The Minkowski dimension and critical effects in fractal evolution of defects

Creation of the programs for determination of the Minkowski dimension and other corresponding fractal characteristics are the main result of this paper. The programs were created also for the one-side fractal dimension. The stable numerical algorithm of multiplicative differentiate is given in this paper. One confirms that the critical effect occurs and it is corresponding to creation of the macroscopic crack through the breaking of the early formed microdefects. The singularity occurs at the moment at which the percolation is simultaneously observed. A character of the singularity is well visible – the jump for tens of orders occurs. The critical effect was observed in the computer simulations of the fatigue process.     

The acoustic theory of the interaction of a shock wave,having an exponential relaxation zone,with a porous medium

An analytic solution of the problem of the reflection of a modulated pressure pulse in the form of an “instantaneous jump and exponential relaxation” in a liquid from a plane boundary of a porous medium of infinite extent, saturated with the same liquid, is constructed. Using the analytic solutions obtained, a numerical analysis is given of the development of distinctive features of the reflected and transmitted waves depending on the porosity and permeability of the porous medium, and also the length of the pulse signal.     

A simple method for estimating the fractal dimension from digital images: The compression dimension

The fractal structure of real world objects is often analyzed using digital images. In this context, the compression fractal dimension is put forward. It provides a simple method for the direct estimation of the dimension of fractals stored as digital image files. The computational scheme can be implemented using readily available free software. Its simplicity also makes it very interesting for introductory elaborations of basic concepts of fractal geometry, complexity, and information theory. A test of the computational scheme using limited-quality images of well-defined fractal sets obtained from the Internet and free software has been performed. Also, a systematic evaluation of the proposed method using computer generated images of the Weierstrass cosine function shows an accuracy comparable to those of the methods most commonly used to estimate the dimension of fractal data sequences applied to the same test problem.     

Group theoretic method for analyzing interaction of a discontinuity wave with a strong shock in an ideal gas

A group theoretic method is used to obtain an exact particular solution to the system of partial differential equations, describing one-dimensional unsteady planar, cylindrically and spherically symmetric motions in an ideal gas, involving shock waves. It is interesting to remark that the exact solution obtained here is precisely the blast wave solution obtained earlier using a different method of approach. Further, the evolution of a discontinuity wave and its interaction with the strong shock are studied within the state characterized by the exact particular solution. The properties of reflected and transmitted waves and the jump in the shock acceleration are completely characterized, and certain observations are noted in respect to their contrasting behavior.     

The Minkowski dimension of the bivariate fractal interpolation surfaces

We present a new construction of continuous bivariate fractal interpolation surface for every set of data. Furthermore, we generalize this construction to higher dimensions. Exact values for the Minkowski dimension of the bivariate fractal interpolation surfaces are obtained.     

On a class of fractal functions with graph Hausdorff dimension 2

We prove that the graph of the continuous function

has Hausdorff dimension 2, where λ > 1, β >  > 1, (x) = 2x, 0  x  1/2, (−x) = (x) and (x + 1) = (x).     

The diffraction of a shock wave by an inclined body of revolution

Summary The interaction between a plane normal shock wave of arbitrary strength and a slender body of revolution, held with its axis at a small angle to the direction of propagation of the shock, is considered. The pressure field is determined by using the method of matched asymptotic expansions and the general results are applied to the problem of diffraction by an inclined circular cone.

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Zusammenfassung Es wird die Beeinflussung einer ebenen Stosswelle beliebiger Stärke durch einen schlanken Rotationskörper untersucht, dessen Achse einen kleinen Winkel mit der Ausbreitungsrichtung des Stosses bildet. Das Druckfeld wird mit Hilfe der abgestimmten asymptotischen Entwicklungen bestimmt, und die allgemeinen Resultate werden auf das Problem der Diffraktion durch einen angestellten Kreiskegel angewendet.

     

Propagation of a shock wave in an isentropic gas with small viscosity

A new method is considered for constructing the front and the dynamics of motion of a shock wave propagating in an isentropic gas with small viscosity with a speed close to the speed of sound. The proposed algorithm of solution makes it possible to considerably raise the accuracy and effectiveness of numerical computations on the computer as compared with known universal numerical methods based on the use of difference schemes.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 273–308, 1977.     

The decay of a magnetohydrodynamic shock wave

Résumé On utilise l'analyse linéaire pour étudier les faibles perturbations non-isentropiques des équations relatives aux écoulements unidimensionnels et non-stationnaires d'un fluide non visqueux idéal, parfaitement conducteur de l'électricité et compressible, soumis à l'action d'un champ magnétique transversal. On utilise la solution générale de la perturbation non-isentropique d'un écoulement par ondes simples centrées pour déterminer la perturbation qui se manifeste lorsqu'une onde de choc magnétohydrodynamique, tout d'abord uniforme et de force arbitraire, rencontre le régime d'onde simple.Dans le cas limité d'un champ magnétique nul, la solution se réduit exactement à celle du problème correspondant de la dynamique classique des gaz. C'est la une confirmation de la validité de la théorie.

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This research was supported by National Science Foundation grant GP 87190.     

The combined compression and shear of a rectangular rubber block

For long rectangular rubber blocks with metal plates bonded to their upper and lower surfaces, approximate force-deflection relations are obtained for the problem of combined compression (or tension) and shear. The results obtained generalise that of Klingbeil and Shield for the case of pure compression alone, as well as incorporating the well known simple shear result. However, unlike the pure compression solution, horizontal and vertical force resultants on the free surfaces cannot both be set to zero, which is a well known characteristic of simple shearing. Here the strategy is adopted of equating to zero only the horizontal force resultants on the free surfaces so that the pure compression approximation of Klingbeil and Shield emerges in the event of zero shear. The force-deflection relations so obtained are illustrated graphically.     

Nonlinear responses from the interaction of two progressing waves at an interface

For scalar semilinear wave equations, we analyze the interaction of two (distorted) plane waves at an interface between media of different nonlinear properties. We show that new waves are generated from the nonlinear interactions, which might be responsible for the observed nonlinear effects in applications. Also, we show that the incident waves and the nonlinear responses determine the location of the interface and some information of the nonlinear properties of the media. In particular, for the case of a jump discontinuity at the interface, we can determine the magnitude of the jump.     

Quick estimation of escape rate with the help of fractal dimension

The non-linearity caused by the application of digital control may lead to transient chaotic behaviour. In the present paper, we analyse a simple model of a digitally controlled mechanical system with dry friction, which may perform transient chaotic vibrations. As a consequence of the digital effects, the behaviour of this system can be described by a 1D piecewise linear map. The fractal dimension of the so-called chaotic repeller set is calculated and the results are used for the quick estimation of the mean lifetime of chaotic transients.     

Wavefront evolution of a plane wave passing through an area with heterogeneities

A two-dimensional eikonal equation with the right-hand side tending to unity as the distance from the origin of coordinates tends to infinity is considered. Some formulas describing the wavefront in such a medium are obtained.     

The motion of a non-uniform hydromagnetic shock wave

Sommaire Nous considérons un écoulement, unidimensionnel, non stationaire, non isentropique, d'un fluide parfait, parfaitement conducteur de l'électricité, soumis à l'action d'un champ magnétique pour une orientation transversale du champ et le mouvement non continu d'une onde de choc magnetohydrodynamique.

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This research was supported by a National Aeronautics and Space Administration general research grant to the University of Wisconsin.     

Modeling the interaction of a wave front with a forest

We investigate the processes that arise when a wave front hits a natural obstacle in the form of a forest. The modeling is carried out in the framework of a single methodological approach that uses the Euler equation to describe the motion of the air mass both over an open area and inside the forest. In the latter case the equations include mass forces associated with the vegetation. The numerical solution is obtained by Godunov’s method using parallel programming techniques. Two types of incident wave front are investigated: a plane shockwave and a nonlinear acoustic impulse modeling a spherical explosion wave at a large distance from the source. The specific features of the interaction process, including penetration of the wave front into the forest, partial reflection from the near boundary, and diffraction above the top boundary, are investigated for different types of vegetation (coniferous and deciduous forests). The numerical results reveal the formation of a pair of ascending and descending currents in the upper part of the forest (inside the tree crowns). The existence of this structure is confirmed by experimental findings. __________ Translated from Prikladnaya Matematika i Informatika, No. 21, pp. 48–71, 2005.     

Estimation of the critical viscous sub-layer in shock wave boundary layer interaction

Zusammenfassung Wenn eine kompressible Grenzschicht (laminar oder turbulent) einer Störung unterworfen wird, sei es durch eigene Instabilität oder von aussen aufgeprägt, so macht sich der Einfluss dieser Störung sowohl stromaufwärts wie stromabwärts in der Strömungsgrenzschicht bemerkbar.Lighthill nimm in seiner mathematischen Behandlung des Mechanismus der Fortpflanzung dieser Störung stromaufwärts und stromabwärts an, dass die laminare Unterschicht der Grenzschicht genug ist, um Kompressibilitätseffekte darin vernachlässigen zu können.In vorliegender Veröffentlichung wurde die Dicke dieser kritischen laminaren Unterschicht abgeschätzt, ausgehend von ähnlichkeitsbetrachtungen und unter Verwendung vorhandener Messresultate. Es wurde festgestellt, dass das Verhältnis zwischen der kritischen, zähen Unterschicht zur totalen Grenzschtdicke (ungestört) im laminaren Fall von der Grössenordnung 110 und im turbulenten Fall etwa 1100 ist.