multiresolution preconditioner · equation (efie) and the method of moments (mom) • junction...
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MULTIRESOLUTION
PRECONDITIONER
Francesca Vipiana–March 2015
March 2015 2 Copyright © 2015 ISMB
OUTLINE
• Background and motivations
• Introduction to the Multi-Resolution (MR) basis functions
• Computational complexity
• Numerical results and properties of MR-MoM system
• Conclusions and perspectives
March 2015 3 Copyright © 2015 ISMB
BACKGROUND
• Analysis of a fully arbitrary 3D conductor (PEC) via the Electric Field Integral
Equation (EFIE) and the Method of Moments (MoM)
• Junction basis functions(3) for
modeling connections between
wires and surfaces
• Piecewise linear (PWL)
functions(2) for wires modeled
via line segments
• Rao-Wilton-Glisson (RWG)
functions(1) for surfaces
modeled with (flat) triangles
(1) S.M.Rao, D.R.Wilton, A.W.Glisson, “Electromagnetic Scattering by Surface of Arbitrary Shape”, IEEE Trans. Antennas Propagation, Vol.30, No.3, pp.409-418, May 1982. (2) C. M. Butler and D. R. Wilton, “Analysis of various numerical techniques applied to thin wire scatterers,” IEEE Trans. Antennas Propagation, Vol. 23, pp. 534–540, July 1975. (3) S. U. Hwu, D. R. Wilton, and S. M. Rao, “Electromagnetic scattering and radiation by arbitrary conducting wire/surface configurations”, Proc. IEEE Int. Symp. Antennas and Propagation, vol. 26, pp. 890–893, June 1988
March 2015 4 Copyright © 2015 ISMB
MOTIVATIONS AND PROBLEMS OF INTEREST:
MULTI-SCALE STRUCTURES
• Electrically large structures with fine details
• Structures modeled by surfaces and wires
• Dense and non-uniform meshes
• Disparate mesh cell sizes
Typical examples:
• antenna placement on complex platforms
• circuits and packaging problems
• scattering from complex structures
Very high conditioning of the MoM matrix
and slow convergence speed of iterative solvers
using standard basis functions
March 2015 5 Copyright © 2015 ISMB
MULTI-RESOLUTION (MR) BASIS FUNCTIONS:
BUILDING BLOCKS
• Multi-level cell grouping scheme
• Generalized cells
• Generalized basis functions
• Multi-resolution (MR) non-solenoidal and solenoidal basis functions
Selected references:
• M. A. Echeverri Bautista, M. A. Francavilla, F. Vipiana, G. Vecchi, “A Hierarchical Fast Solver for EFIE-MoM Analysis of Multiscale Structures at Very Low Frequencies”, IEEE Trans. on Antennas and Propagation, Vol. 62, No. 03, March 2014, pp. 1523-1528.
• M. A. Francavilla, F. Vipiana, G. Vecchi, D. R. Wilton, "Hierarchical Fast MoM Solver for the Modeling of Large Multi-Scale Wire-Surface Structures," IEEE Ant. and Wireless Propag. Letters, Vol. 11, 2012, pp. 1378-1381.
• F. Vipiana, M. A. Francavilla, G. Vecchi, “EFIE Modeling of High-Definition Multi-Scale Structures”, IEEE Transactions on Antennas and Propagation, Vol. 58, no. 7, July 2010, pp. 2362-2374.
• F. Vipiana, G. Vecchi, D. R. Wilton, “A Multi-Resolution Moment Method for Wire-Surface Objects”, IEEE Transactions on Antennas and Propagation, Vol. 58, No. 5, May 2010, pp. 1807-1813.
• F. Vipiana, G. Vecchi, “A novel, symmetrical solenoidal basis for the MoM analysis of closed surfaces”, IEEE Transactions on Antennas and Propagation, Vol. 57, No. 4, Apr. 2009, pp. 1294-1299.
• F. Vipiana, F. P. Andriulli, G. Vecchi, "Two-tier non-simplex grid hierarchic basis for general 3D meshes", Waves in Random and Complex Media, Vol. 19, No. 1, Feb. 2009, pp. 126-146.
• F.P. Andriulli, F. Vipiana, G. Vecchi, “Hierarchical bases for non-hierarchic 3D triangular meshes”, IEEE Transactions on Antennas and Propagation, Vol. 56, No. 8, Part 1, Aug. 2008, pp. 2288-2297.
March 2015 6 Copyright © 2015 ISMB
MULTI-LEVEL CELL GROUPING SCHEME
Level 1 Level 2
Level 3
Level 4 Level 5 (last)
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LEVEL-J GENERALIZED CELLS
Example of a pair of adjacent level-j GENERALIZED cells
1
2
surface part (triangles) wire part (segments)
contact boundaries:
polygonal lines &
points
F. Vipiana, G. Vecchi, D. R. Wilton, “A Multi-Resolution Moment Method for Wire-Surface Objects”, IEEE Trans. on Antennas and Propagation, Vol. 58, No. 5, May 2010.
March 2015 8 Copyright © 2015 ISMB
LEVEL-J GENERALIZED FUNCTIONS
Charge (scalar) function level j generalized function (g-fnc)
Each level j
generalized function
The level j-1
generalized functions
The level 0 standard
functions (initial mesh)
linear combination linear combination
,
1 j
mm
m
n n
jj fff rr ,
0,0 jj
m n
m
n mfff r r
March 2015 9 Copyright © 2015 ISMB
HIERARCHICAL NON-SOLENOIDAL FUNCTIONS
-
-
- +
+ 0
- 0
NO current
-
+ 0 - + +
+
+ Charge (scalar)
functions:
Level j (vector)
non-solenoidal
functions:
For each level j mesh, on each level j+1 cell: level j+1 cell
F.P. Andriulli, F. Vipiana, G. Vecchi, “Hierarchical bases for non-hierarchic 3D triangular meshes”, IEEE Transactions on Antennas and Propagation, Vol. 56, No. 8, Part 1, Aug. 2008, pp. 2288-2297.
Each level j non-solenoidal function is a linear combination of the level-j generalized functions
March 2015 10 Copyright © 2015 ISMB
HIERARCHICAL SOLENOIDAL FUNCTIONS
For each level j mesh,
inside each pair of adjacent level (j+1) cells,
the level j solenoidal functions
(Local Singular Value Loops) are generated
F. Vipiana, G. Vecchi, “A novel, symmetrical solenoidal basis for the MoM analysis of closed surfaces”, IEEE Transactions on Antennas and Propagation, Vol. 57, No. 4, Apr. 2009, pp. 1294-1299.
Each level j solenoidal function is a linear combination
of the level j generalized functions
March 2015 11 Copyright © 2015 ISMB
COMPUTATIONAL COMPLEXITY: NLOGN
Intel Core Duo COU T7250 2GHz – RAM 2GB
[T] = change-of-basis matrix from conventional to MR basis
infinite ground plane
NlogN NlogN
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NUMERICAL RESULTS: EV55 EVEKTOR (MORPHED)
AIRCRAFT
No. of unknowns: 172.113
(171.458 RWG + 647 PWL + 8 junctions)
Excitation: incident plane wave
Iterative solver: BiCGStab (tolerance = 10-4)
~18.5 m
cable network: wire ends attached to
the closest PEC surface
March 2015 13 Copyright © 2015 ISMB
EV55 EVEKTOR (MORPHED) AIRCRAFT:
CONVERGENCE OF BICGSTAB SOLVER AT LOW AND VERY
LOW FREQUENCIES
March 2015 14 Copyright © 2015 ISMB
EV55 EVEKTOR (MORPHED) AIRCRAFT:
SURFACE CURRENT DENSITY
0.1 mHz
1 MHz
March 2015 15 Copyright © 2015 ISMB
NUMERICAL RESULTS:
ELECTRICALLY LARGE MULTI-SCALE STRUCTURE
15 15
No. of unknowns = 621,973
~30 l @ 150 MHz
March 2015 16 Copyright © 2015 ISMB
NUMERICAL RESULTS:
ELECTRICALLY LARGE MULTI-SCALE STRUCTURE
• No. of unknowns = 621,973
• frequency = 150 MHz
• Iterative solver: BiCGStab
• MoM fast-solver: GIFFT(1) (complexity O(N1.5))
• 64-bits workstation DELL Precision T7400
Intel Xeon CPU E5440 @ 2.83GHz
32GB of RAM one-core (double precision)
(1) B. J. Fasenfest, F. Capolino, D. R. Wilton, D. R. Jackson, and N. J. Champagne, “A fast MoM solution for large arrays: Green’s function interpolation with FFT,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 161–164, 2004.
Other commercial code MLFMA + Sparse Approximate
Inverse (SPAI)
MR approach
Final residual 1.6∙10-3 7.0∙10-5
Iteration count 3000 198
Iteration time 78 s 42 s
Solution time 64h 46m 4h 47m
Total run time ~ 6 days ~ 15 h
Memory occupation
Zstrong (0.2l) matrix 15.8 GB
Precond. matrix 1.2 GB
GIFFT matrices 1.0 GB
Total RAM Memory 18.0 GB
March 2015 17 Copyright © 2015 ISMB
CONCLUSIONS
• EM tool to analyze 3-D arbitrary shaped multi-scale wire-surface objects
• Structures discretized with strong non-uniform meshes (fine geometrical
details)
• Systematic multi-scale representation of the solution
• Different preconditioning of different scales
• Low and stable condition number of the resulting MR-MoM matrix
• Fast convergence of iterative solvers
• Stable and fast solution at (very) low frequencies
• Broad-band analysis: same mesh for a wide frequency sweep
• The MR basis is a linear combination of the MoM conventional bases
• Easy interfaced with (fast-)MoM codes, purely multiplicative preconditioner
March 2015 18 Copyright © 2015 ISMB
PERSPECTIVES
• Application of the MR preconditioner to PEC
and (finite) dielectric objects
• Generation of the MR basis functions in the
case of Discontinous Galerkin discretizations
(non conformal meshing) [*]
• Combination of the MR approach with Domain
Decomposition techniques [**] (e.g. in the
solution of the sub-problems)
• Extension of the MR basis function to
curvilinear meshing.
• Integrated Circuit Package EM modeling
[*]Z. Peng, K. H. Lim, and J. F. Lee, “A discontinuous galerkin surface integral equation method for electromagnetic wave scattering from nonpenetrable targets,” IEEE Trans. Antennas Propag., vol. 61, pp. 3617–3628, July 2013. [**]Z. Peng, X. C. Wang, and J. F. Lee, “Integral equation based domain decomposition method for solving electromagnetic wave scattering from non-penetrable objects,” IEEE Trans. Antennas Propag., vol. 59, pp. 3328–3338, Sept. 2011.
Prof. Francesca Vipiana, PhD Antenna and EMC Lab (LACE) Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino (TO), Italy
e: [email protected] w: http://www.polito.it
CONTACTS
www.ismb.it
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