Torsional wave dispersion relations in a pre-stressed bi-material compounded cylinder with an imperfect interface

This paper studies the influence of the imperfectness of the contact condition on the torsional wave propagation in the initially stressed (stretched) bi-material compounded circular cylinder. The investigation is carried out within the scope of the piecewise homogeneous body model with the use of the Three-dimensional Linearized Theory of Elastic Waves in Initially Stresses Bodies. The mathematical formulation of the corresponding eigen-value problem is formulated and the solution method for that is developed. The two cases considered are the bi-material compounded cylinder consists of the solid inner and surrounding hollow outer cylinders (Case 1); the bi-material compounded cylinder consists of the hollow inner and surrounding hollow outer cylinders (Case 2). The mechanical relations of the cylinders’ materials are written through the Murnaghan potential. It is proven that the imperfectness of the contact condition does not influence the asymptotic-limit values of the wave propagation velocity. Moreover, the numerical results on the effects of the imperfectness of the boundary condition on the influence of the initial stresses on the wave propagation velocity are presented and discussed.     

The propagation of electromagnetic waves in parabolic wave guides

Résumé Dans cet exposé, l'auteur examine les propriétés de canaux qui propagent les ondes électromagnétiques. Les parois de ces canaux sont faites d'une paire de paraboles orthogonales. L'auteur donne des expressions obtenues pour les fréquences critiques dans le cas de canaux parfaitement conducteurs, ainsi que des formules nouvelles par lesquelles on peut évaluer les constantes d'atténuation dans le cas de canaux imparfaitement conducturs.     

Multiple TE-modes for planar, self-focusing wave guides

We study in this paper the propagation of electro-magnetic (optical) waves, guided by a medium consisting of layers of dielectric material. This problem can be reduced to an eigenvalues problem of the form Lu+ f(u)u=;u, s H¹ (R){0}, where L is a linear, second order differential operator and f(u) u is a nonlinear term. We are interested in solutions (;_{n, d}, u_{n, d}) where u_n,d has exactly n–1 zeroes and where . (Remark that d² is related to the power of the beam.) Under suitable conditions, such solutions exist, n N^*; tin certain cases, we establish the solution for all values of d>0, while in other cases such solutions exist only when d is large enough. These results agree with what is expected from a physical point of view: either guidance occurs at all powers, or guidance occurs only at high powers, or no guidance at all occurs. Beside the existence of solutions, we analyze their behavior as d varies; tin the case where guidance occurs at all powers we show that all the families of solutions have a common bifurcation behavior. as d0.This research has been supported by the Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung.     

On dispersion relations of waves in multilayered magneto-electro-elastic plates

The dispersion behavior of waves in a layered magneto-electro-elastic plate is studied. Formulations are given for open-circuit surface and short-circuit surface, respectively. Legendre orthogonal polynomial series approach is employed to solve the coupling magneto-electro-elastic controlling equations. The obtained dispersion curves for the open-circuit surface are compared with the published data to confirm the validity of the method. Through numerical examples, the convergence of the method is discussed. The differences of the two boundaries are illustrated. The influences of the piezoelectricity and piezomagnetism are illustrated and the influential factors of the piezoelectric and piezomagnetic effect are discussed. Finally, the influences of the volume fraction and stacking sequence on the piezoelectric effect and piezomagnetic effect are discussed.     

Numerical computation of stress induced microstructure

The mesh transformation method is applied on a two-dimensional elastic crystal model to study the formation of laminated microstructure in austenite-martensite phase transition when certain external loads are applied. Numerical experiments show that simple laminated microstructures with various volume fractions and twin width can be obtained by varying the loads. Numerical experiments also show that second order laminated microstructure with branched needle-like laminates can also be obtained by certain loads.     

Numerical computation of stress induced microstructure

The mesh transformation method is applied on a two-dimensional elastic crystal model to study the formation of laminated microstructure in austenite-martensite phase transition when certain external loads are applied. Numerical experiments show that simple laminated microstructures with various volume fractions and twin width can be obtained by varying the loads. Numerical experiments also show that second order laminated microstructure with branched needle-like laminates can also be obtained by certain loads.     

Numerical approaches for computation of fronts

Moving fronts and pulses appear in many engineering applications like flame propagation and a falling liquid film. Standard computation methods are inappropriate since the problem is defined over an infinite domain and a steady-state solution exists only for a certain front velocity. This work presents a transformation that converts the original problem into a boundary-value problem within a finite domain, in a way that preserves the behavior at the boundaries. Good low-order approximations can be obtained as demonstrated by two examples. In another approach, a central element of adjustable length is incorporated into a three-element structure where the edge-elements obey known asymptotic solutions. That yields multiplicity of travelling fronts in an infinite domain but it successfully approximates standing wave solutions in a finite domain. The approximate solutions are shown to obey the qualitative features known for the exact solutions, like asymptotic solutions or the bifurcation set–the boundary where a new solution emerges or disappears.     

Numerical computation of the coefficients in exponential fitting

We show that a direct numerical computation of the coefficients of any method based on the exponential fitting is possible. This makes unnecessary the knowledge of long sets of analytical expressions for the coefficients, as usually presented in the literature. Consequently, the task of any potential user for writing his/her own code becomes much simpler. The approach is illustrated on the case of the Numerov method for the Schrödinger equation, on a version for which the analytic expressions of coefficients are not known.

     

Numerical computation of branch points in nonlinear equations

Summary The numerical computation of branch points in systems of nonlinear equations is considered. A direct method is presented which requires the solution of one equation only. The branch points are indicated by suitable testfunctions. Numerical results of three examples are given.     

Double dispersion relations in quantum Statistical Mechanics—II

Double dispersion relations are given for the functions $$\gamma (12;3) = \frac{1}{i}\langle T\left( {\psi (1)\psi ^\dag (2)B (3)} \right)\rangle$$ common in many electron problems. Here ψ and ψ^T refer to electron destruction and creation operators, and B (3) is a boson or boson-like operator such as particle density, ion displacement, local spin, particle current density, or electron spin density. Consequences of Hermiticity properties and time reversal invariance as well as sum rules are established for the spectral density functions entering the double dispersion relations. A subtracted double dispersion relation is suggested for the corresponding vertex function. Details of the derivation and examples are given which were not included in an earlier brief report of this work.     

A fast algorithm for repeated computation of linear recurrence relations

Let the calculation of thenth term of a general recurrence relation requireO(q(n)) arithmetical operations. The derivation is given of a new algorithm that performsm repetitions of this calculation with different initial conditions requiring onlyO(q(n)) +O(m) operations. The application of this algorithm to accelerate the calculation of continued fractions and the solution of three- and five-diagonal systems of linear equations, is also described.     

Numerical computation of branch points in ordinary differential equations

Summary This paper deals with the computation of branch points in ordinary differential equations. A direct numerical method is presented which requires the solution of only one boundary value problem. The method handles the general case of branching from a nontrivial solution which is a-prioriunknown. A testfunction is proposed which may indicate branching if used in continuation methods. Several real-life problems demonstrate the procedure.     

Numerical computation of forced oscillations in coupled duffing equations

Oscillating phenomena in non-linear mechanical systems with two degrees of freedom described by coupled Duffing equations are studied from the computational view point. Galerkin approximations of order 7 are computed with a very high precision on an electronic computer by applying a numerical approximation method of Urabe for the Galerkin method. The existence of an exact isolated periodic solution in a small neighborhood of these Galerkin approximations is proved and the error bound of these Galerkin approximations is given. The stability of Stierel's integration method in combination with Galerkin approximations is shown.     

Long distance wave computation using nonlinear solitary waves

A recently developed method is described to propagate short wave equation pulses over indefinite distances and through regions of varying indices of refraction, including multiple reflections. The method, “Wave Confinement”, utilizes a newly developed nonlinear partial differential equation (pde) that propagates basis functions according to the wave equation. These basis functions are generated as stable solitary waves where the discretized equation can be solved without any numerical dissipation. The method can also be used to solve for harmonic waves in the high frequency (Eikonal) limit, including multiple arrivals. The solution involves discretizing the wave equation on a uniform Eulerian grid and adding a simple nonlinear “Confinement” term. This term does not change the amplitude (integrated through each point on the pulse surface) or the propagation velocity, or arrival time, and yet results in capturing the waves as thin surfaces that propagate as thin nonlinear solitary waves and remain ∼2-3 grid cells in thickness indefinitely with no numerical spreading. A new feature described in this paper involves computing scattering of short pulses from complex objects such as complete aircraft. A simple “immersed surface” approach is used, that utilizes the same uniform grid as the propagation and avoids complex, body fitted or adaptive grid schemes.The new method should be useful in areas of wave propagation, from radar scattering and long distance communications to cell phone transmission.     

On the computation of a wave field in a multilayer medium

In this paper, theoretical seismograms of waves propagating in a medium comprising four layers that rest on a half-space are presented. The seismograms are computed by the method of summation of kinematic and dynamic wave analogs. On the basis of these computations, some conclusions concerning the behavior of the wave field at large distances are drawn. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 243–252. Translated by N. S. Smirnova.     

Numerical computation of connecting orbits on a manifold

In this paper we propose a numerical method for approximating connecting orbits on a manifold and its bifurcation parameters. First we extend the standard nondegeneracy condition to the connecting orbits on a manifold. Then we construct a well-posed system such that the nondegenerate connecting orbit pair on a manifold is its regular solution. We use a difference method to discretize the ODE part in this well-posed system and we find that the numerical solutions still remain on the same manifold. We also set up a modified projection boundary condition to truncate connecting orbits on a manifold onto a finite interval. Then we prove the existence of truncated approximate connecting orbit pairs and derive error estimates. Finally, we carry out some numerical experiments to illustrate the theoretical estimates.