A Robin Domain Decomposition Algorithm for Contact Problems: Convergence Results

Ipopa, Mohamed Ali; Sassi, Taoufik

doi:10.1007/978-3-642-02677-5_14

Cited by 3 publications

(4 citation statements)

References 7 publications

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“…Suppose that γ ≥ 0. Then, taking into account the coercivity of bilinear form (21), property (54), and the previous inequality, we obtain…”

Section: Iterative Methods For Nonlinear Variation Equationsmentioning

confidence: 99%

“…Among the domain decomposition methods for unilateral two-body contact problems obtained on the continuous level, one should mention Dirichlet-Neumann [13,14,15], Neumann-Neumann [16,17] and optimization based [18] iterative algorithms. A generalization of Lions' Robin-Robin domain decomposition algorithm to a two-body contact problem was proposed in works [19,20,21]. All of these methods in each iteration require to solve a nonlinear one-sided contact problem with a rigid body (Signorini problem) for one of the bodies, and a linear elasticity problem with Neumann [13,14,15] or Dirichlet [16,17,18] boundary conditions on the possible contact area for the other body, or require to solve nonlinear Signorini problems for both of the bodies [19,20,21].…”

Section: Introductionmentioning

confidence: 99%

“…A generalization of Lions' Robin-Robin domain decomposition algorithm to a two-body contact problem was proposed in works [19,20,21]. All of these methods in each iteration require to solve a nonlinear one-sided contact problem with a rigid body (Signorini problem) for one of the bodies, and a linear elasticity problem with Neumann [13,14,15] or Dirichlet [16,17,18] boundary conditions on the possible contact area for the other body, or require to solve nonlinear Signorini problems for both of the bodies [19,20,21]. Moreover, to increase the convergence rate of Neumann-Neumann and Robin-Robin algorithms, it is recommended to perform an additional iteration, in which the linear elasticity problems with Neumann boundary conditions have to be solved for both of the bodies [16,20].…”

Section: Introductionmentioning

confidence: 99%

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Convergence of penalty Robin-Robin domain decomposition methods for unilateral multibody contact problems of elasticity

Dyyak¹,

Prokopyshyn²,

Prokopyshyn³

2012

Preprint

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The paper is devoted to the penalty Robin-Robin domain decomposition methods (DDMs), proposed by us for the solution of unilateral multibody contact problems of elasticity. These DDMs are based on the penalty method for variational inequalities and some stationary and nonstationary iterative methods for nonlinear variational equations. The main result of the paper is that we give the mathematical justification of proposed DDMs and prove theorems on their convergence. We also investigate the numerical efficiency of these methods using the finite element approximations.

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“…Suppose that γ ≥ 0. Then, taking into account the coercivity of bilinear form (21), property (54), and the previous inequality, we obtain…”

Section: Iterative Methods For Nonlinear Variation Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Convergence of penalty Robin-Robin domain decomposition methods for unilateral multibody contact problems of elasticity

Dyyak¹,

Prokopyshyn²,

Prokopyshyn³

2012

Preprint

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“…Zooming methods has been used in the finite elements analysis of the linear case and proved to be helpful in well capturing the localized singularities. We recommend [3,4,7,8,11,15,16,20,22,28,29,31,34] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Computational Zooming in Near Unilateral Cracks by Schwarz Method with Total Overlap

Belgacem¹

2019

JMS

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We focus on the numerical solver of unilateral cracks by the Schwarz Method with Total Overlap. The aim is to isolate the treatment at the vicinity of the cracks from other regions of the computational domain. This avoids any direct interaction between specific approximations one may use around the singularities born at the tips of the cracks and more standard methods employed away from the cracks. We apply an iterative sub-structuring technique to capture the small structures by insulating the cracks into patches and making a zoom around each of them. The macro-problem is in turn set on the whole domain. As for the classical Schwarz method, the communication between the micro (local) and macro (global) levels is achieved iteratively through some suitable boundary conditions. The micro problem is fed by Dirichlet data along the (outer) boundary of the patches. The specificity of our approach is that the macro problem inherits transmission conditions. Although they are expressed across the cracks, the final algebraic system to invert is blind to the discontinuities of the solution. In fact, the stiffness matrix turns out to be the one related to a safe domain, as if cracks were closed or the unilateral singularities were switched off. Only the right hand side is affected by what happens at the vicinity of the cracks. This enables users to run one of many efficient algorithms found in the literature to solve the linear macro-problem. In the other hand side, in spite of the still bad conditioning and the non-linearity of the unilateral micro problems, they are reduced in size and may be inverted properly by convex optimization algorithms. A successful convergence analysis of this variant of the Schwarz Method is performed after adapting to the unilateral non-linearity the variational tools developed by P. L. Lions.

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A domain decomposition method for two-body contact problems with Tresca friction

et al. 2013

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A Robin Domain Decomposition Algorithm for Contact Problems: Convergence Results

Cited by 3 publications

References 7 publications

Convergence of penalty Robin-Robin domain decomposition methods for unilateral multibody contact problems of elasticity

Convergence of penalty Robin-Robin domain decomposition methods for unilateral multibody contact problems of elasticity

Computational Zooming in Near Unilateral Cracks by Schwarz Method with Total Overlap

A domain decomposition method for two-body contact problems with Tresca friction

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