User Area > Advice
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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