A finite element solution for plasticity with strain-hardening
Some equilibrium and mixed models in the finite element method
Jak řešit úlohy s nejistými vstupními daty?
On a conjugate semi-variational method for parabolic equations
A mixed finite element method for plate bending with a unilateral inner obstacle
A unilateral problem of an elastic plate above a rigid interior obstacle is solved on the basis of a mixed variational inequality formulation. Using the saddle point theory and the Herrmann-Johnson scheme for a simultaneous computation of deflections and moments, an iterative procedure is proposed, each step of which consists in a linear plate problem. The existence, uniqueness and some convergence analysis is presented.
Reissner-Mindlin model for plates of variable thickness. Solution by mixed-interpolated elements
Hard clamped and hard simply supported elastic plate is considered. The mixed finite element analysis combined with some interpolation, proposed by Brezzi, Fortin and Stenberg, is extended to the case of variable thickness and anisotropic material.
Reliable solutions of problems in the deformation theory of plasticity with respect to uncertain material function
Maximization problems are formulated for a class of quasistatic problems in the deformation theory of plasticity with respect to an uncertainty in the material function. Approximate problems are introduced on the basis of cubic Hermite splines and finite elements. The solvability of both continuous and approximate problems is proved and some convergence analysis presented.
Shape optimization by means of the penalty method with extrapolation
A model shape optimal design in is solved by means of the penalty method with extrapolation, which enables to obtain high order approximations of both the state function and the boundary flux, thus offering a reliable gradient for the sensitivity analysis. Convergence of the proposed method is proved for certain subsequences of approximate solutions.
Weight minimization of an elastic plate with a unilateral inner obstacle by a mixed finite element method
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.
Reliable solution of an elasto-plastic Reissner-Mindlin beam for Hencky's model with uncertain yield function
We apply the method of reliable solutions to the bending problem for an elasto-plastic beam, considering the yield function of the von Mises type with uncertain coefficients. The compatibility method is used to find the moments and shear forces. Then we solve a maximization problem for these quantities with respect to the uncertain input data.
Uncertain input data problems and the worst scenario method
An introduction to the worst scenario method is given. We start with an example and a general abstract scheme. An analysis of the method both on the continuous and approximate levels is discussed. We show a possible incorporation of the method into the fuzzy set theory. Finally, we present a survey of applications published during the last decade.
Weight minimization of elastic plates using Reissner-Mindlin model and mixed-interpolated elements
The problem to find an optimal thickness of the plate in a set of bounded Lipschitz continuous functions is considered. Mean values of the intensity of shear stresses must not exceed a given value. Using a penalty method and finite element spaces with interpolation to overcome the “locking” effect, an approximate optimization problem is proposed. We prove its solvability and present some convergence analysis.
Mixed finite element analysis of semi-coercive unilateral contact problems with given friction
A unilateral contact 2D-problem is considered provided one of two elastic bodies can shift in a given direction as a rigid body. Using Lagrange multipliers for both normal and tangential constraints on the contact interface, we introduce a saddle point problem and prove its unique solvability. We discretize the problem by a standard finite element method and prove a convergence of approximations. We propose a numerical realization on the basis of an auxiliary “bolted” problem and the algorithm of...
Post-buckling range of plates in axial compression with uncertain initial geometric imperfections
The method of reliable solutions alias the worst scenario method is applied to the problem of von Kármán equations with uncertain initial deflection. Assuming two-mode initial and total deflections and using Galerkin approximations, the analysis leads to a system of two nonlinear algebraic equations with one or two uncertain parameters-amplitudes of initial deflections. Numerical examples involve (i) minimization of lower buckling loads and (ii) maximization of the maximal mean reduced stress.
Finite element analysis of a static contact problem with Coulomb friction
A unilateral contact problem with a variable coefficient of friction is solved by a simplest variant of the finite element technique. The coefficient of friction may depend on the magnitude of the tangential displacement. The existence of an approximate solution and some a priori estimates are proved.
In memoriam Professor Jindřich Nečas
Sur quelques théorèmes variationnels dans la théorie du fluage linéaire
Par la méthode de transformation en problèmes équivalentes de la théorie d'élasticité, on établit deux théorèmes variationnels analogues aux principes du minimum d'énergie potentielle et de Castigliano dans la théorie d'élasticité, pour les corps homogénes isotropes tenant compte à l'hérédité et de l'âge du matérial.
Dual finite element analysis for semi-coercive unilateral boundary value problems
Penalty method and extrapolation for axisymmetric elliptic problems with Dirichlet boundary conditions
A second order elliptic problem with axisymmetric data is solved in a finite element space, constructed on a triangulation with curved triangles, in such a way, that the (nonhomogeneous) boundary condition is fulfilled in the sense of a penalty. On the basis of two approximate solutions, extrapolates for both the solution and the boundary flux are defined. Some a priori error estimates are derived, provided the exact solution is regular enough. The paper extends some of the results of J.T. King...
Optimization of the domain in elliptic problems by the dual finite element method
An optimal part of the boundary of a plane domain for the Poisson equation with mixed boundary conditions is to be found. The cost functional is (i) the internal energy, (ii) the norm of the external flux through the unknown boundary. For the numerical solution of the state problem a dual variational formulation - in terms of the gradient of the solution - and spaces of divergence-free piecewise linear finite elements are used. The existence of an optimal domain and some convergence results are...
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