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Vibrations of a beam between obstacles. Convergence of a fully discretized approximation

Yves Dumont, Laetitia Paoli (2006)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider mathematical models describing dynamics of an elastic beam which is clamped at its left end to a vibrating support and which can move freely at its right end between two rigid obstacles. We model the contact with Signorini's complementary conditions between the displacement and the shear stress. For this infinite dimensional contact problem, we propose a family of fully discretized approximations and their convergence is proved. Moreover some examples of implementation are presented....

Von Kármán equations. III. Solvability of the von Kármán equations with conditions for geometry of the boundary of the domain

Július Cibula (1991)

Applications of Mathematics

Solvability of the general boundary value problem for von Kármán system of nonlinear equations is studied. The problem is reduced to an operator equation. It is shown that the corresponding functional of energy is coercive and weakly lower semicontinuous. Then the functional of energy attains absolute minimum which is a variational solution of the problem.

Weak solvability and numerical analysis of a class of time-fractional hemivariational inequalities with application to frictional contact problems

Mustapha Bouallala (2024)

Applications of Mathematics

We investigate a generalized class of fractional hemivariational inequalities involving the time-fractional aspect. The existence result is established by employing the Rothe method in conjunction with the surjectivity of multivalued pseudomonotone operators and the properties of the Clarke generalized gradient. We are also exploring a numerical approach to address the problem, utilizing both spatially semi-discrete and fully discrete finite elements, along with a discrete approximation of the fractional...

Weight minimization of an elastic plate with a unilateral inner obstacle by a mixed finite element method

Ivan Hlaváček (1994)

Applications of Mathematics

Unilateral deflection problem of a clamped plate above a rigid inner obstacle is considered. The variable thickness of the plate is to be optimized to reach minimal weight under some constraints for maximal stresses. Since the constraints are expressed in terms of the bending moments only, Herrmann-Hellan finite element scheme is employed. The existence of an optimal thickness is proved and some convergence analysis for approximate penalized optimal design problem is presented.

Weight minimization of elastic bodies weakly supporting tension. I. Domains with one curved side

Ivan Hlaváček, Michal Křížek (1992)

Applications of Mathematics

Shape optimization of a two-dimensional elastic body is considered, provided the material is weakly supporting tension. The problem generalizes that of a masonry dam subjected to its own weight and to the hydrostatic presure. Existence of an optimal shape is proved. Using a penalty method and finite element technique, approximate solutions are proposed and their convergence is analyzed.

Weight minimization of elastic bodies weakly supporting tension. II. Domains with two curved sides

Ivan Hlaváček, Michal Křížek (1992)

Applications of Mathematics

Extending the results of the previous paper [1], the authors consider elastic bodies with two design variables, i.e. "curved trapezoids" with two curved variable sides. The left side is loaded by a hydrostatic pressure. Approximations of the boundary are defined by cubic Hermite splines and piecewise linear finite elements are used for the displacements. Both existence and some convergence analysis is presented for approximate penalized optimal design problems.

Young-measure approximations for elastodynamics with non-monotone stress-strain relations

Carsten Carstensen, Marc Oliver Rieger (2004)

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

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density φ . Their time-evolution leads to a nonlinear wave equation u t t = div S ( D u ) with the non-monotone stress-strain relation S = D φ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very...

Young-Measure approximations for elastodynamics with non-monotone stress-strain relations

Carsten Carstensen, Marc Oliver Rieger (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density ϕ. Their time-evolution leads to a nonlinear wave equation u t t = div S ( D u ) with the non-monotone stress-strain relation S = D φ plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding...

Currently displaying 521 – 540 of 584