Application of Computational Fluid Dynamics (CFD) Based Technology to Computational Electromagnetics Ramesh K. Agarwal

Transcription

1 Application of Computational Fluid Dynamics (CFD) Based Technology to Computational Electromagnetics Ramesh K. Agarwal IEEE Distinguished Lecturer The William Palm Professor of Engineering Washington University in St. Louis

2 Equations of Mathematical Physics Maxwell equations Schroedinger equation Boltzmann equation Einstein equations of general relativity Hydrodynamic device simulation equations Equations of Elasticity Navier-Stokes equations Nonlinear transport equations with complex constitutive equations

3 GOVERNING EQUATIONS OF ELECTROMAGNTICS

4 Maxwell s Equations in Conservation Form

5 Major Components for CEM Analysis (Material Surface)

6 SCATTERING MECHANICS

7 REGION OF APPLICABILITY

8 REGION OF APPLICABILITY

9 THE SCATTERING PROBLEM

10 TWO-DIMENSIONAL GOVERNING EQUATIONS

11 TIME DOMAIN

12 FREQUENCY DOMAIN

13 SCATTERED FORMULATION

14 NUMERICAL METHOD Spatial discretization and resolution characteristics Stability of explicit/point-implicit time integration Filtering Time integration Boundary conditions Post processing

15 SPATIAL DISCRETIZATION Vertex-based control volume

16 Spatial Discretization (Continued)

17 Filtering GOAL: To efficiently annihilate wave modes that are not realizable by the spatial discretization.

18 SPECTRAL FUNCTION

19 PHASE VELOCITY ERROR

20 TIME INTEGRATION Four-stage point implicit Runge-Kutta method:

21 TIME STEP CALCULATION

22 STABILITY

23 Comparison of Convergence Histories

24 Numerical Analysis: 1D analysis for model scalar equation with periodic bc Semi-discrete form using compact differencing

25 Fourier Analysis (continued): Dispersion relationship Analytic dispersion relationship A nondispersive system has become dispersive due to finite discretization Use dispersion relationship to analyze resolution characteristics

26 Fourier Analysis: u is composed of discrete Fourier modes substitution yields

27 Dispersion-Relation-Preserving (DRP) Higher-order Finite-Difference Schemes Fourier-transorm and its inverse are given by:

28 Dispersion-Relation-Preserving (DRP) Higher-order Finite-Difference Schemes

29 Dispersion-Relation-Preserving (DRP) Higher-order Finite-Difference Schemes

30 Dispersion-Relation-Preserving (DRP) Higher-order Finite-Difference Schemes

31 Dispersion-Relation-Preserving (DRP) Higher-order Finite-Difference Schemes Consider a compact fourth-order scheme: Take the Fourier-transform and get where

32 Comparison of Resolution Characteristics

33 BOUNDARY CONDITIONS Perfect Electric Conductor Farfield Dielectric Zonal

34 PHYSICAL BOUNDARY CONDITIONS Perfect electric conductor boundary Material interface boundary Radiation boundary

35 PERFECT CONDUCTOR

36 Dielectric Interface

37 Dielectric Interface Boundary Condition

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39 RADIATION BOUNDARY CONDITION Objective: model an infinite domain Approach: identify incoming wave modes at the radiation boundary and set them to zero Recast the equations into cylindrical coordinates Derive eigenvectors to compute 1D polar characteristics FFT polar characteristics

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41 EXACT FARFIELD BC S

42 Bayliss-Turkel Far-Field Boundary Condition It is based on an asymptotic expansion of the convective wave equation. The second-order operator is given as, where

43 Boundary Conditions The far field boundary condition is based on the secondorder Engquist and Majda absorbing boundary condition: or where

44 TE Scattering from a Cylinder

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49 Perfectly Conducting Circular Cylinder

50 TM Scattering from a PEC Circular Cylinder

51 Coated Conducting Circular Cylinder

52 TM Scattering from a Coated Circular Cylinder

53 Perfectly Conducting Airfoil

54 TE Scattering from a PEC NACA 0012 Airfoil

55 Lossy Homogeneous Circular Cylinder

56 Coated Conducting Airfoil

57 TM Scattering from a Coated NACA 0012 Airfoil

58 Rectangular Cavity

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61 PEC Sphere (ka=1.25) Frequency Domain

62 Lossless Coated Sphere Frequency Domain

63 Meter NASA Almond at 2 GHz Contour Plots of Surface Fields Vertical Polarization Horizontal Polarization

64 Meter NASA Almond at 2 GHz RCS Plots Top Side

65 100 cm x 50 cm Cylinder 1 GHz

66 Monostatic RCS for a square inlet

67 FEM CFD FOD Buster 250 MHz

68 FEM CFD FOD Buster 1 GHz

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70 Monopole Antenna

71 Photonic Band Structure Simulation for MMIC Transmission Coefficient Instantaneous Electric Field Contours Frequency (GHz) Geometry of the Structure

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73 Conclusions CFD based technology (geometry modeling, gridgeneration, numerical algorithms etc.) can be effectively employed to compute scattering from complex electromagnetically large objects in low to moderate frequency range. The numerical Maxwell equations solvers based on this technology are accurate, efficient and robust.