A note on eigenfrequency sensitivities and structural eigenfrequency optimization based on local sub-domain frequencies

Sensitivity (gradient) of a structural eigenfrequency with respect to a change in density (thickness) of a sub-domain is derived in a simple explicit form. The sub-domain is often an element of a finite element (FE) model, but may be a broader sub-domain, say with a group of elements. This simple result has many applications. It is therefore presented before specific use in optimization examples. The engineering approach of fully stressed design is a practical tool with a theoretical foundation. The analog approach to structural eigenfrequency optimization is presented here with its theoretical foundation. A numerical heuristic redesign procedure is proposed and illustrated with examples. For the ideal case, an optimality criterion is fulfilled if the design have the same sub-domain frequency (local Rayleigh quotient). Sensitivity analysis shows an important relation between squared system eigenfrequency and squared local sub-domain frequency for a given eigen-mode. Higher order eigenfrequencies may also be controlled in this manner. The presented examples are based on 2D finite element models with the use of subspace iteration for analysis and a heuristic recursive design procedure based on the derived optimality condition. The design that maximize a frequency depend on the total amount of available material and on a necessary interpolation as illustrated by different design cases. In this note we have assumed a linear and conservative eigenvalue problem without multiple eigenvalues. The presence of multiple, repeated eigenvalues would require extended sensitivity analysis.