Nitsche-based models for the unilateral contact of plates

Fabre, Mathieu; Pozzolini, Cédric; Renard, Yves

doi:10.1051/m2an/2020063

Cited by 2 publications

(2 citation statements)

References 28 publications

(46 reference statements)

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“…For mathematical analysis of elastic plates we refer to contact problems with obstacles [17] and inclusions [18], to history-dependent models [19], analysis of thickness dependence [20], inverse coefficient problems [21] and to the references therein. For the numerical solution of unilateral problems for plates, refer to [22,23].…”

Section: Introductionmentioning

confidence: 99%

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle

Kovtunenko,

Lazarev

2024

Phil. Trans. R. Soc. A.

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A class of variational inequalities describing the equilibrium of elastic Timoshenko plates whose boundary is in contact with the side surface of an inclined obstacle is considered. At the plate boundary, mixed conditions of Dirichlet type and a non-penetration condition of inequality type are imposed on displacements in the mid-plane. The novelty consists of modelling oblique interaction with the inclined obstacle which takes into account shear deformation and rotation of transverse cross-sections in the plate. For proposed problems of equilibrium of the plate contacting the inclined obstacle, the unique solvability of the corresponding variational inequality is proved. Under the assumption that the variational solution is smooth enough, optimality conditions are obtained in the form of equilibrium equations and relations revealing the mechanical properties of integrated stresses, moments and generalized displacements on the contact part of the boundary. Accounting for complementarity type conditions owing to the contact of the plate with the inclined obstacle, a primal-dual variational formulation of the obstacle problem is derived. A semi-smooth Newton method based on a generalized gradient is constructed and performed as a primal-dual active-set algorithm. It is advantageous for efficient numerical solution of the problem, provided by a super-linear estimate for the corresponding iterates in function spaces. This article is part of the theme issue ‘Non-smooth variational problems with applications in mechanics’.

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Section: Introductionmentioning

confidence: 99%

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle

Kovtunenko,

Lazarev

2024

Phil. Trans. R. Soc. A.

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“…These problems are stated in the form of variational inequalities [21][22][23][24], and the following numerical methods were used to solve them: Lagrange multipliers [25][26][27], penalty functions [28; 29] and their combinations [30][31][32]. And other methods using contact finite elements [32][33]; quadratic programming approach [34][35][36]; finite element methods (Spigot-algorithms) [37][38][39]; and other [40][41][42][43][44][45][46].…”

Section: Introductionmentioning

confidence: 99%

Algorithm for calculating the problem of unilateral frictional contact with an increscent external load parameter

Popov,

Lovtsov

2023

Struct. Mech. of Eng. Const. and Build

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The subject of the study is the contact interaction of deformable elements of linear complementarity problem (LCP). To solve the linear complementarity problem, the Lemke method with the introduction of an increasing parameter of external loading is used. The proposed approach solves the degenerated matrix in a finite number of steps, while the dimensionality of the problem is limited to the area of contact. To solve the problem, the initial table of the Lemke method is generated using the contact matrix of stiffness and the contact load vector. The unknowns in the problem are mutual displacements and interaction forces of contacting pairs of points of deformable solids. The proposed approach makes it possible to evaluate the change in working schemes as the parameter of external load increases. The features of the proposed formulation of the problem are shown, the criteria for stopping the stepwise process of solving such problems are considered. Model examples for the proposed algorithm are given. The algorithm has shown its efficiency in application, including for complex model problems. Recommendations on the use of the proposed approach are given.

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Nitsche-based models for the unilateral contact of plates

Cited by 2 publications

References 28 publications

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle

Algorithm for calculating the problem of unilateral frictional contact with an increscent external load parameter

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