Elliptic Cylindrical Coordinates Research Articles

An Elliptical Bipolar Cylindrical Coordinates Based Mixed Domain Near-Field Imaging Algorithm for Scanning 1-D Nonuniform Sparse MIMO Arrays

An Elliptical Bipolar Cylindrical Coordinates Based Mixed Domain Near-Field Imaging Algorithm for Scanning 1-D Nonuniform Sparse MIMO Arrays

Sound radiation from semi-infinite elliptical ducts with uniform subsonic jets: An analytical approach

This paper presents an analytical method for modeling the acoustic field radiation from a semi-infinite elliptical duct with a uniform subsonic jet. Elliptical ducts are often used as inlets for turbofan engines to take full advantage of the pre-compression effect of the fuselage and improve the stealth performance of the aircraft. The method accounts for the instability wave inside the vortex sheet induced by the shearing of jet and ambient flow and its effect to sound radiation. The method uses Mathieu functions to describe the incident and scattered sound in the elliptic cylindrical coordinates. A semi-analytical Wiener-Hopf approach with low computational cost is applied to obtain the near and far field solutions. The near field is illustrated by acoustic pressure maps at different modes and circumferential angles. The far field is shown by directivity patterns at single tones. Numerical simulations based on a finite element method are conducted to validate the accuracy of the analytical method. The results show good agreement between the analytical and numerical solutions. The proposed method can be used to analyze the acoustic characteristics of elliptical ducts with jets, which are relevant for noise control and optimization of turbofan engine inlets.

Experimental generation of helical Mathieu–Gauss vector modes

Vector modes represent the most general state of light in which the spatial and polarisation degrees of freedom are coupled in a non-separable way. Crucially, while polarisation is limited to a bi-dimensional space, the spatial degree of freedom can take any spatial profile. However, most generation and application techniques are mainly limited to spatial modes with polar cylindrical symmetry, such as Laguerre– and Bessel–Gauss modes. In this paper we put forward a novel class of vector modes whose spatial degree of freedom is encoded in the helical Mathieu–Gauss beams of the elliptical cylindrical coordinates. We first introduce these modes theoretically and outline their geometric representation on the higher-order Poincaré sphere. Later on, we demonstrate their experimental generation using a polarisation-insensitive technique comprising the use of a digital micromirror device. Finally, we provide a qualitative and a quantitative characterisation of the same using modern approaches based on quantum mechanics tools. It is worth mentioning that non-polar vector beams are highly desirable in various applications, such as optical trapping and optical communications.

Field and earth‐return impedance for conductors inside an elliptic tunnel

An electrodynamic boundary value problem is solved in the elliptic cylindrical coordinates to give the earth-return impedance of conductors inside an elliptic tunnel. The authors commence the analysis with the eigenfunction expansion of the source in the elliptic cylindrical coordinates, and the series solutions are obtained from the field ansatzes consisting of the modified Mathieu functions Ke m (ξ, q) and Ko m (ξ, q) with imaginary q, for which the detailed descriptions of the numerical implementations are provided, especially for the sign determination rule of Fourier coefficients of the expressions of modified Bessel function products. The proposed method is verified by the numerical results compared with those of finite-element method and good agreement is achieved in all cases. In addition, the presented method is further compared with those developed for the special cases in the literature, from which it is confirmed that proposed method is a valid generalisation of existing ones.

Analysis of forced convection from a horizontal flat plate for liquid metals

ABSTRACT Numerical simulation of the flow and forced convective heat transfer past a horizontal flat plate for liquid metals has b een performed in elliptical cylindrical coordinates. The two dimensional, steady incompressible Navier–Stokes equations in stream function vorticity formulation together with energy equation are solved using fourth order compact finite difference scheme. A stable and efficient method is introduced to solve the fourth order discretizations of governing partial differential equations through iterations in pseudo time. The fourth order convergence of the present scheme is demonstrated. The drag coefficient, average Nusselt number, variation of skin friction and local Nusselt numbers on the plate surface are shown graphically for Reynolds number up to and Prandtl numbers 0.004, 0.020 and 0.065, respectively for liquid sodium, liquid gallium and liquid lithium. It is found that skin friction coefficient and local Nusselt number approaches to Blasius solution and Pohlhaunsen solution (), respectively as Reynolds number approaches to infinity.

Inline Delayed-shot Migration in Tilted Elliptical-cylindrical Coordinates

Riemannian wavefield extrapolation, a wave-equation imaging method for efficient one-way propagation of 3D wavefields, is extended to 3D seismic surveys. I formulate an inline delayed-shot migration procedure in 3D tilted elliptical-cylindrical (TEC) coordinate systems. When inline coordinate tilt angles are well matched to the inline source ray parameters, the TEC coordinate extension affords accurate propagation of both steep-dip and turning-wave components important for successfully imaging complex geologic structure. I show that wavefield extrapolation in TEC coordinates is no more complicated than propagation in elliptically anisotropic media, which is easily handled by current finitedifference practice. I apply this approach to a realistic 3D wide-azimuth synthetic derived from a field Gulf of Mexico data set. The resulting images demonstrate the imaging advantages made possible through 3D RWE implementations, including the improved imaging of steeply dipping salt flanks, at reduced computational cost.

Wavefronts and caustics associated with Mathieu beams.

In this work we compute the wavefronts and the caustics associated with the solutions to the scalar wave equation introduced by Durnin in elliptical cylindrical coordinates generated by the function A(ϕ)=ceν(ϕ,q)+iseν(ϕ,q), with ν being an integral or nonintegral number. We show that the wavefronts and the caustic are invariant under translations along the direction of evolution of the beam. We remark that the wavefronts of the separable Mathieu beams generated by A(ϕ)=ceν(ϕ,q) and A(ϕ)=seν(ϕ,q) are cones and their caustic is the z axis; thus, they are not structurally stable. However, in general, the Mathieu beam generated by A(ϕ)=ceν(ϕ,q)+iseν(ϕ,q) is stable because locally its caustic has singularities of the fold and cusp types. To show this property, we present the wavefronts and the caustics for the Mathieu beams with characteristic value aν=0 and q=0,0.2,0.3,0.5. For q=0, we obtain the Bessel beam of order zero; in this case, the wavefronts are cones and the caustic coincides with the z axis. For q≠0, the wavefronts are deformations of conical ones, and the caustic surface, for some values of q, has singularities of the cusp ridge type. Furthermore, we remark that the set of Mathieu beams with characteristic value aν=0 and 0≤q<1 has associated a caustic with singularities of the swallowtail type, which is structurally stable. Therefore, we conclude that this type of Mathieu beam is more stable than plane waves, Bessel beams, parabolic beams, and those generated by A(ϕ)=ceν(ϕ,q) and A(ϕ)=seν(ϕ,q). To support this conclusion, we present experimental results showing the pattern obtained after obstructing a plane wave, the Bessel beam of order m=5, and the Mathieu beam of order m=5 and q=50 with complex transversal amplitude given by Ce5(ξ,50)ce5(η,50)+iSe5(ξ,50)se5(η,50), where (ξ, η) are the elliptical coordinates on the plane.

Improvised Asymptotic Boundary Conditions for Magnetostatic Field Problems in Ellipsoidal and Elliptic Cylindrical Domains

Improvised Asymptotic Boundary Conditions (IABCs) in ellipsoidal and elliptic cylindrical coordinates have been derived. The IABCs are among the simplest techniques to emulate the open boundary condition, which can be improvised in virtually any finite-element program. The IABC was first proposed with the circular in 2-D, and spherical in axisymmetric 3-D domains. This work extends the IABC formulation to ellipsoidal and elliptic cylindrical domains.

Modal analysis and dispersion curves of an elliptical W-type single mode fiber

In this paper, the propagation of electromagnetic waves in W-type elliptical dielectric optical fiber having various cladding layers is presented. The presented fiber has concentric core and cladding which have elliptical cross section and the refractive index of one of the inner cladding (the cladding between first clad and last clad) maximum. Using elliptic cylindrical coordinates, boundary conditions are derived and longitudinal field components for the even and odd modes are obtained. The characteristic equation for the fiber to be studied is determined by solving the Mathieu (q > 0) and the modified Mathieu functions (q < 0). In order to study the fundamental mode, the modal index m is put as m = 1. Finally, the cutoff frequencies for several lower order modes have been calculated and their dispersion characteristics are plotted. The effects of elliptical eccentricity e on the mode cutoff frequencies and mode transmissions are also addressed. The analysis shows that one can control the propagation property of optical fiber by increasing the number of inner claddings. These claddings provide additional degree of freedom to control the modes.

J1050502 非コンパクトグリーン関数に関するモデリング

This paper describes verification and examination in applying Mathieu functions for constructing Non-compact Green functions for wing-like body. First the paper reviews composition of plane wave and scattering wave by Bessel functions for cylindrical body. Key idea is introduction of Bessel functions and Hankel functions in the composition. Similarity of Mathieu functions in elliptic cylindrical coordinates with Bessel functions in circular cylindrical coordinates is utilized for constructing the Non-compact Green's function for wing-like bodies. Adequacy of Mathieu Functions are examined in their behavior in neighborhood of body surface. Lastly the paper discusses application of the new Non-compact Green's functions to designing low-noise wing with porous surface.

Heat Transfer inside Elliptic Cylindrical Reactor: Effect of Bed Porosity

The study of heat transfer phenomenon in porous media by fluids percolated in the axial direction has been of interest to many researchers in various branches of science and technology. Applications are directed to different process such as filtration, distillation, absorption and adsorption in columns, drying and catalytic reactions in fixed beds. The literature has presented several solutions of the heat diffusion / convection equation in fixed bed reactors, but these studies are limited to a cylindrical geometry. In this sense, this work aim to present a pseudo-homogeneous three-dimensional model to describe the steady-state heat transfer within a fixed bed reactor with elliptic cylindrical geometry by considering variable porosity. The energy equation written in elliptical cylindrical coordinates and applied to the porous medium (particulate system) is discretized numerically using the finite volume method. Results of the temperature distribution within the bed are presented analyzed. It was verified that with increased porosity heat transfer inside the reactor tends to be more intense and thus, lower temperature gradients are found in all cross section of the reactor.

Note on the use of localized beam models for light scattering theories in spherical coordinates

Localized beam models provide the most efficient and enlightening ways to evaluate beam shape coefficients of electromagnetic arbitrary shaped beams for use in light scattering theories. At the present time, they are valid in spherical and (circular and elliptical) cylindrical coordinates. A misuse of localized beam models in spherical coordinates recently appeared several times in the literature. We therefore present a warning to avoid the propagation of an incorrect use of localized beam models.

Mathieu Progressive Waves

A new family of exact solutions to the wave equation representing relatively undistorted progressive waves is constructed using separation of variables in the elliptic cylindrical coordinates and one of the Bateman transforms. The general form of this Bateman transform in an orthogonal curvilinear cylindrical coordinate system is discussed and a specific problem of physical feasibility of the obtained solutions, connected with their dependence on the cyclic coordinate, is addressed. The limiting case of zero eccentricity, in which the elliptic cylindrical coordinates turn into their circular cylindrical counterparts, is shown to correspond to the focused wave modes of the Bessel-Gauss type.

Inline Delayed-shot Migration in Tilted Elliptical-cylindrical Coordinates

Inline Delayed-shot Migration in Tilted Elliptical-cylindrical Coordinates

Multiscale modelling of a composite electroactive polymer structure

Danfoss PolyPower has developed a tubular actuator comprising a dielectric elastomersheet with specially shaped compliant electrodes rolled into a tube. This paper is concernedwith the modelling of this kind of tubular actuator. This is a challenging task due to thesystem’s multiscale nature which is caused by the orders of magnitude difference betweenthe length and thickness of the sheets as well as the thickness of the electrodes and theelastomer in the sheets. A further complication is the presence of passive parts at both endsof the actuator, i.e. areas without electrodes which are needed in order to avoid shortcircuits between negative and positively charged electrodes on the two sides of the sheet.Due to the complexities in shape and size it is necessary to introduce some simplifyingassumptions. This paper presents a set of models where the three-dimensional problem hasbeen reduced to two-dimensional problems, ensuring that the resulting models can behandled numerically within the framework of the finite element method. Thesemodels have been derived by expressing Navier’s equation in elliptical cylindricalcoordinates in order to take full advantage of the special shape of these actuators.Emphasis is placed on studying the passive parts of the actuator, as these degrade theeffectiveness of the actuator. Two approaches are used here to model the passive parts: aspring-stiffness analogy model and a longitudinal section model of the actuator. Themodels have been compared with experimental results for the force–elongationcharacteristics of the commercially available PolyPower ‘InLastor push’ actuator. Thecomparison shows good agreement between model and experiments for the casewhere the passive parts were taken into account. One of the models developed issubsequently used to study geometric effects—specifically the effect of changing theellipticity of the tubular actuator on the actuator’s performance is investigated.

Inline delayed-shot migration in tilted elliptical-cylindrical coordinates

Riemannian wavefield extrapolation, a one-way wave-equation method for propagating seismic data on generalized coordinate systems, is extended to inline delayed-shot migration using 3D tilted elliptical-cylindrical (TEC) coordinate meshes. Compared to Cartesian geometries, TEC coordinates are more conformal to the shape of inline delayed-source impulse response, which allows the bulk of wavefield energy to propagate at angles lower to the extrapolation axis, thus improving global propagation accuracy. When inline coordinate tilt angles are well matched to the inline source ray parameters, the TEC coordinate extension affords accurate propagation of both steep-dip and turning-wave components important for successfully imaging complex geologic structure. Wavefield extrapolation in TEC coordinates is no more complicated than propagation in elliptically anisotropic media and can be handled by existing implicit finite-difference methods. Impulse response tests illustrate the phase accuracy of the method and show that the approach is free of numerical anisotropy. Migration tests from a realistic 3D wide-azimuth synthetic derived from a field Gulf of Mexico data set demonstrate the imaging advantages afforded by the technique, including the improved imaging of steeply dipping salt flanks at a reduced computational cost.

Exact analytical approaches for elliptic cylindrical invisibility cloaks

Elliptic cylindrical cloaks are investigated analytically in elliptic cylindrical coordinates. It is shown that both the linear and higher-order transformations produce an imperfect cloaking due to the poor symmetry of the coordinate system. The imperfection being in concordance with that obtained by numerical simulations, it cannot be eliminated by improving the computer techniques. The cloaking becomes almost perfect in the limit case of nearly circular cloaks with the advantage that none of the parameters is singular in the cloak shell. In circular cylindrical coordinates instead, a perfect cloaking is achieved with elliptic cloaks, as has been proven in other studies by numerical simulations. Analytic solutions to Maxwell's equations are provided in these coordinates only in the limit case of nearly circular cloaks. Simple general expressions for material parameters are given also in circular cylindrical coordinates that can be then transformed in Cartesian coordinates.

A three-dimensional ice-sheet flow solution

AbstractAn accurate three-dimensional reduced model (shallow-ice approximation) flow with velocity depending on all three spatial coordinates is constructed for the commonly adopted isotropic viscous law with temperature-dependent rate factor. The solution is for steady flow with a prescribed temperature distribution, but can be extended to flow with a coupled energy balance, and to unsteady flow. The accuracy hinges on the reduction to a two-point ordinary differential equation problem for the surface profile, on an unknown span, for which established accurate numerical methods are available. This is achieved by setting one horizontal velocity component in elliptic cylindrical coordinates to zero, but the other two components depend on all three spatial variables. While not of direct physical interest, such an ‘exact’ solution is valuable as a test solution for the large-scale numerical codes commonly used in ice-sheet modelling, which have not yet been subjected to such a comparison.

Elliptical beams

A very general beam solution of the paraxial wave equation in elliptic cylindrical coordinates is presented. We call such a field an elliptic beam (EB). The complex amplitude of the EB is described by either the generalized Ince functions or the Whittaker-Hill functions and is characterized by four parameters that are complex in the most general situation. The propagation through complex ABCD optical systems and the conditions for square integrability are studied in detail. Special cases of the EB are the standard, elegant, and generalized Ince-Gauss beams, Mathieu-Gauss beams, among others.

Steady State Model for Heat Transfer in a Packed Bed Reactor of Elliptic Cylindrical Shape

The fixed-bed reactors of circular cylindrical geometry with heated or cooled walls are frequently used to carry out heterogeneous reactions of solid-gas type in engineering applications. The design of a fixed bed reactor requires an extensive knowledge of heat transfer characteristics within the packed bed. In this sense, this work presents a three-dimensional mathematical model to predict the heat transfer inside a fixed bed elliptical cylinder heat exchanger. The model considers uniform velocity and temperature profiles of the fluid phase at the entrance of the reactor, and constant thermo-physical properties. The surface of the equipment convective boundary condition is assumed to be constant. The energy equation, written in the elliptical cylindrical coordinates, was discretized using a finite-volume method considering a fully implicit formulation, and WUDS interpolation scheme. Numerical results of the dimensionless temperature profiles inside the packed bed reactor at a steady state are presented and temperature distribution is interpreted. To validate the model, numerical results obtained for the circular cylinder are compared with analytical results from literature and a good agreement was obtained.