Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments Miloslav ˇ Capek 1 , Luk´ ınek 1 , Mats Gustafsson 2 aˇ s Jel´ 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 July 10, 2019 AP-S/URSI 2019 Atlanta, GA, US Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 1 / 19
Outline 1. Problem Parametrization 2. Inversion-free Solution of Linear System 3. Graph Representation 4. Monte Carlo Analysis (Q-factor Optimization) 5. Heuristically Restarted Topology Sensitivity 6. Concluding Remarks (Sub-)optimal solution of Q-factor minimization over triangularized grid, 753 DOF. 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. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 2 / 19
Problem Parametrization Degrees of Freedom Ω Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 3 / 19
Problem Parametrization Degrees of Freedom Ω → { T t } Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 3 / 19
Problem Parametrization Degrees of Freedom Ω → { T t } → { ψ n ( r ) } Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 3 / 19
Problem Parametrization Degrees of Freedom Ω → { T t } → { ψ n ( r ) } → g ◮ g ∈ { 0 , 1 } N × 1 is characteristic vector (discretized characteristic function) Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 3 / 19
Problem Parametrization Shape Optimization Capability to effectively remove/add a degree of freedom. 1 1 M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” IEEE Trans. Antennas Propag. , vol. 67, no. 6, pp. 3889 –3901, 2019. doi : 10.1109/TAP.2019.2902749 Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 4 / 19
Problem Parametrization Shape Optimization Capability to effectively remove/add a degree of freedom. 1 7 6 23 22 74 73 198 192 192 198 73 74 22 23 6 7 ◮ Perfectly compatible with 27 85 186 186 85 27 7 2 − 1 5 − 4 5 − 18 − 478 − 478 − 18 5 − 4 5 − 1 2 7 method of moments; 26 3 1 − 937 − 937 1 3 26 11 0 2 − 1 5 − 1 11 − 854 − 854 11 − 1 5 − 1 2 0 11 ◮ basis functions used as DOF. 0 − 1 9 − 1164 9 − 1 0 11 1 0 1 − 4 5 − 8 − 13 − 13 − 8 5 − 4 1 0 1 11 30 3 2 − 1 − 1 2 3 30 − 4 − 3 − 1 − 1 − 3 − 4 10 1 0 0 0 0 0 0 1 10 − 1 − 10 − 5 − 10 − 1 0 0 − 1 − 1 − 1 − 1 9 0 0 0 0 0 0 0 0 0 0 9 19 1 1 0 0 1 1 19 5 1 0 1 0 1 0 1 1 0 1 0 1 0 1 5 14 26 37 43 37 26 14 4 4 9 8 14 13 18 17 17 18 13 14 8 9 4 4 Example of topology sensitivity, ka = 1 / 2, plate fed in the middle. 1 M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” IEEE Trans. Antennas Propag. , vol. 67, no. 6, pp. 3889 –3901, 2019. doi : 10.1109/TAP.2019.2902749 Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 4 / 19
Problem Parametrization Shape Optimization Capability to effectively remove/add a degree of freedom. 1 7 6 23 22 74 73 198 192 192 198 73 74 22 23 6 7 ◮ Perfectly compatible with 27 85 186 186 85 27 7 2 − 1 5 − 4 5 − 18 − 478 − 478 − 18 5 − 4 5 − 1 2 7 method of moments; 26 3 1 − 937 − 937 1 3 26 11 0 2 − 1 5 − 1 11 − 854 − 854 11 − 1 5 − 1 2 0 11 ◮ basis functions used as DOF. 0 − 1 9 − 1164 9 − 1 0 ◮ Inversion-free for the smallest 11 1 0 1 − 4 5 − 8 − 13 − 13 − 8 5 − 4 1 0 1 11 30 3 2 − 1 − 1 2 3 30 perturbations; − 4 − 3 − 1 − 1 − 3 − 4 10 1 0 0 0 0 0 0 1 10 − 1 − 10 − 5 − 10 − 1 0 0 ◮ gradient-based shape − 1 − 1 − 1 − 1 9 0 0 0 0 0 0 0 0 0 0 9 optimization possible 19 1 1 0 0 1 1 19 5 1 0 1 0 1 0 1 1 0 1 0 1 0 1 5 deterministically. 14 26 37 43 37 26 14 4 4 9 8 14 13 18 17 17 18 13 14 8 9 4 4 Example of topology sensitivity, ka = 1 / 2, plate fed in the middle. 1 M. Capek, L. Jelinek, and M. Gustafsson, “Shape synthesis based on topology sensitivity,” IEEE Trans. Antennas Propag. , vol. 67, no. 6, pp. 3889 –3901, 2019. doi : 10.1109/TAP.2019.2902749 Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 4 / 19
Inversion-free Solution of Linear System Removing and Adding DOF 2 DOF removal: DOF addition: � � �� � � �� y f − Y fb + x fb y f x b � � I = C T I = l f V 0 , l f V 0 , y b 0 − 1 Y bb z b Admittance matrix update: Admittance matrix update: � � � z b Y + x b x T � Y − 1 Y = 1 − x b � � Y = C T y b y T C T b C , C , − x T b 1 Y bb z b b � 0 g n = b z b = � z T ⇔ x b = Y � z b , Z bb − � b x b C nn = 1 otherwise ⇔ � 1 g n = S ( m ) ⇔ C mn = 0 otherwise ⇔ ◮ All columns of C matrix containing solely zeros are eliminated before use. 2 M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of moments,” , 2019, In press (AWPL). doi : 10.1109/LAWP.2019.2912459 Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19
Inversion-free Solution of Linear System Removing and Adding DOF 2 DOF removal: DOF addition: � � �� � � �� y f − Y fb + x fb y f x b � � I = C T I = l f V 0 , l f V 0 , y b 0 − 1 Y bb z b Admittance matrix update: Admittance matrix update: � � � z b Y + x b x T � Y − 1 Y = 1 − x b � � Y = C T y b y T C T b C , C , − x T b 1 Y bb z b b � 0 g n = b z b = � z T ⇔ x b = Y � z b , Z bb − � b x b C nn = 1 otherwise ⇔ � 1 g n = S ( m ) ⇔ C mn = 0 otherwise ⇔ ◮ All columns of C matrix containing solely zeros are eliminated before use. 2 M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of moments,” , 2019, In press (AWPL). doi : 10.1109/LAWP.2019.2912459 Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19
Inversion-free Solution of Linear System Removing and Adding DOF 2 DOF removal: DOF addition: � � �� � � �� y f − Y fb + x fb y f x b � � I = C T I = l f V 0 , l f V 0 , y b 0 − 1 Y bb z b Admittance matrix update: Admittance matrix update: � � � z b Y + x b x T � Y − 1 Y = 1 − x b � � Y = C T y b y T C T b C , C , − x T b 1 Y bb z b b � 0 g n = b z b = � z T ⇔ x b = Y � z b , Z bb − � b x b C nn = 1 otherwise ⇔ � 1 g n = S ( m ) ⇔ C mn = 0 otherwise ⇔ ◮ All columns of C matrix containing solely zeros are eliminated before use. 2 M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of moments,” , 2019, In press (AWPL). doi : 10.1109/LAWP.2019.2912459 Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19
Inversion-free Solution of Linear System Removing and Adding DOF 2 DOF removal: DOF addition: � � �� y f � � �� y f − Y fb + x fb x b � � I = C T I = l f V 0 , l f V 0 , y b 0 − 1 Y bb z b Admittance matrix update: Admittance matrix update: � � � z b Y + x b x T � Y − 1 Y = 1 − x b � � Y = C T y b y T C T b C , C , − x T b 1 Y bb z b b � 0 g n = b z b = � z T ⇔ x b = Y � z b , Z bb − � b x b C nn = 1 otherwise ⇔ � 1 g n = S ( m ) ⇔ C mn = 0 otherwise ⇔ ◮ All columns of C matrix containing solely zeros are eliminated before use. 2 M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of moments,” , 2019, In press (AWPL). doi : 10.1109/LAWP.2019.2912459 Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19
Inversion-free Solution of Linear System Removing and Adding DOF 2 DOF removal: DOF addition: � � �� y f � � �� y f − Y fb + x fb x b � � I = C T I = l f V 0 , l f V 0 , y b 0 − 1 Y bb z b Admittance matrix update: Admittance matrix update: � � � z b Y + x b x T � Y − 1 Y = 1 − x b � � Y = C T y b y T C T b C , C , − x T b 1 Y bb z b b � 0 g n = b z b = � z T ⇔ x b = Y � z b , Z bb − � b x b C nn = 1 otherwise ⇔ � 1 g n = S ( m ) ⇔ C mn = 0 otherwise ⇔ ◮ All columns of C matrix containing solely zeros are eliminated before use. 2 M. Capek, L. Jelinek, and M. Gustafsson, “Inversion-free evaluation of nearest neighbors in method of moments,” , 2019, In press (AWPL). doi : 10.1109/LAWP.2019.2912459 Miloslav ˇ Capek, et al. Inversion-Free Evaluation of Small Geometry Perturbation in Method of Moments 5 / 19
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