Buckling calculations

To calculate buckling loads routine arpackbu.c is called. The following steps are needed in a buckling calculation:

The buckling mode in ARPACK (Mode 4, cf [43]) is used to solve a system of the form

$\displaystyle \begin{bmatrix}K \end{bmatrix} \begin{Bmatrix}U \end{Bmatrix} = \lambda \begin{bmatrix}KG \end{bmatrix} \begin{Bmatrix}U \end{Bmatrix}$ (684)

where $ \begin{bmatrix}K \end{bmatrix}$ is symmetric and positive definite and $ \begin{bmatrix}KG \end{bmatrix}$ is symmetric but indefinite. The iterative procedure to find the eigenvalues requires routines to solve

$\displaystyle \begin{bmatrix}K - \sigma KG \end{bmatrix} \begin{Bmatrix}U \end{Bmatrix} = \begin{Bmatrix}X_1 \end{Bmatrix}$ (685)

and to calculate

$\displaystyle \begin{Bmatrix}Y \end{Bmatrix} = \begin{bmatrix}K \end{bmatrix} \begin{Bmatrix}X_2 \end{Bmatrix}.$ (686)

Similar to the frequency calculations, the LU decomposition (SPOOLES) to solve Equation (685) is performed before the loop determining the buckling factor, since the left hand side of the equation does not vary. The matrix multiplication in Equation (686) is taken care of by routine op.f.

A major difference with the frequency calculations is that an additional iteration loop is necessary to guarantee that the value of the buckling factor is right. Indeed, experience has shown that the value of $ \sigma$ matters here and that the inequality $ 5 \sigma <$   buckling factor$ < 50000 \sigma$ should be satisfied. If it is not, the whole procedure starting with the LU decomposition is repeated with a new value of $ \sigma=$buckling factor$ /500$. If necessary, up to four such iterations are allowed.