scholarly journals Numerical analysis of doubly-history dependent variational inequalities in contact mechanics

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wei Xu ◽  
Cheng Wang ◽  
Mingyan He ◽  
Wenbin Chen ◽  
Weimin Han ◽  
...  
Keyword(s):  
Numerical Analysis ◽  
Variational Inequalities ◽  
Contact Mechanics ◽  
Optimal Order ◽  
Left Endpoint ◽  
Linear Element ◽  
Order Error ◽  
Discrete Method ◽  
Fully Discrete ◽  
Convergence Orders

AbstractThis paper is devoted to numerical analysis of doubly-history dependent variational inequalities in contact mechanics. A fully discrete method is introduced for the variational inequalities, in which the doubly-history dependent operator is approximated by repeated left endpoint rule and the spatial variable is approximated by the linear element method. An optimal order error estimate is derived under appropriate solution regularities, and numerical examples illustrate the convergence orders of the method.

Numerical analysis of stationary variational-hemivariational inequalities with applications in contact mechanics

2017 ◽  
Vol 23 (3) ◽  
pp. 279-293 ◽  
Author(s):  
Weimin Han
Keyword(s):  
Finite Element ◽  
Numerical Analysis ◽  
Contact Mechanics ◽  
Type Inequality ◽  
Convergence Result ◽  
Optimal Order ◽  
Hemivariational Inequalities ◽  
Order Error ◽  
Finite Element Approximations ◽  
Solution Regularity

This paper is devoted to numerical analysis of general finite element approximations to stationary variational-hemivariational inequalities with or without constraints. The focus is on convergence under minimal solution regularity and error estimation under suitable solution regularity assumptions that cover both internal and external approximations of the stationary variational-hemivariational inequalities. A framework is developed for general variational-hemivariational inequalities, including a convergence result and a Céa type inequality. It is illustrated how to derive optimal order error estimates for linear finite element solutions of sample problems from contact mechanics.

scholarly journals Variational and numerical analysis of the Signorini′s contact problem in viscoplasticity with damage

2003 ◽  
Vol 2003 (2) ◽  
pp. 87-114 ◽  
Author(s):  
J. R. Fernández ◽  
M. Sofonea
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Mechanical Damage ◽  
Parabolic Type ◽  
Approximate Solutions ◽  
Damage Function ◽  
Optimal Order ◽  
Order Error ◽  
Fully Discrete ◽  
Solution Regularity

We consider the quasistatic Signorini′s contact problem with damage for elastic-viscoplastic bodies. The mechanical damage of the material, caused by excessive stress or strain, is described by a damage function whose evolution is modeled by an inclusion of parabolic type. We provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution of the model. We then introduce and study a fully discrete scheme for the numerical solutions of the problem. An optimal order error estimate is derived for the approximate solutions under suitable solution regularity. Numerical examples are presented to show the performance of the method.

scholarly journals Numerical Analysis of a Sliding frictional contact problem with Normal Compliance and Unilateral Contact

2020 ◽  
Author(s):  
Yahyeh Souleiman ◽  
Mikael Barboteu
Keyword(s):  
Numerical Analysis ◽  
Contact Problem ◽  
Frictional Contact ◽  
Nonlinear Partial Differential Equations ◽  
Viscoelastic Body ◽  
Sliding Contact ◽  
Memory Term ◽  
Optimal Order ◽  
Normal Compliance ◽  
Order Error

Abstract This paper represents a continuation of [15] and [18]. Here, we consider the numerical analysis of a non trivial frictional contact problen in a form of a system of evolution nonlinear partial differential equations. The model describes the equilibrium of a viscoelastic body in sliding contact with a moving foundation. The contact is modeled with a multivalued normal compliance condition with memory term restricted by a unilateral constraint, and is associated to a sliding version of Coulomb's law of dry friction. After a description of the model and some assumptions, we derive a variational formulation of the problem, which consists of a system coupling a variational inequality for the displacement field and a nonlinear equation for the stress field. Then, we introduce a fully discrete scheme for the numerical approximation of the sliding contact problem. Under certain solution regularity assumptions, we derive an optimal order error estimate and we provide numerical validation of this result by considering some numerical simulations in the study of a two-dimensional problem.

scholarly journals The Backward Euler Fully Discrete Finite Volume Method for the Problem of Purely Longitudinal Motion of a Homogeneous Bar

2012 ◽  
Vol 2012 ◽  
pp. 1-23
Author(s):  
Ziwen Jiang ◽  
Deren Xie
Keyword(s):  
Finite Volume Method ◽  
Finite Volume ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Optimal Order ◽  
Longitudinal Motion ◽  
Order Error ◽  
Fully Discrete ◽  
Backward Euler ◽  
Volume Method

We present a linear backward Euler fully discrete finite volume method for the initial-boundary-value problem of purely longitudinal motion of a homogeneous bar and an give optimal order error estimates inL2andH1norms. Furthermore, we obtain the superconvergence error estimate of the generalized projection of the solutionuinH1norm. Numerical experiment illustrates the convergence and stability of this scheme.

scholarly journals Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ailing Zhu
Keyword(s):  
Error Analysis ◽  
Error Estimate ◽  
Optimal Order ◽  
Triangular Meshes ◽  
Element Space ◽  
Order Error ◽  
First Order ◽  
Fully Discrete ◽  
Mixed Element

The semidiscrete and fully discrete discontinuous mixed covolume schemes for the linear parabolic integrodifferential problems on triangular meshes are proposed. The error analysis of the semidiscrete and fully discrete discontinuous mixed covolume scheme is presented and the optimal order error estimate in discontinuousH(div)and first-order error estimate inL2are obtained with the lowest order Raviart-Thomas mixed element space.

scholarly journals Discontinuous Mixed Covolume Methods for Parabolic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ailing Zhu ◽  
Ziwen Jiang
Keyword(s):  
Error Analysis ◽  
Error Estimate ◽  
Error Estimates ◽  
Parabolic Problems ◽  
Optimal Order ◽  
Order Error ◽  
First Order ◽  
Fully Discrete ◽  
Optimal Order Error Estimates ◽  
Backward Euler

We present the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume schemes for parabolic problems on triangular meshes. We give the error analysis of the discontinuous mixed covolume schemes and obtain optimal order error estimates in discontinuousHdivand first-order error estimate inL2.

Numerical analysis of the energy-dependent radiative transfer equation

2018 ◽  
Vol 39 (3) ◽  
pp. 1529-1562
Author(s):  
Kenneth Czuprynski ◽  
Joseph Eichholz ◽  
Weimin Han
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Optimal Order ◽  
Well Posedness ◽  
Order Error ◽  
Fully Discrete ◽  
Dependent Form ◽  
Energy Dependent ◽  
Theoretical Results

Abstract The energy-dependent form of the radiative transfer equation (RTE) is important in a variety of applications. In a previous paper the well-posedness and an energy discretization for the energy-dependent RTE were investigated. In this paper the fully discrete scheme is introduced and analysed. Optimal-order error estimates and a well-posedness analysis of the discrete system are provided. The theoretical results are validated through numerical examples.

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

A New Weak Galerkin Finite Element Scheme for the Brinkman Model

2016 ◽  
Vol 19 (5) ◽  
pp. 1409-1434 ◽  
Author(s):  
Qilong Zhai ◽  
Ran Zhang ◽  
Lin Mu
Keyword(s):  
Finite Element ◽  
Three Dimensional ◽  
Optimal Order ◽  
Brinkman Model ◽  
Weak Galerkin ◽  
Order Error ◽  
Brinkman Equations ◽  
Generalized Distributions ◽  
Variational Form ◽  
Weak Derivatives

AbstractThe Brinkman model describes flow of fluid in complex porous media with a high-contrast permeability coefficient such that the flow is dominated by Darcy in some regions and by Stokes in others. A weak Galerkin (WG) finite element method for solving the Brinkman equations in two or three dimensional spaces by using polynomials is developed and analyzed. The WG method is designed by using the generalized functions and their weak derivatives which are defined as generalized distributions. The variational form we considered in this paper is based on two gradient operators which is different from the usual gradient-divergence operators for Brinkman equations. The WG method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order error estimates are established for the corresponding WG finite element solutions in various norms. Some computational results are presented to demonstrate the robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman equations.

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