Elliptical Coordinate System Research Articles

Large Deflection of a Single-Phase-Lag Non-Simple Hygrothermoelastic Elliptic Plate Under Heat-Moisture Load

A novel mathematical model in hygrothermoelastic theory has developed to describe heat and moisture distribution in a nonsimple medium. The model incorporates non-Fourier and non-Fick laws using a single phase-lag model to establish a relation between hygrothermoelasticity and unstable conduction processes. The model is applied to an ellipse-shaped composite material subjected to humid-heat loading. The Laplace and Mathieu integral transform was employed to obtain an exact solution of linearly coupled partial differential equations in an elliptical coordinate system. The Gaver–Stehfest algorithm is utilized for the inversion of Laplace functions. Berger’s approximation equation was employed to simplify the calculation of significant deformations for elliptic plates. The model also provides quantitative results for the composition of T300/5208, which consists of epoxy resin reinforced with graphite fibers. The hygrothermal behaviors due to the single-phase-lag time, geometry dimensions, material type, and the two temperatures are explored. The results reveal that nonclassical theories with single-phase-lag time and two-temperature have significant influences on the hygrothermoelastic behaviors in elliptic-shaped objects. The investigation can be effectually led using micro- and nano-elliptic shape resonators in future research.

A new method with C-means segmentation for non-uniform image coordinate system definition in panoramic imaging employing Ladybug2 camera

A panorama ensures a stunning wide-angle field of view up to 360° representation of a scene, exceeding the limits of a normal photograph. Panoramic cameras satisfy the single-viewpoint characteristic. There are several types of panoramic cameras for 360-degree imaging. Multi-camera panoramic imaging systems pose a difficulty in obtaining a single projection center for the cameras. In a variety of practical implementations of panoramic cameras, it is possible to calculate three-dimensional coordinates from a panoramic image, especially using the Direct Linear Transformation (DLT) method. In this study, not only a defining method of the non-uniform image coordinate system is presented by utilizing the C-Means algorithm for a single panoramic image, captured with a Ladybug2 panoramic camera in a panoramic calibration room but also the use of an elliptical panoramic projection coordinate system is defined by the Singular Value Decomposition method in a panoramic view. The results of the suggested method have been compared with the DLT algorithm for a single panoramic image which defined a conventional photogrammetric image coordinate system. It has been observed that the proposed method provides more accurate results for the 3D coordinate definition.

Scattering of SH wave in an elastic half-space by a semi-elliptical crater with surface elasticity

Cratering occurs inevitably on the surface of materials at all scales due to comprehensive factors. The presence of craters may have a significant influence on the mechanical performance of the surface of an elastic body. In this paper, we study the SH wave propagation in an elastic semi-infinite medium whose surface has a semi-elliptical crater. We explore the scattering of the SH wave by the crater when surface elasticity is incorporated in the model of deformation to account for possible small-scale effects of the crater. We obtain a series solution for the scattered wave function by employing the Mathieu functions in the corresponding elliptical coordinate system. We discuss in some numerical examples the dynamic stress concentration around the crater relative to the wave frequency and few geometric parameters.

Efficient Analysis of Unlined Elliptic Tunnels: An Approximate Method for Dynamic Response to Incident Plane SH Waves

ABSTRACT Dynamic analysis of underground structures, such as tunnels, is crucial to ensure their safety and stability during seismic loading. This paper proposes a more general method of large elliptic arc assumption-based wave function expansion, which is attributed to the important role of the large circular arc assumption-based wave function expansion method in the elastic wave scattering problem. The study is conducted within elliptic coordinate systems, employing the Mathieu function and Mathieu function addition theorem. The applicability and limitations of this method are analysed by comparing it with the existing exact analytical solutions obtained through the mirror-based wave function expansion method.

On the anisotropic coalescence of elliptic cylindrical voids considering the geometric and distributive properties

The geometric and distributive properties of voids significantly influence anisotropic coalescence behaviour. However, this problem has received little attention owing to the complexity of considering all the properties in the current analytical framework of limit analysis. To address this issue, this study proposes an analytical framework based on an elliptic coordinate system, including the determination of the ligament zone, characterization of plastic flow, and derivation of the void coalescence criterion, for porous materials with various geometric and distributive properties, including size, shape, spacing, and orientation. This framework is motivated by our observations that the evolution of the void geometry and surrounding plastic flow can be well characterized by the grid of the elliptic coordinate system. Subsequently, an analytical function is proposed to determine the ligament zone and coalescence direction with various void properties. A hollow nonaxisymmetric cylindrical unit cell is proposed to describe this ligament zone, and the corresponding trial velocity field is derived by extending the previous Gurson-like velocity field into the elliptic cylindrical coordinate system. The rationality of the field is validated by comparing its equivalent strain rate field with numerical simulations. Finally, a coalescence criterion is derived via the limit analysis of the proposed unit cell undergoing internal necking. Two heuristic adjustments are formulated for the overflow phenomenon in the rigid zone and outer ligament zones. Numerical assessments with various void properties confirm the accuracy of the analytical model. The coalescence criterion can predict the independent and coupling effects of geometric and distributive properties on anisotropic void coalescence. This study provides possible solutions to future plasticity problems of ellipsoidal inclusions.

Theory of electric field effect on the optical properties of elliptical quantum wire

In the approximation of the effective mass, the electric field effect on the optical properties of the elliptical quantum wire (EQW) was investigated. In an elliptic coordinate system, exact solutions of the Schrödinger equation for an electron in an EQW with hard walls are obtained. The electron energy spectrum consists of even and odd energies states, whose wave functions are expressed through even and odd Mathieu functions of the first kind. Using these solutions, an orthonormal basis was constructed. The influence of the electric field perpendicular to the quantum wire and parallel to the ellipse’s major axis on the energies and oscillator strength of quantum electron transitions was calculated using the matrix method. It is shown that the ellipticity of the quantum wire leads to the removal of the degeneracy of the energy spectrum of quasiparticles since the energies of even and odd electronic states depend differently on the ratio of the EQW semi-axes. The electron’s ground state is the even state, which is nondegenerate because there is no corresponding odd state. The electric field has a greater effect on energy states with a lower value of the magnetic quantum number. As the electric field strength increases, the energies of even and odd states shift to the region of lower energies.

Innovative K-Means based machine learning method for determination of non-uniform image coordinate system in panoramic imaging: a case study with Ladybug2 camera.

Currently, the practical implementations of panoramic cameras range from vehicle navigation to space studies due to their 360-degree imaging capability in particular. In this variety of uses, it is possible to calculate three-dimensional coordinates from a panoramic image, especially using the Direct Linear Transformation (DLT) method. There are several types of omnidirectional cameras which can be classified mainly as central and non-central cameras for 360-degree imaging. The central omnidirectional cameras are those which satisfy the single-viewpoint characteristic. Multi-camera systems are usually developed for applications for which two-image stereo vision is not flexible enough to capture the environment surrounding a moving platform. Although the technology based on multi-view geometry is inexpensive, accessible, and highly customizable, multi-camera panoramic imaging systems pose a difficulty in obtaining a single projection center for the cameras. In this study, not only a defining method of the non-uniform image coordinate system is suggested by means of the K-Means algorithm for a single panoramic image, captured with a Ladybug2 panoramic camera in the panoramic calibration room but also the use of an elliptical panoramic projection coordinate system definition by Singular Value Decomposition (SVD) method in panoramic view. The results of the suggested method have been compared with the DLT algorithm for a single panoramic image which defined a conventional photogrammetric image coordinate system.

Fast Factorized Backprojection Algorithm in Orthogonal Elliptical Coordinate System for Ocean Scenes Imaging Using Geosynchronous Spaceborne–Airborne VHF UWB Bistatic SAR

Geosynchronous (GEO) spaceborne–airborne very high-frequency ultra-wideband bistatic synthetic aperture radar (VHF UWB BiSAR) can conduct high-resolution and wide-swath imaging for ocean scenes. However, GEO spaceborne–airborne VHF UWB BiSAR imaging faces some challenges such as the geometric configuration, huge amount of echo data, serious range–azimuth coupling, large spatial variance, and complex motion error, which increases the difficulty of the high-efficiency and high-precision imaging. In this paper, we present an improved bistatic fast factorization backprojection (FFBP) algorithm for ocean scene imaging using the GEO satellite-unmanned aerial vehicle (GEO-UAV) VHF UWB BiSAR, which can solve the above issues with high efficiency and high precision. This method reconstructs the subimages in the orthogonal elliptical polar (OEP) coordinate system based on the GEO satellite and UAV trajectories as well as the location of the imaged scene, which can further reduce the computational burden. First, the imaging geometry and signal model of the GEO-UAV VHF UWB BiSAR are established, and the construction of the OEP coordinate system and the subaperture imaging method are proposed. Moreover, the Nyquist sampling requirements for the subimages in the OEP coordinate system are derived from the range error perspective, which can offer a near-optimum tradeoff between precision and efficiency. In addition, the superiority of the OEP coordinate system is analyzed, which demonstrates that the angular dimensional sampling rate of the subimages is significantly reduced. Finally, the implementation processes and computational burden of the proposed algorithm are provided, and the speed-up factor of the proposed FFBP algorithm compared with the BP algorithm is derived and discussed. Experimental results of ideal point targets and natural ocean scenes demonstrate the correctness and effectiveness of the proposed algorithm, which can achieve near-optimal imaging performance with a low computational burden.

Wave interaction with an elliptic disc submerged in a two-layer fluid

Wave interaction with a horizontal elliptic disc submerged in a two-layer fluid is studied here assuming linear theory. An elliptic coordinate system is used to solve the three-dimensional problem analytically. The fluid region is divided into a number of subregions and by means of separation of variables, velocity potential in each region is expressed in terms of series of Mathieu and modified Mathieu functions of real argument. Matching conditions along with the orthogonal properties of eigenfunctions provide solutions for the fluid velocity potentials. Numerical results are presented for the wave-induced forces and moments. The effect of variations in the depth of submergence of the disc, angle of incidence of the incoming wave, density ratio of the fluid and aspect ratio of the disc on these physical quantities have been analyzed. In particular, sudden amplification of forces and moments have been observed at certain frequencies when the disc is situated at close proximity to the interface. Various other behaviors of the solution are investigated in some detail.

Temperature distribution in an elliptical body without internal heat sources under boundary conditions of the third kind with partial adiabatic isolation

The article deals with the distribution of the temperature field in an elliptical body without internal heat sources. In this case, the boundary conditions are boundary conditions of the third kind. The solution is found when moving to an elliptic coordinate system. The author has obtained an analytical solution for the distribution of the temperature field in a body with an elliptical cross section of infinite length at a given ambient temperature with partial adiabatic isolation in the form of a functional series using hypergeometric functions.

Thermal calculation of a flat plate solar collector

The article is devoted to the thermal calculation of a flat plate solar collector. In this case, the boundary conditions are boundary conditions of the third kind. The solution is found in the transition to an elliptical coordinate system. The author has obtained an analytical solution for the distribution of the temperature field of a collector with an elliptical cross-section at zero ambient temperature with partial adiabatic isolation in the form of a functional series using hypergeometric functions.

Temperature distribution in an elliptical body without internal heat sources under boundary conditions of the third kind with partial adiabatic isolation

The article deals with the distribution of the temperature field in an elliptical body without internal heat sources. In this case, the boundary conditions are boundary conditions of the third kind. The solution is found when moving to an elliptic coordinate system. The author has obtained an analytical solution for the distribution of the temperature field in a body with an elliptical cross section of infinite length at a given ambient temperature with partial adiabatic isolation in the form of a functional series using hypergeometric functions.

Determination of the optimal cross-sectional size of a flat plate solar collector

The article is devoted to the determination of thermal stresses in a flat plate solar collector and the calculation of its optimal size. The solution is found when moving to an elliptical coordinate system. The obtained result is useful because by varying the geometric shape of the cross-section, it is possible to reduce the level of thermal stresses that occur, and, consequently, to increase the reliability and durability of the solar collector design.

Numerical Study of Forced Convection Between Two Elliptical Cylinders with Variable ThermoPhysical Properties

This study presents a numerical simulation of the three dimensional laminar forced convection between two elliptical horizontal cylinders with variable thermophysical properties using a program generated in FORTRAN language. The inner cylinder is uniformly heated whereas the outer cylinder is adiabatic. The flow and thermal fields are modeled by the continuity, momentum and energy equations with appropriate initial and boundary conditions using an elliptical coordinate system. The model equations are numerically solved by a finite volume numerical method with second order accurate spatiotemporal discretization. The variations of axial velocity, temperature and Nusselt number have been studied in three cases of forced convection, which are 100, 200 and 300 of Reynolds values. The results obtained show that the change in flow regime with increasing Reynolds number value has a significant effect on the value of Nusselt number, rather the heat exchange coefficient. Where Re=300 gives the best heat exchange.

Bifurcation and Multiplicity of Solutions of the Navier–Stokes Equations in Driven Semi-Elliptical Cavity Flow

In this paper, bifurcations in the solution of the Navier–Stokes equations are studied and multiple solutions of the driven semi-elliptical cavity flow are presented. The two-dimensional steady incompressible driven viscous flow in a semi-elliptical cavity is solved numerically. To this end, the problem is formulated using an elliptic coordinate system that transforms the geometry conformally and provides a body fitted coordinate system. The presented results show that above a bifurcation Reynolds number the solution of the governing flow equations bifurcates and there exist multiple solutions for a particular Reynolds number when the aspect ratio of the semi-elliptical cavity geometry is 0.26 ⩽D⩽ 0.8. The bifurcation Reynolds numbers for different aspect ratios and also multiple solutions at different Reynolds numbers are presented in detail.

Reduced-quaternionic Mathieu functions, time-dependent Moisil-Teodorescu operators, and the imaginary-time wave equation

We construct a one-parameter family of generalized Mathieu functions, which are reduced quaternion-valued functions of a pair of real variables lying in an ellipse, and which we call λ-reduced quaternionic Mathieu functions. We prove that the λ-RQM functions, which are in the kernel of the Moisil-Teodorescu operator D+λ (D is the Dirac operator and λ∈R∖{0}), form a complete orthogonal system in the Hilbert space of square-integrable λ-metamonogenic functions with respect to the L2-norm over confocal ellipses. Further, we introduce the zero-boundary λ-RQM-functions, which are λ-RQM functions whose scalar part vanishes on the boundary of the ellipse. The limiting values of the λ-RQM functions as the eccentricity of the ellipse tends to zero are expressed in terms of Bessel functions of the first kind and form a complete orthogonal system for λ-metamonogenic functions with respect to the L2-norm on the unit disk. A connection between the λ-RQM functions and the time-dependent solutions of the imaginary-time wave equation in the elliptical coordinate system is shown.

Analytical and numerical investigation of Poiseuille flow through semi-elliptic annulus

A fully developed laminar flow through semi-elliptic annulus formed between two confocal elliptical ducts, driven by a constant pressure-gradient, has been analyzed. The elliptic cylindrical coordinate system has been used to determine the exact solutions for “wide” and “narrow” semi-elliptic annuli with cross sections being symmetric about the minor and major axes of the confocal elliptic boundaries, respectively. For both configurations, exact analytical expressions have been obtained for velocity distribution, volume flow rate, shear stress, and Poiseuille number. The results are expressed in terms of two non-dimensional physical parameters: the ratio of the length of the semi-minor axis to the semi-major axis of the outer boundary, ro, 0 ≤ro<1, and the ratio of the length of semi-major axes of inner and outer elliptic boundaries, rma, c ≤rma<1, with c being the non-dimensional focal distance of the elliptic boundaries. Based on the analytical expressions, the graphical and tabulated results of the flow fields are presented for representative values of ro and rma to illustrate the characteristic features of the flow. Numerical evaluation of the analytical expressions shows that the flow field and the corresponding distributions of velocity and shear stresses are characteristically different for wide and narrow semi-elliptic annuli. In addition to the analytical results, a bivariate Chebyshev pseudospectral method is formulated in the elliptic-cylindrical coordinate system for obtaining the numerical solution of the problem. The numerical results show that the proposed method yields “exponential convergence” or “infinite order of accuracy,” as expected from a spectral method; exact agreement has been observed between the analytical and numerical results.

Two-dimensional sound field reproduction based on Mathieu function expansion.

In this study, a two-dimensional sound field reproduction method based on Mathieu function expansion (MFE) is proposed. Mathieu functions are orthogonal solutions of the Helmholtz equation in an elliptical coordinate system. A MFE, which is similar to the conventional circular harmonic expansion, is applied to the sound field. The MFE-based method introduces elliptical properties such that the listening area is an ellipse. Three methods are described: an analytical sound field reproduction method for elliptical loudspeaker arrays, a method applying transformation (i.e., stretch, rotation, and translation) to the listening area, and a numerical approach for arbitrarily shaped arrays. A suitable truncation order for the MFE is also derived. The performance of all methods is tested in computer simulations with examples.

Verified simulation of the stationary polymer fluid flows in the channel with elliptical cross-section

This work is aimed at the numerical analysis of the stationary non-isothermal flows of an incompressible viscoelastic polymer fluid in the channels with elliptical cross-sections. The description of such flows is done on the basis of the mesoscopic approach, and the resolving equations are derived. For solving them three algorithms, which use different techniques of constructing the approximate solutions, are designed: the least-squares collocation method based on the piecewise polynomial approximations, which lead to the overdetermined systems of linear algebraic equations; the finite element method, which uses weak formulations; and the non-local method without saturation, which operates with the global approximations in the elliptical coordinate system and with the matrix Sylvester equations. The proposed algorithms are verified by solving the test problem with the known analytical solution. Further we use them for the numerical analysis of the polymer fluid flows with its parameters varying in wide ranges. Comparison of the results obtained by the different algorithms shows their high performance and confirms that the solution of the considered non-linear problem exists and that it was computed accurately. The singularities of the obtained stationary solutions are analyzed. Taking them into account within the proposed algorithms enables us to increase the accuracy and the speed of simulations.

Energy calculation of the temperature field of an elliptical body without internal heat sources under boundary conditions of the third kind

The article presents an energy calculation of an elliptical body without internal heat sources under boundary conditions of the third kind. The solution is obtained by decomposition into a trigonometric series when moving to an elliptic coordinate system. A special case was also considered for given specific sizes. The reliability of the obtained result is confirmed by the fact that the temperature distribution does not depend on the boundary conditions and on the size of the body, but is completely determined by the ambient temperature.