Three-dimensional structured illumination microscopy (3D-SIM) is an essential tool for volumetric fluorescence imaging, which improves both axial and lateral resolution by down-modulating high-frequency information of the sample into the passband of optical transfer function (OTF). And when combining with the 4Pi structure, the performance of 3D-SIM can be further improved. The reconstruction results of generally used linear 3D algorithm, however, are lack of high-fidelity and proneess to generate artifacts. In this paper, we proposed a novel iterative algorithm based on gradient descent combined with a nonlinear optimizer, which can be applied to all 3D-SIM setups (including I5S setup). We verified through simulation that the proposed solution, termed as nonlinear gradient descent structured illumination microscopy (NGD-SIM), achieves more fidelity results which can reach the limitation of theoretical resolution improvement of SIM. Moreover, it can be firmly validated on simulation that this algorithm can effectively reduce the amount of raw data in the case of sinusoidal-pattern illumination, i.e., the algorithm doesn't need five-step phase shifting; data with any number of phases can theoretically be reconstructed. Our method also provides the possibility to extend the application of sinusoidal-pattern illumination to any kind of interference fringe, which is generated by diversified types of illumination mode.