IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 3, MAY2004 1521 Sensitivity Analysis With Full-Wave Electromagnetic Solvers Based on Structured Grids Shirook M. Ali, Student Member, IEEE, Natalia K. Nikolova, Member, IEEE, and Mohamed H. Bakr, Member, IEEE Abstract—Recently, we proposed a novel technique for design sensitivity analysis of high-frequency structures with respect to localized perturbations in conductive parameters. Here, we gen- eralize the technique to include shape and material variations and utilize the response sensitivities in gradient-based optimization. Our technique belongs to the class of adjoint-variable methods. Thus, it computes the response and its gradient with only two electromagnetic (EM) simulations—of the original and the adjoint problems—regardless of the number of design parameters. For the first time, adjoint sensitivities with respect to conductive, di- electric-magnetic material and shape perturbations are computed via EM solvers on structured grids. Our approximate sensitivity analysis does not require analytical derivatives of the system matrix. This makes the technique versatile and easy to implement. The technique defaults to exact sensitivities with analytical system matrix derivatives when global design parameters are being per- turbed. We discuss the accuracy of the approximate sensitivities, as well as the practicality of the exact sensitivities in specific design problems. We also discuss implementations in gradient-based optimization and illustrate them through simulation and design with the frequency domain transmission line method (FD-TLM). Index Terms—Design methodology, frequency domain TLM, sensitivity. I. INTRODUCTION G RADIENT-BASED optimization needs not only the ob- jective (or response) function of the system but also its gradient in the design parameter space. The gradient may also be used for tolerance and yield analysis. With the adjoint vari- able method, the response and its gradient are computed with at most two system analyses regardless of the number of design parameters. Adjoint-based sensitivity analysis has been used in control theory [1], structural design [2], circuit theory [3], etc. The implementation of the adjoint approach with electromagnetic (EM) solvers for high-frequency problems is only recent [4]–[10]. Exact sensitivities with the finite-element method (FEM) are considered in [4] and [5], where analytical deriva- tives of the FEM system matrix with respect to the Cartesian coordinates of the mesh vertices are available. With method of moments (MoM) solvers, exact sensitivities are often not practical due to the complexity of the utilized Green’s func- Manuscript received July 29, 2003; revised February 24, 2004. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under grants 227660-2000, 227660-2003, OGP0249780-2002, and STPGP 269760. The authors are with the Department of Electrical and Computer Engi- neering, McMaster University, Hamilton, ON L8S 4K1, Canada (e-mail: alis5@mcmaster.ca; talia@mcmaster.ca; mbakr@mail.ece.mcmaster.ca). Digital Object Identifier 10.1109/TMAG.2004.827173 tions, expansion, and weighting functions. A feasible technique for adjoint-based sensitivity analysis (EM-FAST) with the MoM is suggested in [6], where regeneration of the mesh discretizing a slightly perturbed structure is used to compute the system matrix derivatives with finite differences. In [7] and [8], unstructured grids are used to obtain exact sensitivities with the finite-difference time-domain discrete surface integral (FDTD-DSI) method. In practice, the technique requires a finite-element meshing procedure. All of these approaches are either impractical or inapplicable with solvers on structured grids such as the FDTD method, the time-domain transmission line method (TD-TLM), and the frequency-domain transmis- sion line method (FD-TLM). We propose techniques through which the adjoint variable method can be easily implemented in practical sensitivity analysis with structured-grid EM computational algorithms. Exact and approximate sensitivity expressions are obtained and discussed. For the exact case, we utilize the analytical derivatives of the system matrix. These are available when global design parameters—such as the constitutive parameters of the material filling the whole computational domain or shape parameters stretching across the structure—are perturbed. We show that the application of the exact adjoint technique with EM solvers on structured grids is limited. This is because local shape perturbations result in system matrices that are not analytical functions of the coordinates of the mesh nodes. The local shape perturbations are restricted to multiples of the grid size in the respective direction. When perfect conducting boundaries are perturbed, the size of the system matrix changes, and, therefore, a meaningful definition of a derivative is not possible. When dielectric-magnetic interface boundaries are perturbed, the size of the system matrix is preserved but its coefficients, affected by the stepwise change, assume values from a predefined discrete set. The lack of continuity of the allowed matrix coefficient values forbids reliable approximations of the system matrix derivatives, and thus makes the exact adjoint-based analysis inapplicable. Here, we propose an approximate adjoint technique for sensi- tivity analysis, which does not require analytical system matrix derivatives. We utilize the differences between perturbed and original system matrices. These differences can be as large as the original coefficients themselves. The change in the matrix coefficients corresponding to a perturbation is a “binary” switch between two known values related to the original and the per- turbed boundary/interface. Our technique, unlike the finite-dif- ference EM-FAST technique [6], puts no restrictions on the amount of change in the perturbed system matrix by preserving the second-order term in the proposed sensitivity expression. 0018-9464/04$20.00 © 2004 IEEE