Design and optimization of electromagnetic devices require models that converge reliably and quickly and can represent intense localized magnetic saturation. This paper presents a parameterized 2D nonlinear magnetic equivalent circuit (MEC) and proposes solution methods for it: an approximate Jacobian that is symmetric positive definite, so each Newton-Raphson iteration can use sparse Cholesky factorization with fill-reducing reordering, and Anderson mixing as step control. These methods are compared with alternative nonlinear iterations, linear solvers, and step-control methods in convergence, computational cost, and prediction accuracy. Radial-flux magnetic gears (RFMGs) with structural bridges serve as a challenging test case, because their thin bridges exhibit intense local saturation. The methods are evaluated on three bridged base designs and on 1,024 bridged designs, each paired with the same design without the bridge. On the base designs, Newton-Raphson iteration with the approximate Jacobian reached much lower residuals than fixed-point iteration, which converged for none of them. Sparse Cholesky factorization with reordering was the fastest linear solver tested. Across the 1,024 design pairs, full-step Newton-Raphson converged for 193 bridged and 564 bridge-free designs, whereas Anderson mixing converged for 1,003 and 1,022 and required fewer iterations. Bridged designs converged less often and more slowly under every method. The nonlinear MEC was 2.17 and 1.84 times faster than fixed-mesh nonlinear finite element analysis (FEA) in total computation time for the bridged and bridge-free populations. The median outer-rotor torque discrepancy relative to FEA was 0.22%, and the Spearman rank correlations of the design rankings were at least 0.9996.

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