Domain Decomposition Methods for Analysis of Structures with Repeatable Components

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• Domain Decomposition Methods for Analysis of
  Structures with Repeatable Components
• Including Large Multifunctional Arrays
• John L. Volakis
• Rick Kindt Kubilay Sertel
• volakis.1_at_osu.edu

volakis_at_ece.osu.edu
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References

• Book
• J.L. Volakis, A. Chatterjee and L. Kempel, Finite
  Element Method for Electromagnetics, IEEE Press,
  1998 (also obtainable from Wiley)
• Finite Element-Boundary Integral Overview paper
• J. Volakis, T. Ozdemir and J. Gong Hybrid Finite
  Element methodologies for antennas and
  scattering, (invited paper) IEEE Trans. Antenna
  Propagat., pp. 493-507, March 1997
• Fast Mulitpole Method with Curvilinear Elements
  Papers
• K. Sertel and J.L. Volakis, Method of moments
  solution of volume integrals equation using
  parametric geometry modeling, Radio Sci., Vol.
  37, No. 1, 2002.
• K. Sertel and J.L. Volakis, Multilevel fast
  multipole method solution of volume integral
  equations using parametric geometry modeling,
  IEEE Trans. Antenna Propagat., July 2004
• J. L. Volakis, K. Sertel, Eric Jorgensen, and R.
  W. Kindt, Hybrid Finite Element and Volume
  Integral Methods for Scattering Using Parametric
  Geometry, Computer Modeling in Engineering and
  Sciences (CMES), Vol.5, No. 5, pp. 463-476, 2004
• S. Bindiganavale(Navale) and J.L. Volakis,
  Comparison of three FMM techniques for solving
  hybrid FE-BI systems, IEEE Antennas and
  Propagation Magazine, Vol. 39, no. 4, Au. 1997
• Array and Finite Array Analysis Papers
• T. Eibert and J.L. Volakis, Fast Spectral Domain
  Algorithm for Hybrid Finite Element/Boundary
  Integral Modeling of Doubly Periodic Structures,
  IEE Proceedings, Microwaves, Antennas and
  Propagation, Vol. 147, No. 5, pp. 329-334,
  October 2000
• R. Kindt, Kubilay Sertel, Erdem Topsakal and John
  Volakis, A domain decomposition of the finite
  element-boundary integral method for finite array
  analysis, 2002 ACES conference, Monterey, CA
• R. W. Kindt, K. Sertel, E. Topsakal, and J. L.
  Volakis, "Array Decomposition Method for the
  Accurate Analysis of Finite Arrays," IEEE Trans.
  Antennas Propagat.,, vol. 51, pp. 1364-1372,
  2003.
• R. Kindt, K. Sertel, E. Topsakal, and J. L.
  Volakis, "An extension of the array decomposition
  method for large finite-array analysis,"
  Microwave and Optical Technology Letters, vol.
  38, pp. 323-328, 2003.
• R. W. Kindt and J. L. Volakis, "Array
  Decomposition-Fast Multipole Method for finite
  array analysis," Radio Science, vol. 39, 2004.
• R. W. Kindt, "Rigorous Analysis of Composite
  Finite Array Structures," in Ph.D. Dissertation
  EECS Dept., University of Michigan, Ann Arbor,
  2004.

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AMFIA - Analysis Methods for Infinite and Finite
Arrays, including structural interactions
What it does Large array analysis code for inner
and intra array coupling and performance
evaluation on treated treated platforms How

• Finite Element Volume Methods for antenna volume
  modeling
• MLFMM/boundary integral for aperture modeling
• New supercell/decomposition technique for exact
  large
• finite
  array evaluations
• Interfaces with commercial gridders for geometry
  generation

Aircraft Conformal Arrays
Original Sponsorship by DDX (ONR)
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Geometric Representation

• Curvilinear hexahedral elements for closed object
  modeling
• Exact geometric representation with far fewer
  basis functions
• Internal E-fields modeled with Finite Element
  Method
• Surface currents formulated with integral
  equations (bi-quadratic quadrilaterals)

FEM Analysis
V
S
Boundary Integral Analysis
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Curvilinear Hexahedral Basis functions

• The basis and testing functions are chosen as the
  tangential field component along one of the 12
  edges of a hexahedral element.

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IE Discretization/Moment Method Using
Curvilinear Elements

• The EFIE is solved over the PEC surfaces of the
  arbitrary object
• CFIE is solved over open surfaces
• 1st Order Curvilinear basis functions over
  quadrilateral biquadratic elements are used in
  discretizing the integral equation

K. Sertel and J.L. Volakis, Radio Sci., Vol. 37,
No. 1, 2002.
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Finite Element-Boundary Integral Method (FE-BI)

• Field continuity enforced inside volume with FEM
• Integral equations model currents at volume
  surface to enforce boundary conditions and
  satisfy radiation conditions external to volume
• Combined formulation provides unique solution
  both inside and outside volume

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Math Details for hexahedral FE-BI

• In the FE-BI formulation, the matrix is
  constructed by setting ?F(E)/?E 0 with the
  function F(E) given as see book by Volakis,
  et.al., IEEE Press, 1998

Here the surface integral is taken at the
boundary enclosing the volume. Upon
discretization, the following system is solved
where Ev are the electric fields internal to the
volume and Es are the electric fields on the
surface of the volume.
aCFIE factor
T 1- ß/4p.
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1. Standard FE-BI Matrix Assembly
Hybrid Finite Element-Boundary Integral and Fast
Multipole Method
Assembly by geometric type
fields, excitations, grouped by geometric type as
well - internal, surface, external
surface FEM operators
interior FEM operators
BI surface operators
Same matrix assembly for any problem Typical
modeling of arrays with FE-BI individual
operator sub-matrices get larger for larger
arrays---not practical for large arrays
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Requirements for Standard FE-BI Analysis

• Electrically small element, 1.6l length
• 1100 unknowns (7MB)

Computations performed on Itanium server, 16GB
main memory
estimated

• Conclusions
• Rigorous but expensive
• NOT reasonable for array analysis need to
  reduce storage requirements

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Boundary Integral Computational Requirements
Nnm nnumber of domains m unknowns per domain

• For N unknowns
• Matrix Storage O(N2)
• Direct Solution (LU or GE) O(N3)
• Iterative Solution (CG, QMR, GMRES, etc) O(N2)
  per iteration

Surface Integral Equations
hence
Surface Integral Equation

• Matrix Storage O(f4)
• Direct Solution O(f6)
• Iterative Solution O(f4) per iteration

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Memory and CPU Reduction Require use of Iterative
Solvers
Consider the Matrix System
Start with an initial guess (for m0)
Find the residual
Test for convergence
if converged, STOP else
Generate the next guess using the error
Represents CG, CGS, BiCG, QMR, GMRES, Jacobi,
Gauss-Seidel, SOR
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How does the fast multipole method (FMM) work?
Key approach to FMM is the matrix decomposition
into near and far zone components
Indirectly evaluated
Sparse/narrowband matrix
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FMM details for 2D analysis
2D scattering requires the evaluation of the
integral
To speed-up the integral evaluation, we proceed
to separate the prime and unprimed coordinates
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Translation Operator
This operator only depends on the Separation
distance between the groups So, they can be
pre-computed and stored for reuse
for large group separation
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Example 2D Scattering
2D Example Scattering Example
Translation Operator
Mbno. of elements in a group
Nnumber of groups
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(No Transcript)
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3D fast multipole method
Separate r and r via Gegenbauers addition
theorem for spherical harmonics
Observation point
Plane-wave expansion
Results in
Source point
for point sources, where
Translation operator
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Group interactions within the FMM Formulation
Free-space Greens Function
Translation operator
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Evaluation of matrix elements and the
matrix-vector product
Spherical wave expansion and subsequent plane
wave expansion of free-space Greens function
leads to FMM algorithm.
Far-field signature functions
Matrix-vector product in FMM picture
For near basis functions the original moment
method/ boundary integral matrix elements are
retained. What determines accuracy ?
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Single Level FMM Algorithm
FMM is a fast computation of the matrix-vector
product in the iterative solver of the matrix
system.

• Matrix-vector product for far-field terms is
  done indirectly leading to O(N1.5) flops.
• Near-field matrix-vector product is done in a
  conventional manner.

• Far-field matrix-vector product is done in 3
  sweeps
• Aggregations
• Translations
• Disaggregations

near-field terms
aggregations
translations
disaggregations
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2nd Level FMM for further CPU reduction
A single group is divided in 4 groups
Phase center corrections
Need spectral samples at Use Interpolation
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Interpolation Operators
Interpolation for signature functions
1-D example
Lagrange interpolators
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Multilevel FMM(MLFMM) Algorithm and Evaluation of
the Matrix-Vector Product
Step 1- Do near-field matrix-vector
product Step 2- Do aggregations and
translations for far clusters in level 2 but
near clusters in level 1 Step 3- Interpolate
level 2 aggregations onto level 1 Step 4-
Do translations for far clusters in
level 1 Step 5- Anterpolate level 1
disaggregations onto level 2 Step 6- Do
disaggregations in level 2 to compute
matrix-vector product
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Clustering Approach Oct-tree Grouping
Zeroth level cube
1st level Subdivision into 8(oct) children
Empty cubes discarded
Why oct-tree ? Rectangular symmetry provides
significant memory savings
2nd level subdivision
3rd level subdivision
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Multilevel FMM (MLFMM) Performance
CPU Time
Memory
MLFMM provides 3 orders of magnitude in memory
reduction for 1,000,000 unknowns
MLFMM is 2 orders of magnitude faster for
1,000,000 unknowns
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Fast Multipole Method for the Matrix Subsystem
for the Surface Integral in the FE-BI formulation

• Form clusters of basis functions
• Consider two clusters, m,n
• Compute signature functions of the groups
• Form of conventional near-zone testing -
  convolutional
• Form of FMM far-zone testing - diagonal

cluster m
cluster n
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Still, FMM and MLFMM is still not adequate

• FMM approach to array analysis
• Noticeable reduction in storage
• Same convergence characteristics as conventional
  FE-BI

• Benefits
• Sparse BI matrix array storage O(m1.5
  n1.5) vs. O(m2 n2) much larger problem analysis
  possible
• Drawbacks
• No change in solution convergence behavior

Conclusion FMM alone not adequate for array
analysis need to improve solution approach
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2. Step by Step Array Modeling
Example Geometry-
Surface Wave Absorber (SWA) For Coupling Reduction
LTSA Array elements in this geometry
are designed to operate at 9GHz
Cavity Wall
Subarray-1
Subarray-2
Actual element with balun feed
with absorber
Measurement article at NRL
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Antenna Element Detail

• single array element is complex
• material and fine details
• 1001 ratios among elements cause
• significant deterioration in preconditioning
• detailed feed modeling is crucial
• element itself is very large
• finite arrays modeling necessary for
• coupling evaluation and scan performance

Use Finite Element-Boundary Integral Method to
model
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Computational and Modeling Challenges

• Full scale airframes require millions of unknowns
  for modeling at frequencies greater than 1GHz
  (70ft long metallic fighter)

• a 30x30 finite TSA array requires 1 million
  unknowns, and a 100x100 TSA array requires 6.5
  million unknowns (small TSA element)
• Solution of finite array problems are therefore
  as large as those of classical full scattering
  problems, where research is still very active

• Large basis functions can alleviate unknown
  count, but lead to poorer conditioned systems

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Array Decomposition Method (ADM)

• Main concept
• Divide-up array into groups consisting of
  repeating elements (supercells)---take advantage
  of inherent redundancies
• Each supercell is modeled independently with
  method of choice (Hybrid finite element-boundary
  integral/FE-BI method is used here)
• No requirement for periodicity or connectivity
  of the nearby elements

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Rotational ADM Example

• Aircraft fuselage
• 1.5 million unknowns
• 100l in circumference, 30l tall
• Conformal 10x40 slot array
• 25 Terabytes
• 10 Gigabytes over 3 orders magnitude storage
  savings

Near zone decomposition alone
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Multi-Dimensional Decomposition Example Metal
Shields

• What a normal person sees two metal boxes
• What it is two five dimensional arrays
  accelerated via 5D FFT

Array 1
Array 2
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Multiple Supercells Example

• 2 x (3x4) arrays
• Vertical walls
• Radome
• Ground plane
• 8 supercells
• Desktop PC problem
• Less than 1 GB memory
• 1000 unk. per element
• 100k unknowns
• 6 min. per scan angle
• (with no supercell100GB)

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Advantages of Multi-Dimensional Formulation

• Arbitrary dimensional definition no coordinate
  system limitation
• As many dimensions as necessary for maximum
  re-use of geometric features
• Implementable with simultaneous translational and
  rotational dimensions with some caveats
• Selective multi-dimensional FFT acceleration
  some dimensions can be truncated or treated via
  alternative methods (MLFMM)

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Multi-Cell Array Decomposition

• Concept Most of problem decomposed into
  equivalent cells in a given dimension
• Cross-terms between systems with a common
  dimension Toeplitz form
• Example system solved with a 5-point FFT, 9-point
  FFT, two 7-point FFTs
• Scalable to multiple dimensions

1
2
3
1
2
3
4
5
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Array Decomposition Mathematics
2.1 Expand matrix system grouped by element
interactions
- self-coupling terms along diagonal give
complete FE-BI representation for each element of
array - mutual-coupling sub-matrices for
interactions between all elements - fields and
excitations grouped by element
no approximations - expanded matrix contains
same information as conventional FE-BI matrix
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2.2 Preconditioning
Self-coupling terms along block-diagonal
identical - LU decompose self-cell and use as
block-diagonal preconditioner
Highly effective preconditioning based on the
physics of the problem Relatively insignificant
storage and computation requirements FEM
restricted to self-cells - preconditioned out of
solution process
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2.3 Taking advantage of element repeatability
Conventional Storage n2 terms
conventional solution
block-convolution
block-FFT solution
(1)
(2)
Toeplitz Storage
2n-1 terms
unique terms given by difference in element
indexes
(3)
(4)
Significant storage reductions

• FEM Storage O(mFEM) vs. O(nmFEM)
• BI Storage O(nm2BI) vs. O(n2m2BI)

m unknowns/array element
n number of array elements
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2.4 ADM Projected Storage and Fill-Time Savings

• FEM Storage O(mFEM) (vs. O(n mFEM))
• BI Storage O(nm2BI) (O(n2m2BI))
• O(mBInlogn) computational cost
• FEM computational cost can be circumvented

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Example Validations

• ADM can be equally used for input impedance,
  radiation and scattering calculations
• Good agreement for analysis of broadband antenna
  elements

Tapered slot element
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Tested Array Configurations
30x30 array one million unknowns
Element has 1103 unknowns (507 FEM, 596 BI) 6x6
limit for conventional FE-BI on 800MHz 16GB
Itanium
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Overall Savings with ADM
Over 3 orders of magnitude speed-up
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Storage is still a bottleneck for large arrays

• ADM storage still includes much overhead from far
  zone element interactions
• As the array becomes larger and larger (and more
  so for array to array interactions), storage
  continues to increase, even though linearly
• AD-FMM incorporates the FMM for reducing storage
  and for increasing CPU speed in the matrix vector
  products of the solver

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3. AD-FMM Overview

• Exact decomposition method for finite array-type
  structures
• Fixed near-zone matrix storage for any-sized
  finite array
• Single set of signature functions for element
• Translation operator storage is less than the
  storage for solution vector O(N) storage for
  large arrays
• Acceleration of far-zone interactions via
  explicit FFT
• Effective preconditioning based on unit-cell
  decomposition
• very few iterations for convergence

AD-FMM (near- far-zone decomp.)
near-zone decomp.
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3.1 AD-FMM for far zone interactions
Main concept

• Treat distant array element interactions in
  far-zone via the FMM
• Elements in near-zone treated via ADM

• Identical array elements - single set of far-zone
  signatures to compute
• - n times less than conventional FMM

near-zone

• Toeplitz property for translation terms

Far-zone translation storage

• Far-zone interactions can be accelerated via FFT

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Main Contribution
ADM and FMM working together

• Complementary Toeplitz storage sets
• Near-zone storage fixed - FOR ANY
  SIZED ARRAY

near-zone
Handled in near-zone
Handled in far-zone
Near-zone storage
Far-zone storage
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AD-FMM summary

• Benefits
• O(mFEM) FEM storage
• O(m2BI) BI matrix storage
• Excellent preconditioning, few iterations for
  convergence
• FFT solution acceleration of far-zone
  interactions
• Remaining limitations
• O(m2BI) BI matrix storage requirements still a
  bottleneck
• Potential for large near-zone influences

near-zone
Near-zone storage
Translation storage
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Storage Cost Analysis
Storage reduction with AD-FMM

• Four main storage constructs
• Near-zone matrix storage
• Far-zone signature storage
• Translation matrix storage
• Solution/excitation and work vectors

• Cost of methods
• FE-BI storage
• O(N2 ) (dominated by 1.)
• FMM storage
• O(N1.5 ) (dominated by 1.,2.,3.)
• AD-FMM storage
• O(N) (dominated by 4.)

m unknowns/array element K far-zone angles to
represent element
in the limit
fixed storage
Near-zone storage
Translation storage
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3.2 AD-FMM Bridge Systems

• AD-FMM uses unique bridge systems to model
  current flow between elements
• Minimal unknown requirements
• Very low overhead
• No approximations

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Comparison with reference results

• No surprises

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Theoretical Cost Comparison
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projected results
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What does AD-FMM allows?single array example

• 621,516 unknowns
• High basis density

Pay attention to benefit of multi-dimensional
specification
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Single-cell test model

• Strongest diagonal form for fastest solution and
  convergence
• Single-cell 4-8 iterations for convergence
• Multi-cell 100

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Intra-Array Coupling-Validation

• Excite single element of array
• Measure received signal at other ports of array

element index
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Pipe scattering via ADFMM

• Pipe modeled as array of rings using AD-FMM
  analysis
• 3GHz example
• 12,000 unknowns
• 11MB matrix storage
• Iterative solution approach
• 12 iterations/scan angle
• 15 sec. / iteration
• (1 AMD XP2800 CPU)

Monostatic scattering, VV, HH 3GHz
22 current bridge elements
23 array ring elements
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AD-FMM for large scatterers

• 36 inch pipe example
• 61l long, 7l diameter at 20GHz
• 128 ring elements to form pipe
• 127 bridge elements connect rings
• 543,568 unknowns total
• 250MB matrix storage
• 171MB preconditioner storage
• 350MB basis Fourier coefficient storage
• 439MB translation matrix storage
• 1.2GB total storage
• Compare to a conventional storage
• of 3.4TB

61l
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Embedded Array Patterns-Validation

• Excite single polarization, center of array

Co-pol.
X-pol.
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Array to Array Coupling
Array 1

• Coupling between identical arrays can be analyzed
  as a single, three-dimensional array (11x10x2)

Array 2
Reasonable numbers for analysis
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Intra-Array Coupling

• Same matrix storage requirements as single array
  problem
• Many platform-type interactions can be carried
  out rigorously, if required

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Array Covered by a finite FSS
Interior view

• Non-commensurate FSS modeling
• FSS and array of difference periodicities
• Small Array example
• N37,844
• 169MB storage vs. 11.3GB
• 28 iterations
• 80sec./iteration
• 2445sec. solve

FSS
See Paper Radiation and Coupling Among Large
Multiple Antenna Arrays on Ships, by
Burkholder, Sertel, Pathak, Volakis, Navale (APS
2003)
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ADFMM for Evaluating Propagation in indoor
Wireless Communication Channels
Wall thickness 0.07 m and dielectric constant
(?r2.0) with ?0.01, windows added along the
walls. 20 chairs with dielectric constant
(?r2.0). Chair dimension (0.5mX0.5mX0.8m) Anten
na located at (0, 0, 1.8), 81 measurement points
evenly distributed in the room. About 5 million
unknowns per run
Power vs Time