In this article, a fast and accurate method for computing the nonlinearities, such as third-order harmonic generation and third-order intermodulation distortion, in complex superconducting structures based on the ring resonators is proposed. First, a recap of the nonlinearity in the superconductors and its physical concept are mentioned. Then, the nonlinear circuit model of superconducting ring resonators (SRRs) is reviewed and developed to be applicable to complex structures, such as multipole SRR-based reaction-type Chebyshev filters. In the third section, it is shown that the combination of the harmonic balance method with the nonlinear circuit model computes the nonlinearity quickly and accurately. The method is applied to several SRR-based structures, and the computed nonlinearity is compared with the measurements in order to prove the claim.