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
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 DISCRETIZATION Vertex-based control volume
Spatial Discretization (Continued)
Filtering GOAL: To efficiently annihilate wave modes that are not realizable by the spatial discretization.
SPECTRAL FUNCTION
PHASE VELOCITY ERROR
TIME INTEGRATION Four-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 Analytic dispersion relationship A nondispersive system has become dispersive due to finite discretization Use dispersion relationship to analyze resolution characteristics
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-Field Boundary 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 secondorder 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 Buster 250 MHz
FEM CFD FOD Buster 1 GHz
Monopole Antenna
Photonic Band Structure Simulation for MMIC Transmission Coefficient Instantaneous Electric Field Contours Frequency (GHz) Geometry of the Structure
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.