Derivatives of the Eigenpairs
2.1 Analytical Derivatives
For simplicity, we eliminate the subscripts k (mode shapes) and j (design variables). Thus, the eigenproblem of Eq. (3) is expressed as
The eigenvector is often normalized such that
®TM Ф =1 (22)
To evaluate the derivatives of the eigenpairs (5Ф/дX and dX/dX), we differentiate Eqs. (21), (22) with respect to a design variable x and rearrange to obtain
nniwsd Ф SA _ d K d M
(K – XM)—————- МФ = – (———– Л—— )Ф
dX dX dX dX
_T_.dФ 1 T dM
Ф2 M = — Ф2———- Ф
2 or, in matrix form
Note that this is only correct if the eigenvalue X is distinct.
Several methods have been proposed to solve Eq. (25). In general, the solution involves much computational effort. Specifically, a matrix of the order (m+1), m being the number of degrees of freedom, must be factorized for each of the p considered mode shapes. In addition, the matrices d K / dX, д M / dX must be calculated and forward and backward substitutions must be carried out for each design variable.
2.2 Finite-Difference Derivatives
In the forward-difference method, the derivatives are approximated from the exact displacements at the original point X and at the perturbed point X+5X by
д Ф _ Ф(X + SX) – Ф(X) dX ~ SX
where 5X is a predetermined step-size. The accuracy can be improved by adopting the central- difference approximation, where the derivatives are computed from the exact displacements at the two points X – 5X and X+ 5X by
дФ _ Ф(X + SX)-Ф(X-SX) dX ~ 2SX
Finite-difference methods are the easiest to implement and therefore they are attractive in many applications. When Ф(Ж) is known, application of Eq. (27) involves only one additional calculation of the displacements at X+ 5X whereas Eq. (28) requires calculation at the two points X-8X and X+5X. For a problem with n design variables, finite difference derivative calculations require repetition of the analysis for n+1 [Eq. (27)] or 2n+1 [Eq. (28)] different design points. This procedure is usually not efficient compared to, for example, analytical and semi-analytical methods. An efficient solution procedure using the CA approach is described below.
As noted earlier, finite-difference approximations might have accuracy problems. The following two sources of errors should be considered whenever these approximations are used:
a. The truncation error, which is a result of neglecting terms in the Taylor series expansion of the perturbed response.
b. The condition error, which is the difference between the numerical evaluation of the function and its exact value. Examples for this type of error include round-off error in calculating 5Ф/ d X from the original and perturbed values of Ф, and calculation of the response by approximate analysis. The latter can also be the result of a finite number of iterations being used within an iterative procedure.
These are two conflicting considerations. That is, a small step size <5X will reduce the truncation error, but may increase the condition error. In some cases there may not be any step size which yields an acceptable error. Some considerations for choosing the forward-difference step-size are discussed elsewhere (Burton, 1992). In certain applications, truncation errors are not of major importance since it is often sufficient to find the average rate of change in the structural response and not necessarily the accurate local rate of change at a given point. Therefore, to eliminate round-off errors due to approximations it is recommended to increase the step-size.
It is well known that relatively small response values are not calculated as accurately as large response values (Haftka and Gurdal, 1993). The same applies to derivatives. Thus, it would be difficult to evaluate accurately small response derivatives by finite difference or other approximations.