Helle Hein

Optimization of clamped plastic shallow shells subjected to initial impulsive loading

An optimization technique is suggested for shallow spherical shells made of a ductile material and subjected to initial impact loading. The shell under consideration is pierced with a central hole and clamped at the outer edge. The optimal design of the shell of piece-wise constant thickness is established under the condition that the maximal residual deflection attains its minimal value for given total weight. The material of the shell is assumed to obey the Tresca yield condition and associated flow law. By the use of the method of mode form motions the problem is transformed into a particular problem of non-linear programming and solved numerically.

Application of the Haar wavelet method for solution the problems of mathematical calculus

Abstract In recent times the wavelet methods have obtained a great popularity for solving differential and integral equations. From different wavelet families we consider here the Haar wavelets. Since the Haar wavelets are mathematically most simple to be compared with other wavelets, then interest to them is rapidly increasing and there is a great number of papers,where thesewavelets are used tor solving problems of calculus. An overview of such works can be found in the survey paper by Hariharan and Kannan [1] and also in the text-book by Lepik and Hein [2]. The aim of the present paper is more narrow: we want to popularize our method of solution, which is published in 19 papers and presented in the text-book [2]. This method is quite universal, since a large group of problems can be solved by a unit approach. The paper is organised as follows. In Section 1 fundamentals of the wavelet method are described. In Section 2 the Haar wavelet method and solution algorithms are presented. In Sections 3-9 different problems of calculus and structural mechanics are solved. In Section 10 the advantageous features of the Haar wavelet method are summed up.

Optimization of clamped rigid-plastic shallow shells of piecewise constant thickness

Abstract The minimum weight problem is studied under the condition that the considered shell has a piecewise constant thickness. The shell with free internal edge and clamped outer edge is subjected to uniformly distributed internal pressure. Moderately large deflections are taken into account and a deformation-type theory of plasticity is employed. The optimization problem includes the additional restriction, which demands that the maximal deflections of the shell of piecewise constant thickness and of the reference shell, of constant thickness, coincide. Employing the variational methods of the optimal control theory, necessary optimality conditions are established. The results obtained are used to establish the optimal parameters for the shell of piecewise constant thickness.

Crack Identification in Vibrating Beams Using Haar Wavelets and Neural Networks

This study investigates the depth and location of cracks in homogeneous Euler-Bernoulli beams with free vibrations. The problem is frequently encountered in industrial design and modeling, where an exact model requires the frequency output to be calibrated with a physical measure. The crack is simulated by a line spring model. The boundary value problem is solved using the Haar wavelets. The characteristic parameters are predicted with the aid of neural networks. The proposed method is compared to an alternative approach based on neural networks and several frequencies only. The significance of the complex approach of Haar wavelets and neural networks lies in its ability to make fast accurate model-independent predictions calculating only one natural frequency and training the network only once.

Optimization of rigid-plastic shallow curved beams

The minimum weight problem of a shallow circular beam is studied in the case when the beam has a piece-wise constant thickness. The minimum of the weight is sought under the condition that the deflections of the beam of piece-wise constant thickness do not exceed those of the reference beam of constant thickness for given values of the external loading. The beam is subjected to uniformly distributed transverse pressure and to axial dead load. The material of the beam is assumed to be ideally rigid-plastic. Moderately large deflections are taken into account. Necessary optimality conditions are derived and used in order to establish the optimal values of the design parameters.

Optimization of rigid-plastic shallow spherical shells of piecewise constant thickness

An approximate method developed earlier for the investigation of large plastic deflections of circular and annular plates is accommodated for shallow spherical shells. The material of the shells is assumed to obey Trescas yield condition and the associated deformation law. The minimum weight problem concerning shells operating in the post-yield range is posed under the conditions that (i) the thickness of the structure is piece-wise constant and (ii) the maximal deflections of the optimized shell and a reference shell of constant thickness, respectively, coincide. Necessary optimality conditions are derived with the aid of the variational methods of the optimal control theory. The set of equations obtained is solved numerically.

Applying Haar Wavelets in the Optimal Control Theory

In this chapter, the Haar wavelet method for solving problems of constrained optimal control is applied. To begin with, we are reminded of some of the necessary conditions for the optimal control.

Optimization of shallow shells with different yield stresses in tension and compression

The minimum weight problem of thin rigid-plastic shallow spherical shells is studied. The thickness of the shell is piece-wise constant and the material has different yield stresses in tension and compression. The flow theory of plasticity is employed. Both solid and sandwich shells are considered. Necessary optimality conditions are derived with the aid of optimal control theory.

Applying Haar Wavelets in Damage Detection Using Machine Learning Methods

The basic idea of the present Chapter is to establish directly an input-output relationship between the modal responses and the delamination locations/sizes using back-propagation neural networks.

Vibrations of Cracked Euler-Bernoulli Beams

In this Chapter, the Haar wavelet method is applied for analysing bending and vibrations of elastic Euler-Bernoulli beams.