Detecting the causes of ill-conditioning in structural finite element models

• FE models can contain errors that can lead to a class of ill-conditioned stiffness matrices.
• The extremal eigenpairs of such matrices can be used to identify causes of ill conditioning.
• New technique for diagnosing the causes is developed and implemented in a commercial FE package.
• The technique improves the conditioning when applied to a variety of structural models.

In 2011, version 8.6 of the finite element-based structural analysis package Oasys GSA was released. A new feature in this release was the estimation of the 1-norm condition number κ1(K)=∥K∥1∥K-1∥1κ1(K)=∥K∥1∥K-1∥1 of the stiffness matrix K   of structural models by using a 1-norm estimation algorithm of Higham and Tisseur to estimate ∥K-1∥1∥K-1∥1. The condition estimate is reported as part of the information provided to engineers when they carry out linear/static analysis of models and a warning is raised if the condition number is found to be large. The inclusion of this feature prompted queries from users asking how the condition number impacted the analysis and, in cases where the software displayed an ill conditioning warning, how the ill conditioning could be “fixed”. We describe a method that we have developed and implemented in the software that enables engineers to detect sources of ill conditioning in their models and rectify them. We give the theoretical background and illustrate our discussion with real-life examples of structural models to which this tool has been applied and found useful. Typically, condition numbers of stiffness matrices reduce from O(1016)O(1016) for erroneous models to O(108)O(108) or less for the corrected model.