Localization phenomena in structural dynamics

A survey of vibration localization phenomena in the context of structural dynamics and vibrations is presented. The review covers the more common and relevant cases where mode localization and vibration confinement are likely to occur in engineering structures. Examples considered include periodic or nearly periodic multi-span beams and multi-bay trusses, large space structures, space antennas, and almost periodic (a.p.) structures with circular symmetry, e.g., bladed disks in turbomachines. Both analytical and numerical methods for analyzing and predicting localization in finite and infinite systems are discussed. In this paper, we show how the problem of mode localization and vibration confinement can be formulated as a problem in the theory of stability of differential equations with a.p. coefficients. Using stability theory, new definitions of mode localization can be established for both linear and nonlinear structures. The possibility of stabilizing certain nonconservative fluid-structure systems using structural disorder is demonstrated, and stability theorems are given for aeroelastic systems governed by normal operators. We also illustrate how the results from localization theory and the associated stability theory can be applied to the vibration control problem, by triggering vibration confinement by active or passive means.     

On the convexity of equilibrium plasma configurations

We investigate the problem of plasma confinement by a magnetic field in an infinite cylinder. We show that if the cylinder has convex cross-section, then there exists an equilibrium plasma configuration with convex cross-section.     

Gradient bounds for plasma confinement

The problem of plasma confinement in a Tokomak machine has recently attracted considerable mathematical attention. In the present article, gradient bounds for the solution are obtained through the use of differential inequalities and the Hopf maximum principle. These estimates, which take slightly different forms in the plasma and vacuum regions, become exact in the one-dimensional case. An estimate is also given for the distance from the boundary of the interior point where the maximum of the solution is attained.     

Numerical Investigations of the Dynamics of Development of Two-Dimensional Perturbations in Magnetic Galathea Traps

Differential Equations - Studying the stability of traps for magnetic plasma confinement is one of the main tasks in solving the problem of controlled thermonuclear fusion. The main trends and...     

On the Uniqueness of Solutions of a Nonlinear Elliptic Problem Arising in the Confinement of a Plasma in a Stellarator Device

We study the uniqueness of solutions of a semilinear elliptic problem obtained from an inverse formulation when the nonlinear terms of the equation are prescribed in a general class of real functions. The inverse problem arises in the modeling of the magnetic confinement of a plasma in a Stellarator device. The uniqueness proof relies on an -estimate on the solution of an auxiliary nonlocal problem formulated in terms of the relative rearrangement of a datum with respect to the solution. Accepted 21 March 1997     

On a nonlocal elliptic problem arising in the confinement of a plasma in a current carrying stellarator

This paper deals with a nonlocal free boundary problem arising in the study of the dynamics of the confinement of a plasma in a Stellarator device. The free boundary represents the separation between the plasma and vacuum regions, and the nonlocal term involves the notions of relative rearrangement and monotone rearrangement. We establish some new properties on the decreasing rearrangement that can be used to prove the convergence of the approximate problem, and then prove the existence of solutions by Galerkin method. Copyright © 2013 John Wiley & Sons, Ltd.     

On the two and one-half dimensional Vlasov–Poisson system with an external magnetic field: Global well-posedness and stability of confined steady states

The time evolution of a two-component collisionless plasma is modelled by the Vlasov–Poisson system. In this work, the setting is two and one-half dimensional, that is, the distribution functions of the particles species are independent of the third space dimension. We consider the case that an external magnetic field is present in order to confine the plasma in a given infinitely long cylinder. After discussing global well-posedness of the corresponding Cauchy problem, we construct stationary solutions whose support stays away from the wall of the confinement device. Then, in the main part of this work we investigate the stability of such steady states, both with respect to perturbations of the initial data, where we employ the energy-Casimir method, and also with respect to perturbations of the external magnetic field.     

Sopra un problema di interazione di onde elettromagnetiche in un plasma non lineare a geometria sferica

We consider the problem of the approximate determination of the electromagnetic field generated by two oscillating electric dipoles in the center of a spherical cavity imbedded in a nonlinear plasma. The influence of the nonlinearities of the plasma upon the field are investigated by means of a perturbative solution of the relevant equations, deduced from the ‘quasi-hydrodynamic’ approach. Analytical and numerical results, which illustrate several nonlinear effects, as well as the influence of the rarefaction and ionization of the gas, are included. The main result of the paper is the prediction of an observable second harmonic field.     

Higher Critical Points in an Elliptic Free Boundary Problem

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary problems have already been studied extensively. But because the functionals are not smooth, standard variational methods cannot be used directly to prove the existence of higher critical points. Here we find a nontrivial critical point of mountain pass type and prove many of the same estimates known for minimizers, including Lipschitz continuity and nondegeneracy. We then show that the free boundary is smooth in dimension 2 and prove partial regularity in higher dimensions.     

A new look at quarks confinement

Quarks confinement is an experimental fact. ‘tHooft and later on Gross, Wilczek and Politzer have contributed in various ways to our present considerable theoretical understanding of the problem. However, an exact water proof theoretical derivation of the problem is at best still in progress. The present note argues that to understand quarks confinement, a deeper understanding of the Planck scale physics is indispensable and shows using analytical topological arguments that absolute confinement is a result of a phase transition of quantum spacetime at the Planck scale.     

Optimal numerical control of single-well to cross-well chaos transition in mechanical systems

The optimal numerical control of nonlinear dynamics and chaos is investigated by means of a technique based on removal of the relevant homo/heteroclinic bifurcations, to be obtained by modifying the shape of the excitation. To highlight how the procedure works, the analysis is accomplished by referring to the Duffing equation, although the method is general and holds, at least in principle, for whatever nonlinear system. Attention is focused on the single-well to cross-well chaos transition due to a homoclinic bifurcation of an appropriate period 3 saddle [Int. J. Bifur. Chaos 4 (1994) 933]. It is shown how it is possible to eliminate this bifurcation simply by adding a single superharmonic correction to the basic harmonic excitation. Successively, the problem of the optimal choice of the superharmonic is addressed and solved numerically. The optimal solutions are determined in the two cases of symmetric (odd) and asymmetric (even) excitations, and it is shown how they entail practical, though variable, effectiveness of control in terms of confinement and regularization of system dynamics.     

Essential spectrum of linearized operator for MHD plasma in cylindrical region

Linearized operators for ideal magnetohydrodynamics related to the plasma confinement are considered for a cylindrical geometry. The essential spectrum for the Fourier decomposed operator is determined by an analysis of its inverse. In particular, the mathematically rigorous treatment of the magnetic axis is presented.

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Résumé On considère les opérateurs linéarisés de la magnetohydrodynamique idéale en géométrie cylindrique par rapport au confinement du plasma. Le spectre essential pour l'opérateur Fourierdécomposé est déterminé par l'analyse de son inverse. En particulier, on présente le traitement mathématiquement rigoureux de l'axe magnétique.

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This work was supported by the Swiss National Foundation of Sciences during the stay of the author at the Swiss Federal Institute of Technology. Lausanne.     

等离子体反应器流场的数值模拟

本文采用颗粒轨道模型对等离子体反应器进行数值模拟.数值格式用SIMPLER法.湍流模型用简单的次网格模型.计算结果给出速度场、温度场及颗粒轨道,对工程设计和工艺过程参数控制有参考价值.     

On the existence of the free-boundary for some problems arising in plasma physics

We study some conditions for the existence of a free-boundary for two different bidimensional models arising in the magnetic confinement of a plasma. We derive estimates on the size and location of the region surrounded by the free-boundary (known as plasma region) for both models. We also derive sufficient conditions for the non-existence of the free-boundary.     

A perturbation solution for bacterial growth and bioremediation in a porous medium with bio-clogging

We investigate a flow problem of relevance in bioremediation and develop a mathematical model for transport of contamination by groundwater and the spreading, confinement, and remediation of chemical waste. The model is based on the fluid mass and momentum balance equations and simultaneous transport and consumption of the pollutant (hydrocarbon) and nutrient (oxygen). Particular emphasis is placed on the study of processes involving the full coupling of reaction, transport and mechanical effects. Dimensional analysis and asymptotic reduction are used to simplify the governing equations, which are then solved numerically.     

Numerical solution of a free boundary problem by continuation

An algorithm is described for the numerical solution of a free boundary problem using continuation. The problem considered is a quasi-steady problem arising in electrochemical machining. The domain of the problem is mapped onto a square and the governing nonlinear equations are discretised using finite differences. A Newton-like iteration is employed for the solution of the nonlinear algebraic system and global convergence is achieved by means of continuation. Numerical results are included.     

Riemann–Hilbert Problems,Matrix Orthogonal Polynomials and Discrete Matrix Equations with Singularity Confinement

In this paper, matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not.     

Flatness-based Feed-forward Control for an Elastic Heavy Rope

Considered is the feed-forward control design for an elastic heavy rope that is model by a quasilinear partial differential equation of hyperbolic type. The problem is solved by means of a flatness-based approach that involves the solution of a Cauchy problem associated with the model equation. Simulation results are given to illustrate the results. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariance of means

We give a survey of results dealing with the problem of invariance of means which, for means of two variables, is expressed by the equality \(K\circ \left( M,N\right) =K\). At the very beginning the Gauss composition of means and its strict connection with the invariance problem is presented. Most of the reported research was done during the last two decades, when means theory became one of the most engaging and influential topics of the theory of functional equations. The main attention has been focused on quasi-arithmetic and weighted quasi-arithmetic means, also on some of their surroundings. Among other means of great importance Bajraktarevi? means and Cauchy means are discussed.