Continuum sensitivity analysis and improved Nelson's method for beam shape eigensensitivities

Gradient-based optimization techniques require accurate and efficient sensitivity or design derivative analysis. In general, numerical sensitivity methods such as finite differences are easy to implement but imprecise and computationally inefficient. In contrast, analytical sensitivity methods are highly accurate and efficient. Although these methods have been widely evaluated for static problems or dynamic analysis in the time domain, no analytical sensitivity methods have been developed for eigenvalue problems. In this paper, two different analytical methods for shape eigensensitivity analysis have been evaluated: the Continuum Sensitivity Analysis (CSA) and an enhanced version of Nelson's method. They are both analytical techniques but differ in how the analytical differentiation is performed: before and after the discretization, respectively. CSA has been applied to eigenvalue problems for the first time, while Nelson's method has been improved and adapted to shape optimizations. Both methods have been applied to different cases involving shape optimization of beams. Both vibration and buckling problems were analysed considering the eigenvalue as a design variable. Both methods have been successfully applied, and Nelson's method proved to be more convenient for this kind of problem.