behaviour of solutions to a class of fully history-dependent variational-hemivariational inequalities with respect to the perturbation of the data. First, the existence and uniqueness of the solution to a class of fully history-dependent variational-hemivariational inequalities is obtained by using a fixed point theorem. Second, we obtain continuous dependence result of solutions with respect to all the data of variational-hemivariational inequalities. Meanwhile, the convergence results of the solutions for the special case of abstract variational inequalities are also given. Finally, to illustrate our main results, we consider a class of viscoelastic contact problem with a long memory. By using our abstract result, we get the continuous dependence of the solutions to frictional contact problem with respect to all the data. [ABSTRACT FROM AUTHOR]

In this study, a novel approach is proposed by integrating the finite element tearing and interconnecting (FETI) method into the B-differentiable equations (BDEs) method for the analysis of 3D elastic frictional contact problem with small deformations. The contact blocks are divided into several nonoverlapping substructures with nonconforming meshes on the contact surface and the interface between two adjacent substructures. The enforcement of contact conditions and interface continuity conditions is achieved by using dual Lagrange multipliers discretized on the slave surface, typically defined with fine meshes. The modified Boolean transformation matrix is utilized to convert the contact stress into the equivalent nodal force. For large-scale elastic contact problems, the equilibrium equations for substructures and the relationship between the relative displacements and contact stresses on the contact surfaces and interfaces (i.e., the contact flexibility matrix) are efficiently computed using the FETI method. Subsequently, the governing equations consisting of the contact equations, interface continuity equations, and equilibrium equations for each floating substructure are uniformly formulated as the BDEs. These BDEs can be solved using the B-differentiable damped Newton method (BDNM). The proposed method harnesses the parallel scalability of the FETI method and extends the applicability of the BDEs algorithm, benefiting from its ability to precisely satisfy the contact constraints and theoretically ensure convergence when solving large-scale contact problems. The Hilber/Hughes/Taylor (HHT) time integration scheme is employed to investigate elastic dynamic contact problems. Numerical examples demonstrate the accuracy, convergence rate, and parallel scalability of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper we investigate a class of time-dependent quasi-variational-hemivariational inequalities (TDQVHVIs) of elliptic type in a reflexive separable Banach space, which is characterized by a constraint set depending on a solution. The solvability of the TDQVHVIs is obtained by employing a measurable selection theorem for measurable set-valued mappings, while the uniqueness of solution to the TDQVHVIs is guaranteed by enhancing the assumptions on the data. Then, under additional hypotheses, we deliver a continuous dependence result when all the data are subjected to perturbations. Finally, the applicability of the abstract results is illustrated by a frictional elastic contact problem with locking materials for which the existence and stability of the weak solutions is proved. [ABSTRACT FROM AUTHOR]

The mathematical model of multilayer viscoelastic system based on the actual structure of asphalt pavement and the properties of its numerical solutions are studied. Firstly, based on the partial differential equations, the variational inequality model of the multilayer viscoelastic system is derived, and the existence and uniqueness of its solutions are further proved. Then, based on the finite element theory, the convergence property and error analysis of the semi‐discrete numerical solutions of the variational inequalities are further studied. Finally, the convergence and error analysis of the fully discrete numerical solutions of the variational inequalities are derived by the forward difference method. The conclusion of the above error analysis also confirms the feasibility of studying the mechanical response of asphalt pavement based on variational inequality. [ABSTRACT FROM AUTHOR]

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element f ∈ X . Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence { u n } ⊂ X , which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws. [ABSTRACT FROM AUTHOR]

In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities arising in contact problems. A fully discrete scheme is introduced for the inequality problem, in which the history-dependent operator is approximated by the trapezoidal rule and the spatial variable is approximated by the lowest-order virtual element method. An optimal order error estimate is derived under appropriate solution regularity assumptions. Numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element f∈X. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence {un}⊂X, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws.

• New methods for frictional contact problem were developed and validated. • A model smoothing method for non-smooth dynamics is proposed. • A normalized method is employed to facilitate convergence. • A co-simulation method is presented to set the dynamic model of overhead crane. • Stick-slip and detachment behavior are detected under strong earthquake. Stacked multibody systems, exemplified by overhead cranes in nuclear power plants (OCNPP), play a crucial role in engineering applications. Under strong earthquakes, non-smooth behaviors, including detachment and stick-slip between wheels and rails, may induce uncertain motions, potentially triggering catastrophic accidents. In this study, a co-simulation method using ANSYS and MATLAB is proposed for modeling the stacked components. The wheel-rail relationship is modeled with multiple frictional unilateral constraints, which are delineated by a set-valued force law. The resultant contact-impact problem is formulated as a nonlinear complementarity problem, and then reconstructed and rescaled using a model smoothing method combined with a normalization technique to facilitate convergence. Lastly, numerical simulations are conducted, taking into account various wheel locking conditions, to explore the nonlinear behavior and stress distribution. The results illustrate the non-smooth characteristics of frictional contact within the stacked components. The presented work is expected to guide the structural optimization of stacked multibody systems. [ABSTRACT FROM AUTHOR]

This article studies a boundary element method for dynamic frictional contact between linearly elastic bodies. We formulate these problems as a variational inequality on the boundary, involving the elastodynamic Poincaré–Steklov operator. The variational inequality is solved in a mixed formulation using boundary elements in space and time. In the model problem of unilateral Tresca friction contact with a rigid obstacle we obtain an a priori estimate for the resulting Galerkin approximations. Numerical experiments in two space dimensions demonstrate the stability, energy conservation and convergence of the proposed method for contact problems involving concrete and steel in the linearly elastic regime. They address both unilateral and two-sided dynamic contact with Tresca or Coulomb friction. [ABSTRACT FROM AUTHOR]

In the present paper, we study inverse problems for a class of nonlinear hemivariational inequalities. We prove the existence and uniqueness of a solution to inverse problems. Finally, we introduce an inverse problem for an electro-elastic frictional contact problem to illustrate our results. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion. [ABSTRACT FROM AUTHOR]

In this paper, we consider a penalty method for a class of differential hemivariational inequalities in reflexive Banach spaces, governed by a set of constraints. First, we recall the existence and uniqueness result of the differential hemivariational inequality. Further, we introduce a penalized problem without constraints and we prove that, as a penalty parameter tends to zero, the solution of the original inequality can be approached to the solution of the penalized problem. Finally, we illustrate our results by an example of a contact problem for which our abstract results can be applied. [ABSTRACT FROM AUTHOR]

Abstract Frictional contact problems in linear elasticity are considered in this paper. The contact constraint is imposed in the weak sense using the fixed point method, which leads to a variational equation problem. For solving such a nonlinear variational problem, we study two projection methods using different self-adaptive rules. Based on the self-adaptive projection method, we propose a modified self-adaptive rule which is more effective to update the parameter. The methods can be implemented easily in conjunction with the boundary element method for the solution. Numerical experiments are reported to illustrate theoretical results.

This paper is devoted to the study of a new class of implicit state-dependent sweeping processes with history-dependent operators. Based on the methods of convex analysis, we prove the equivalence of the history/state dependent implicit sweeping process and a nonlinear differential equation, which, through a fixed point argument for history-dependent operators, enables us to prove the existence, uniqueness, and continuous dependence of the solution in a very general framework. Moreover, we present some new convergence results with respect to perturbations in the data, including perturbations of the associated moving sets. Finally, the theoretical results are applied to prove the well-posedness of a history-dependent quasi-static contact problem. [ABSTRACT FROM AUTHOR]

The main contributions of this article are twofold. Firstly, we develop a reliable and efficient a posteriori error estimator for a class of discontinuous Galerkin methods for the frictional contact problem with reduced normal compliance (which is modeled as a quasi-variational inequality). Secondly, we establish medius analysis (a priori error analysis under the minimal regularity assumption on the exact solution) in the energy norm for the underlying problem. Numerical results substantiate the convergence behavior of the error over uniform mesh and performance of a posteriori error estimator over adaptive mesh. [ABSTRACT FROM AUTHOR]

Abstract The purpose of this paper is to investigate a class of generalized mixed hemivariational–variational inequalities of elliptic type in a Banach space. A well-posedness result for the inequality is obtained, including existence, uniqueness, and stability of solution. The approach is based on a set-valued fixed point theorem combined with the theory of nonsmooth analysis. An elastic contact problem with the constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, in which the contact boundary conditions are described by a unilateral frictionless contact condition, and a bilateral multivalued frictional contact law. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a class of sweeping processes with history-dependent operators. A well-posedness result is obtained, including the existence, uniqueness, and stability of the solution. Our approach is based on the variable time step-length discrete approximation method combined with a fixed point principle for history-dependent operators. Then, a quasi-static frictional contact problem for viscoelastic materials with unilateral constraints in velocity is considered. The abstract result is applied in the study of this problem in order to provide its unique weak solvability as well as the continuous dependence of the solution with respect to the initial data. [ABSTRACT FROM AUTHOR]

Abstract The goal of the paper is to study the behavior of solutions to the history-dependent variational–hemivariational inequalities under perturbation of the data. First, we give an existence and uniqueness result which has been obtained by using previous works. Second, we deliver a continuous dependence result when all the data are subjected to perturbations. Finally, a quasistatic viscoelastic frictional contact problem is presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

A system which couples an abstract hemivariational inequality of hyperbolic type and an evolution equation in a Banach space is studied. The global existence of the system is established by exploiting the Rothe method. An application to a dynamic adhesive viscoelastic contact problem with friction is provided for which results on existence and regularity of weak solutions are proved. [ABSTRACT FROM AUTHOR]

The large deformation frictional contact between shell structures is commonplace in engineering applications like the sheet metal forming process. Due to the specific assumptions of shell structures, the numerical calculation of shell contact is more complicated than the computation of contact between normal two or three dimensional solids. In recent years, the Kirchhoff–Love shell has become one of the most popular mathematical shell models under the context of isogeometric analysis. In this paper, we detailedly formulate the frictional contact contributions for Kirchhoff–Love shell structures considering both stick and slide states. The variation and linearization of contact variables considered to be sophisticated are explicitly derived and plainly organized. The general nonlinear hyperelastic materials are utilized to describe the constitutive relations of shell structures undergoing large deformation. The normal and tangential contact constraints are enforced using the classical penalty method. Based on the precise representation of shell structures using NURBS functions, we developed an effective contact detection algorithm to accurately capture the contact areas between shells. A computation framework for solving Kirchhoff–Love shell frictional contact problems is established in the context of NURBS-based isogeometric analysis. Several shell contact examples are calculated, and the obtained results are compared with those of the classical finite element method. The convergence and accuracy of the presented method are verified in these examples. [Display omitted] • Formulas for shell's large deformation frictional contact in IGA have been derived. • The contact detection algorithm based on the shell's actual surfaces is explored. • Numerical examples of shell contact using the penalty method are analyzed in IGA. • FEA results from ABAQUS are also calculated as comparisons with IGA results. [ABSTRACT FROM AUTHOR]

In this paper we study a class of hyperbolic variational inequalities without a term depending on the first order derivative. Results on existence, uniqueness and regularity of a solution to the variational inequality are provided through the Rothe method. A frictional dynamic contact problem for viscoelastic material with noncoercive viscosity and subdifferential boundary conditions is studied as an illustrative application. [ABSTRACT FROM AUTHOR]