Application of Computational Fluid Dynamics (CFD)Based Technology to Computational Electromagnetics

Ramesh K. Agarwal

IEEE Distinguished LecturerThe William Palm Professor of Engineering

Washington University in St. Louis

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

GOVERNING EQUATIONS OF ELECTROMAGNTICS

Maxwell’s Equations in Conservation Form

Major Components for CEM Analysis

(Material Surface)

SCATTERING MECHANICS

REGION OF APPLICABILITY

REGION OF APPLICABILITY

THE SCATTERING PROBLEM

TWO-DIMENSIONAL GOVERNING EQUATIONS

TIME DOMAIN

FREQUENCY DOMAIN

SCATTERED FORMULATION

NUMERICAL METHOD

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

SPATIAL DISCRETIZATIONVertex-based control volume

Spatial Discretization (Continued)

FilteringGOAL: To efficiently annihilate wave modes that

are not realizable by the spatial discretization.

SPECTRAL FUNCTION

PHASE VELOCITY ERROR

TIME INTEGRATIONFour-stage point implicit Runge-Kutta method:

TIME STEP CALCULATION

STABILITY

Comparison of Convergence Histories

Numerical Analysis:

•1D analysis for model scalar equation with periodic bc

•Semi-discrete form using compact differencing

Fourier Analysis (continued):

•Dispersion relationship

•A nondispersive system has become dispersive due to finite discretization

•Use dispersion relationship to analyze resolution characteristics

•Analytic dispersion relationship

Fourier Analysis:

•u is composed of discrete Fourier modes

•substitution yields

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

Fourier-transorm and its inverse are given by:

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

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

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

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

Consider a compact fourth-order scheme:

Take the Fourier-transform and get

where

Comparison of Resolution Characteristics

BOUNDARY CONDITIONS

• Perfect Electric Conductor• Farfield• Dielectric• Zonal

PHYSICAL BOUNDARY CONDITIONS

•Perfect electric conductor boundary

•Material interface boundary

•Radiation boundary

PERFECT CONDUCTOR

Dielectric Interface

Dielectric Interface Boundary Condition

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

EXACT FARFIELD BC’S

Bayliss-Turkel Far-FieldBoundary Condition

It is based on an asymptotic expansion of the convective wave equation. The second-order operator is given as,

where

Boundary Conditions

The far field boundary condition is based on the second-order Engquist and Majda absorbing boundary condition:

or

where

TE Scattering from a Cylinder

Perfectly Conducting Circular Cylinder

TM Scattering from a PEC Circular Cylinder

Coated Conducting Circular Cylinder

TM Scattering from a Coated Circular Cylinder

Perfectly Conducting Airfoil

TE Scattering from a PEC NACA 0012 Airfoil

Lossy Homogeneous Circular Cylinder

Coated Conducting Airfoil

TM Scattering from a Coated NACA 0012 Airfoil

Rectangular Cavity

PEC Sphere (ka=1.25)

Frequency Domain

Lossless Coated Sphere

Frequency Domain

Meter NASA Almond at 2 GHz

Contour Plots of Surface Fields

Vertical Polarization Horizontal Polarization

Meter NASA Almond at 2 GHz

RCS Plots

Top Side

100 cm x 50 cm Cylinder 1 GHz

Monostatic RCS for a square inlet

FEM

CFD FOD Buster250 MHz

FEM

CFD FOD Buster1 GHz

Monopole Antenna

Transmission Coefficient

Frequency (GHz)Geometry of the Structure

Photonic Band Structure Simulation for MMIC

Instantaneous Electric Field Contours

Conclusions

• CFD based technology (geometry modeling, grid-generation, 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.