Minimum-weight design of multi-purpose cylindrical bars

We solve the problem of minimizing the weight of elastic cylindrical bars under given constraints on their torsional and bending rigidities. The possible types of solutions are classed in the space of the design parameters. The necessary optimality conditions are derived and then used in the formulation of a closed-form boundary-value problem for a region with an unspecified boundary—the unknown cross-sectional shape of the optimal bar. An analytical solution of this and other related problems in terms of the given design parameters is obtained.     

Optimal Damping of Perturbations of Moving Thermoelastic Panel

The translational motion of a thermoelastic web subject to transverse vibrations caused by initial perturbations is considered. It is assumed that a web moving with a constant translational velocity is described by the model of a thermoelastic panel simply supported at its ends. The problem of optimal damping of vibrations when applying active transverse actions is formulated. For solving the optimization problem, modern methods developed in control theory for systems with distributed parameters described by partial differential equations are used.     

Mathematical modelling of the axially moving panels subjected to thermomechanical actions

The paper is devoted to a stability and out-of-plane deformation analysis of an axially moving elastic web modelled as a panel (a plate undergoing cylindrical deformation). The panel is under homogeneous pure mechanical in-plane tension and thermal strains corresponding to the thermal tension and bending. In accordance with the static approach of stability analysis the problem of out-of-plane thermomechanical divergence (buckling) is reduced to an eigenvalue problem which is analytically solved. This problem corresponds to the case of in-plane thermomechanical tension and zero thermal bending. The general case of deformations induced by combined thermomechanical bending and tension is reduced to nonhomogeneous boundary-value problem and analyzed with the help of Fourier series.     

On the instability of an axially moving elastic plate

Problems of stability of an axially moving elastic band travelling at constant velocity between two supports and experiencing small transverse vibrations are considered in a 2D formulation. The model of a thin elastic plate subjected to bending and tension is used to describe the bending moment and the distribution of membrane forces. The stability of the plate is investigated with the help of an analytical approach. In the frame of a general dynamic analysis, it is shown that the onset of instability takes place in the form of divergence (buckling). Then the static forms of instability are investigated, and critical regimes are studied as functions of geometric and mechanical problem parameters. It is shown that in the limit of a narrow strip, the 2D formulation reduces to the classical 1D model. In the limit of a wide band, there is a small but finite discrepancy between the results given by the 1D model and the full 2D formulation, where the discrepancy depends on the Poisson ratio of the material. Finally, the results are illustrated via numerical examples, and it is observed that the transverse displacement becomes localised in the vicinity of free boundaries.     

Some problems of optimizing shell shape and thickness distribution on the basis of a genetic algorithm

We consider problems related to designing axisymmetric shells of minimal weight (mass) and the development of efficient nonlocal optimization methods. The optimization problems under study consist in simultaneous search for the optimal geometry and the shell thickness optimal distribution from the minimal weight condition under strength constraints and additional geometric constraints imposed on the thickness function, the transverse cross-section radii distribution, and the volume enclosed by the shell. Using the method of penalty functions, we reduce the above optimal design problem to a nonconvex minimization problem for the extended Lagrange functional. To find the global optimum, we apply an efficient genetic algorithm. We present the results of numerical solution of the optimal design problem for dome-like shells of revolution under the action of gravity forces. We present some data characterizing the convergence of the method developed here.     

Shape Design Sensitivity Analysis for Optimization Problems with Local and Global Functionals

ABSTRACT

In this paper, problems of sensitivity analysis and shape design for elastic solids are investigated. Optimization criteria are provided by integral and local functionals defined on the internal region and the boundary of the body. New sensitivity analysis relations are derived and implemented in successive optimization algorithms. Important aspects of effective shape optimization algorithms are discussed.     

Probabilistic Approaches for Optimal Beam DesignBased on Fracture Mechanics

The questions raised in this paper are related to an important class of problems concerning the shape optimization of brittle or quasibrittle elastic bodies with cracks. The optimization problems taken into consideration consist in finding the boundary of a body in such a way that the cost functional (volume of the body) reaches a minimum, while satisfying prescribed bounds on stress intensity factors. These problems are characterized by incomplete information concerning crack size, location and orientation. In this context the paper presents some possible formulations of optimal structural design problems based on probabilistic approaches. Using these approaches optimal designs for beams with surface and internal cracks have been found.Sommario.Il presente lavoro riguarda un'importante classe di problemi di ottimizzazione di forma di corpi elastici fragili o quasi fragili con fratture. I problemi di ottimizzazione considerati consistono nel trovare la forma di un corpo in modo tale che il suo volume sia minimo nel rispetto di alcune condizioni riguardanti il fattore di intensità di sforzo. Questi problemi sono caratterizzati da informazioni incomplete riguardanti le dimensioni della frattura, la sua posizione ed il suo orientamento. In questo contesto il presente lavoro presenta alcune possibili formulazioni del problema di ottimizzazione strutturale basato sull'approccio probabilistico. Usando tali approcci vengono ricavate le forme di travi ottime con fratture interne e di superficie.     

On the game theory approach to problems of optimization of elastic bodies : PMM vol. 37, n6, 1973, pp. 1098–1108

Problems of optimization of elastic bodies are considered usually in deterministic formulation, and for their solution the methods of variational calculus and the theory of optimal control are applicable (c.f., e.g., [1] and [2–4]). In the present paper there are considered those cases when either the complete information concerning the applied loads is not available,or it is known that the structure may be subjected subsequently to various loads of a certain class. The formulation is given of the problem of the determination of the shape of the elastic body, optimal for a class of loads, and there is indicated a general scheme for its solution based on the “minimax” approach used in the theory of games. Problems of optimization of elastic beams are considered and as a result of their solution certain features of optimal shapes are exhibited.