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Optimal Currents and Shape Synthesis in Electromagnetism Part II Topology Sensitivity Miloslav Capek 1 1 Department of Electromagnetic Field, Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz January 16,

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SLIDE 1

Optimal Currents and Shape Synthesis in Electromagnetism

Part II – Topology Sensitivity Miloslav ˇ Capek1

1Department of Electromagnetic Field,

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

January 16, 2019 Departmental seminar Chalmers University of Technology

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 1 / 36

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SLIDE 2

Outline

  • 1. Shape Synthesis
  • 2. Discretization of a Model
  • 3. Shape Synthesis Techniques
  • 4. Topology Sensitivity: Motivation
  • 5. Topology Sensitivity: Derivation
  • 6. Topology Sensitivity: Examples
  • 7. Conversion to a Graph: Greedy Algorithm
  • 8. Concluding Remarks and Future Work

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

Topology sensitivity of a PIFA.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 2 / 36

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SLIDE 3

Shape Synthesis

Analysis × Synthesis

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 3 / 36

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SLIDE 4

Shape Synthesis

Analysis × Synthesis

Analysis (A) ◮ Shape Ω is given, BCs are known, determine EM quantities. g = L {J (r)} = Af f ≡

  • Ω, Ei

, g ≡ {p}

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 3 / 36

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SLIDE 5

Shape Synthesis

Analysis × Synthesis

Analysis (A) ◮ Shape Ω is given, BCs are known, determine EM quantities. g = L {J (r)} = Af ? Synthesis (S ≡ A−1) ◮ EM behavior is specified, neither Ω nor BCs are known. f = Sg = A−1g f ≡

  • Ω, Ei

, g ≡ {p}

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 3 / 36

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SLIDE 6

Shape Synthesis

Synthesis

How to get f = A−1g? Questions inherently related to synthesis are1 (f ≡

  • Ω, Ei

, g ≡ {pi})

  • 1. Can g be chosen arbitrary?
  • 2. If g is such that there exists a solution f, is that solution unique?
  • 3. If g is known only approximately, which is always the case, is the corresponding solution

for f close to the exact one?

  • 4. If f is not exactly realized what effect will this have on Af?
  • 1G. Deschamps and H. Cabayan, “Antenna synthesis and solution of inverse problems by regularization

methods,” IEEE Transactions on Antennas and Propagation, vol. 20, no. 3, pp. 268–274, 1972. doi: 10.1109/tap.1972.1140197. [Online]. Available: https://doi.org/10.1109/tap.1972.1140197

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 4 / 36

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SLIDE 7

Shape Synthesis

Synthesis

How to get f = A−1g? Questions inherently related to synthesis are1 (f ≡

  • Ω, Ei

, g ≡ {pi})

  • 1. Can g be chosen arbitrary? No.
  • 2. If g is such that there exists a solution f, is that solution unique? No.
  • 3. If g is known only approximately, which is always the case, is the corresponding solution

for f close to the exact one? No.

  • 4. If f is not exactly realized what effect will this have on Af? Potentially huge.

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

  • 1G. Deschamps and H. Cabayan, “Antenna synthesis and solution of inverse problems by regularization

methods,” IEEE Transactions on Antennas and Propagation, vol. 20, no. 3, pp. 268–274, 1972. doi: 10.1109/tap.1972.1140197. [Online]. Available: https://doi.org/10.1109/tap.1972.1140197

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 4 / 36

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SLIDE 8

Discretization of a Model

Discretization

Ω σ → ∞

(PEC) Original problem.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 5 / 36

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SLIDE 9

Discretization of a Model

Discretization

Ω σ → ∞

(PEC) Original problem.

Ω ǫ0, µ0

Equivalent problem.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 5 / 36

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SLIDE 10

Discretization of a Model

Discretization

Ω σ → ∞

(PEC) Original problem.

Ω ǫ0, µ0

Equivalent problem.

ΩT

Triangularized domain ΩT .

Structure Ω → ΩT , current density in vacuum J (r), r ∈ ΩT .

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 5 / 36

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SLIDE 11

Discretization of a Model

Operators Represented In RWG Basis Functions

Starting point in this work is a given discretization into T triangles ti, ΩT =

T

  • i=1

ti.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 6 / 36

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SLIDE 12

Discretization of a Model

Operators Represented In RWG Basis Functions

Starting point in this work is a given discretization into T triangles ti, ΩT =

T

  • i=1

ti. RWG basis functions {ψn (r)} are applied as J (r) ≈

N

  • n=1

Inψn (r) , where N is the number of all inner edges. P +

n

P −

n

ρ+

n

ρ−

n

A+

n

A−

n

ln T +

n

T −

n

O r y z x ψn (r) = ln 2A±

n

ρ±

n

RWG basis function ψn (r).

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 6 / 36

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SLIDE 13

Discretization of a Model

Operators Represented In RWG Basis Functions

Starting point in this work is a given discretization into T triangles ti, ΩT =

T

  • i=1

ti. RWG basis functions {ψn (r)} are applied as J (r) ≈

N

  • n=1

Inψn (r) , where N is the number of all inner edges. Matrix representation of the operators used J, AJ = [I∗

mψm, AψnIn] = IHAI.

P +

n

P −

n

ρ+

n

ρ−

n

A+

n

A−

n

ln T +

n

T −

n

O r y z x ψn (r) = ln 2A±

n

ρ±

n

RWG basis function ψn (r).

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 6 / 36

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SLIDE 14

Shape Synthesis Techniques

Shape Synthesis: Properties and Approaches

  • 1. Designers’ skill and knowledge.
  • 2. Parametric sweeps.
  • 3. Heuristic algorithms (global optimization).
  • 4. Topology optimization (local optimization).

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 7 / 36

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SLIDE 15

Shape Synthesis Techniques

Shape Synthesis: Properties and Approaches

  • 1. Designers’ skill and knowledge.

◮ Nonintuitive/complex design?

  • 2. Parametric sweeps.

◮ What parameters? How many?

  • 3. Heuristic algorithms (global optimization).

◮ Convergence. No-free-lunch. “Solution.”

  • 4. Topology optimization (local optimization).

◮ This talk. . . partly.

Optimal solution: ◮ Combination of all approaches.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 7 / 36

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SLIDE 16

Shape Synthesis Techniques

Topology Optimization

minimize f =

  • Ω

F (ρ (r)) dV subject to

  • Ω

ρ dV − V0 ≤ 0 ◮ min. compliance → max. stiffness ◮ solved within FEM ◮ mesh dependence ◮ instability (chess board)

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 8 / 36

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SLIDE 17

Shape Synthesis Techniques

Topology Optimization

minimize f =

  • Ω

F (ρ (r)) dV subject to

  • Ω

ρ dV − V0 ≤ 0 ◮ min. compliance → max. stiffness ◮ solved within FEM ◮ mesh dependence ◮ instability (chess board)

1216 × 3456 × 256 ≈ 1.1 · 109 unknowns, FEM2.

  • 2N. Aage, E. Andreassen, B. S. Lazarov, et al., “Giga-voxel computational morphogenesis for structural

design,” Nature, vol. 550, pp. 84–86, 2017. doi: 10.1038/nature23911

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 8 / 36

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SLIDE 18

Shape Synthesis Techniques

Topology Optimization in EM

State-of-the-art in mechanics, serious problems in EM3 ◮ “gray” elements, rounding yields different results, ◮ numerical oscillation (chessboard), ◮ more sensitive to local minima (current paths?), ◮ threshold function for MoM. Fundamental difference between EM vector field and stiffness in mechanics?

Histogram of the best candidates found for minI Q, NSGA-II.

  • 3S. Liu, Q. Wang, and R. Gao, “A topology optimization method for design of small GPR antennas,”
  • Struct. Multidisc. Optim., vol. 50, pp. 1165–1174, 2014. doi: 10.1007/s00158-014-1106-y

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 9 / 36

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SLIDE 19

Topology Sensitivity: Motivation

Topology Sensitivity

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

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 10 / 36

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SLIDE 20

Topology Sensitivity: Motivation

Topology Sensitivity

Idea behind this work Let us accept NP-hardness of the problem, do brute force, but do it cleverly. . . ◮ Inspired by pixeling4, but RWG functions are the unknowns (T vs. N unknowns). ◮ Fixed mesh grid ΩT : operators calculated once, results comparable with the bounds. ◮ Woodbury identity employed: get rid of repetitive matrix inversion! ◮ Feeding is specified at the beginning.

  • 4Y. Rahmat-Samii, J. M. Kovitz, and H. Rajagopalan, “Nature-inspired optimization techniques in

communication antenna design,” Proc. IEEE, vol. 100, no. 7, pp. 2132–2144, 2012. doi: 10.1109/JPROC.2012.2188489

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 10 / 36

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SLIDE 21

Topology Sensitivity: Motivation

Comparison of Pixeling Techniques

Pixel removal

T1 T2 T3 T4 T5 T6 T7 T8

Z11 Z12 Z13 · · · Z1N Z21 Z22 Z23 · · · Z2N Z31 Z32 Z33 · · · Z3N . . . . . . . . . ... . . . ZN1 ZN2 ZN3 · · · ZNN                       Classical pixeling removes metallic patches5.

(ZG + ZL) I = ZI = V

  • 5Y. Rahmat-Samii, J. M. Kovitz, and H. Rajagopalan, “Nature-inspired optimization techniques in

communication antenna design,” Proc. IEEE, vol. 100, no. 7, pp. 2132–2144, 2012. doi: 10.1109/JPROC.2012.2188489

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 11 / 36

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SLIDE 22

Topology Sensitivity: Motivation

Comparison of Pixeling Techniques

Pixel removal

T1 T2 T3 T4 T5 T6 T7 T8

Z11 Z12 Z13 · · · Z1N Z21 Z22 Z23 · · · Z2N Z31 Z32 Z33 · · · Z3N . . . . . . . . . ... . . . ZN1 ZN2 ZN3 · · · ZNN                       Classical pixeling removes metallic patches5.

(ZG + ZL) I = ZI = V

Edge removal

ψ1 ψ2 ψ3 ψ4 ψ5 ψ6 ψ7 ψ8 ψ9

· · · Y22 Y23 · · · Y2N Y32 Y33 · · · Y2N . . . . . . . . . ... . . . YN2 YN3 · · · YNN                       Proposed basis function removal.

I = Z−1V = YV

  • 5Y. Rahmat-Samii, J. M. Kovitz, and H. Rajagopalan, “Nature-inspired optimization techniques in

communication antenna design,” Proc. IEEE, vol. 100, no. 7, pp. 2132–2144, 2012. doi: 10.1109/JPROC.2012.2188489

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 11 / 36

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SLIDE 23

Topology Sensitivity: Motivation

Pixeling and Edge Removal

Comparison of the longest meander possible for classical pixeling and edge removal.

◮ “Infinitesimally” small perturbation of a structure ΩT is a removal of RWG edge6.

  • 6M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” , 2018,

submitted, arxiv: 1808.02479. [Online]. Available: https://arxiv.org/abs/1808.02479

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 12 / 36

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SLIDE 24

Topology Sensitivity: Motivation

Shape Synthesis: Rigorous Definition

For a given impedance matrix Z ∈ CN×N, matrices A, {Bi}, {Bj}, given excitation vector V ∈ CN, found 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 ◮ structure perturbation ◮ combinatorial optimization ◮ A =

  • x ∗ xT

⊗ A

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 13 / 36

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SLIDE 25

Topology Sensitivity: Motivation

Shape Synthesis: Rigorous Definition

For a given impedance matrix Z ∈ CN×N, matrices A, {Bi}, {Bj}, given excitation vector V ∈ CN, found 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 ◮ structure perturbation ◮ combinatorial optimization ◮ A =

  • x ∗ xT

⊗ A minimize IHA (x) I subject to IHBi (x) I = pi IHBj (x) I ≤ pj Z (x) I = V x ∈ [0, 1]N ◮ material perturbation ◮ relaxation of the combinatorial approach ◮ Aii = Aii + xiR0

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 13 / 36

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SLIDE 26

Topology Sensitivity: Derivation

Incorporation of Lumped Element R∞

(ZG + ZL) I = ZI = V

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 14 / 36

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SLIDE 27

Topology Sensitivity: Derivation

Incorporation of Lumped Element R∞

(ZG + ZL) I = ZI = V Lumped element with resistivity R∞ ZL,nn = R∞ ⇔ n ∈ B Example: B = {1, 3}

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 14 / 36

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SLIDE 28

Topology Sensitivity: Derivation

Incorporation of Lumped Element R∞

(ZG + ZL) I = ZI = V Lumped element with resistivity R∞ ZL,nn = R∞ ⇔ n ∈ B CB,nn = ⇔ n ∈ B 1 ⇔

  • therwise

(All columns containing only zeros are removed.) Example: B = {1, 3} CB = 1 · · · 1 · · · T

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 14 / 36

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SLIDE 29

Topology Sensitivity: Derivation

Incorporation of Lumped Element R∞

(ZG + ZL) I = ZI = V Lumped element with resistivity R∞ ZL,nn = R∞ ⇔ n ∈ B CB,nn = ⇔ n ∈ B 1 ⇔

  • therwise

(All columns containing only zeros are removed.) ZL = CBR∞CT

B,

Example: B = {1, 3} CB = 1 · · · 1 · · · T CBR∞CT

B =

       R∞ · · · · · · R∞ · · · . . . . . . . . . ... . . . · · ·       

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 14 / 36

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SLIDE 30

Topology Sensitivity: Derivation

Utilization of Woodbury Formula

Definitions Z = ZG + ZL = ZG + CBR∞CT

B,

I = Z−1V = YV.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 15 / 36

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SLIDE 31

Topology Sensitivity: Derivation

Utilization of Woodbury Formula

Definitions Z = ZG + ZL = ZG + CBR∞CT

B,

I = Z−1V = YV. Sherman-Morrison-Woodbury formula (A + EBF)−1 = A−1 − A−1E

  • B−1 + FA−1E

−1 FA−1

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 15 / 36

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SLIDE 32

Topology Sensitivity: Derivation

Utilization of Woodbury Formula

Definitions Z = ZG + ZL = ZG + CBR∞CT

B,

I = Z−1V = YV. Sherman-Morrison-Woodbury formula (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

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 15 / 36

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SLIDE 33

Topology Sensitivity: Derivation

Utilization of Woodbury Formula

Definitions Z = ZG + ZL = ZG + CBR∞CT

B,

I = Z−1V = YV. Sherman-Morrison-Woodbury formula (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.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 15 / 36

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SLIDE 34

Topology Sensitivity: Derivation

Simplification With Indexing Property of Matrix CB

Y = YG − YGCB

  • CT

BYGCB

−1CT

BYG.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 16 / 36

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SLIDE 35

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 Optimal Currents and Shape Synthesis in Electromagnetism 16 / 36

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SLIDE 36

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 Optimal Currents and Shape Synthesis in Electromagnetism 16 / 36

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SLIDE 37

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 Optimal Currents and Shape Synthesis in Electromagnetism 17 / 36

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SLIDE 38

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 Optimal Currents and Shape Synthesis in Electromagnetism 17 / 36

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SLIDE 39

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 Optimal Currents and Shape Synthesis in Electromagnetism 17 / 36

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SLIDE 40

Topology Sensitivity: Derivation

Topology Sensitivity

All potential removals at once: IfB =

  • If + ζf1I1

· · · If + ζfNIN

  • .

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 18 / 36

slide-41
SLIDE 41

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

  • .

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 18 / 36

slide-42
SLIDE 42

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

  • .

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

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 18 / 36

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SLIDE 43

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, of three

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 19 / 36

slide-44
SLIDE 44

Topology Sensitivity: Examples

Example: Thin-strip Dipole – Q-factor

1 2 3 4 5 6 7 8 1 2 3 4 5 6 A B

A B

kℓ Q/Qlb

center-fed dipole

  • ptimized shape

Radiation Q-factor of center-fed dipole Ωdip, discretized into N = 79 basis functions.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 20 / 36

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SLIDE 45

Conversion to a Graph: Greedy Algorithm

Greedy Step

A discretization establishes a graph. G (V, E) = G (P, E) → {Ti} → {ψn (r)}

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 21 / 36

slide-46
SLIDE 46

Conversion to a Graph: Greedy Algorithm

Graph Representation: Reduction to Tree

1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000

6 2

  • 6
  • 5

8

  • 9
  • 7

10 4

  • 1
  • 9

4

  • 9
  • 9
  • 8

6 6 4

  • 7

3 9 3 6

  • 1
  • 1

7

  • 8
  • 7

Synthesis for N = 4 as a directional binary tree.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 22 / 36

slide-47
SLIDE 47

Conversion to a Graph: Greedy Algorithm

Graph Representation: Reduction to Tree

1111 1110 1101 1100 1011 1010 1001 1000 0111 0110 0101 0100 0011 0010 0001 0000

6 2

  • 6
  • 5

8

  • 9
  • 7

10 4

  • 1
  • 9

4

  • 9
  • 9
  • 8

6 6 4

  • 7

3 9 3 6

  • 1
  • 1

7

  • 8
  • 7

Synthesis for N = 4 as a directional binary tree.

28 25 29 24 28 18 28 28 24 22 28 29 23 20 24 28

Greedy algorithm in directional graph.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 22 / 36

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SLIDE 48

Conversion to a Graph: Greedy Algorithm

Greedy Algorithm

One gradient-based search through the entire tree (the most pessimistic run): ◮ max (N − 1) series ◮ N (N − 1) (N − 2) · · · = N! evaluations Shermann-Morrison-Woodbury: N − n speed-up at every tree level Note of solvability of the problem Problem is not convex → combination of global and local algorithms.

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 23 / 36

slide-49
SLIDE 49

Conversion to a Graph: Greedy Algorithm

Greedy Algorithm – Example: Rectangular Plate

Miloslav ˇ Capek Optimal Currents and Shape Synthesis in Electromagnetism 24 / 36

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

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SLIDE 50

Conversion to a Graph: Greedy Algorithm

Compression of the Problem

50 100 150 200 250 300 350 20 40 60 80

p = ∞ 79.7 s p = 1 84.9 s p = 5 66.4 s p = 50 63.3 s dynamic 44.8 s

iteration i total comp. time, t(1 : i) [s]

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SLIDE 51

Conversion to a Graph: Greedy Algorithm

Number of Evaluated Antennas and Computational Time

plate (8 × 4) plate (14 × 7) sphere electrical size (ka) 0.5 0.5 0.5 basis functions (N) 180 567 900 number of iterations (I) 71 279 380 evaluated antennas 10332 119420 270129 realized Q/Qlb 1.57 1.45 1.51 edge removal (p = ∞) 0.30 s 23.5 s 79.7 s edge removal (p = 50) 0.28 s 19.4 63.6 s edge removal (p = 1) 0.43 s 23.3 s 84.9 s classical pixel removal 10 s 1437 s 10500 s

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SLIDE 52

Conversion to a Graph: Greedy Algorithm

Moving in the Solution Space, Part #1

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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SLIDE 53

Conversion to a Graph: Greedy Algorithm

Moving in the Solution Space, Part #1

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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SLIDE 54

Conversion to a Graph: Greedy Algorithm

Shape Reconstruction

Ω Ω− Ω+ Adding and removing DOF.

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SLIDE 55

Conversion to a Graph: Greedy Algorithm

Shape Reconstruction

◮ Basis functions can be added back (shape reconstruction). [IE∪B] = CE∪b   yf + xf1 z1 x1 · · · yf + xfb zb xb · · · −xf1 z1 · · · −xfb zb · · ·   lfV0 where xb = Yzb, zb = Zbb − zT

b xb.

Ω Ω− Ω+ Adding and removing DOF.

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SLIDE 56

Conversion to a Graph: Greedy Algorithm

Price to Pay for Reconstruction

100 101 102 103 10−3 10−2 10−1 100

  • topo. removal
  • topo. removal (all vectors)
  • topo. addition

M operations (matrix size 2 · 103) computational time

Comparison of basis function removal and addition.

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SLIDE 57

Conversion to a Graph: Greedy Algorithm

Moving in the Solution Space, Part #2

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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SLIDE 58

Conversion to a Graph: Greedy Algorithm

Moving in the Solution Space, Part #2

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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SLIDE 59

Conversion to a Graph: Greedy Algorithm

Moving in the Solution Space, Part #2

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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SLIDE 60

Conversion to a Graph: Greedy Algorithm

Nearest Neighbor (NN) Search

An initial sample of topology sensitivity investigation.

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SLIDE 61

Conversion to a Graph: Greedy Algorithm

Nearest Neighbor (NN) Search

An initial sample of topology sensitivity investigation. The final sample resulting from a (NN) search.

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SLIDE 62

Conversion to a Graph: Greedy Algorithm

Live demonstration in MATLAB.

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SLIDE 63

Concluding Remarks and Future Work

All Approaches to Synthesis at Once

Do not find an approximative solution of the exact model but, instead, find an exact solution of the approximate model.

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SLIDE 64

Concluding Remarks and Future Work

Reduction of the Complexity

Full grid of 21 × 11 pixels (N = 1354).

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SLIDE 65

Concluding Remarks and Future Work

Reduction of the Complexity

Truncated grid of 21 × 11 pixels (N = 954).

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SLIDE 66

Concluding Remarks and Future Work

Reduction of the Complexity

Truncated grid of 21 × 11 pixels with modified mesh (N = 115).

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SLIDE 67

Concluding Remarks and Future Work

Concluding Remarks

What has been done7. . . ◮ Inversion-free structure perturbation (removal/addition). ◮ Evaluation of topology sensitivity, greedy algorithm. ◮ Vectorization and parallelization friendly algebraic derivation.

  • 7M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” , 2018,

submitted, arxiv: 1808.02479. [Online]. Available: https://arxiv.org/abs/1808.02479

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SLIDE 68

Concluding Remarks and Future Work

Concluding Remarks

What has been done7. . . ◮ Inversion-free structure perturbation (removal/addition). ◮ Evaluation of topology sensitivity, greedy algorithm. ◮ Vectorization and parallelization friendly algebraic derivation. Topics of ongoing research ◮ Analysis of existing designs – can they be improved? ◮ Add topology sensitivity into heuristic optimization as a local step. ◮ Utilization for “data mining” (machine learning). ◮ Further study of graph representation and formal synthesis problem. ◮ Admittance matrix pivots (big data, graph clustering).

  • 7M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” , 2018,

submitted, arxiv: 1808.02479. [Online]. Available: https://arxiv.org/abs/1808.02479

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SLIDE 69

Questions

Questions?

Miloslav ˇ Capek miloslav.capek@fel.cvut.cz January 16, 2019 version 1.1 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.

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SLIDE 70

Questions

Moving in the Solution Space

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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SLIDE 71

Questions

Moving in the Solution Space

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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SLIDE 72

Questions

Moving in the Solution Space

11 · · · 11 11 · · · 00 00 · · · 00 00 · · · 11

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SLIDE 73

Questions

Synthesis – Generalized Framework

Complete and general description of synthesis. Desired quantity: ˆ I (source current), given quantity: YΩ (source region). ˆ I =

  • 1 − YGCB
  • Z−1

L

+ CT

BYGCB

−1 CT

B

  • YGCFvV0

ˆ I = (1 − P) YΩV YΩ initial system to be optimized V excitation (external/boundary condition) I solution to original (arbitrarily shaped) structure Ω P (any) modification of the initial (arbitrarily shaped) structure Ω

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SLIDE 74

Questions

Computational Complexity

Characterization of the synthesis problem

Number of inner edges N Levels of the tree N + 1 Total number of solutions 2N Number of connections down N − n Number of connections up n Number of nodes at the n-th level N! n! (N − n)! = N n

  • Number of connections down from the n-th level

n N n

  • Miloslav ˇ

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