Perturbation theory of multi-cell superconducting radio-frequency (SRF) cavities, which is widely used in analysis of field flatness adjustment, is briefly described, and theoretical calculation method of field flatness is deduced. Multi-field coupled simulation calculations are performed for resonant cavities with minor longitudinal deformation, the field flatness and its trend with deformation quantity are obtained. For two widely adopted methods for SRF cavity development, including ‘first-single-then-the-whole pre-tuning’ and ‘whole-cavity stretching/squeezing’ during tuning process, theoretical analysis is carried out based on the capacitively coupled LC oscillator model of perturbation theory for multi-cell cavities, and meanwhile further verifying multi-field coupled simulation calculations are performed by combination of CST MWS and ANSYS for three typical SRF 9-cell cavities - for main Linac of XFEL, TRIUMF e-Linac, and for injector of SHINE as a candidate. It is shown that theoretical analysis results and simulation results are perfectly consistent with each other. The validity and feasibility of the two widely adopted methods above are verified and proved to be correct in the scope of perturbation. Cavity shape structure with excellent acceleration performance and field flatness can be obtained just by adding multi-physics field analysis to the optimization design of single-cell, which makes the design process of cavity shape more efficient. It is illustrated that the tuning sensitivity factors of optimized single-cells including end-cells and mid-cell should be as close or equal as possible to maintain field-flatness during cavity deformation.