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On a 1-D model of stress relaxation in an annealed glass

Vladimír Janovský, David Just (2002)

Applications of Mathematics

A 1-D model of a slab of glass of a small thickness is considered. The governing equations are those of the classical 1-D linear viscoelasticity. A load due to the temperature gradients is assumed. The aim is to model the process called annealing. It is shown that an additional load due to structural strain is crucial for the success of the model. Algorithms of a numerical solution of the governing equations are proposed. Numerical results are presented and commented.

On a computational approach to multiple contacts / impacts of elastic bodies

Vala, Jiří, Rek, Václav (2023)

Programs and Algorithms of Numerical Mathematics

The analysis of dynamic contacts/impacts of several deformable bodies belongs to both theoretically and computationally complicated problems, because of the presence of unpleasant nonlinearities and of the need of effective contact detection. This paper sketches how such difficulties can be overcome, at least for a model problem with several elastic bodies, using i) the explicit time-discretization scheme and ii) the finite element technique adopted to contact evaluations together with iii) the...

On a new computational algorithm for impacts of elastic bodies

Hynek Štekbauer, Ivan Němec, Rostislav Lang, Daniel Burkart, Jiří Vala (2022)

Applications of Mathematics

Computational modelling of contact problems is still one of the most difficult aspects of non-linear analysis in engineering mechanics. The article introduces an original efficient explicit algorithm for evaluation of impacts of bodies, satisfying the conservation of both momentum and energy exactly. The algorithm is described in its linearized 2-dimensional formulation in details, as open to numerous generalizations including 3-dimensional ones, and supplied by numerical examples obtained from...

On a superconvergent finite element scheme for elliptic systems. II. Boundary conditions of Newton's or Neumann's type

Ivan Hlaváček, Michal Křížek (1987)

Aplikace matematiky

A simple superconvergent scheme for the derivatives of finite element solution is presented, when linear triangular elements are employed to solve second order elliptic systems with boundary conditions of Newton’s or Neumann’s type. For bounded plane domains with smooth boundary the local O ( h 3 / 2 ) -superconvergence of the derivatives in the L 2 -norm is proved. The paper is a direct continuations of [2], where an analogous problem with Dirichlet’s boundary conditions is treated.

On a superconvergent finite element scheme for elliptic systems. III. Optimal interior estimates

Ivan Hlaváček, Michal Křížek (1987)

Aplikace matematiky

Second order elliptic systems with boundary conditions of Dirichlet, Neumann’s or Newton’s type are solved by means of linear finite elements on regular uniform triangulations. Error estimates of the optimal order O ( h 2 ) are proved for the averaged gradient on any fixed interior subdomain, provided the problem under consideration is regular in a certain sense.

On a superconvergent finite element scheme for elliptic systems. I. Dirichlet boundary condition

Ivan Hlaváček, Michal Křížek (1987)

Aplikace matematiky

Second order elliptic systems with Dirichlet boundary conditions are solved by means of affine finite elements on regular uniform triangulations. A simple averagign scheme is proposed, which implies a superconvergence of the gradient. For domains with enough smooth boundary, a global estimate O ( h 3 / 2 ) is proved in the L 2 -norm. For a class of polygonal domains the global estimate O ( h 2 ) can be proven.

On a type of Signorini problem without friction in linear thermoelasticity

Jiří Nedoma (1983)

Aplikace matematiky

In the paper the Signorini problem without friction in the linear thermoelasticity for the steady-state case is investigated. The problem discussed is the model geodynamical problem, physical analysis of which is based on the plate tectonic hypothesis and the theory of thermoelasticity. The existence and unicity of the solution of the Signorini problem without friction for the steady-state case in the linear thermoelasticity as well as its finite element approximation is proved. It is known that...

On an interaction of two elastic bodies: analysis and algorithms

Ivona Svobodová (2012)

Applications of Mathematics

The paper deals with existence and uniqueness results and with the numerical solution of the nonsmooth variational problem describing a deflection of a thin annular plate with Neumann boundary conditions. Various types of the subsoil and the obstacle which influence the plate deformation are considered. Numerical experiments compare two different algorithms.

On an inversion formula of certain Laplace transforms in dissipative wave propagation

Pasquale Renno (1981)

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

Si determina una formula di inversione di alcune trasformate di Laplace che intervengono nell’analisi formale di problemi al contorno relativi ad una classe di mezzi dissipativi. Le espressioni esplicite proposte definiscono funzioni analitiche a decrescenza rapida dotate di numerose proprietà di massimo, utili anche all’analisi di problemi unilaterali.

On coupled thermoelastic vibration of geometrically nonlinear thin plates satisfying generalized mechanical and thermal conditions on the boundary and on the surface

Hans-Ullrich Wenk (1982)

Aplikace matematiky

The vibration problem in two variables is derived from the spatial situation (a plate as a three-dimensional body) on the basis of geometrically nonlinear plate theory (using Kármán's hypothesis) and coupled linear thermoelasticity. That leads to coupled strongly nonlinear two-dimensional equilibrium and heat conducting equations (under classical mechanical and thermal boundary conditions). For the generalized problem with subgradient conditions on the boundary and in the domain (including also...

On equilibrium finite elements in three-dimensional case

Sergey Korotov (1997)

Applications of Mathematics

The space of divergence-free functions with vanishing normal flux on the boundary is approximated by subspaces of finite elements that have the same property. The easiest way of generating basis functions in these subspaces is considered.

On finite element uniqueness studies for Coulombs frictional contact model

Patrick Hild (2002)

International Journal of Applied Mathematics and Computer Science

We are interested in the finite element approximation of Coulomb's frictional unilateral contact problem in linear elasticity. Using a mixed finite element method and an appropriate regularization, it becomes possible to prove existence and uniqueness when the friction coefficient is less than Cε^{2}|log(h)|^{-1}, where h and ε denote the discretization and regularization parameters, respectively. This bound converging very slowly towards 0 when h decreases (in comparison with the already known...

On Ishlinskij's model for non-perfectly elastic bodies

Pavel Krejčí (1988)

Aplikace matematiky

The main goal of the paper is to formulate some new properties of the Ishlinskii hysteresis operator F , which characterizes e.g. the relation between the deformation and the stress in a non-perfectly elastic (elastico-plastic) material. We introduce two energy functionals and derive the energy inequalities. As an example we investigate the equation u ' ' + F ( u ) = 0 describing the motion of a mass point at the extremity of an elastico-plastic spring.

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