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Error estimation and adaptivity for nonlinear FE analysis

Antonio Huerta, Antonio Rodríguez-Ferran, Pedro Díez (2002)

International Journal of Applied Mathematics and Computer Science

An adaptive strategy for nonlinear finite-element analysis, based on the combination of error estimation and h-remeshing, is presented. Its two main ingredients are a residual-type error estimator and an unstructured quadrilateral mesh generator. The error estimator is based on simple local computations over the elements and the so-called patches. In contrast to other residual estimators, no flux splitting is required. The adaptive strategy is illustrated by means of a complex nonlinear problem:...

Estimation of vibration frequencies of linear elastic membranes

Luca Sabatini (2018)

Applications of Mathematics

The free motion of a thin elastic linear membrane is described, in a simplyfied model, by a second order linear homogeneous hyperbolic system of partial differential equations whose spatial part is the Laplace Beltrami operator acting on a Riemannian 2-dimensional manifold with boundary. We adapt the estimates of the spectrum of the Laplacian obtained in the last years by several authors for compact closed Riemannian manifolds. To make so, we use the standard technique of the doubled manifold to...

Eulerian formulation and level set models for incompressible fluid-structure interaction

Georges-Henri Cottet, Emmanuel Maitre, Thomas Milcent (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

This paper is devoted to Eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straightforward way complex 3D systems. We first analyze the level set model of immersed membranes proposed in [Cottet and Maitre, Math. Models Methods Appl. Sci.16 (2006) 415–438]. We in particular show that this model can be interpreted as a generalization of so-called Korteweg fluids. We then extend this model...

Exact boundary controllability of a hybrid system of elasticity by the HUM method

Bopeng Rao (2001)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by one partial differential equation and two ordinary differential equations. Using the HUM method, we prove that the hybrid system is exactly controllable in an arbitrarily short time in the usual energy space.

Exact Boundary Controllability of a Hybrid System of elasticity by the HUM Method

Bopeng Rao (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by one partial differential equation and two ordinary differential equations. Using the HUM method, we prove that the hybrid system is exactly controllable in an arbitrarily short time in the usual energy space.

Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions

Scott W. Hansen, Oleg Imanuvilov (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Exact controllability results for a multilayer plate system are obtained from the method of Carleman estimates. The multilayer plate system is a natural multilayer generalization of a classical three-layer “sandwich plate” system due to Rao and Nakra. The multilayer version involves a number of Lamé systems for plane elasticity coupled with a scalar Kirchhoff plate equation. The plate is assumed to be either clamped or hinged and controls are assumed to be locally distributed in a neighborhood...

Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions

Scott W. Hansen, Oleg Imanuvilov (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Exact controllability results for a multilayer plate system are obtained from the method of Carleman estimates. The multilayer plate system is a natural multilayer generalization of a classical three-layer “sandwich plate” system due to Rao and Nakra. The multilayer version involves a number of Lamé systems for plane elasticity coupled with a scalar Kirchhoff plate equation. The plate is assumed to be either clamped or hinged and controls are assumed to be locally distributed in a neighborhood...

Exact controllability of an elastic membrane coupled with a potential fluid

Scott Hansen (2001)

International Journal of Applied Mathematics and Computer Science

We consider the problem of boundary control of an elastic system with coupling to a potential equation. The potential equation represents the linearized motions of an incompressible inviscid fluid in a cavity bounded in part by an elastic membrane. Sufficient control is placed on a portion of the elastic membrane to insure that the uncoupled membrane is exactly controllable. The main result is that if the density of the fluid is sufficiently small, then the coupled system is exactly controllable....

Exact controllability of shells in minimal time

Paola Loreti (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We prove an exact controllability result for thin cups using the Fourier method and recent improvements of Ingham type theorems, given in a previous paper [2].

Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order

Guo Wang Chen, Shu Bin Wang (1995)

Commentationes Mathematicae Universitatis Carolinae

The existence and uniqueness of classical global solution and blow up of non-global solution to the first boundary value problem and the second boundary value problem for the equation u t t - α u x x - β u x x t t = ϕ ( u x ) x are proved. Finally, the results of the above problem are applied to the equation arising from nonlinear waves in elastic rods u t t - a 0 + n a 1 ( u x ) n - 1 u x x - a 2 u x x t t = 0 .

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