Non-dispersive wave propagation in nonlinear dielectrics

Zusammenfassung Unter der Voraussetzung, dass der Zusammenhang zwischen elektrischer Polarisation und elektrischer Feldstärke nichtlinear, aber von der Frequenz unabhängig ist, wird gezeigt, dass im allgemeinen nach einer endlichen, vom Anfangszustand abhängigen Zeit in der Wellenausbreitung Mehrdeutigkeiten auftreten. Als Spezialfall wird auch die einfache Wellenausbreitung untersucht.     

On wave propagation along a weakly laterally inhomogeneous, weakly bent anisotropic elastic layer

The propagation of a waveguide mode along a weakly bent elastic anisotropic layer is considered. Analytic expressions for the amplitude and the Berry phase of the corresponding oscillation are obtained.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 12–32.     

Hybrid boundary element formulation for acoustic wave propagation

The so‐called hybrid stress boundary element method (HSBEM) is introduced in a frequency domain formulation for the computation of acoustic radiation and scattering in closed and in finite domains. Different from other boundary element formulations, the HSBEM is based on an extended Hellinger‐Reissner variational principle and leads to a Hermitian, frequency‐dependent stiffness equation. Due to this, the method is very well suited for treating fluid structure interaction problems since the effort for the coupling the structure, discretized by a finite elements, and the fluid, discretized by the HSBEM is strongly reduced. To arrive at a boundary integral formulation, the field variables are separated into boundary variables, which are approximated by piecewise polynomial functions, and domain variables, which are approximated by a superposition of singular fundamental solutions weighed by source strength. This approximation cancels the domain integral over the equation of motion in the hybrid principle and leads to a boundary integral formulation, incorporating singular integrals. Comparing to previous results published by the authors, new considerations concerning the interpretation of singular contributions in the stiffness matrix for exterior domain problems are communicated here.     

Mathematical modeling of electromagnetic wave propagation in nonlinear electrodynamics

The eikonal method for an electromagnetic wave propagating according to the laws of non-linear electrodynamics in vacuum in external electromagnetic and gravitational fields is developed. A mathematical model of the propagation of electromagnetic signals in the parameterized post-Maxwellian electrodynamics in vacuum is constructed. As an example of using the proposed method, the angles of the nonlinear electrodynamical and gravitational curvature of the normal wave rays propagating in the field of a charged collapsar are calculated.     

Effective wave propagation in a prestressed nonlinear elastic composite bar

The problem of determining the effective incremental responseof nonlinearly elastic composite materials given some initialprestress is of interest in numerous application areas. In particular,the case when small-amplitude elastic waves pass through a prestressedinhomogeneous structure is of great importance. Of specificinterest is how the initial finite deformation affects the microstructureand thus the subsequent response of the composite. Modellingthis effect is in general extremely difficult. In this article,we consider the simplest problem of this type where the materialis a one dimensional composite bar consisting of two distinctphases periodically distributed. Neglecting lateral contractions,the initial deformation is thus piecewise homogeneous and wecan therefore determine the incremental behaviour semi-analytically,given the constitutive behaviour (strain energy function) ofthe phases in question. We apply asymptotic homogenization theoryin the deformed configuration in order to find the effectiveresponse of the deformed material in the low-frequency limitwhere the wavelength of the propagating waves is much longerthan the characteristic length scale of the microstructure.We close by considering the arbitrary frequency case and illustratehow the initial deformation affects the location of stop bandsand pass bands of the material. Work is under way to confirmthese results experimentally.     

Variational formulations for nonlinear wave propagation and unsteady transonic flow

A variational principle developed recently for constrained vector fields is applied to nonlinear waves and to unsteady, transonic flow. The mathematical conditions on the admissible functional form of the speed of propagation is compatible with physical considerations for the first systems and the treatment of the mixed derivative is indicative for a proper way to discretize the second for numerical calculations. The variational formulation provides a framework for stability analysis and finite element approximations for the nonlinear systems considered.

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Zusammenfassung Ein Variationsprinzip, das in letzter Zeit für Vektorfelder mit einschränkenden Bedingungen entwickelt worden ist, wird auf nichtlineare Wellen und auf nichtstationäre transonische Strömung angewendet. Die mathematische Bedingung für die zulässige funktionale Form der Fortpflanzungsgeschwindigkeit ist verträglich mit den physikalischen Bedingungen für das erste System, und die Behandlung der gemischten Ableitung deutet an, wie das zweite System für numerische Behandlung diskretisiert werden soll. Die variationale Formulierung gibt einen Rahmen für eine Stabilitätsuntersuchung und für eine Näherung durch finite Elemente für das nichtlineare System.

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Work partially supported by Grant AFOSR-73-2561.     

Homogenization approach for optimal design of perforated layer in acoustic wave propagation

Homogenization provides a reduced model of the acoustic wave propagation through a periodically perforated layer immersed in an inviscid fluid. The shape of the perforation influences some parameters of the non-local transmission condition. We developed and implemented numerically the sensitivity analysis for an optimal design of the perforation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An explicit finite-difference scheme for wave propagation in nonlinear optical structures

In this paper, we present an algorithm that solves a time-domain nonlinear coupled system arising in nonlinear optics. The algorithm is an explicit nonlinear finite-difference method (NFDM) based on the exact solution of the nonlinear discrete equations. It enables simulations that preserve the characteristics of nonlinearity as well as coupling, and can be extended to arbitrary input waveform conditions.     

Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping

We consider a wave equation with nonlinear acoustic boundary conditions. This is a nonlinearly coupled system of hyperbolic equations modeling an acoustic/structure interaction, with an additional boundary damping term to induce both existence of solutions as well as stability. Using the methods of Lasiecka and Tataru for a wave equation with nonlinear boundary damping, we demonstrate well-posedness and uniform decay rates for solutions in the finite energy space, with the results depending on the relationship between (i) the mass of the structure, (ii) the nonlinear coupling term, and (iii) the size of the nonlinear damping. We also show that solutions (in the linear case) depend continuously on the mass of the structure as it tends to zero, which provides rigorous justification for studying the case where the mass is equal to zero.     

Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions

We consider a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Well-posedness in the Hadamard sense for strong and weak solutions is proved by using the theory of nonlinear semigroups.