Surface integral methods for high-frequency electromagnetic scattering

Surface integral equation based methods are advantageous for simulation of electromagnetic waves scattered by three dimensional obstacles, because they efficiently reduce the dimension of the problem and are robust for high-frequency problems. However, the cost of setting up the associated discretized dense linear systems is prohibitive due to evaluation of highly oscillatory magnetic and electric dipole surface integral operators using standard cubatures. The computational complexity of evaluating such integrals depends on the incident wave frequency, and the size and shape of the obstacles. In this work we discuss a surface integral reformulation of the scattering problem that involves evaluation of surface integrals with a highly oscillatory physical density, and discuss methods for efficient evaluation of such integrals for a class of smooth three dimensional scatterers whose diameter is a large multiple of the incident wavelength. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of surface smectic states of liquid crystals

The Landau–de Gennes model of liquid crystals is a functional acting on wave functions (order parameters) and vector fields (director fields). In a specific asymptotic limit of the physical parameters, we construct critical points such that the wave function (order parameter) is localized near the boundary of the domain, and we determine a sharp localization of the boundary region where the wave function concentrates. Furthermore, we compute the asymptotics of the energy of such critical points along with a boundary energy that may serve in localizing the director field. In physical terms, our results prove the existence of a surface smectic state.     

Quantized Riemann surfaces and semiclassical spectral series for a non-self-adjoint Schrödinger operator with periodic coefficients

We consider a non-self-adjoint Schrödinger operator describing the motion of a particle in a one-dimensional space with an analytic potential iV (x) that is periodic with a real period T and is purely imaginary on the real axis. We study the spectrum of this operator in the semiclassical limit and show that the points of its spectrum asymptotically belong to the so-called spectral graph. We construct the spectral graph and evaluate the asymptotic form of the spectrum. A Riemann surface of the particle energy-conservation equation can be constructed in the phase space. We show that both the spectral graph and the asymptotic form of the spectrum can be evaluated in terms of integrals of the pdx form (where x ∈31 ?/T? and p ∈, ? are the particle coordinate and momentum) taken along basis cycles on this Riemann surface. We use the technique of Stokes lines to construct the asymptotic form of the spectrum.     

Geometric properties of root vectors for equation of nonhomogeneous damped string: transformation operators method

The spectral decomposition theorem for a class of nonselfadjoint operators in a Hilbert space is obtained in the paper. These operators are the dynamics generators for the systems governed by 1–dim hyperbolic equations with spatially nonhomogeneous coefficients containing first order damping terms and subject to linear nonselfadjoint boundary conditions. These equations and boundary conditions describe, in particular, a spatially nonhomogeneous string subject to a distributed viscous damping and also damped at the boundary points. The main result leading to the spectral decomposition is the fact that the generalized eigenvectors (root vectors) of the above operators form Riesz bases in the corresponding energy spaces. The proofs are based on the transformation operators method. The classical concept of transformation operators is extended to the equation of damped string. Originally, this concept was developed by I. M. Gelfand, B. M. Levitan and V. A. Marchenko for 1–dim Schrödinger equation in connection with the inverse scattering problem. In the classical case, the transformation operator maps the exponential function (stationary wave function of the free particle) into the Jost solution of the perturbed Schrödinger equation. For the equation of a nonhomogeneous damped string, it is natural to introduce two transformation operators (outgoing and incoming transformation operators). The terminology is motivated by an analog with the Lax—Phillips scattering theory. The transformation operators method is used to reduce the Riesz bases property problem for the generalized eigenvectors to the similar problem for a system of nonharmonic exponentials whose complex frequencies are precisely the eigenvalues of our operators. The latter problem is solved based on the spectral asymptotics and known facts about exponential families. The main result presented in the paper means that the generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of non—selfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.     

Transport equations for waves in a half Space

We derive boundary conditions for the phase space energy density of acoustic waves in a half space, in the high frequency limit. These boundary conditions generalize the usual reflection—transmission relations for plane waves and are well suited for the study of wave propagation in bounded randed random media in the radiative transport approximation[15]. The high frequency analysis is based on direct calculations with Fourier integrals in the case of constant coefficients and Wigner measures in general, and it is presented in detail     

Cascade of energy in turbulent flows

A starting point for the conventional theory of turbulence [12–14] is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes [19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade of energy, a fundamental mechanism used to explain the Kolmogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rigorous manner, by working directly with the Navier–Stokes equations and stationary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some discussions and partly intuitive proofs appear in the literature. See, e.g., [1,2,10,11,16,17,21], and [22]. It is noteworthy that a mathematical framework can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier–Stokes equations. A similar result concerning the transfer of energy is valid in space dimension two. Here, however, due to vorticity constraints not present in the three-dimensional case, such energy transfer is accompanied by a similar transfer of enstrophy to higher modes. Moreover, at low wave numbers, in the spectral region below that of injection of energy, an inverse (from high to low modes) transfer of energy (as well as enstrophy) takes place. These results are directly related to the mechanisms of direct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy [1,15]. In a forthcoming article [9] we will discuss conditions for the actual existence of the inertial range in dimension three.     

Efficient evaluation of highly oscillatory acoustic scattering surface integrals

We consider the approximation of some highly oscillatory weakly singular surface integrals, arising from boundary integral methods with smooth global basis functions for solving problems of high frequency acoustic scattering by three-dimensional convex obstacles, described globally in spherical coordinates. As the frequency of the incident wave increases, the performance of standard quadrature schemes deteriorates. Naive application of asymptotic schemes also fails due to the weak singularity. We propose here a new scheme based on a combination of an asymptotic approach and exact treatment of singularities in an appropriate coordinate system. For the case of a spherical scatterer we demonstrate via error analysis and numerical results that, provided the observation point is sufficiently far from the shadow boundary, a high level of accuracy can be achieved with a minimal computational cost.     

Control,observation and polynomial decay for a coupled heat-wave system

This Note is devoted to study the control, observation and polynomial decay of a linearized 1-d model for fluid–structure interaction, where a wave and a heat equation evolve in two bounded intervals, with natural transmission conditions at the point of interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. The controllability and observability of the system through the wave component are derived from sidewise energy estimate and Carleman inequalities. As for the control and observation through the heat component, we need to develop first a careful spectral high frequency analysis for the underlying semigroup, which yields a new Ingahm-type inequality. It is shown that the controllable/observable subspace for both cases are quite different. Also, we obtain a sharp polynomial decay rate for the energy of smooth solutions. To cite this article: X. Zhang, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

The multi–phase averaging techniques and the modulation theory of the focussing and defocusing nonlinear schrodinger equations

Multi–phase averaging techniques have been applied successfully in the investigations of the modulational and generalized Benjamine–Feir instabilities for the quasi–periodic, N–phase, inverse spectral solutions of KdV [1], sine–Gordon (s–G) [2,3,4], and focussing and defocusing nonlinear Schrodinger equation [5,10], The key is that the multi–phase averagings, as the N–fold integrals, can be transferred to the N–iterated integrals, and therefore, can be evaluated, which is essential in the analysis of PDE perturbations analyzed by the averaging methods. In this paper, the transformations from cerain N–fold integrals to the N–iterated integrals for NLS are developed rigorously, and made to be numerically computable. Those integrals are also closely related to KdV and s–G. As an application, the modulation theory of the modulating N–phuse NLS solutions are Presented, a result given by Forest and Lee in [5,10].     

The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

     

Wave localization and conversion phenomena in multi-coupled multi-span beams

The linear dynamics of nearly periodic disordered multi-span beams resting on flexible supports are investigated. A wave transfer matrix methodology is chosen to examine the propagation of waves and the transmission of vibration along the structure. The spans are bi-coupled through the rotation and the transverse displacement at the supports and thus the beam motion is made up of two independent wave types. While for the ordered infinite beam there exists frequency passbands for which the free harmonic waves propagate without attenuation, the introduction of a slight disorder among the span lengths results in the localization of the vibration energy to few spans and in the conversion of the energy from one type of wave to the other. The energy conversion phenomenon renders the mechanism of localization much more complex than in mono-coupled periodic systems. The contribution of each type of wave to the global beam motion is analyzed in terms of frequency. It is observed that the spatial decay of each wave type is mainly governed by an exponential envelope. The corresponding exponential decay constants define a measure of localization for each wave and are found to be equal to the Lyapunov exponents of the product of random wave transfer matrices. It is also found that at frequencies which belong to a passband for both wave types, the decay rate of an incident wave vector is bounded by the two Lyapunov exponents, while at frequencies which belong to a passband for one wave type and a stopband for the other, localization effects are best predicted by the smallest of the two Lyapunov exponents.     

Generation and dissipation of ocean surface waves

Ocean surface waves constitute one of the most important sources of external forces that act on ships and offshore structures. Most ocean waves are generated by wind, but various other effects such as currents, ground and coastal topology, breaking and wave–wave interaction have an influence on the growth and dissipation of wave energy at specific frequency ranges. These water waves are inherently random in nature and their exact shape is difficult to describe, even when confining the described area to a small range. While many different approaches exist to describe the spectral characteristics of ocean waves, some of the processes which affect the generation of waves are still poorly understood. This paper addresses some techniques which serve to describe seaway spectra with respect to the subsequent analysis of dynamic mechanical systems in the ocean such as ships, platforms and pipelines. Advantages and limits of the different approaches are discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The steady motions of a disc on an absolutely rough plane

The branching of the steady motions of a heavy circular disc on an absolutely rough horizontal plane is investigated. The motions corresponding to critical points of the energy integral at fixed levels of two other integrals having the form of hypergeometric series are considered.     

Fast,numerically stable computation of oscillatory integrals with stationary points

We present a numerically stable way to compute oscillatory integrals. For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the amount of work needed decreases as the frequency increases. Moreover, we can modify the method for computing oscillatory integrals with stationary points. This is the first stable algorithm for oscillatory integrals with stationary points which does not lose accuracy as the frequency increases and does not require deformation into the complex plane.     

On error estimates of an exponential wave integrator sine pseudospectral method for the Klein–Gordon–Zakharov system

In this article, we propose an exponential wave integrator sine pseudospectral (EWI‐SP) method for solving the Klein–Gordon–Zakharov (KGZ) system. The numerical method is based on a Deuflhard‐type exponential wave integrator for temporal integrations and the sine pseudospectral method for spatial discretizations. The scheme is fully explicit, time reversible and very efficient due to the fast algorithm. Rigorous finite time error estimates are established for the EWI‐SP method in energy space with no CFL‐type conditions which show that the method has second order accuracy in time and spectral accuracy in space. Extensive numerical experiments and comparisons are done to confirm the theoretical studies. Numerical results suggest the EWI‐SP allows large time steps and mesh size in practical computing. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 266–291, 2016     

On the extendability of Noether’s integrals for orbifolds of constant negative curvature

This paper is concerned with the problem of the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure).     

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

This paper introduces a novel boundary integral equation (BIE) method for the numerical solution of problems of planewave scattering by periodic line arrays of two-dimensional penetrable obstacles. Our approach is built upon a direct BIE formulation that leverages the simplicity of the free-space Green function but in turn entails evaluation of integrals over the unit-cell boundaries. Such integrals are here treated via the window Green function method. The windowing approximation together with a finite-rank operator correction—used to properly impose the Rayleigh radiation condition—yield a robust second-kind BIE that produces superalgebraically convergent solutions throughout the spectrum, including at the challenging Rayleigh–Wood anomalies. The corrected windowed BIE can be discretized by means of off-the-shelf Nyström and boundary element methods, and it leads to linear systems suitable for iterative linear algebra solvers as well as standard fast matrix–vector product algorithms. A variety of numerical examples demonstrate the accuracy and robustness of the proposed methodology.     

Polynomial decay and control of a 1−d model for fluid–structure interaction

We consider a linearized and simplified 1?d model for fluid–structure interaction. The domain where the system evolves consists in two bounded intervals in which the wave and heat equations evolve respectively, with transmission conditions at the point of interface. First, we develop a careful spectral asymptotic analysis on high frequencies. Next, according to this spectral analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Finally, we prove the null-controllability of the system when the control acts on the boundary of the interval where the heat equation holds. The proof is based on a new Ingham-type inequality, which follows from the spectral analysis we develop and the null controllability result in Zuazua (in: J.L. Menaldi et al. (Eds.), Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 198–210) where the control acts on the wave component. To cite this article: X. Zhang, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Exact solutions of the linear problem of the steady-state waves created by a dipole in a flow of a stratified fluid

The problem of the steady-state waves which are formed when there is uniform flow of a non-viscous, incompressible, vertically stratified fluid round a dipole is considered in a linear formulation. Using the analytical properties of the solutions, two formulae are obtained for the vertical displacement field in the form of series of single integrals taken over the spectral curves. These formulae are simpler than those which have been previously proposed /1/ since the integrands do not contain special functions with logarithmic singularities and enable one to simplify the numerical analysis of the close domain of the wave field in which the asymptotic forms /2–4/ are applicable /5/.