Topology Sensitivity in Method of Moments Miloslav ˇ Capek 1 , Luk´ ınek 1 , Mats Gustafsson 2 , and V´ y 1 aˇ s Jel´ ıt Losenick´ 1 Department of Electromagnetic Field, Czech Technical University in Prague, Czech Republic miloslav.capek@fel.cvut.cz 2 Department of Electrical and Information Technology, Lund University, Sweden April 2, 2019 the 13th European Conference on Antennas and Propagation Krak´ ow, 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 (Sub-)optimal solution of Q-factor minimization over triangularized grid, 753 dofs, GA+TS. This talk concerns: ◮ electric currents in vacuum, ◮ time-harmonic quantities, i.e. , A ( r , t ) = Re { A ( r ) exp (j ωt ) } . 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. Ω, E i � � p = L J ( r ) = A Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 3 / 20
Shape Synthesis Analysis × Synthesis ? Synthesis ( S ≡ A − 1 ) Analysis ( A ) ◮ Shape Ω is given, BCs are known, ◮ EM behavior is specified, neither Ω nor determine EM quantities. BCs are known. Ω, E i � Ω, E i � � = A − 1 p = S p p = L J ( r ) = A � Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 3 / 20
Shape Synthesis Shapes Known to Be Optimal 1 (In Certain Sense) (a) (b) A possible parametrization (unknowns: with of the strip, width of the gap). 1 M. 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 Optimal 1 (In Certain Sense) 1000 Q lb , TM rad 500 Q rad 200 p ≡ Q rad 100 50 20 10 (a) (b) 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 A possible parametrization (unknowns: ka with of the strip, width of the gap). Objective function p is Q-factor Q rad compared to its bound Q lb , TM . rad 1 M. 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 Ω, E i � � = A − 1 p = S p user ? How to get There are serious obstacles on the way: 1. Objectives p user cannot be set freely. Ω, E i � � 2. Solution reaching p user is non-unique. Ω, E i � � 3. If is not realized exactly, the effect on p user is potentially huge. 4. If p user is known only approximately, which is always the case, the corresponding solution for Ω, E i � � 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 Ω, E i � � = A − 1 p = S p user ? How to get There are serious obstacles on the way: 1. Objectives p user cannot be set freely. Ω, E i � � 2. Solution reaching p user is non-unique. Ω, E i � � 3. If is not realized exactly, the effect on p user is potentially huge. 4. If p user is known only approximately, which is always the case, the corresponding solution for Ω, E i � � 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 min I 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 min I Q , NSGA-II. A histogram of the best candidates found. ◮ Numerical oscillation (chessboard), ◮ “Gray” elements, ◮ sensitive to local minima. ◮ 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 ∈ C N × N , matrices A , { B i } , { B j } , a given excitation vector V ∈ C N , find a vector x such that I H A ( x ) I minimize I H B i ( x ) I = p i subject to I H B j ( x ) I ≤ p j 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 ∈ C N × N , matrices A , { B i } , { B j } , a given excitation vector V ∈ C N , find a vector x such that I H A ( x ) I minimize I H B i ( x ) I = p i subject to I H B j ( x ) I ≤ p j Z ( x ) I = V x ∈ { 0 , 1 } N ◮ Combinatorial optimization (suffers from curse of dimensionality, 2 N 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. . . Edge removal ◮ Each of N basis functions (dofs) is taken as { 0 , 1 } unknown (not present/present). 0 0 0 · · · 0 ψ 7 ◮ Feeding ( E i ) is specified at the beginning. 0 Y 22 Y 23 · · · Y 2 N ψ 2 ψ 6 0 Y 32 Y 33 · · · Y 2 N ◮ Fixed mesh grid Ω T : matrix operators ψ 1 ψ 3 ψ 9 calculated the only time. . . . . ... . . . . . . . . ψ 5 ψ 8 ◮ Woodbury identity employed: get rid of 0 Y N 2 Y N 3 · · · Y NN ψ 4 repetitive matrix inversion! 2 A basis function removal, I = Z − 1 V = YV . 2 R. 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 Ω T . Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 9 / 20
Topology Sensitivity: Derivation Initial Setup Initial shape Ω . A discretized region Ω T . Basis functions { ψ n } applied. Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 9 / 20
Topology Sensitivity: Derivation Utilization of Woodbury Formula � 1 � T 0 0 · · · 0 Z = Z G + Z L = Z G + C B R ∞ C T I = Z − 1 V = YV . B , C B = , 0 0 1 · · · 0 Miloslav ˇ Capek, et al. Topology Sensitivity in Method of Moments 10 / 20
Topology Sensitivity: Derivation Utilization of Woodbury Formula � T � 1 0 0 · · · 0 Z = Z G + Z L = Z G + C B R ∞ C T I = Z − 1 V = YV . B , C B = , 0 0 1 · · · 0 Sherman-Morrison-Woodbury formula 3 ( A + EBF ) − 1 = A − 1 − A − 1 E � − 1 FA − 1 B − 1 + FA − 1 E � 3 W. 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 � T � 1 0 0 · · · 0 Z = Z G + Z L = Z G + C B R ∞ C T I = Z − 1 V = YV . B , C B = , 0 0 1 · · · 0 Sherman-Morrison-Woodbury formula 3 ( A + EBF ) − 1 = A − 1 − A − 1 E � − 1 FA − 1 B − 1 + FA − 1 E � � 1 � − 1 Y = Z − 1 = Z − 1 G − Z − 1 1 D + C T B Z − 1 C T B Z − 1 G C B G C B G R ∞ 3 W. 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
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