1. An Efficient Numerical Technique for Gradient Computation with Full-Wave EM Solvers * e-mail: alis5@mcmaster.caalis5@mcmaster.ca tel: (905) 525 9140 ext. 27762 fax: (905) 523 4407 Shirook M. Ali * and Natalia K. Nikolova McMaster University Department of Electrical and Computer Engineering Computational Electromagnetics Laboratory

2. Objectives and Outline  Applications with the frequency-domain TLM  Conclusions  Adjoint variable method in full-wave analysis computational efficiency, feasibility, and accuracy  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

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

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

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

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

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

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

11. Applications with the FD-TLM 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. 33.23.43.63.844.2 x 10 9 -120 -100 -80 -60 -40 -20 0 frequency (Hz)  /  L (m - 1 ) FFD with one cell perturbation Sensitivities with the AVM approach f

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