Topology Sensitivity in Method of Moments Miloslav Capek 1 , Luk - - PowerPoint PPT Presentation

Topology Sensitivity in Method of Moments

Miloslav ˇ Capek1, Luk´ aˇ s Jel´ ınek1, Mats Gustafsson2, and V´ ıt Losenick´ y1

1Department of Electromagnetic Field,

Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz

2Department of Electrical and Information Technology,

Lund University, Sweden

April 2, 2019 the 13th European Conference on Antennas and Propagation Krak´

• w, Poland

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 1 / 20

Outline

• 1. Shape Synthesis
• 2. Topology Sensitivity: Derivation
• 3. Topology Sensitivity: Examples
• 4. Shape Reconstruction
• 5. Computational Time – Comparison
• 6. Concluding Remarks

This talk concerns: ◮ electric currents in vacuum, ◮ time-harmonic quantities, i.e., A (r, t) = Re {A (r) exp (jωt)}.

(Sub-)optimal solution of Q-factor minimization

• ver triangularized grid, 753 dofs, GA+TS.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 2 / 20

Shape Synthesis

Analysis × Synthesis

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 3 / 20

Shape Synthesis

Analysis × Synthesis

Analysis (A) ◮ Shape Ω is given, BCs are known, determine EM quantities. p = LJ (r) = A

• Ω, Ei

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 3 / 20

Shape Synthesis

Analysis × Synthesis

Analysis (A) ◮ Shape Ω is given, BCs are known, determine EM quantities. p = LJ (r) = A

• Ω, Ei

? Synthesis (S ≡ A−1) ◮ EM behavior is specified, neither Ω nor BCs are known.

• Ω, Ei

= A−1p = Sp

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 3 / 20

Shape Synthesis

Shapes Known to Be Optimal1 (In Certain Sense)

(a) (b) A possible parametrization (unknowns: with of the strip, width of the gap).

• 1M. Capek, L. Jelinek, K. Schab, et al., “Optimal planar electric dipole antenna,” , 2018, submitted,

arXiv:1808.10755. [Online]. Available: https://arxiv.org/abs/1808.10755

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 4 / 20

Shape Synthesis

Shapes Known to Be Optimal1 (In Certain Sense)

(a) (b) A possible parametrization (unknowns: with of the strip, width of the gap).

0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 20 50 100 200 500 1000 ka p ≡ Qrad Qlb,TM

rad

Qrad Objective function p is Q-factor Qrad compared to its bound Qlb,TM

rad

.

• 1M. Capek, L. Jelinek, K. Schab, et al., “Optimal planar electric dipole antenna,” , 2018, submitted,

arXiv:1808.10755. [Online]. Available: https://arxiv.org/abs/1808.10755

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 4 / 20

Shape Synthesis

Synthesis – Difficulties

How to get

• Ω, Ei

= A−1p = Spuser? There are serious obstacles on the way:

• 1. Objectives puser cannot be set freely.
• 2. Solution
• Ω, Ei

reaching puser is non-unique.

• 3. If
• Ω, Ei

is not realized exactly, the effect on puser is potentially huge.

• 4. If puser is known only approximately, which is

always the case, the corresponding solution for

• Ω, Ei

is not necessarily close to the exact one.

A discretized meanderline antenna “M1” from [Best, IEEE APM 2015].

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 5 / 20

Shape Synthesis

Synthesis – Difficulties

How to get

• Ω, Ei

= A−1p = Spuser? There are serious obstacles on the way:

• 1. Objectives puser cannot be set freely.
• 2. Solution
• Ω, Ei

reaching puser is non-unique.

• 3. If
• Ω, Ei

is not realized exactly, the effect on puser is potentially huge.

• 4. If puser is known only approximately, which is

always the case, the corresponding solution for

• Ω, Ei

is not necessarily close to the exact one.

A discretized meanderline antenna “M1” from [Best, IEEE APM 2015].

Generally, infinitely many possibilities and local minima → need for shape discretization.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 5 / 20

Shape Synthesis

Topology Optimization in EM

A particular solution found for minI Q, NSGA-II. A histogram of the best candidates found.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 6 / 20

Shape Synthesis

Topology Optimization in EM

A particular solution found for minI Q, NSGA-II.

◮ Numerical oscillation (chessboard), ◮ sensitive to local minima.

A histogram of the best candidates found.

◮ “Gray” elements, ◮ threshold function for MoM.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 6 / 20

Shape Synthesis

Shape Synthesis: Rigorous Definition

For a given impedance matrix Z ∈ CN×N, matrices A, {Bi}, {Bj}, a given excitation vector V ∈ CN, find a vector x such that minimize IHA (x) I subject to IHBi (x) I = pi IHBj (x) I ≤ pj Z (x) I = V x ∈ {0, 1}N

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 7 / 20

Shape Synthesis

Shape Synthesis: Rigorous Definition

For a given impedance matrix Z ∈ CN×N, matrices A, {Bi}, {Bj}, a given excitation vector V ∈ CN, find a vector x such that minimize IHA (x) I subject to IHBi (x) I = pi IHBj (x) I ≤ pj Z (x) I = V x ∈ {0, 1}N ◮ Combinatorial optimization (suffers from curse of dimensionality, 2N possible solutions), ◮ vector x serves as a characteristic function (structure perturbation), ◮ there is no “good” algorithm (working in polynomial time).

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 7 / 20

Shape Synthesis

Shape Synthesis: Combinatorial Optimization Approach

Idea behind this work Let us accept NP-hardness of the problem, do brute force, but do it cleverly. . .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 8 / 20

Shape Synthesis

Shape Synthesis: Combinatorial Optimization Approach

Idea behind this work Let us accept NP-hardness of the problem, do brute force, but do it cleverly. . . ◮ Each of N basis functions (dofs) is taken as {0, 1} unknown (not present/present). ◮ Feeding (Ei) is specified at the beginning. ◮ Fixed mesh grid ΩT : matrix operators calculated the only time. ◮ Woodbury identity employed: get rid of repetitive matrix inversion!2

Edge removal

ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9

· · · Y22 Y23 · · · Y2N Y32 Y33 · · · Y2N . . . . . . . . . ... . . . YN2 YN3 · · · YNN                       A basis function removal, I = Z−1V = YV.

• 2R. Kastner, “An “add-on” method for the analysis of scattering from large planar heterostructures,” IEEE
• Trans. Antennas Propag, vol. 37, no. 3, pp. 353–361, 1989. doi: 10.1109/8.18732

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 8 / 20

Topology Sensitivity: Derivation

Initial Setup

Initial shape Ω.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 9 / 20

Topology Sensitivity: Derivation

Initial Setup

Initial shape Ω. A discretized region ΩT .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 9 / 20

Topology Sensitivity: Derivation

Initial Setup

Initial shape Ω. A discretized region ΩT . Basis functions {ψn} applied.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 9 / 20

Topology Sensitivity: Derivation

Utilization of Woodbury Formula

Z = ZG + ZL = ZG + CBR∞CT

B,

CB = 1 · · · 1 · · · T , I = Z−1V = YV.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 10 / 20

Topology Sensitivity: Derivation

Utilization of Woodbury Formula

Z = ZG + ZL = ZG + CBR∞CT

B,

CB =

• 1

· · · 1 · · · T , I = Z−1V = YV. Sherman-Morrison-Woodbury formula3 (A + EBF)−1 = A−1 − A−1E

• B−1 + FA−1E

−1 FA−1

• 3W. W. Hager, “Updating the inverse of a matrix,” SIAM Review, vol. 31, pp. 221–239, 1989. doi:

10.1137/1031049

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 10 / 20

Topology Sensitivity: Derivation

Utilization of Woodbury Formula

Z = ZG + ZL = ZG + CBR∞CT

B,

CB =

• 1

· · · 1 · · · T , I = Z−1V = YV. Sherman-Morrison-Woodbury formula3 (A + EBF)−1 = A−1 − A−1E

• B−1 + FA−1E

−1 FA−1 Y = Z−1 = Z−1

G − Z−1 G CB

1 R∞ 1D + CT

BZ−1 G CB

−1 CT

BZ−1 G

• 3W. W. Hager, “Updating the inverse of a matrix,” SIAM Review, vol. 31, pp. 221–239, 1989. doi:

10.1137/1031049

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 10 / 20

Topology Sensitivity: Derivation

Utilization of Woodbury Formula

Z = ZG + ZL = ZG + CBR∞CT

B,

CB =

• 1

· · · 1 · · · T , I = Z−1V = YV. Sherman-Morrison-Woodbury formula3 (A + EBF)−1 = A−1 − A−1E

• B−1 + FA−1E

−1 FA−1 Y = Z−1 = Z−1

G − Z−1 G CB

1 R∞ 1D + CT

BZ−1 G CB

−1 CT

BZ−1 G

For Z−1

G = YG and R∞ → ∞

Y = YG − YGCB

• CT

BYGCB

−1 CT

BYG.

• 3W. W. Hager, “Updating the inverse of a matrix,” SIAM Review, vol. 31, pp. 221–239, 1989. doi:

10.1137/1031049

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 10 / 20

Topology Sensitivity: Derivation

Simplification With Indexing Property of Matrix CB

Y = YG − YGCB

• CT

BYGCB

−1CT

BYG.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 11 / 20

Topology Sensitivity: Derivation

Simplification With Indexing Property of Matrix CB

Y = YG − YGCB

• CT

BYGCB

−1CT

BYG.

For one (n-th) edge removed, |B| = 1: YGCB = yG,n,

• CT

BYGCB

−1 = 1 Ynn , CT

BYG = yT G,n.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 11 / 20

Topology Sensitivity: Derivation

Simplification With Indexing Property of Matrix CB

Y = YG − YGCB

• CT

BYGCB

−1CT

BYG.

For one (n-th) edge removed, |B| = 1: YGCB = yG,n,

• CT

BYGCB

−1 = 1 Ynn , CT

BYG = yT G,n.

Notice CB is indexing matrix (MATLAB) only. . . Y = YG − yG,nyT

G,n

Ynn .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 11 / 20

Topology Sensitivity: Derivation

Incorporation of Fixed and Localized Feeding

Imagine further, that only one (f-th) edge is fed Vf = V0

• · · ·

lf · · · T .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 12 / 20

Topology Sensitivity: Derivation

Incorporation of Fixed and Localized Feeding

Imagine further, that only one (f-th) edge is fed Vf = V0

• · · ·

lf · · · T . Ifn =

• YG −

yG,nyT

G,n

Ynn

• Vf = · · · = If −

lfln l2

n

Yfn Ynn

• V0lnyG,n = If + ζfnIn,

with If = YGVf and ζij = −lilj l2

j

Yij Yjj = −Zin,jj Zin,ij .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 12 / 20

Topology Sensitivity: Derivation

Incorporation of Fixed and Localized Feeding

Imagine further, that only one (f-th) edge is fed Vf = V0

• · · ·

lf · · · T . Ifn =

• YG −

yG,nyT

G,n

Ynn

• Vf = · · · = If −

lfln l2

n

Yfn Ynn

• V0lnyG,n = If + ζfnIn,

with If = YGVf and ζij = −lilj l2

j

Yij Yjj = −Zin,jj Zin,ij . This is equivalent to a specific two-port feeding V = V0

• . . .

lf . . . ζfnln . . . T .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 12 / 20

Topology Sensitivity: Derivation

Topology Sensitivity

All potential removals at once: IfB =

• If + ζf1I1

· · · If + ζfNIN

• .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 13 / 20

Topology Sensitivity: Derivation

Topology Sensitivity

All potential removals at once: IfB =

• If + ζf1I1

· · · If + ζfNIN

• .

An antenna observable defined as quadratic form x (I) = IHAI IHBI. is calculated with a Hadamard product (vectorization) x (IfB) = diag

• IH

fBAIfB

• ⊘ diag
• IH

fBBIfB

• ≡
• I∗

fB ⊗ (AIfB)

• ⊘
• I∗

fB ⊗ (BIfB)

• .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 13 / 20

Topology Sensitivity: Derivation

Topology Sensitivity

All potential removals at once: IfB =

• If + ζf1I1

· · · If + ζfNIN

• .

An antenna observable defined as quadratic form x (I) = IHAI IHBI. is calculated with a Hadamard product (vectorization) x (IfB) = diag

• IH

fBAIfB

• ⊘ diag
• IH

fBBIfB

• ≡
• I∗

fB ⊗ (AIfB)

• ⊘
• I∗

fB ⊗ (BIfB)

• .

Finally, topology sensitivity is defined here as τ fB (x, ΩT ) = x (IfB) − x (If) ≈ ∇x (If) .

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 13 / 20

Topology Sensitivity: Examples

Example: Thin-strip Dipole – Input Reactance

−0.4 −0.2 0.2 0.4 500 1000 ξ/ℓ τfS (|Xin|, Ωdip)

kℓ = 3π/4 kℓ = π kℓ = 3π/2 Topology sensitivity τ fB (|Xin|) of a center-fed dipole Ωdip, discretized into N = 79 basis functions.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 14 / 20

Shape Reconstruction

Shape Reconstruction

Basis functions can be added back (shape reconstruction).

Ω Ω− Ω+ Adding and removing DOF.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 15 / 20

Shape Reconstruction

Shape Reconstruction

Basis functions can be added back (shape reconstruction).

Ω Ω− Ω+ Adding and removing DOF.

◮ Current vector update4:

• IfB+
• = CT

B+

• · · · ,
• yf
• + xfb

zb

• xb

−1

• ,

· · ·

• lfV0,

where xb = Yzb, zb = Zbb − zT

b xb.

◮ Matrix update: YE∪b = 1 zb CT

B+

• zbY + xbxT

b

−xb −xT

b

1

• CB+.
• 4M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of

moments,” , 2019, submitted, arxiv: 1902.05975. [Online]. Available: https://arxiv.org/abs/1902.05975

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 15 / 20

Shape Reconstruction

Greedy Algorithm – Example: Rectangular Plate

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 16 / 20

4 4 14 1 19 5 8 9 26 1 1 13 14 37 1 1 17 18 43 1 18 17 37 1 14 13 26 1 1 9 8 14 1 1 4 4 5 19 1 9 1 30 10 −1 3 −10 −4 2 −1 −1 −5 −1 −1 −3 −1 −1 −10 −3 −1 −1 −1 2 −4 3 9 10 30 1 1 11 26 11 1 −1 −1 3 2 5 −4 9 −1 1 5 −13 −8 −1164 −854 −937 11 −8 −13 9 11 −937 −854 −4 5 −1 5 1 −1 1 2 3 −1 11 1 11 26 2 7 27 6 7 5 −1 85 22 23 5 −4 186 73 74 −478 −18 192 198 −18 −478 186 198 192 −4 5 85 74 73 −1 5 27 23 22 7 2 7 6

Shape Reconstruction Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 17 / 20

i = 0, Q/QTM

lb

= 119.35

Computational Time – Comparison

Number of Evaluated Antennas and Computational Time

plate 4 × 8 6 × 12 8 × 16 dofs, N 180 414 744 runs, I 5 · 104 5 · 104 1 · 103

• comp. time, T [s]

2.4 · 103 5.8 · 104 1.2 · 104 evaluated shapes 7.2 · 108 3.9 · 109 2.6 · 108 shapes per second 3 · 105 7 · 104 2 · 104 Qmin/QTM

lb

1.18 1.12 1.11

Computer: CPU Threadripper 1950 (3.4 GHz), 128 GB RAM.

◮ Algorithm fully parallelized, 16 physical cores used (almost linear speed-up).

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 18 / 20

Concluding Remarks

Concluding Remarks

What has been done5. . . ◮ Deterministic inversion-free structure perturbation (removal/addition). ◮ Evaluation of topology sensitivity, i.e., consequence of removing one dof. ◮ Vectorization and parallelization possible since friendly algebraic derivation.

• 5M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” , 2019, to

appear in IEEE Trans. AP. [Online]. Available: https://arxiv.org/abs/1808.02479

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 19 / 20

Concluding Remarks

Concluding Remarks

What has been done5. . . ◮ Deterministic inversion-free structure perturbation (removal/addition). ◮ Evaluation of topology sensitivity, i.e., consequence of removing one dof. ◮ Vectorization and parallelization possible since friendly algebraic derivation. Topics of ongoing research ◮ Greedy search over all nearest neighbors. ◮ Incorporation of heuristic algorithms (local & global steps). ◮ Adjustments of actual designs, sub-structure optimization. ◮ Further study of graph representation and properties of synthesis.

• 5M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” , 2019, to

appear in IEEE Trans. AP. [Online]. Available: https://arxiv.org/abs/1808.02479

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 19 / 20

Questions

Questions?

Miloslav ˇ Capek miloslav.capek@fel.cvut.cz April 2, 2019 version 1.0 The presentation is available at

◮ capek.elmag.org Acknowledgment: This work was supported by the Ministry of Education, Youth and Sports through the project CZ.02.2.69/0.0/0.0/16 027/0008465 and by the Czech Science Foundation under project No. 19-06049S.

Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 20 / 20