Choosing the finite difference interval

The objective of the heuristic algorithm is to select the perturbation sizes that lead to accurately computed derivatives. This is done on an element to element basis, so it is possible for a given design parameter that the perturbation size for one element will be different from that of another element. The selection of the perturbation sizes is based on the behavior of a scalar metric $s_p$ for each design parameter $h_p$. This metric must be both convenient and representative of the terms that are numerically differenced. It is chosen as follows:

• Static steps. For static steps the scalar metric is chosen as the norm of the element pseudoload:

  $$s_p=\left|\right|(d{F}_{el}^M/d{h}_p){|}_{\Delta u_{el}^N\ne \Delta u_{el}^N\left(h_p\right)}\left|\right|.$$

  Since an accurate pseudoload is necessary to obtain an accurate sensitivity solution, the norm of the pseudoload is an appropriate choice for $s_p$. This choice is also convenient since the pseudoload has to be computed in any case.

• Frequency steps. For frequency steps the scalar metric is chosen as the projection of the matrix $\left(\frac{D{K}^{NM}}{D{h}_p}-{\lambda }_{\alpha }\frac{D{M}^{NM}}{D{h}_p}\right)$ onto an eigenvector ${\overline{\varphi }}_{\alpha }^N$:

  $$s_p={\overline{\varphi }}_{\alpha }^N\left(\frac{D{K}^{NM}}{D{h}_p}-{\lambda }_{\alpha }\frac{D{M}^{NM}}{D{h}_p}\right){\overline{\varphi }}_{\alpha }^M.$$

  The choice of ${\overline{\varphi }}_{\alpha }^N$ depends on whether a given mode $\alpha$ has a distinct eigenvalue or is associated with a repeated eigenvalue.

  If mode $\alpha$ has a distinct eigenvalue, ${\overline{\varphi }}_{\alpha }^N$ is taken as ${\varphi }_{\alpha }^N$. Consequently, $s_p$ becomes simply the numerator of Equation 5. Therefore, $s_p$ is a direct measure of the magnitude of the eigenvalue sensitivity and is also convenient since this term already must be calculated to obtain the eigenvalue sensitivity.

  Unlike the distinct eigenvalue case, the sensitivities of a repeated eigenvalue cannot be treated independently. The sensitivities of a repeated eigenvalue are extracted simultaneously from the same reduced eigenvalue system, and this system is obtained by numerical differencing (recall Equation 7 and Equation 8). Consequently a single perturbation size (for each design parameter) must be used to obtain all sensitivities of a repeated eigenvalue. To calculate the single perturbation size, a single scalar metric is obtained by taking ${\overline{\varphi }}_{\alpha }^N$ as the sum of the eigenvectors associated with the repeated eigenvalue. The calculation of $s_p$ is similar to the calculation of the matrix $B$ (the only term in the reduced eigenvalue system obtained by numerical differencing); therefore, this choice $s_p$ is both representative of the repeated eigenvalue case and involves differencing calculations that are already being done to obtain the reduced eigenvalue system.

The basic idea of the algorithm is compute the scalar metric $s_p$ for a range of perturbation sizes $\delta h_p^I$, $I=1,2,\dots$, which vary by orders of magnitude. For each $\delta h_p^I$, a corresponding $s_p^I$ is computed. The relative change in $s_p$ between consecutive perturbation sizes, calculated as $e_p^I=\left|s_p^I-s_p^{I-1}\right|/s_p^I$, is used to measure how close a perturbation size is to optimum. In this range of perturbation sizes the one that yields the smallest relative change, denoted as $e_p^a$, is identified as the best perturbation size, $\delta h_p^a$. If $e_p^a$ is not less than a specified tolerance, the range of perturbation sizes is expanded (up to a certain limit), and the testing continues. It is important to realize that $e_p^I$ is not used directly in assessing the accuracy of the numerical differentiation but rather is intended as a means for determining the optimum perturbation size. Thus, a tighter tolerance on $e_p^a$ causes the algorithm to expend more effort in finding an optimum perturbation size but does not guarantee more accurate sensitivities.