Domain Decomposition Methods for Solving Large Electromagnetic Problems

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Domain Decomposition Methods for Solving Large
Electromagnetic Problems
Marinos Vouvakis, Seung-Cheol Lee, Kezhong Zhao
and Jin-Fa Lee Computational Science Group, ESL,
ECE Dept. The Ohio State University http//esl.eng
.ohio-state.edu/csg
Sponsored jointly by Ansoft Corp. Pittsburgh And
Temasek Lab., NUS, Singapore and Northrop Grumman
Corp.
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Vector Finite Element Methods

• Capability of brute-force FEM in a single PC
  using

• p-type MUltiplicative Schwarz (pMUS)
  preconditioner

• h-adaptive refinement

1 Jin-Fa Lee, Din Kow Sun, pMUS (p-type
Multiplicative Schwarz) Method with Vector Finite
Elements for Modeling Three-Dimensional Waveguide
Discontinuities, IEEE Trans. Microwave Theory
Tech., March, 2004. 2 D. K. Sun, J. F. Lee,
and Z. J. Cendes, Construction of Nearly
Orthogonal Nedelec Bases for Rapid Convergence
with Multilevel Preconditioned Solvers, SIAM J.
Sci. Comput., vol. 23, no. 4, pp. 1053-1076.
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Dual band operation
7 elements antenna array, circular polarized
The 7 elements array, coated with Radome, sitting
on top of a large finite ground plane
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Smoothed finite ground plane
Smoothed finite ground plane
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Outline

• Non-overlapping Conformal Domain Decomposition
• Conventional Formulation
• Alternative Formulation with Largrange
  Multiplier
• Duality Pairing
• Non-Overlapping Non-Conformal Domain
  Decomposition
• Mortar Technique
• 1st Order Transmission Condition Between Domains
• Fourier Analysis
• Convergence Plots for Dirichlet-Neumann mapping
  and Robin Transmission Condition
• Numerical Results
• Vivaldi Arrays
• Monopole Antenna Arrays

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Implementing ABC in FEM
B.V.P.
1 Traditional Formulation
Imposing the natural boundary condition results in
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2 Alternative Formulation Introduce an
additional set of unknowns on the boundary
, and the BVP reads
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Galerkin Formulation of the Alternative BVP
Note the matrix is Symmetric and There is no zero
diagonal block!!!
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DDM with non-matching grids and Mortar Technique
And,
Do these two BVPs imply the needed Physics,
namely ?
YES, though weakly
Galerkin treatments result in
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Non-overlap and non-matching Grids DDMs with
Dual-Primal Unknowns

• Primal Unknowns (FEM variables) Ei, ei
• Dual Unknowns (Mortar variables) Ji
• Need T.C. (Transmission Conditions) between
  domains to enforce tangential continuities of
  electric and magnetic fields
• Proper T.C. leads to CONVERGING domain
  decomspotion method

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Usual Dirichlet-Neumann Mapping
NOT WORKING
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Works for Radition and Scattering Problems
1st Order Transmission Condition Robin-Robin
Mapping
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Conforming DDMs
At nth iteration, let us assume that
are available everywhere. Then,
and

• Gauss-Siedel iteration scheme is adopted, and
  therefore denote the most
  update values of the variables
• The two BVPs are well-posed. Existence and
  uniqueness are trivially established
• For simplicity, ABC is used for mesh truncation
  in the formulation. Other mesh truncation methods
  can be subsituted into the formulation with
  straightforward modifications.

Can be employed to update variables

• Common variable e is used to strongly enforced
  tangential component of electric field being
  continuous
• Tangential component of magnetic field continuous
  is imposed weakly through natural boundary
  condition in FEM formulation

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Boundary Value Problem for Single Building Block
?back
?left
?top
?bottom
?front
?right
non-conforming triangular meshes
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Non-conforming DDM and Mortar Technique
And,
Do these two BVPs imply the needed Physics,
namely ?
YES, though weakly
Galerkin treatments result in
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Boundary Value Problem for Two Domains (cont.)
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Boundary Value Problem for Two Domains (cont.)
Mortar Elements (non-Galerkin testing)
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DD Viewed as Operator Splitting
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Mortar DDM
where
Back (3)
Back (3)
Left (0)
Right (1)
Domain (i-1, j)
Left (0)
Right (1)
Domain (i1, j)
Front (2)
Front (2)
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Comparisons With brute-force FEBI
7?7 Vivaldi Array
10?10 Vivaldi Array
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Beam Scanning
15x10 vivaldi array with scanning angle ? 45,?
0
?
?
Scaning
? 45,? 0
? 0 plane
broadside
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50?50 Vivaldi Array-Beam Scanning
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Conclusions

• Mortar technique is a powerful machinary to to
  implement non-conforming domain decomposition
  method
• Using Robin transmission condition results in
  slower convergence for high frequency Fourier
  modes and fast rate of damping for low
  frequency Fourier modes.
• The MortarDDM can be further improved to model
  large electromagnetic problems by employing FETI
  algorithm

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Mesh truncation ABC, PML, BIE
Galerkin statement
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Notes
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Mortar Technique for DDM with Non-matching Grids

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Galerkin Finite Element Formulation
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Classical Schwarz Algorithm without Overlap
Classical Schwarz, Dirichlet-to-Neumann Mapping
for 2 Non-overlapping Domains

• Starts with arbitrary initial solutions
• Iterates until the relative changes between
  iteration are small ? Converged

Does NOT work.

• Overlap domains It works to damp the evanecent
  modes in the error but still not affecting the
  propagation modes
• Robin conditions It works for over-lapping
  domains but the evanecent modes in the error will
  not be damped if non-overlapping domains are used

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y
Fourier Analysis
TE modes hz component
N
D
x
Also, the radiation boundary conditions at x?
Take a Fourier transform in the y direction,
namely H(x, ky) components, then
Take into account the radiation B.C. at x?, the
solutions are
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Evaluate H at x0 employing the Dirichlet-Neumann
Mappings, we have
Define the convergence rate for each Fourier
component
Classical Schwarz Dirichlet-Neumann mapping has a
convergence rate of 1 for all evanescent and
propagating modes
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Fourier Analysis of 1st Order Transmission
Condition
Fourier Transform
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Applying Silver-Muller radiation condition in ?1
and ?2
TE
TM
TE and TM Fourier field representations on the
two subdomains
1st order generalized Robin Transmission Condition
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TE
TM
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DDM as an Operator Splitting Preconditioner
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Coarse discretization
Fine discretization
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Periodic Structures
i1M (Many repeated domains)
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Keeping only the variables on interface
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Recovering of the Primal Unknowns
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11?11 Jandayala Patch Array
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11?11 Jandayala Patch Array
E at Near Field (Top)
E at Near Field (Bottom)
Bistatic RCS (xz-plane)
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16?16 Dual Polarized Exponential Tapered
(Vivaldi) Horn Array
0.98 ?
7.89 ?
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16?16 Dual Polarized Exponential Tapered
(Vivaldi) Horn Array
RCS for q0º f0º Incidence
xz-plane RCS
E at Near Zone
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16?16 Dual Polarized Exponential Tapered
(Vivaldi) Horn Array
RCS for q30º f0º Incidence
xz-plane RCS
yz-plane RCS
E at Near Zone
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Complete Flow Chart of ODDMortarFETI Algorithm
Adaptive mesh refinement for every building block
Partitioning the problem domain
FETI using pMUS CG method
Multiple domains glued together through mortar
Construct the domains using the building blocks
Outer loop DDM Iteration using Gauss-Siedel or
Krylov methods
Engineering Info. Antenna Pattern, Impedances, Q
factor, etc
Recover Primal Variables for Each Domain from
Dual-Variable Solution
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DP-FETI vs. Direct DDM
27.23 Million Unknowns
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Comparison of MortarDDM and MortarDDMFETI
Solution time only
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Mortar DDM FETI
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Dual-Primal Finite Element Tearing and
Interconecting like Algorithm (DP-FETI)
Let
Pre-processing
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Large Finite Array
15x15
50x50
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18?18 Dual Polarized Exponential Tapered
(Vivaldi) Horn Array
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PBG Nanocavity
Small PBG laser nanocavity
Few mm at 200THz
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7-Layer PBG Nanocavity
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14-Layer PBG Nanocavity
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PBG Optical Channel Waveguidef170 THz
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ODD with Multiple Building Blocks
Partition 1 (9 blocks)
Partition 2 (3 blocks)
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Partitioning
Partition 1 (9 blocks)
Partition 2 (3 blocks)
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9?12 Monopole Array
Directivity D022.03 dbi (full FEM-PML) D022.02
dbi (DD-FEM)
Scan direction
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DDM 120 deg 23.46dBi
Azimuthal Gain by DDM
Elevation Gain by DDM
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Rotman Lens for Switch Beam Arrays
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RCS from Large Finite ArraysPatch Antenna Array
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Plane Wave Scattering from a dual-polarized Notch
Antenna Array (16x16)
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Radiation from a dual-polarized Notch Antenna
Array (16x16)
Broad Side
Broad Side
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Active Reflection Coeff. With Broad and 300 Scan
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Electromagnetic Bandgap (EBG) material
Artificial Magnetic Conductor