An Efficient Numerical Technique for Gradient Computation with Full-Wave EM Solvers

1. McMaster University An Efficient Numerical Technique for Gradient Computation with Full-Wave EM Solvers Shirook M. Ali* and Natalia K. Nikolova * e-mail:alis5@mcmaster.ca tel: (905) 525 9140 ext. 27762 fax: (905) 523 4407 Department of Electrical and Computer Engineering Computational Electromagnetics Laboratory

2. Objectives and Outline • Optimization using gradient-based methods • adjoint-sensitivity analysis: objectives • obtain the response and its gradient with two full- wave analyses for all the design parameters, re- meshing is not necessary • Adjoint variable method in full-wave analysis • computational efficiency, feasibility, and accuracy • Applications with the frequency-domain TLM • Conclusions

3. Optimization via gradient-based methods - scalar objective function - design parameters - state variables objective The design problem

4. The optimization process K+1 analyses 2 analyses Fig. 1. Shape optimization process.

5. Adjoint Sensitivities of Linear Systems response function sensitivity: the adjoint variable method (AVM) [E.J. Haug et al., Design Sensitivity Analysis of Structural Systems, 1986], [J.W. Banler, Optimization, vol. 1, 1994]

6. Adjoint Sensitivities of Linear Systems

7. Adjoint Sensitivities of Linear Systems (b) (a) Fig. 2. Deformation and unwanted perturbations. (a) (b) Fig. 3. Discrete perturbations. feasibility and accuracy of the AVM with solvers on structured grids

8. Applications with the FD-TLM 12 4 z 7 y 2 11 x 3 6 10 8 9 1 5 Fig. 5 (a). The SCN. Fig. 4 (a). The initial cavity structure. Fig. 4 (b). The perturbed cavity. Fig. 5 (b). The perturbed SCN. Cavity

9. Applications with the FD-TLM Cavity Fig. 7. The cost function during the optimization process of the cavity. Fig. 6. Sensitivities of the cavity with respect to its length.

10. Applications with the FD-TLM (b) perturbed filter Single resonator filter (SRF) (a) initial filter Fig. 8. The SRF structure.

11. Applications with the FD-TLM 0 -20 ) -40 1 - (m L ¶ -60 / f ¶ -80 -100 FFD with one cell perturbation Sensitivities with the AVM approach -120 3 3.2 3.4 3.6 3.8 4 4.2 frequency (Hz) 9 x 10 Single resonator filter (SRF) Fig. 9. Sensitivities of the SRF with respect to the length of the septa. Fig. 10. The cost function during the optimization process of the SRF.

12. Conclusions • The AVM is implemented into a feasible technique for frequency domain DSA of HF structures • Reduction in the CPU time requirement by a factor of K • Feasibility: does not require re-meshing during the optimization process • Improved accuracy and convergence • Factors affecting the accuracy • Perturbation step size • Finite differences for the computation of the gradients