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A nonlinear model of a turbine blade by asymptotic analysis

José Rodríguez (2002)

International Journal of Applied Mathematics and Computer Science

In this paper we obtain a limit model for a turbine blade fixed to a 3D solid. This model is a three-dimensional linear elasticity problem in the 3D part of the piece (the rotor) and a two-dimensional problem (the nonlinear shallow shell equations) in the 2D part (the turbine blade), with junction conditions in the part of the turbine blade fixed to the rotor. To obtain this model, we perform an asymptotic analysis, starting with the nonlinear three-dimensional elasticity equations on all the pieces...

An elastic membrane with an attached non-linear thermoelastic rod

Werner Horn, Jan Sokołowski (2002)

International Journal of Applied Mathematics and Computer Science

We study a thermo-mechanical system consisting of an elastic membrane to which a shape-memory rod is glued. The slow movements of the membrane are controlled by the motions of the attached rods. A quasi-static model is used. We include the elastic feedback of the membrane on the rods. This results in investigating an elliptic boundary value problem in a domain Ω ⊂ R^2 with a cut, coupled with non-linear equations for the vertical motions of the rod and the temperature on the rod. We prove the existence...

Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1

Tiziana Durante, Taras A. Mel’nyk (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider quasilinear optimal control problems involving a thick two-level junction Ωε which consists of the junction body Ω0 and a large number of thin cylinders with the cross-section of order 𝒪(ε2). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls given on their lateral surfaces and bases respectively. In addition, the quasilinear boundary conditions depend on parameters ε, α, β and the...

Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1∗

Tiziana Durante, Taras A. Mel’nyk (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider quasilinear optimal control problems involving a thick two-level junction Ωε which consists of the junction body Ω0 and a large number of thin cylinders with the cross-section of order 𝒪(ε2). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls given on their lateral surfaces and bases respectively. In addition, the quasilinear boundary...

Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1∗

Tiziana Durante, Taras A. Mel’nyk (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We consider quasilinear optimal control problems involving a thick two-level junction Ωε which consists of the junction body Ω0 and a large number of thin cylinders with the cross-section of order 𝒪(ε2). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls given on their lateral surfaces and bases respectively. In addition, the quasilinear boundary...

Junction of elastic plates and beams

Antonio Gaudiello, Régis Monneau, Jacqueline Mossino, François Murat, Ali Sili (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the linearized elasticity system in a multidomain of 𝐑 3 . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius r ε . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and r ε tend to zero simultaneously, with r ε ≫ ε 2 , we identify the limit problem. This limit problem involves six junction conditions.

Scalar boundary value problems on junctions of thin rods and plates

R. Bunoiu, G. Cardone, S. A. Nazarov (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative...

Self-similarly expanding networks to curve shortening flow

Oliver C. Schnürer, Felix Schulze (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form 120 degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.

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