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Sensitivity analysis of a nonlinear obstacle plate problem

Isabel N. Figueiredo, Carlos F. Leal (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9, 10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that...

Sensitivity Analysis of a Nonlinear Obstacle Plate Problem

Isabel N. Figueiredo, Carlos F. Leal (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9,10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that...

Shakedown theorems in poroplastic dynamics

Giuseppe Cocchetti, Giulio Maier (2002)

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

The constitutive model assumed in this Note is poroplastic two-phase (solid-fluid) with full saturation and stable in Drucker’s sense. A solid or structure of this material is considered, subjected to dynamic external actions, in particular periodic or intermittent, in a small deformation regime. A sufficient condition and a necessary one are established, by a «static» approach, for shakedown (or adaptation), namely for boundedness in time of the cumulative dissipated energy.

Shape and topological sensitivity analysis in domains with cracks

Alexander Khludnev, Jan Sokołowski, Katarzyna Szulc (2010)

Applications of Mathematics

The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural...

Shape and topology optimization of the robust compliance via the level set method

François Jouve, Grégoire Allaire, Frédéric de Gournay (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute it....

Shape and topology optimization of the robust compliance via the level set method

Frédéric de Gournay, Grégoire Allaire, François Jouve (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute...

Shape optimization for dynamic contact problems

Andrzej Myśliński (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The paper deals with shape optimization of dynamic contact problem with Coulomb friction for viscoelastic bodies. The mass nonpenetrability condition is formulated in velocities. The friction coefficient is assumed to be bounded. Using material derivative method as well as the results concerning the regularity of solution to dynamic variational inequality the directional derivative of the cost functional is calculated and the necessary optimality condition is formulated.

Shape optimization in contact problems based on penalization of the state inequality

Jaroslav Haslinger, Pekka Neittaanmäki, Timo Tiihonen (1986)

Aplikace matematiky

The paper deals with the approximation of optimal shape of elastic bodies, unilaterally supported by a rigid, frictionless foundation. Original state inequality, describing the behaviour of such a body is replaced by a family of penalized state problems. The relation between optimal shapes for the original state inequality and those for penalized state equations is established.

Shape optimization of an elasto-perfectly plastic body

Ivan Hlaváček (1987)

Aplikace matematiky

Within the range of Prandtl-Reuss model of elasto-plasticity the following optimal design problem is solved. Given body forces and surface tractions, a part of the boundary, where the (two-dimensional) body is fixed, is to be found, so as to minimize an integral of the squared yield function. The state problem is formulated in terms of stresses by means of a time-dependent variational inequality. For approximate solutions piecewise linear approximations of the unknown boundary, piecewise constant...

Shape optimization of an elasto-plastic body for the model with strain- hardening

Vladislav Pištora (1990)

Aplikace matematiky

The state problem of elasto-plasticity (for the model with strain-hardening) is formulated in terms of stresses and hardening parameters by means of a time-dependent variational inequality. The optimal design problem is to find the shape of a part of the boundary such that a given cost functional is minimized. For the approximate solutions piecewise linear approximations of the unknown boundary, piecewise constant triangular elements for the stress and the hardening parameter, and backward differences...

Shape optimization of elastic axisymmetric bodies

Ivan Hlaváček (1989)

Aplikace matematiky

The shape of the meridian curve of an elastic body is optimized within a class of Lipschitz functions. Only axisymmetric mixed boundary value problems are considered. Four different cost functionals are used and approximate piecewise linear solutions defined on the basis of a finite element technique. Some convergence and existence results are derived by means of the theory of the appropriate weighted Sobolev spaces.

Shape optimization of elastic axisymmetric plate on an elastic foundation

Petr Salač (1995)

Applications of Mathematics

An elastic simply supported axisymmetric plate of given volume, fixed on an elastic foundation, is considered. The design variable is taken to be the thickness of the plate. The thickness and its partial derivatives of the first order are bounded. The load consists of a concentrated force acting in the centre of the plate, forces concentrated on the circle, an axisymmetric load and the weight of the plate. The cost functional is the norm in the weighted Sobolev space of the deflection curve of radius....

Shape optimization of elasto-plastic axisymmetric bodies

Ivan Hlaváček (1991)

Applications of Mathematics

A minimization of a cost functional with respect to a part of a boundary is considered for an elasto-plastic axisymmetric body obeying Hencky's law. The principle of Haar-Kármán and piecewise linear stress approximations are used to solve the state problem. A convergence result and the existence of an optimal boundary is proved.

Shape optimization of elasto-plastic bodies

Zuzana Dimitrovová (2001)

Applications of Mathematics

Existence of an optimal shape of a deformable body made from a physically nonlinear material obeying a specific nonlinear generalized Hooke’s law (in fact, the so called deformation theory of plasticity is invoked in this case) is proved. Approximation of the problem by finite elements is also discussed.

Shape optimization of elastoplastic bodies obeying Hencky's law

Ivan Hlaváček (1986)

Aplikace matematiky

A minimization of a cost functional with respect to a part of the boundary, where the body is fixed, is considered. The criterion is defined by an integral of a yield function. The principle of Haar-Kármán and piecewise constant stress approximations are used to solve the state problem. A convergence result and the existence of an optimal boundary is proved.

Shape optimization of materially non-linear bodies in contact

Jaroslav Haslinger, Raino Mäkinen (1997)

Applications of Mathematics

Optimal shape design problem for a deformable body in contact with a rigid foundation is studied. The body is made from material obeying a nonlinear Hooke’s law. We study the existence of an optimal shape as well as its approximation with the finite element method. Practical realization with nonlinear programming is discussed. A numerical example is included.

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