Resonator design method
The following is the general method for designing ultrasonic resonators. The particulars can be adjusted as needed.
 Specify design goals, typically with the following descending priority 

Primary resonant mode and frequency
 If the resonator will be heat treated, make allowance for possible frequency change.
 Frequency separation (between primary frequency and adjacent frequencies). May depend on mode of adjacent frequencies.
 Gain
 Amplitude (if needed for stress calculations)
 Maximum transverse amplitude (as % of axial amplitude)
 Maximum transducer node angle (may be related to the transverse amplitude)
 Life
 Number of ultrasonic cycles (may be infinite)
 Other (wear, cavitation erosion, ceramic heating, ceramic cracking, ceramic shifting, joints)
 Maximum fatigue stress for each component
 Maximum loss (may be related to the node angle)
 Specify design restrictions.
 Static loads
 Dimensional (e.g., wrench flats, stud parameters, housing interference)
 Machining methods
 Specify materials based on 
 Fatigue strength
 Loss
 Wear (including cavitation erosion)
 Machinability
 Biological compatibility
 Cost
 Heat transfer
 Decide design technique (empirical  e.g., trialanderror, analytical (FEA), or combination).
 For analytical design:
 Specify material properties (Young's modulus, density, Poisson's ratio, endurance strength).
 Decide model type (axisymmetric, 1/2 3D, full 3D).
 Build the model.
 Defeature as needed to reduce meshing problems and analysis time.
 Apply materials.
 For FEA 
 Specify a mesh size (global, for each part, locally refined).
 Apply boundary conditions.
 Run the analysis.
 Record the results.
 See if the results have converged for each relevant performance parameter.
 The results for some performance parameters may be far outside their window of interest in which case convergence for that performance parameter can be ignored. These might include adjacent resonances and stresses.
 Stress convergence may require localized mesh refinements at stress concentrations. This will generally be done after other performance goals have been achieved in order to minimize modeling and run times.
 When the results have converged, compare to goals. The goals should be achieved in the following order:
 Primary resonant frequency
 Frequency separation (between primary frequency and adjacent frequencies). May depend on mode of adjacent frequencies. Consider if certain modes are not significant because they probably can't be excited (e.g., torsional) or may result from simplified geometry (e.g., oblate mode of end cap where the end cap has been approximated as a cylinder).
 Gain
 Transverse amplitude (as % of axial amplitude)
 Transducer node angle
 Stress for each component
 Ultrasonic stress
 Ultrasonic stress + static stress (if static load is applied)
 If a goal has not been achieved 
 Determine the cause.
 Determine a likely solution for that goal, considering the likely effects on other goals. (Interpolation/extrapolation, possibly by graphing, may be helpful.)
 Iterate until all goals have been achieved. If particular goals can't be achieved then adjust these goals or ignore these goals (i.e., some goals may have to be adjusted or sacrificed in order to achieve higher priority goals).
 If not analyzing full 3D, consider periodically increasing the model state if modes that have been suppressed by the limited model might cause a problem.
 If axisymmetric, consider 1/2 3D or full 3D.
 If 1/2 3D, consider full 3D.

Machine a prototype.
 Allow extra material for tuning.
 Heat treat if necessary.
 Tune to desired frequency.
 Laboratory test. Compare results to specifications and predicted performance. Redesign if needed.
 Life test if predicted stresses are high. Redesign if needed.
 Field test. Redesign if needed.