Topic

# Paraboloid

About: Paraboloid is a research topic. Over the lifetime, 1774 publications have been published within this topic receiving 14827 citations.

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01 Jan 2006

TL;DR: In this article, the authors define the notion of conformal antennas as follows: 1.1 Linear Arrays 2.2 Discrete Elements 2.3 Directional Radiators 2.4 Surface Waves 3.3.4 Finite Difference Time Domain Methods (FDTD) 3.4.5 Finite Element Method (FEM).

Abstract: Preface. Abbreviations and Acronyms. 1 INTRODUCTION. 1.1 The Definition of a Conformal Antenna. 1.2 Why Conformal Antennas? 1.3 History. 1.4 Metal Radomes. 1.5 Sonar Arrays. References. 2 CIRCULAR ARRAY THEORY. 2.1 Introduction. 2.2 Fundamentals. 2.2.1 Linear Arrays. 2.2.2 Circular Arrays. 2.3 Phase Mode Theory. 2.3.1 Introduction. 2.3.2 Discrete Elements. 2.3.3 Directional Elements. 2.4 The Ripple Problem in Omnidirectional Patterns. 2.4.1 Isotropic Radiators. 2.4.2 Higher-Order Phase Modes. 2.4.3 Directional Radiators. 2.5 Elevation Pattern. 2.6 Focused Beam Pattern. References. 3 THE SHAPES OF CONFORMAL ANTENNAS. 3.1 Introduction. 3.2 360- Coverage. 3.2.1 360- Coverage Using Planar Surfaces. 3.2.2 360- Coverage Using a Curved Surface. 3.3 Hemispherical Coverage. 3.3.1 Introduction. 3.3.2 Hemispherical Coverage Using Planar Surfaces. 3.3.3 Half Sphere. 3.3.4 Cone. 3.3.5 Ellipsoid. 3.3.6 Paraboloid. 3.3.7 Comparing Shapes. 3.4 Multifaceted Surfaces. 3.5 References. 4 METHODS OF ANALYSIS. 4.1 Introduction. 4.2 The Problem. 4.3 Electrically Small Surfaces. 4.3.1 Introduction. 4.3.2 Modal Solutions. 4.3.2.1 Introduction. 4.3.2.2 The Circular Cylinder. 4.3.2.3 A Unit Cell Approach. 4.3.3 Integral Equations and the Method of Moments. 4.3.4 Finite Difference Time Domain Methods (FDTD). 4.3.4.1 Introduction. 4.3.4.2 Conformal or Contour-Patch (CP) FDTD. 4.3.4.3 FDTD in Global Curvilinear Coordinates. 4.3.4.4 FDTD in Cylindrical Coordinates. 4.3.5 Finite Element Method (FEM). 4.3.5.1 Introduction. 4.3.5.2 Hybrid FE-BI Method. 4.4 Electrically Large Surfaces. 4.4.1 Introduction. 4.4.2 High-Frequency Methods for PEC Surfaces. 4.4.3 High-Frequency Methods for Dielectric Coated Surfaces. 4.5 Two Examples. 4.5.1 Introduction. 4.5.2 The Aperture Antenna. 4.5.3 The Microstrip-Patch Antenna. 4.6 A Comparison of Analysis Methods. Appendix 4A-Interpretation of the ray theory. 4A.1 Watson Transformation. 4A.2 Fock Substitution. 4A.3 SDP Integration. 4A.4 Surface Waves. 4A.5 Generalization. References. 5 GEODESICS ON CURVED SURFACES. 5.1 Introduction. 5.1.1 Definition of a Surface and Related Parameters. 5.1.2 The Geodesic Equation. 5.1.3 Solving the Geodesic Equation and the Existence of Geodesics. 5.2 Singly Curved Surfaces. 5.3 Doubly Curved Surfaces. 5.3.1 Introduction. 5.3.2 The Cone. 5.3.3 Rotationally Symmetric Doubly Curved Surfaces. 5.3.4 Properties of Geodesics on Doubly Curved Surfaces. 5.3.5 Geodesic Splitting. 5.4 Arbitrarily Shaped Surfaces. 5.4.1 Hybrid surfaces. 5.4.2 Analytically Described Surfaces. References. 6 ANTENNAS ON SINGLY CURVED SURFACES. 6.1 Introduction. 6.2 Aperture Antennas on Circular Cylinders. 6.2.1 Introduction. 6.2.2 Theory. 6.2.3 Mutual Coupling. 6.2.3.1 Isolated Mutual Coupling. 6.2.3.2 Cross Polarization Coupling. 6.2.3.3 Array mutual coupling. 6.2.4 Radiation Characteristics. 6.2.4.1 Isolated-Element Patterns. 6.2.4.2 Embedded-Element Patterns. 6.3 Aperture Antennas on General Convex Cylinders. 6.3.1 Introduction. 6.3.2 Mutual Coupling. 6.3.2.1 The Elliptic Cylinder. 6.3.2.2 The Parabolic Cylinder. 6.3.2.3 The Hyperbolic Cylinder. 6.3.3 Radiation Characteristics. 6.3.3.1 The Elliptic Cylinder. 6.3.3.2 End Effects. 6.4 Aperture Antennas on Faceted Cylinders. 6.4.1 Introduction. 6.4.2 Mutual Coupling. 6.4.3 Radiation Characteristics. 6.5 Aperture Antennas on Dielectric Coated Circular Cylinders. 6.5.1 Introduction. 6.5.2 Mutual Coupling. 6.5.2.1 Isolated Mutual Coupling. 6.5.2.2 Array Mutual Coupling. 6.5.3 Radiation Characteristics. 6.5.3.1 Isolated-Element Patterns. 6.5.3.2 Embedded-Element Patterns. 6.6 Microstrip-Patch Antennas on Coated Circular Cylinders. 6.6.1 Introduction. 6.6.2 Theory. 6.6.3 Mutual Coupling. 6.6.3.1 Single-Element Characteristics. 6.6.3.2 Isolated and Array Mutual Coupling. 6.6.4 Radiation Characteristics. 6.6.4.1 Isolated-Element Patterns. 6.6.4.2 Embedded-Element Patterns. 6.7 The Cone. 6.7.1 Introduction. 6.7.2 Mutual Coupling. 6.7.2.1 Aperture Antennas. 6.7.2.2 Microstrip-Patch Antennas. 6.7.3 Radiation Characteristics. 6.7.3.1 Aperture Antennas 248 6.7.3.2 Microstrip-Patch Antennas. References. 7 ANTENNAS ON DOUBLY CURVED SURFACES. 7.1 Introduction. 7.2 Aperture Antennas. 7.2.1 Introduction. 7.2.2 Mutual Coupling. 7.2.2.1 Isolated Mutual Coupling. 7.2.2.2 Array Mutual Coupling. 7.2.3 Radiation Characteristics. 7.3 Microstrip-Patch Antennas. 7.3.1 Introduction. 7.3.2 Mutual Coupling. 7.3.2.1 Single-Element Characteristics. 7.3.2.2 Isolated Mutual Coupling. 7.3.3 Radiation Characteristics. References. 8 CONFORMAL ARRAY CHARACTERISTICS. 8.1 Introduction. 8.2 Mechanical Considerations. 8.2.1 Array Shapes. 8.2.2 Element Distribution on a Curved Surface. 8.2.3 Multifacet Solutions. 8.2.4 Tile Architecture. 8.2.5 Static and Dynamic Stress. 8.2.6 Other Electromagnetic Considerations. 8.3 Radiation Patterns. 8.3.1 Introduction. 8.3.2 Grating Lobes. 8.3.3 Scan-Invariant Pattern. 8.3.4 Phase-Scanned Pattern. 8.3.5 A Simple Aperture Model for Microstrip Arrays. 8.4 Array Impedance. 8.4.1 Introduction. 8.4.2 Phase-Mode Impedance. 8.5 Polarization. 8.5.1 Polarization Definitions. 8.5.2 Cylindrical Arrays. 8.5.2.1 Dipole Elements. 8.5.2.2 Aperture elements. 8.5.3 Polarization in Doubly Curved Arrays. 8.5.3.1 A Paraboloidal Array. 8.5.4 Polarization Control. 8.6 Characteristics of Selected Conformal Arrays. 8.6.1 Nearly Planar Arrays. 8.6.2 Circular Arrays. 8.6.3 Cylindrical Arrays. 8.6.4 Conical Arrays. 8.6.5 Spherical Arrays. 8.6.6 Paraboloidal Arrays. 8.6.7 Ellipsoidal Arrays. 8.6.8 Other Shapes. References. 9 BEAM FORMING. 9.1 Introduction. 9.2 A Note on Orthogonal Beams. 9.3 Analog Feed Systems. 9.3.1 Vector Transfer Matrix Systems. 9.3.2 Switch Matrix Systems. 9.3.3 Butler Matrix Feed Systems. 9.3.4 RF Lens Feed Systems. 9.3.4.1 The R-2R Lens Feed. 9.3.4.2 The R-kR Lens Feed. 9.3.4.3 Mode-Controlled Lenses. 9.3.4.4 The Luneburg Lens. 9.3.4.5 The Geodesic Lens. 9.3.4.6 The Dome Antenna. 9.4 Digital Beam Forming. 9.5 Adaptive Beam Forming. 9.5.1 Introduction. 9.5.2 The Sample Matrix Inversion Method. 9.5.3 An Adaptive Beam Forming Simulation Using a Circular Array. 9.6 Remarks on Feed Systems. References. 10 CONFORMAL ARRAY PATTERN SYNTHESIS. 10.1 Introduction. 10.2 Shape Optimization. 10.3 Fourier Methods for Circular Ring Arrays. 10.4 Dolph-Chebysjev Pattern Synthesis. 10.4.1 Isotropic Elements. 10.4.2 Directive Elements. 10.5 An Aperture Projection Method. 10.6 The Method of Alternating Projections. 10.7 Adaptive Array Methods. 10.8 Least-Mean-Squares Methods (LMS). 10.9 Polarimetric Pattern Synthesis. 10.10 Other Optimization Methods. 10.11 A Synthesis Example Including Mutual Coupling. 10.12 A Comparison of Synthesis Methods. References. 11 SCATTERING FROM CONFORMAL ARRAYS. 11.1 Introduction. 11.2 Definitions. 11.3 Radar Cross Section Analysis. 11.3.1 General. 11.3.2 Analysis Method for an Array on a Conducting Cylinder. 11.3.3 Analysis Method for an Array on a Conducting Cylinder with a Dielectric Coating. 11.4 Cylindrical Array. 11.4.1 Analysis and Experiment-Rectangular Grid. 11.4.2 Higher-Order Waveguide Modes. 11.4.3 Triangular Grid. 11.4.4 Conclusions from the PEC Conformal Array Analysis. 11.5 Cylindrical Array with Dielectric Coating. 11.5.1 Single Element with Dielectric Coating. 11.5.2 Array with Dielectric Coating. 11.6 Radiation and Scattering Trade-off. 11.6.1 Introduction. 11.6.2 Single-Element Results. 11.6.3 Array Results. 11.7 Discussion. References. Subject Index. About the Authors.

524 citations

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TL;DR: In this paper, a simple method for a second-order structural reliability approximation is presented, which is based on an approximating paraboloid which is fitted to the limit-state surface at discrete points around the point with minimal distance from the origin.

Abstract: A simple method is presented for a second‐order structural reliability approximation. The method is based on an approximating paraboloid which is fitted to the limit‐state surface at discrete points around the point with minimal distance from the origin. An expression for the second‐order error in the approximation is derived, and the error is shown to be small, even for large dimensions and dispersed curvatures. In comparison to the existing approximation method, the proposed method is simpler and requires less computation. It is insensitive to noise in the limit‐state surface, approximately accounts for higher‐order effects, and facilitates the use of an existing formula for the probability content of parabolic sets.

469 citations

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TL;DR: In this paper, the probability of the parabolic failure domain is computed exactly by inversion of the characteristic function for the Parabolic quadratic form, and the exact results for the probability content of the failure domain obtained from the full second-order Taylor expansion of a failure function at the design point are presented.

Abstract: In second-order reliability methods the failure surface in the standard normal space is approximated by a parabolic surface at the design point. The corresponding probability is computed by asymptotic formulas and by approximation formulas. In this paper the probability content of the parabolic failure domain is computed exactly by inversion of the characteristic function for the parabolic quadratic form. Also, the exact results for the probability content of the failure domain obtained from the full second-order Taylor expansion of the failure function at the design point is presented. The approximating parabola does not depend on the formulation of the failure function as long as this preserves the original failure surface. This invariance characteristic is in general not shared by the approximation obtained using the full second-order Taylor expansion of the failure function at the design point. The exact results for the probability content of the approximating quadratic domains significantly extend the class of problems that can be treated by approximate methods.

252 citations

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TL;DR: In this paper, the authors adapted the argument of Wolff to also handle subsets of elliptic surfaces such as paraboloids, and obtained a sharp L 2 bilinear restriction theorem for bounded subsets in general dimension.

Abstract: Recently Wolff [W3] obtained a sharp L
2 bilinear restriction theorem
for bounded subsets of the cone in general dimension. Here we adapt
the argument of Wolff to also handle subsets of “elliptic surfaces” such
as paraboloids. Except for an endpoint, this answers a conjecture
of Machedon and Klainerman, and also improves upon the known
restriction theory for the paraboloid and sphere.

240 citations

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TL;DR: In this article, the effects of magnetic field, nonlinear thermal radiation and homogeneous-heterogeneous quartic autocatalysis chemical reaction on an electrically conducting (36nm) alumina-water nanofluid containing gyrotactic-microorganism over an upper horizontal surface of a paraboloid of revolution is presented.

Abstract: In this paper, the effects of magnetic field, nonlinear thermal radiation and homogeneous-heterogeneous quartic autocatalysis chemical reaction on an electrically conducting (36 nm) alumina-water nanofluid containing gyrotactic-microorganism over an upper horizontal surface of a paraboloid of revolution is presented. The case of unequal diffusion coefficients of reactant A (bulk-fluid) and reactant B (catalyst at the surface) in the presence of bioconvection is presented. In this article, a new buoyancy induced model for nanofluid flow along an upper horizontal surface of a paraboloid of revolution is introduced. The viscosity and thermal conductivity are assumed to vary with volume fraction and suitable models for the case 0% ≤ ϕ ≤ 0.8% are adopted. The transformed governing equations are solved numerically using Runge-Kutta fourth order along with shooting technique (RK4SM). Good agreement is obtained between the solutions of RK4SM and MATLAB bvp5c for a limiting case. The influence of pertinent parameters are illustrated graphically and discussed. It is found that at any values of magnetic field parameter, the local skin friction coefficient is larger at high values of thickness parameter while local heat transfer rate is smaller at high values of temperature parameter.

217 citations