Computational Electromagnetics

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Computational Electromagnetics

• Daniel S. Katz
• Director for Cyberinfrastructure Development, CCT
• Adjunct Associate Professor,
• Electrical and Computer Engineering
• Louisiana State University

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Electromagnetics

• Maxwells Equations
• Lots of versions, pick the right set for your
  problem and methodology
• Wavelength and frequency are inversely related
• An object of size 1 m is one wavelength long at
  300 MHz or 2 wavelengths long at 150 MHz
• Either frequency or size can be scaled as needed
  
• Radar Cross Section (RCS) as an example problem
• A plane wave at some incident angle and some
  frequency illuminates a target.
• Monostatic or Backscatter RCS what energy comes
  back?
• Bistatic RCS what energy goes off in another
  direction?
• Toy problem sphere
• Radius r volume 4/3 P r3 surface area 4 P
  r2

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CEM methods

• Maxwells Equations-based methods
• Optics-based methods

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• sdfsdf

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Outline

• Method of Moments (MoM)
• Finite Difference Time Domain (FDTD)

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Corner Reflector

• Surface currents displayed
• Tone burst has hit reflector
• Burst incident at 45 to left, right sides 10
  to bottom
• Frequency 10 GHz
• Dimension 5 cm on edge

Photograph of computer monitor, 1989
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Dielectric Lens

• Dielectric lenses can be made in different
  materials with different properties
• Above quantum well infrared photodetector (QWIP),
  can increase QWIP sensitivity by 14x
• 20 THz plane wave incident downward

Scanning Electron Microscope (SEM) Image of a
portion of the dielectric lens (credit Dan
Wilson, JPL Microdevices Laboratory)
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Method of Moments

• Maxwells Equations in Integral Form
  (Stratton-Chu Equations)
• Total field (to which boundary conditions are
  applied) is summation of incident field (defined
  by incident wave only) and scattered field (the
  result of the incident wave interacting with the
  scattering object)

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Method of Moments (2)

• Electric Field Integral Equation (EFIE)
• Continuity relationship for current and charge
  density

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Method of Moments (3)

• For a perfect electric conductor, tangential
  component of E on the surface is zero boundary
  condition
• And M is zero
• Can rewrite EFIE as operator on

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Method of Moments (4)

• Represent surface current as summation of element
  currents and basis functions
• Enforce the boundary condition on the surface (at
  N unique locations match points)
• At each (N) match point, the total field is the
  sum of
• the incident field, and
• the field caused by currents located on the other
  points
• This leads to a coupled system of equations by
  which every part of the body affects every other
  part (Maxwells equations are elliptic)
• Sample the region around the match points and
  average by using a weight function

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Method of Moments (5)

• Define inner product integral as
• (Integration is over the pth surface area which
  surrounds the pth BC match point)

• jn current at location n
• vn forcing function at location n
• znm interaction between locations n and m

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Method of Moments (6)

• Assume incident place wave
• Impedance matrix terms
• Scattered field

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Sequential MoM

• jn current at location n
• vn forcing function at location n
• znm interaction between locations n and m
• Based on geometry, calculate
• LU decompose
• Based on input plane wave, calculate
• Backsolve for
• Use to find far fields, or RCS, using
  equivalence principle

(image from http//www.cgal.org/)
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Sequential MoM

• How big is the matrix? (What is N?)
• Depends on the geometry
• Need to have enough points to define the object
• Also want at least 5 points per linear wavelength
• For a sphere of radius 1 wavelength, N 200
• For a sphere of radius 2 wavelengths, N 8000
• N is O(x2), x is linear size of object
• Matrix is size O(N2) or O(x4)
• Filling matrix is O(N2) or O(x4)
• Solving the matrix is O(N3) or O(x6)
• Backsolve for one RHS is O(N2) or O(x4)
• Calculating one RCS is O(N) or O(x2)

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Sequential MoM vs. problem

• One LU decomposition suffices for an object at a
  frequency
• Multiple incident angles means multiple RHSs
• Multiple scattering angles means using jns
  multiple times
• Change frequency, redo LU decomposition
• Time saving tricks
• Use symmetry 8 way symmetry (sphere) reduces
  matrix size to 1/8th, and solution time to
  1/512th
• Easy to calculate znm and zmn at the same time,
  reduce matrix fill time by almost ½

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Parallel MoM

• Store matrix in block cyclic format for good
  performance in solve
• Good point
• Can use standard parallel linear algebra
  libraries
• ScaLAPACK, PLAPACK, etc.
• Hard to fill matrix and RHS
• Easiest to have all the geometry on all
  processors
• Divide patches by procs, fill rows of matrix that
  go with each procs patches, same for RHS
• Problems
• Still hard to fill the block-cyclic matrix
• Either lots of communication or lots of
  duplicated computation
• Probably have to compute znm and zmn
  independently

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Direct vs. Iterative Matrix Solution

• Given BAX, where A is a matrix and B and X are
  vectors, a direct solution for X is basically
  finding A-1B
• This is the LU decomposition approach, more or
  less
• O(N3)
• Iterative schemes are also possible, where the
  core of the iteration loop is multiplying A times
  a vector y
• Each iteration (matrix vector multiplication)
  is O(N2)
• May need a lot of iterations to converge
• For dense matrices, this usually doesnt make
  sense
• What if A is sparse?
• Matrix vector multiplication with a sparse matrix
  may be low cost, particularly if the number of
  non-zero elements is small and only these
  elements of the matrix are stored
• O(n_non_zeros)

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Fast Multipole Method

• Introduced by Greengard and Rokhlin in mid 1980s
• Group patches that are close to each other
• Leads to errors, but the scale of the errors can
  be predefined
• Similar to some n-body solution methods
• Use an iterative solver for the matrix
• Dont have to store all elements, can build some
  on the fly
• Use FMM hierarchically to get solver thats O(n
  log n)
• Classical MoM is basically dead for large
  problems

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Outline

• Method of Moments (MoM)
• Finite Difference Time Domain (FDTD)

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Maxwells Equations in Curl Form

• Maxwells (curl) Equations
• Electric field vector
• Magnetic flux density vector
• Electric flux density vector
• Magnetic field vector
• In linear, isotropic, non-dispersive media

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Maxwells Equations in Curl Form

• Writing out the vector components

(image from wikipedia)
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1-D Maxwells Equations

• Assume E only has a z component, and that
  everything is constant in y

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Central Differencing

• Consider Taylor series expansion of f(x) expanded
  around x0 with an offset of d/2
• Subtract the second from the first
• Notes
• This is second-order accurate
• No sample at x0

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1-D FDTD

• Apply differencing

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1-D FDTD details

• Non-rigorously
• Energy should not propagate more than one spatial
  step in each temporal step
• Computer implementation

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1-D FDTD Code

• Define media (ca, cb)
• Initialize fields to zero
• Loop over time (n 1 to nmax)
• Loop over space for ez (i0 to imax)
• ezi cai(hyi-hyi-1)
• Loop over space for hy (i1 to imax-1)
• hyi cbi(ezi1-ezi)

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1-D FDTD Code - BC

• What about Ez0 and Ezimax?
• We need boundary conditions to ensure that waves
  propagate past these points without reflecting
• Simple choice, if dt/dxc
• Ezn0 Ezn-1 1
• Mathematic/geometric option in 2d and 3d
• Mur RBC (1981) Mur RBC
• Model absorbing material (virtual range)
• Berenger (1994) Berenger PML

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1-D FDTD Code - Inputs

• How to input energy into the system?
• Use a hard source
• ez10 cssin(omegadttimestep)
• Simple, but leads to reflections
• Use a soft source
• Amperes Law
• Apply finite differences
• Separate into normal update and additive source
• ezi cai(hyi-hyi-1)
• ez10 cssin(omegadttimestep)

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1-D FDTD Code - Scatterers

• How to find scattered field?
• Use a total field / scattered field formulation
  for the main grid
• Compute two 1-D grids, one for the incident field
  and one for the total/scattered field
• Incident grid is homogeneous TF/SF grid has
  scatterer geometry
• Add/subtract incident field on total
  field/scattered field boundaries

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1-D FDTD Code Scatterers (2)

• eztotal50 ca50(hytotal50-hytotal49)
• Correct update from difference equation, but
  doesnt match grid
• hytotal49 hyinc49 hyscat49
• eztotal50 ca50(hytotal50-hyscat49)
  (normal update)
• eztotal50 - ca50hyinc49 (special update
  for TF/SF interface)
• Similar changes needed for hy49 update, and ez
  and hy at TF/SF interface on right side of grid

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Ghost Cells (2D)

• Parallel Implementation
• Need to update these cells on a given processor,
  using second order central differences (one cell
  on each side)
• In order to update outer cells, need cells one
  step further away
• These have to be communicated from neighboring
  processors

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Ghost Cells (2D) saving communication?

• What if we communicate more cells every other
  time step?
• Seems like less communication calls, same amount
  of communication total
• Better for small messages, where latency matters
  more than bandwidth
• But there are problems with this idea

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Load Balancing

• How to divide this domain for 4 procs?
• OpenMP worry about
• Work
• MPI worry about
• Memory
• Work
• Communication

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3-D Decomposition
Standard Domain Decomposition
Required Ghost Cells
One plane of ghost cells must be communicated to
each neighboring processor each time step.
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3-D Decomposition (2)
Standard Domain Decomposition
Boundary Decomposition
Data at four faces must be redistributed twice
each time step!!
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Parallel FDTD Modeling Example Periodic
Plasmonic System
wraparound boundary conditions for side domain
walls. PML for top and bottom boundary
3D FDTD domain of unit cell and domain
decomposition
Credit Tae-Woo Lee
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Frequency Domain Results

• FDTD is time domain how to get frequency domain
  results for scattering calculations?
• Keep track of fields at some locus in the
  scattered field domain
• Use FFT or similar to transform from time domain
  to frequency domain (see first part of notes)
• Use equivalence principal to transform to
  far-field (RCS)
• Can do this for multiple bistatic angles after
  one FDTD run
• Need to run long enough to reach steady state
• Approx. enough time for fields to move from one
  corner of the domain to the opposite corner 4
  times
• What about multiple frequencies?
• Can use incident pulse with good frequency
  content, FFT-1

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FDTD Staircasing

• Want Get
• Can modify update equations near surface for
  smoothing

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Engine Inlet

• Aluminum inlet in box coated with radar absorbing
  material
• 2-D slice of 3-D problem
• Inlet dimensions 5 x 5 x 30 cm
• Scattered electric field magnitude displayed
• 10 GHz plane wave incident from right side
• Slice after 120 cycles of incident wave

Photographs of computer monitor, 1989
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Credits

• MoM material
• Shaeffer, MOM3D Method of Moments Code Manual,
  1992
• Wikipedia
• Umashankar, UIC
• FMM material
• Wikipedia
• FDTD material
• Allen Taflove, Northwestern, John Schneider, WSU,
  Tae-Woo Lee, LSU