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 and simulations of a frictional contact problem with damage and memory

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hailing Xuan ◽  
Xiaoliang Cheng
Keyword(s):  
Contact Problem ◽  
Frictional Contact ◽  
Numerical Solutions ◽  
Deformable Body ◽  
Mechanical Damage ◽  
Coupled System ◽  
Constitutive Law ◽  
Optimal Order ◽  
Order Error ◽  
Nonlinear Parabolic

<p style='text-indent:20px;'>In this paper, we study a frictional contact model which takes into account the damage and the memory. The deformable body consists of a viscoelastic material and the process is assumed to be quasistatic. The mechanical damage of the material which caused by the tension or the compression is included in the constitutive law and the damage function is modelled by a nonlinear parabolic inclusion. Then the variational formulation of the model is governed by a coupled system consisting of a history-dependent hemivariational inequality and a nonlinear parabolic variational inequality. We introduce and study a fully discrete scheme of the problem and derive error estimates for numerical solutions. Under appropriate solution regularity assumptions, an optimal order error estimate is derived for the linear finite element method. Several numerical experiments for the contact problem are given for providing numerical evidence of the theoretical results.</p>

scholarly journals Error estimates for Galerkin finite element methods for the Camassa–Holm equation

2020 ◽  
Author(s):  
DC Antonopoulos ◽  
VA Dougalis ◽  
Dimitrios Mitsotakis
Keyword(s):  
Finite Element ◽  
Error Estimates ◽  
Travelling Waves ◽  
Approximate Solutions ◽  
Optimal Order ◽  
Uniform Mesh ◽  
Galerkin Finite Element ◽  
Fully Discrete ◽  
Semidiscrete Approximation ◽  
Galerkin Finite Element Methods

© 2019, Springer-Verlag GmbH Germany, part of Springer Nature. We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order L2-error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.

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.

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 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 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.

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 Variational and Numerical Analysis for Frictional Contact Problem with Normal Compliance in Thermo-Electro-Viscoelasticity

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Mustapha Bouallala ◽  
El Hassan Essoufi ◽  
Mohamed Alaoui
Keyword(s):  
Contact Problem ◽  
Variational Formulation ◽  
Frictional Contact ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Normal Compliance ◽  
Fully Discrete ◽  
Tresca’S Friction ◽  
Discrete Finite Element ◽  
Finite Element Schemes

In this paper, we consider a mathematical model of a contact problem in thermo-electro-viscoelasticity with the normal compliance conditions and Tresca’s friction law. We present a variational formulation of the problem, and we prove the existence and uniqueness of the weak solution. We also study the numerical approach using spatially semidiscrete and fully discrete finite element schemes with Euler’s backward scheme. Finally, we derive error estimates on the approximate solutions.

scholarly journals Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations

2020 ◽  
Author(s):  
Norikazu Saito
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Parabolic Type ◽  
Finite Element Approximation ◽  
Optimal Order ◽  
Element Approximation ◽  
Step Method ◽  
Order Error ◽  
Time Stepping ◽  
Optimal Order Error Estimates

Abstract The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška–Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it differs from previous works by which the method is formulated as the one-step method. We apply our abstract results to the finite element approximation of a second-order parabolic equation with space-time variable coefficient functions in a polyhedral domain, and derive the optimal order error estimates in several norms.

H1 -Galerkin Expanded Mixed Finite Element Methods for Pseudo-Hyperbolicequations

2011 ◽  
Vol 268-270 ◽  
pp. 908-912 ◽  
Author(s):  
Mei Xia Li
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Optimal Order ◽  
Order Error ◽  
Expanded Mixed Finite Element ◽  
Physical Quantities ◽  
Element Method

H1-Galerkin mixed finite element method combining with expanded mixed element method are discussed for a class of second-order pseudo-hyperbolic equations. The methods possesses the advantage of mixed finite element while avoiding directly inverting the permeability tensor, which is important especially in a low permeability zone. Depended on the physical quantities of interest, the methods are discussed. The existence and uniqueness of numerical solutions of the scheme are derived and an optimal order error estimate for the methods is obtained.

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