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Finite Volume Time Domain (FVTD) Computations for Electromagnetic Scattering Prof. A. Chatterjee Department Of Aerospace Engineering IIT Bombay.

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Outline Of Presentation Part I : Introduction to the finite volume time domain technique Maxwell Equation in Conservative form Finite volume method for conservation laws Numerical Schemes Volume grid generation Boundary Conditions Validation PEC sphere Almond Ogive lossy sphere lossy cone sphere Part II: Applications RCS prediction of low observable aircraft Configurations Intake B2 F117

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Maxwell’s curl Equations with losses: Constitutive Relations Maxwell Equation (differential form)

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3D Maxwell’s Equations in Conservative form: where, Maxwell Equation in Conservative form

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Numerical Technique Finite Volume Time Domain (FVTD) technique. Higher Order Characteristic based technique for spatial discretization based on the Essentially Non-Oscillatory (ENO) method. Multi-stage Runge-Kutta time integration.

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Numerical Formulation Maxwell’s Equations (Operator form) Decomposition of Total Field

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Finite Volume Framework Conservative form can be written as where integrating over an arbitrary control volume,

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discretized form for j th cell, application of divergence theorem gives, Higher order characteristic based technique Runge Kutta time stepping 3D domain divided into hexahedral cells Cell centred formulation Finite Volume Framework …. contd

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Boundary Conditions On Perfectly Conducting (PEC) surface Total tangential electric field, Total normal magnetic field, Far-field boundaries Characteristic boundary conditions (zero scattered field) Numerical Boundary Treatment Surface of the body is a Perfect Electric Conductor (PEC) Tangential component of electric field zero at surface, i.e, In scattered field formulation, implemented as (E inc known) Similarly the boundary condition for the magnetic field, Field values in the ghost cells computed by extrapolating the scattered field values from the interior Normal components of the electric fields and tangential components of the magnetic fields in the ghost cells taken identical to those on the surface of the conductor

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Boundary Conditions …. cont. Numerical Boundary Treatment In the far field, characteristic based boundary treatments are applied Scattered field is taken as zero Fluxes are decomposed along the characteristic directions normal to the cell faces; for the ghost cells outside the domain, incoming characteristic fluxes for the scattered field are taken to be zero Methodology Time domain computations for sinusoidal steady state Complex field in frequency domain from time history of solution using Fourier Transform

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Finite Volume Formulation for Conservation Laws

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Introduction Aim: To solve a set of governing equations describing a set of conservation laws in the integral form in a specified domain with prescribed boundary conditions Equation of Conservation Laws e.g. In Fluid Mechanics Mass conservation Momentum conservation Energy conservation + Equation of state as a closure equation e.g. In Electromagnetics Maxwell’s Curl Equations e.g. In Magnetohydrodynamics Navier Stokes Equations & Maxwell’s Equations

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Conservation Equation Consider generic conservation equation for a conserved variable u, and assume that the corresponding flux vector known Integrating over an arbitrary volume where,

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Conservation Equation for The conservation equation for in a general form: Rate of change of in a control volume Convective flux across the surface Source or sink term Here : = 1 Mass conservation equation = u,v,w Momentum conservation equation = Energy equation

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Finite Volume Method for Conservation Laws Starting point: Integral form of conservation equation. Domain covered by finite number of contiguous control volumes (CV) Conservation equation is applied to each CV Computational node : Center of each CV, where the Variable values are calculated. Interpolation is used to calculate variable values of CV surfaces in terms of nodal (CV center) values. Results in an algebraic equation for each CV per equation Methodology

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Computational Domain for Finite Volume Method Control Volume Surface Grid Volume Grid Computational domain divided into finite control Volumes

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Finite Volume Discretizations Geometry Creation Domain Discretization: Domain is subdivided into a finite number of small control volumes (CV) by a grid which defines the control volume boundaries Grid Terminology Node-based finite volume scheme: u stored at vertex Cell-based finite volume scheme: u stored at cell centroid Typical CV with the notations

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Finite Volume Method … contd. The generic conservation equation for a conserved variable u, reproduced here is applied to each CV Net flux through CV boundary faces = f = component of the convective flux vector in the direction normal to the CV face Note: CVs should not overlap Domain volume should be equal to the sum of volumes of all the CVs Each CV face is unique to two CVs which lie on either side of it

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Finite Volume Method … contd. To evaluate surface integral exactly one should know integrand f everywhere on “S” The above information is not available, only nodal (CV center) values of ‘u’ are calculated. Approximation must be introduced Integral is approximated in terms of the variable values at one or more locations on the cell face. Cell face values are approximated in terms of nodal CV center values

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Finite Volume Method … contd. Global Conservation: Integral conservation equation is applied to each CV Sum equations for all CVs Global conservation is obtained since surface integral over CV faces cancel out AdvantageDis-advantage Simplest method Complex geometries Scheme is conservative Most popular amongst CFD workers ( > 95%) Three levels of approximations Interpolation Differentiation Integration

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Numerical Schemes Differ by how numerical flux f num is evaluated at cell face and by how time integration is performed Differ in spatial and temporal order of accuracy Upwind (Characteristic based) Schemes: Flux Splitting Schemes, Riemann/Godunov Solvers Steger Warming, Van Leer flux vector splitting Roe type solvers Central Difference based Schemes: Lax-Wendroff Scheme Jameson’s scheme

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Time Stepping Space and Time discretization combined e.g. Lax Wendroff Scheme Taylor Series expansion in Time Space and Time seperated Set of ODE’s obtained after space discretization March in time with Runge-Kutta method

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ENO (Essentially Non-Oscillatory)Scheme Higher order accuracy Non-oscillatory resolution of discontinuities (adaptive stenciling) Models discontinuous changes in material properties Numerical flux at right-hand face of j th cell with r th -order accurate ENO scheme The smoothest possible stencil

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Validation for Standard Test Case

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Validation Metallic Sphere: Volume Grid – O-O Topology, Single Block (50×45×20 cells) Frequency for analysis = 0.09 GHz ( Electric Size = 1.4660 ) Volume Grid Discretization (O-O Topology)

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Bistatic RCS (dB) for Metallic Sphere at 0.09 GHz E-planeh-plane Validation …. contd Metallic Sphere:

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Validation - Metallic Ogive Metallic Ogive: Electro-Magnetic Code Consortium (EMCC) benchmark target Geometry – Half angle 22.62 degrees, length 5”, thickness 1” Volume Grid - O-H topology, single block (40×40×138 cells) Monostatic RCS at 1.18 GHz (Electric size 2π) for vertical polarization Excellent agreement with experimental results Maximum RCS of approx -20 dBsm = 10 -2 m 2 Minimum RCS of approx -60 dBsm = 10 -6 m 2

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Validation …. contd Metallic Ogive: Rendered OgiveSurface Grid

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Ogive Volume Grid Cross-Section Validation …. contd Ogive – Volume Grid and Surface Currents Metallic Ogive:

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Validation …. contd Metallic Ogive: Ogive Monostatic RCS Plot (1.18 GHz, VV Polarization)

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Validation …. contd Almond: Electro-Magnetic Code Consortium (EMCC) benchmark target Geometry – length 9.936” Volume Grid - O-O topology, single block (15×121×125 cells) Monostatic RCS at 1.19 GHz (Electric size 2π) for vertical polarization Excellent agreement with experimental results

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Validation …. contd Almond: Rendered AlmondSurface Grid

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Validation …. contd Almond: Almond – Surface Currents Angle of incidence = 0 deg. Angle of incidence = 90 deg.

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Validation …. contd Almond: Almond Monostatic RCS Plot (1.19 GHz, VV Polarization)

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Validation …. contd PEC Sphere with Non-Lossy coating: PEC ka = 2.6858 Coating (t / λ) = 0.05, ka 1 = 3.0, ε‘ = 3.0 and 4.0, μ‘ = 1.0 Volume Grid O-O Topology, Single Block (64×45×32 cells) Bistatic RCS for Sphere with Discontinuous Nonlossy Coating (Backscatter at 180 degree)

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Validation …. contd PEC Sphere with Lossy dielectric coating: PEC ka = 1.5 Coating (t / λ) = 0.05, ε r = 3.0 – j4.0, μ r = 5.0 – j6.0 Volume Grid O-O Topology, Single Block (64×48×32 cells) Monostatic RCS Sphere with Lossy Coating For different orders of accuracy For different discretization

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Validation …. contd PEC Cone Sphere with Lossy Coating: Geometry : vertex angle 90 degree, sphere diameter 0.955 λ Coating (t / λ) = 0.01, ε r = 3.0 – j4.0, μ r = 5.0 – j6.0 Volume Grid O-O Topology, Single Block (80×40×38 cells) Bistatic RCS for cone-sphere with lossy Coating (Backscatter at 180 degree) E-planeh-plane

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Radar Cross Section of Low Observable Aircraft Configurations Applications: Industrial Problems

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Industrial Problems RCS Analysis of Engine intake configurations B2 “Advanced Technology Bomber” F-117 “Nighthawk”

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RCS of Low Observable Aircraft Configurations Introduction RCS, low-observability, air intake configurations Intake geometries and grid generation Results

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RCS Analysis of Engine Intake configurations Introduction: Low-observability: low back-scatter for near-axial incident illumination as well as low returns over a broad angular region Low-observables characterized by greater contribution of traveling and creeping waves to the Radar Cross-Section (RCS) Definition of RCS: the area of an isotropic reflector returning the same power per solid angle as the given body. At far field, it is proportional to the ratio of the power received from the target to the power incident on the target. Engine Intake Configurations studied B2 “Advanced Technology Bomber” and F-117 “Nighthawk” chosen for present study: Low-observable (stealthy) aircraft configurations (RCS of -40 and -25 dBsm respectively) Fine geometric details not available due to military sensitivity

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RCS of some military aircrafts AircraftRCS [dBsm][m 2 ][ft 2 ] F-15 Eagle264054,358 F-4 Phantom II201001,076 B-52 Stratofortress2099.51,071 Su-271215161.4 B-1A10 107.6 F-16 Fighting Falcon7553.82 B-1B Lancer0.091.0210.98 F-18E/F Super Hornet0110.76 BGM-109 Tomahawk-130.050.538 SR-71 Blackbird-18.50.0140.15 F-22 Raptor-220.00650.07 F-117 Nighthawk-250.0030.03 B-2 Spirit-400.00010.01 Boeing Bird of Prey-700.00000010.000008

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Estimation of RCS Computation of first time-domain numerical solution proposed by Yee in 1966 (FDTD, second order in space & time); followed by other FDTD-based algorithms FVTD-based schemes adopted to handle more complicated geometries Hyperbolicity of the Maxwell's equations in their conservative form exploited by characteristic-based algorithms Drawbacks: Requirement of large computational resources (both processor speeds and memory) Lack of theoretical estimates on grid-fineness and minimum distance to the far-field Similar studies: RCS of VFY218 (Conceptual aircraft): approx. 15 dBsm @ 100 MHz, nose-on incidence (monostatic) Not a low-observable RCS of F117 (Stealth Fighter): approx. -20 dBsm @ 215.38 MHz, nose-on incidence (monostatic)

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RCS Calculation Above algorithm used to compute the total fields at the surface of the PEC. Equivalent surface currents are given by Far-zone transform used to calculate the scattered fields E sc and H sc at infinite distance and RCS is calculated as where R is taken to be a sufficiently large number (100,000) (equivalent electric current) (equivalent magnetic current)

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RCS ANALYSIS OF ENGINE INTAKE CONFIGURATIONS

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RCS And Air Intake configurations Width of air intake duct vis-à-vis wavelength of electromagnetic radiation Sr-71 air intake Grilled air intake of f117 Position / profile of intake duct s-shaped wing mounted intake duct of b-2 Wing blended intake duct of yf-23

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Intake centre Body SR-71 CENTRE BODY ANNULAR REGION DUCT WALL

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INTAKE GRILL F-117 AIRCRAFT SOFT FACE HARD FACE GRILL ENG. FACE

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B-2 BOMBER S-shaped duct with coated wall

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YF-23 ADVANCED TACTICAL FIGHTER

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RCS Analysis of Engine Intake configurations Representative models of engine intake cavities considered are Straight cylindrical cavity, open from one end and the other end Terminated by a flat plate Terminated by a hub and a flat plate Terminated by a hub with a set of straight rotor blades and a flat plate Hub of cylindrical shape Hub of sphere-cylindrical shape

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k i j 2 λ 4 λ Multiblock representation of a straight cylindrical cavity terminated by a plate PLATED END ƒ = 15 GHz λ = 20 mm es = 4 π Contd.. 20 λ OPEN END

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Bistatic RCS: Straight Cylindrical Cavity with a Flat Plate Termination

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Monostatic RCS: Straight Cylindrical Cavity with a Flat Plate Termination

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Front view

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Rear view (angle of irradiation φ=28°) Φ

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λ 2 λ 4 λ λ HUB ( SPHERE-CYLINDER) λ = 20 mm es = 4 π i j k PLATED END OPEN END Straight cylindrical cavity with a hub & plate termination

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Bistatic RCS: Straight Cylindrical Cavity with different terminations and that of a Hub

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FRONT VIEW

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Straight cylindrical cavity with a hub blade & plate termination 6λ6λ ƒ=6 GHz λ=50 mm es = 6π 45° 2.5° OPEN END BLADE HUB j i 4 5 6 78 9 10 11 PLATED END 4λ4λ 6λ6λ 3λ3λ k VIEW-OPEN END i

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One block VOL. DISCRETIZATION OF BLOCKS 4-11 (REPRESENTING REGION BETWEEN THE BLADES)

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Bistatic RCS: Straight Cylindrical Cavity with a Hub, Blades and Plate termination

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Monostatic RCS: Straight Cylindrical Cavity with a Hub, Blades and Plate termination 8GHz, Vertical-Polarization

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Monostatic RCS: Straight Cylindrical Cavity with a Hub, Blades and Plate termination 8GHz, Horizontal-Polarization

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Surface Current Distribution on Straight Cylindrical Cavity with Hub, Blades and Plate terminations 6 GHz, Horizontal Polarization, Φ=0°

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Surface Current Distribution on Straight Cylindrical Cavity with Hub, Blades and Plate terminations 8 GHz, Vertical Polarization, Φ=0°

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Straight cylindrical cavity with a hub blade & plate termination 6λ6λ 6λ6λ 3λ3λ ƒ=6 GHz λ=50 mm es = 6π 45° 2.5° OPEN END BLADE HUB j i k 4 5 6 78 9 10 11 PLATED END 4.5 λ VIEW-OPEN END

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Surface discretization of the complete domain 1 2 3 4 -11 1213 14 15 Resolution = 20 grid pts/λ

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Comparative Monostatic RCS: Intake cavities with Cylindrical and Sphere-cylindrical Hub configurations 8GHz Vertical Polarization

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Side view of the cavity (φ =0)

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Front view of the cavity

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B2 – Advanced Technology Bomber Geometry and Grid Generation: Geometrical details taken from public literature - 3-view diagram from Jane's All the World's Aircraft Lack of fine details due to military sensitivity; photographs from the internet used to reconstruct the geometry Basic dimensions: 20.9 m length, 5.1 m height and 52.43 m wingspan Shape of aircraft contributes to its low-observability: blended wing-body, buried engines, lack of external tanks or weapons, etc Surface grid: 25 2-D blocks, 45,840 nodes, formed by piece-wise bilinear surfaces Airfoil section used for the wing: E180 (used in a radio-controlled model of the aircraft) Average cell dimensions: 0.9 mm in longitudinal direction and 1.7 mm in span-wise direction Volume grid: 52 blocks, approx. 1.5 million cells, nodes clustered near the surface Wingspan of the grid: 293.9 mm Minimum wavelength allowable on present grid (satisfying Nyquist's sampling criterion) is 1.8 mm (electric size 1025.4), corresponding to a wavelength of 0.321 m (frequency of approx 1 GHz) for the full scale aircraft

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B2 – Advanced Technology Bomber Image of the B2 in flightRendered Image of B2 Surface Grid

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B2 – Advanced Technology Bomber B2 Surface Grid (close view):

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Surface Currents on B2 – Nose-on incidence @ 300 MHz, VV polarization B2 – Advanced Technology Bomber

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Surface Currents on B2 – Broadside incidence @ 300 MHz, VV polarization B2 – Advanced Technology Bomber

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Surface Currents on B2 – Nose-on incidence @ 300 MHz, HH polarization

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Surface Currents on B2 – Broadside incidence @ 300 MHz, HH polarization B2 – Advanced Technology Bomber

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Bi-static RCS Plot @ 50 MHz (VV) B2 – Advanced Technology Bomber

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Bi-static RCS Plot @ 300 MHz (VV) B2 – Advanced Technology Bomber

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Bi-static RCS Plot @ 300 MHz (HH) B2 – Advanced Technology Bomber

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Monostatic RCS Plot @ 300 MHz B2 – Advanced Technology Bomber

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F-117 “Nighthawk” Rendered Image of F-117 Surface Grid

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Surface Currents at 68 degrees angle of incidence (HH polarization) Surface Currents at 90 degrees angle of incidence (VV polarization) F-117 Surface Currents: F-117 “Nighthawk”

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Bi-static RCS Plot @ 280 MHz (VV)

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Acknowledgement Anuj ShrimalNarendra Deore Manoj VaghelaDebojyoti Ghosh Wing Cdr. A. BhattacharyaSandip Jadhav IITZeus Grid GeneratorProf. G.R.Shevare and his Team

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Thank you

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