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Diagonal Decay

The stiffness matrix is a crucial component in a finite element analysis, but it can be well or poorly conditioned. Poor conditioning may result in round-off error, which is a loss of accuracy in the evaluation of the terms during the reduction process of the solution. This in turn leads to inaccuracies in the predicted displacements and stresses.

LUSAS monitors the round-off error by evaluating the amount of diagonal decay present during the Gaussian reduction process. This criterion is based on the assumption that initially large diagonal terms accumulate errors proportional to their size. As reduction progresses, the diagonal term is reduced, amplifying the errors until they become a maximum when the diagonal term is the pivot. An indication of probable errors may be obtained by examining the change in magnitude of the diagonal term.

The tolerance threshold above which a diagonal decay warning is output (0.1E5) is actually quite conservative. Although a check would always be recommended for any warning of this description, significant effects would not generally be expected until the decay reaches a value of 0.1E8 or greater.

In general, poor conditioning of the stiffness matrix occurs because of large variations in the magnitude of diagonal stiffness terms. This usually occurs because of

  • Large stiff elements being connected to small less stiff elements. An example may be where a stiff beam element is being used to transfer load into the structure. The stiffness of the beam would need to be reduced - typically the beam would only need to be 1000 times the stiffness of the local elements
  • Elements with highly disparate stiffnesses, for example a beam element may have a bending stiffness that is orders of magnitude less than it's axial stiffness. Consider a cantilever beam problem which is notoriously problematic with respect to ill-conditioning because of the potential for large differences between the axial (~EA/L) and shear/rotational stiffness (~12EI/L3) components. The longer the beam, the greater the difference between EA/L and 12EI/L3

For more information on the more frequent reasons for such warnings click here

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