Efficient Finite-Difference Derivatives

2.3 The Reduced Eigenproblem

Eigenproblem reanalysis by the CA method has been discussed in detail in previous studies (Kirsch, 2003b; Kirsch and Bogomolni, 2004). The solution procedure is briefly described in this section. We assume that the corresponding stiffness matrix К is given in the decomposed form

Kc = UqtUq (29)

where Uo is an upper triangular matrix. The initial eigenpair Ф0, X0 is obtained by solving the initial eigenproblem

К Фо = А, оМФо (30)

Assume a perturbation 8X in the design and corresponding changes 5К in the stiffness matrix and 5M in the mass matrix, respectively. The modified matrices are given by

К = К0 + 5К M = M0 + 5M (31)

The object is to estimate efficiently and accurately the requested eigenpair Ф, X, without solving the complete set of modified equations

(К0 + 5К) Ф = A, M Ф (32)

The solution process involves the following steps.

a. Calculate the modified matrices К, M [Eqs. (31)].

b. Calculate the matrix of basis vectors rB

rB = [Г1, Г2, …, rj (33)

where r1, r2, …, rs are the basis vectors, and s is much smaller than the number of degrees of freedom. For any requested eigenpair Ф, X the basis vectors are determined separately, using the steps described in the next section.

c. Calculate the reduced matrices Kr and MR by

КR = rB KrB MR = rT MrB (34)

d. Solve the reduced sxs eigenproblem for the first eigenpair A1, y1

Kr У1 =^M R У1 (35)

where y1 is a vector of unknown coefficients

yiT = {УъУ2, ■■■ ,ys} (36)

Various methods (e. g. inverse vector iteration) can be used for this purpose.

e. Evaluate the requested mode shape Ф by

ф = y 1r1 + y2r2 + … + ysrs = rB y,

The requested eigenvalue is already given from Eq. (35) X =

It was found that high accuracy is often achieved with a very small number of basis vectors. In such cases the above solution procedure is most effective.