Improved Basis Vectors

The effectiveness of the solution approach depends, to a great extent, on the appropriate choice of the basis vectors. Proper selection of the basis vectors is perhaps the most important factor affecting the successful application of the method. It was found that the basis vectors determined by the method described in this section provide accurate results with a small computational effort.

The basis vectors for any requested eigenpair Ф, X, are first calculated by the terms of the binomial series as follows (Kirsch et. al. submitted b). The first basis vector is selected as

r, = K-1 M Ф0 (38)

Additional vectors are calculated by the terms of the binomial series

r* = – B rk_, (39)

where matrix B is given by

B = K ^ 5K (40)

Calculation of each basis vector by Eq. (39) involves only forward and backward substitutions, since K0 is given in the decomposed form of Eq. (29) from the initial analysis.

Substituting Eq. (40) into Eq. (39) yields

rk = – K-15Krk_i (41)

It was found (Barthelemy et. al. 1988; Pedersen et. al. 1989) that the expression of Eq. (41) might cause inaccurate results in calculating sensitivities with respect to shape design variables. To improve the accuracy, it is possible to use the central-difference expression

5 K = K(X+ SX) – K(X – SX) (42)

in Eq. (40), instead of the forward-difference expression [Eq. (31)]

5K = K(X+ SX) – Ko (43)

This modification may reduce significantly the number of basis vectors required to achieve

sufficiently accurate results. In summary, the resulting expressions for calculating the basis vectors [instead of Eqs. (38) – (40)] are

r1 = r1 = K Q1 M Ф0


= – B rk-1



= K 0‘ § K


It should be noted that forward-difference derivatives [only one additional reanalysis for (X+ 5X)] can be used with the central difference expressions of Eqs. (44) – (46).

To improve the accuracy of the results for the higher mode shapes, we use Gram-Schmidt orthogonalizations of the approximate mode shapes and the basis vectors, with respect to the mass matrix (Bogomolni et. al. in press).