Finite Volume Time Domain (FVTD) Computations for Electromagnetic Scattering

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

2. 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 Outline Of Presentation

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

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

5. 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.

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

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

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

9. 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 (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

10. 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

11. Finite Volume Formulation for Conservation Laws

12. 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

13. 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,

14. 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

15. 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

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

17. 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

18. 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

19. 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

20. 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

21. Numerical Schemes • Differ 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

22. 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

23. 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 jthcell with rth-order accurate ENO scheme The smoothest possible stencil

24. Validation for Standard Test Case

25. 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)

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

27. 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

28. Validation ….contd Metallic Ogive: Surface Grid Rendered Ogive

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

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

31. 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

32. Validation ….contd Almond: Surface Grid Rendered Almond

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

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

35. 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)

36. 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) For different orders of accuracy For different discretization Monostatic RCS Sphere with Lossy Coating

37. 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) E-plane h-plane Bistatic RCS for cone-sphere with lossy Coating (Backscatter at 180 degree)

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

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

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

41. 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

42. RCS of some military aircrafts

43. 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)

44. 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) (equivalent electric current) (equivalent magnetic current)

45. RCS ANALYSIS OF ENGINE INTAKE CONFIGURATIONS

46. 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

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

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

49. B-2 BOMBER S-shaped duct with coated wall

50. YF-23 ADVANCED TACTICAL FIGHTER