The measured physical parameters of a superconducting cavity differ from those of the designed ideal cavity. This is due to shape deviations caused by both loose machine tolerances during fabrication and by the tuning process for the accelerating mode. We present a shape determination algorithm to solve for the unknown deviations from the ideal cavity using experimentally measured cavity data. The objective is to match the results of the deformed cavity model to experimental data through least-squares minimization. The inversion variables are unknown shape deformation parameters that describe perturbations of the ideal cavity. The constraint is the Maxwell eigenvalue problem. We solve the nonlinear optimization problem using a line-search based reduced space GaussNewton method where we compute shape sensitivities with a discrete adjoint approach. We present two shape determination examples, one from synthetic and the other from experimental data. The results demonstrate that the proposed algorithm is very effective in determining the deformed cavity shape.

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A nonlinear Rayleigh-Ritz iterative (NRRIT) method for solving nonlinear eigenvalue problems is studied in this paper. It is an extension of the nonlinear Arnoldi algorithm due to Heinrich Voss. The efficiency of the NRRIT method is demonstrated by comparing with the inverse iteration method to solve a highly nonlinear eigenvalue problem arising from finite element electromagnetic simulation in accelerator modeling.

We formulate the problem of designing the low-loss cavity for the International Linear Collider (ILC) as an electromagnetic shape optimization problem involving a Maxwell eigenvalue problem. The objective is to maximize the stored energy of a trapped mode in the end cell while maintaining a specified frequency corresponding to the accelerating mode. A continuous adjoint method is presented for computation of the design gradient of the objective and constraint. The gradients are used within a nonlinear optimization scheme to compute the optimal shape for a simplified model of the ILC in a small multiple of the cost of solving the Maxwell eigenvalue problem.