Ülo Lepik

Numerical solution of evolution equations by the Haar wavelet method

An efficient numerical method for solution of nonlinear evolution equations based on the Haar wavelets approach is proposed. The method is tested in the case of Burgers and sine-Gordon equations. Numerical results, obtained by computer simulation, are compared with other available solutions. These calculations demonstrate that the accuracy of the Haar wavelet solutions is quite high even in the case of a small number of grid points.

Haar wavelet method for nonlinear integro-differential equations

Abstract A numerical method for solving nonlinear integral equations based on the Haar wavelets is presented. The method is applicable for Volterra integral equations and integro-differential equations; it can be used also for solving boundary value problems of ordinary differential equations. The efficiency of the proposed method is tested with the aid of four examples. High accuracy even for a small number of collocation points is stated.

Solving integral and differential equations by the aid of non-uniform Haar wavelets

A modification of the Haar wavelet method, for which the stepsize of the argument is variable, is proposed. To establish the efficiency of the method three test problems, for which exact solution is known, are considered. Computer simulations show clear preference of the suggested method compared with the Haar wavelet method of a constant stepsize.

Dynamic buckling of elastic–plastic beams including effects of axial stress waves

Abstract Buckling of axially compressed elastic–plastic beams is discussed. The load is applied instantaneously and remains unaltered during the motion. The effect of stress waves travelling along the beam is taken into account. It is assumed that the material of the beam has linear-strain hardening. A method of solution, based on the Galerkin technique, is proposed; this method is applicable to an arbitrary number of degrees of freedom. Numerical examples are presented.

Optimal design of plastic structures under impulsive and dynamic pressure loading

Abstract Optimal design of a rigid-plastic stepped beam and circular plate is considered in the first part of the paper assuming the mode form of motion. The form of optimal mode is sought for which a structure of constant volume attains a minimum of local or mean deflection. It is assumed that the constant kinetic energy Ko is attained by the structure through impulsive loading. Differences between optimal static and dynamic solutions are discussed. Non-uniqueness of modes is demonstrated and significance of stable mode motions is emphasized. In the second part of the paper, an optimal design of a rigid-plastic stepped beam loaded by a uniform pressure over a time interval 0 ⩽ t ⩽ t 1 is considered assuming constant beam volume and looking for a design corresponding to minimum of local deflection. The solution presented is valid for moderate dynamic pressures when mode motion occurs during consecutive time intervals and no travelling plastic hinges exist.

On dynamic buckling of elastic–plastic beams

Abstract A theoretical study of dynamic buckling of elastic–plastic beams under compressive axial forces is presented. It is assumed that the loading process is so slow that axial inertia effects may be neglected. Four types of axial loading are considered. It is supposed that the beams material has linear strain hardening. Equations of motion are derived from Hamiltons principle. A numerical method of integrating these equations is presented. The main aim of the paper is to shed light on the problem how the deflection shapes develop in the post-critical stage. For this purpose six examples are presented.

Optimal design of beams with minimum compliance

Abstract An optimization problem of non-linear elastic or viscous beams is discussed. To the beam an additional support is introduced whose location must be selected so as to minimize the compliance of the beam. The problem is solved with the aid of optimal control theory. Both rigid and flexible supports are considered. Some new optimization conditions, which are valid for arbitral compliance criterions, are deduced. A few illustrating examples are given.

On plastic buckling of cylindrical shells struck axially with a mass

Buckling of elastic-plastic cylindrical shells axially struck with a mass is discussed. The effect of stress wave fronts travelling along the shell is taken into account. This allows us to cast some more light on the mechanism of progressive buckling. Only axisymmetric buckling modes are considered. The analysis is based on the Kirchhoff-Love hypotheses. As a constitutive law, the rate independent elasto-plastic relations with linear strain hardening and von Mises yield condition are adopted. A method for calculating bifurcation times and buckling modes is presented. Numerical examples are given.

Elastic-plastic vibrations of a buckled beam

Abstract The paper discusses non-linear vibrations of a buckled beam under harmonic excitation. The material of the beam is elastic-plastic with linear strain-hardening. The equations of motion are integrated by Galerkins method. The validity of the assumption, that the membrane force is constant along the beam, is discussed. It is shown that even in the case of elastic deformations, this hypothesis may cause great discrepancies in the deflection history diagrams to be compared with the real solution. In the elastic case, the Melnikov method has been used for estimating the threshold transverse load at which chaotic motion can take place. In the range of elastic-plastic vibrations, chaotic motion of the beam is discussed.

Vibrations of elastic-plastic fully clamped beams and flat arches under impulsive loading

Abstract The problem of dynamic response of fully clamped beams and flat arches under transverse impulsive loading is solved by Galerkins method. Possibility of chaotic behavior of beams and arches is discussed; for this purpose displacement-time histories, phase portraits and power spectrum diagrams are put together.