Solution Convergence

<< Click to Display Table of Contents >>

Navigation:  Flexcom > Theory > Analysis > Modal Analysis >

Solution Convergence

Previous pageNext page

Theory

Eigenproblem Shift

The basic equation that Modes solves is K v = λ M v (as per Mathematical (Eq.1) presented earlier), where K and M are stiffness and mass matrices respectively, v is an eigenvector, and λ is the corresponding eigenvalue. Shifting by a value μ means transforming this equation into (K + μ M) v = η M v. The eigenvalues of the original and transformed equations are related by ηi = λi + μ for all i eigenvalues. Eigenvectors are identical. Bathe et al., (1976) claim that applying a shift to an eigenproblem can speed convergence and prevent problems when the stiffness matrix is positive semidefinite. By default Modes applies a shift of 1. In a very small number of analyses increasing this value can guarantee convergence which might otherwise not be achieved, but this option should be very rarely used.

Effective Compression and Local Buckling

Occasionally Flexcom can report the following error message during a modal analysis…

Error: Solution stopped during subspace iteration. Stiffness matrix is not positive definite. Non-positive pivot for equation N, pivot=X.Y

Any error message relating to negative pivot terms in the global stiffness matrix generally stems from effective compression in the initial static solution. Where compression is present, this can lead to instability in the subsequent modal analysis, as the riser is susceptible to buckling and the modal solution becomes indeterminate. If this message appears, you should examine the effective tension distribution in the initial static analysis, and check to see if there are regions where the effective tension is negative.

Furthermore, even a successful modal analysis can occasionally present negative eigenvalues in the solution. These are unusual, and again may be caused by effective compression in the preceding static analysis stage.

If either of these issues are a cause of concern, there are a couple of options available to you…

It may be possible to alter the loading on the model in order to avoid effective compression. For example, additional buoyancy material may be used to reduce the apparent weight of the structure under consideration.

It may be possible to alter the structural properties, in order to augment the bending stiffness. The critical buckling load is related to bending stiffness, and if the resistance to bending can be increased, this may prevent local buckling.

Relevant Keywords

*EIGENPAIRS is used to specify the required number of natural frequencies, and also parameters relating to the subspace iteration algorithm, including an Eigenproblem Shift parameter.