On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments

Belgacem, Faker Ben; Bernardi, Christine; Blouza, Adel; Vohralı́k, Martin

doi:10.1093/imanum/drr003

Cited by 11 publications

(24 citation statements)

References 18 publications

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“…being the traditional energy norms. Further information about posteriori error estimation for contact problems can be found in [62,63,64] Appendix B Quadratic second-order cone programming…”

Section: Discussionmentioning

confidence: 99%

Dual finite-element analysis using second-order cone programming for structures including contact

Boustani

Bleyer

Arquier³

et al. 2020

Engineering Structures

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Computation of elastic structures in contact is performed by means of a dual analysis combining displacement-based and equilibrium-based finite elements. Contact conditions are formulated in the framework of second-order cone programmaing (SOCP) and an efficient interior point method (IPM) algorithm is presented for the resolution of the associated optimization problems. The dual approach allows the user to assess the quality of convergence and to efficiently calculate a discretization error estimator which includes a contact error term. An efficient remeshing scheme, based on the local contributions of the elements to the global error, can then be used to efficiently improve the solution accuracy. The whole process is illustrated on some examples and applied to a typical steel assembly. Its efficiency, in particular concerning the IPM solver, is demonstrated in comparison with the industrial finite element code Abaqus.

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Section: Discussionmentioning

confidence: 99%

Dual finite-element analysis using second-order cone programming for structures including contact

Boustani

Bleyer

Arquier³

et al. 2020

Engineering Structures

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“…While a large percentage of the references in this area deal with boundary value problems of partial differential equations, one can also find papers on adaptive solution of variational inequalities arising in contact mechanics (e.g. Ben Belgacem, Bernardi, Blouza and Vohralík 2012, Han 2005, Bostan and Han 2009, Hild and Lleras 2009. A natural step is to introduce a posteriori error estimators for hemivariational inequalities, analyse their reliability and efficiency, and use them to solve related contact problems.…”

Section: Discussionmentioning

confidence: 99%

Numerical analysis of hemivariational inequalities in contact mechanics

Han¹,

Sofonea²

2019

Acta Numerica

105

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Contact phenomena arise in a variety of industrial process and engineering applications. For this reason, contact mechanics has attracted substantial attention from research communities. Mathematical problems from contact mechanics have been studied extensively for over half a century. Effort was initially focused on variational inequality formulations, and in the past ten years considerable effort has been devoted to contact problems in the form of hemivariational inequalities. This article surveys recent development in studies of hemivariational inequalities arising in contact mechanics. We focus on contact problems with elastic and viscoelastic materials, in the framework of linearized strain theory, with a particular emphasis on their numerical analysis. We begin by introducing three representative mathematical models which describe the contact between a deformable body in contact with a foundation, in static, history-dependent and dynamic cases. In weak formulations, the models we consider lead to various forms of hemivariational inequalities in which the unknown is either the displacement or the velocity field. Based on these examples, we introduce and study three abstract hemivariational inequalities for which we present existence and uniqueness results, together with convergence analysis and error estimates for numerical solutions. The results on the abstract hemivariational inequalities are general and can be applied to the study of a variety of problems in contact mechanics; in particular, they are applied to the three representative mathematical models. We present numerical simulation results giving numerical evidence on the theoretically predicted optimal convergence order; we also provide mechanical interpretations of simulation results.

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“…Finally, using Assumption 2.1, (10), and (13) our two-phase flow model with exchange between phases is governed by the following system: find…”

Section: Governing Partial Differential Equations and Nonlinear Complmentioning

confidence: 99%

“…More recently, one approach consists in obtaining the so-called potential and equilibrated flux reconstructions solving auxiliary local problems (see Destuynder and Métivet [30], Braess and Schöberl [17], Ern and Vohralík [35], and the references therein. Concerning a posteriori error estimate for variational inequalities, one can point out the pioneering work of Kornhuber [50], Chen and Nochetto [24], Veeser [72], Repin [67] and Ben Belgacem et al [10]. In particular in [10], a posteriori error estimates are given for exact solvers and recently, in [28] a posteriori error estimates are derived for inexact semismooth solvers and provide adaptive stopping criteria.…”

Section: Introductionmentioning

confidence: 99%

“…Concerning a posteriori error estimate for variational inequalities, one can point out the pioneering work of Kornhuber [50], Chen and Nochetto [24], Veeser [72], Repin [67] and Ben Belgacem et al [10]. In particular in [10], a posteriori error estimates are given for exact solvers and recently, in [28] a posteriori error estimates are derived for inexact semismooth solvers and provide adaptive stopping criteria. The concept of adaptive stopping criteria relies on stopping the nonlinear and linear iterations whenever the associated estimators do not affect significantly the overall error, see [7,28,34,46,56].…”

Section: Introductionmentioning

confidence: 99%

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A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints

et al. 2020

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In this work, we develop an a-posteriori-steered algorithm for a compositional two-phase flow with exchange of components between the phases in porous media. As a model problem, we choose the two-phase liquid-gas flow with appearance and disappearance of the gas phase formulated as a system of nonlinear evolutive partial differential equations with nonlinear complementarity constraints. The discretization of our model is based on the backward Euler scheme in time and the finite volume scheme in space. The resulting nonlinear system is solved via an inexact semismooth Newton method. The key ingredient for the a posteriori analysis are the discretization, linearization, and algebraic flux reconstructions allowing to devise estimators for each error component. These enable to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver whenever the corresponding error components do not affect significantly the overall error. Numerical experiments are performed using the Newton-min algorithm as well as the Newton-Fischer-Burmeister algorithm in combination with the GMRES iterative linear solver to show the efficiency of the proposed adaptive method.

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On the unilateral contact between membranes. Part 2: a posteriori analysis and numerical experiments

Cited by 11 publications

References 18 publications

Dual finite-element analysis using second-order cone programming for structures including contact

Dual finite-element analysis using second-order cone programming for structures including contact

Numerical analysis of hemivariational inequalities in contact mechanics

A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints

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