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

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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

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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

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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.

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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.

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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

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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

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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

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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

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EV55 EVEKTOR (MORPHED) AIRCRAFT:

CONVERGENCE OF BICGSTAB SOLVER AT LOW AND VERY

LOW FREQUENCIES

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EV55 EVEKTOR (MORPHED) AIRCRAFT:

SURFACE CURRENT DENSITY

0.1 mHz

1 MHz

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NUMERICAL RESULTS:

ELECTRICALLY LARGE MULTI-SCALE STRUCTURE

15 15

No. of unknowns = 621,973

~30 l @ 150 MHz

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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

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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

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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: francesca.vipiana@polito.it w: http://www.polito.it

CONTACTS

www.ismb.it

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