finite volume time domain technique (fvtd) computations for

Finite Volume Time Domain (FVTD) Computations for Electromagnetic

Scattering

Prof. A. ChatterjeeDepartment Of Aerospace Engineering

IIT Bombay.

Outline Of PresentationPart I : Introduction to the finite volume time domain technique

Maxwell Equation in Conservative formFinite volume method for conservation lawsNumerical SchemesVolume grid generationBoundary ConditionsValidation

PEC sphereAlmondOgivelossy spherelossy cone sphere

Part II: ApplicationsRCS prediction of low observable aircraft Configurations

Intake B2F117

Maxwell’s curl Equations with losses:

Constitutive Relations

Maxwell Equation (differential form)

HEB

t

EJHD

it

HBED

j

j

r

r

3D Maxwell’s Equations in Conservative form:

where,

Maxwell Equation in Conservative form

shgfu zyxt uuu )()()(

ziz

yiy

xix

z

y

x

x

y

x

y

x

z

x

z

y

z

y

z

z

y

x

z

y

x

EJ

EJEJ

H

HH

s

B

B

D

D

h

B

BD

D

g

BB

DD

f

D

DDB

BB

u

;

0/

/0/

/

;

/0

//

0/

;

//0

//

0

;

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.

Numerical FormulationMaxwell’s Equations (Operator form)

Decomposition of Total Field

suL )(

iii

iss

suLS

SsuL

)(

)(

sisi ssuuL )(

Finite Volume FrameworkConservative form can be written as

where

integrating over an arbitrary control volume,

isz

sy

sx

sst Ssuhugufu )()()(

iz

iy

ix

iit

i suhugufuS )()()(

dVt

VddVss

dVt

Vd

i

v

v

i

v

is

s

v

v

s

)]([)(

)]([

uFu

uFu

discretized form for jth cell,

application of divergence theorem gives,

dVussdSnt

Vd

v

it

is

s

isv

s

)(ˆ)]()([ uFuF

u

)~

~~()}ˆ)]()(({[~

1 dtd

ssVSndt

dV

iji

jsjj

M

mjm

issj

j

uuFuF

u

Higher order characteristic based technique

Runge Kutta time stepping

3D domain divided into hexahedral cells

Cell centred formulation

Finite Volume Framework ….contd

Boundary ConditionsOn Perfectly Conducting (PEC) surface

Total tangential electric field,

Total normal magnetic field,

Far-field boundaries

Characteristic boundary conditions (zero scattered field)

.0En

.0 Bn

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 (Einc 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

.0En

.0 Bn

incscat EnEn

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

Finite Volume Formulation for Conservation Laws

IntroductionAim: 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 conservationMomentum conservationEnergy conservation

+ Equation of state as a closure equation

e.g. In ElectromagneticsMaxwell’s Curl Equations

e.g. In MagnetohydrodynamicsNavier Stokes Equations & Maxwell’s Equations

Conservation EquationConsider generic conservation equation for a conserved variable u, and assume that the corresponding flux vector known

Integrating over an arbitrary volume

where,

shgfu zyxt uuu )()()(

kujuiu ˆ)h(ˆ)g(ˆ)f(F

n̂

dVsdSnt

Vd

vs

v

)(ˆ)]([ uF

u

Conservation Equation for The conservation equation for in a general form:

VV SdVqdSSVdV

t

Rate of changeof in a control

volume

Convective flux acrossthe surface

Source orsink term

Here : = 1 Mass conservation equation = u,v,w Momentum conservation equation = Energy equation

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

Computational Domain for Finite Volume Method

Control Volume

Surface Grid

Volume Grid

Computational domain divided into finite control Volumes

Finite Volume DiscretizationsGeometry CreationDomain Discretization: Domain is subdivided into a finite number of small control volumes (CV) by a grid which defines the control volume boundariesGrid Terminology

Node-based finite volume scheme: u stored at vertexCell-based finite volume scheme: u stored at cell centroid

n̂

Typical CV with the notations

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

dVsdSnt

Vd

vs

v

)(ˆ)]([ uF

u

SdSfk

numS

f

V

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

SdSfk

numS

f

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

Advantage Dis-advantage

Simplest method

Complex geometries

Scheme is conservative

Most popular amongst CFD workers

( > 95%)

Three levels of approximations

• Interpolation

• Differentiation

• Integration

Numerical SchemesDiffer by how numerical flux fnum 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

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

0)( uRdtdu

SV

uR

f1)(

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 jth cell with rth-order accurate ENO scheme

The smoothest possible stencil

),.....,,( 21 kjrkjrkjk xxxS

1

01,2/1

r

llkrj

rlkj ff

Validation for Standard Test Case

ValidationMetallic 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)

Bistatic RCS (dB) for Metallic Sphere at 0.09 GHz

E-plane h-plane

Validation ….contd Metallic Sphere:

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 m2

Minimum RCS of approx -60 dBsm = 10-6 m2

Validation ….contd Metallic Ogive:

Rendered OgiveSurface Grid

Ogive Volume Grid Cross-Section

Validation ….contd

Ogive – Volume Grid and Surface Currents

Metallic Ogive:

Validation ….contd Metallic Ogive:

Ogive Monostatic RCS Plot (1.18 GHz, VV Polarization)

Validation ….contdAlmond:

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

Validation ….contdAlmond:

Rendered AlmondSurface Grid

Validation ….contd Almond:

Almond – Surface Currents

Angle of incidence = 0 deg. Angle of incidence = 90 deg.

Validation ….contd Almond:

Almond Monostatic RCS Plot (1.19 GHz, VV Polarization)

Validation ….contd PEC Sphere with Non-Lossy coating:

PEC ka = 2.6858 Coating (t / λ) = 0.05, ka1 = 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)

Validation ….contd PEC Sphere with Lossy dielectric coating:

PEC ka = 1.5 Coating (t / λ) = 0.05, εr = 3.0 – j4.0, μr = 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

Validation ….contd PEC Cone Sphere with Lossy Coating:

Geometry : vertex angle 90 degree, sphere diameter 0.955 λ Coating (t / λ) = 0.01, εr = 3.0 – j4.0, μr = 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-plane h-plane

Radar Cross Section of Low Observable Aircraft Configurations

Applications: Industrial Problems

Industrial Problems

RCS Analysis of Engine intake configurations

B2 “Advanced Technology Bomber”

F-117 “Nighthawk”

RCS of Low Observable Aircraft Configurations

Introduction

RCS, low-observability, air intake configurations

Intake geometries and grid generation

Results

RCS Analysis of Engine Intake configurationsIntroduction:

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

RCS of some military aircraftsAircraft RCS RCS RCS

[dBsm] [m2] [ft2]

F-15 Eagle 26 405 4,358

F-4 Phantom II 20 100 1,076

B-52 Stratofortress 20 99.5 1,071

Su-27 12 15 161.4

B-1A 10 10 107.6

F-16 Fighting Falcon 7 5 53.82

B-1B Lancer 0.09 1.02 10.98

F-18E/F Super Hornet 0 1 10.76

BGM-109 Tomahawk -13 0.05 0.538

SR-71 Blackbird -18.5 0.014 0.15

F-22 Raptor -22 0.0065 0.07

F-117 Nighthawk -25 0.003 0.03

B-2 Spirit -40 0.0001 0.01

Boeing Bird of Prey -70 0.0000001 0.000008

Estimation of RCSComputation 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)

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 Esc and Hsc at infinite distance and RCS is calculated as

where R is taken to be a sufficiently large number (100,000)

2

22

||||4

inc

sc

EER

En̂Ms

Hn̂Js (equivalent electric current)

(equivalent magnetic current)

RCS ANALYSIS OF ENGINE INTAKE CONFIGURATIONS

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

Intake centre Body

SR-71

CENTRE BODY

ANNULAR REGION

DUCT WALL

INTAKE GRILL

F-117 AIRCRAFT

SOFT

FACEHARD

FACE

GRILL

ENG. FACE

B-2 BOMBER

S-shaped duct with coated wall

YF-23 ADVANCED TACTICAL FIGHTER

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

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

Bistatic RCS: Straight Cylindrical Cavity with a Flat Plate Termination

Monostatic RCS: Straight Cylindrical Cavity with a Flat Plate Termination

Front view

Rear view (angle of irradiation φ=28°)

Φ

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

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

FRONT VIEW

Straight cylindrical cavity with a hub blade & plate termination

6λ

ƒ=6 GHz λ=50 mm es = 6π

45°

2.5°

OPEN END BLADE HUBj

i

4

5

6

78

9

10

11

PLATED END

4λ

6λ

3λ

k

VIEW-OPEN END

i

One block

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

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

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

8GHz, Vertical-Polarization

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

8GHz, Horizontal-Polarization

Surface Current Distribution on Straight Cylindrical Cavity with Hub, Blades and Plate terminations

6 GHz, Horizontal Polarization, Φ=0°

Surface Current Distribution on Straight Cylindrical Cavity with Hub, Blades and Plate terminations

8 GHz, Vertical Polarization, Φ=0°

Straight cylindrical cavity with a hub blade & plate termination

6λ

6λ3λ

ƒ=6 GHz λ=50 mm es = 6π

45°

2.5°

OPEN END BLADE HUBj

i

k4

5

6

78

9

10

11

PLATED END

4.5 λ

VIEW-OPEN END

Surface discretization of the complete domain

1

2

3

4 -11

12 13

14

15

Resolution = 20 grid pts/λ

Comparative Monostatic RCS: Intake cavities with Cylindrical and Sphere-cylindrical Hub configurations

8GHz Vertical Polarization

Side view of the cavity (φ =0)

Front view of the cavity

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

B2 – Advanced Technology Bomber

Image of the B2 in flight

Rendered Image of B2 Surface Grid

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

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


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



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

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


Bi-static RCS Plot @ 50 MHz (VV)


Bi-static RCS Plot @ 300 MHz (HH)


Monostatic RCS Plot @ 300 MHz



Rendered Image of F-117 Surface Grid

Surface Currents at 68 degrees angle of incidence (HH polarization)

Surface Currents at 90 degrees angle of incidence (VV polarization)

F-117 Surface Currents:


Acknowledgement

Anuj Shrimal Narendra Deore

Manoj Vaghela Debojyoti Ghosh

Wing Cdr. A. Bhattacharya Sandip Jadhav

IITZeus Grid Generator Prof. G.R.Shevare and his Team

Thank you

finite volume time domain technique (fvtd) computations for

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