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Error Norms

When using GSA it is important that the user can be confident in the accuracy of the results. There are a number of checks that the user can carry out to check accuracy, however GSA provides some measure of the accuracy of the displacement solution. This is reported in the Analysis Details output. The definition of the error norm is different for static and modal dynamic or buckling results.

Statics​

In static analysis the error norm is not calculated unless the model shows signs of being ill-conditioned. The calculation is as follows

  1. Calculate the residual.
  2. Solve for the displacements resulting from the residual and compare these with the actual displacements.
  3. Calculate the error norm.

Thus

fR=f−KuuR=K−1fRe=∥uR∥∥u∥\begin{aligned}\mathbf{f}_{R} &= \mathbf{f} - \mathbf{Ku}\\ \mathbf{u}_{R} &= \mathbf{K}^{- 1}\mathbf{f}_{R}\\ e &= \frac{\left\| \mathbf{u}_{R} \right\|}{\left\| \mathbf{u} \right\|}\end{aligned}

where

∥u∥=∑ui2\left\| \mathbf{u} \right\| = \sqrt{\sum_{}^{}{u_{i}}^{2}}

Dynamics and buckling​

In a dynamic analysis the error norm is always calculated as follows

e=∥Ku−λMu∥∥Ku∥e = \frac{\left\| \mathbf{Ku} - \lambda\mathbf{Mu} \right\|}{\left\| \mathbf{Ku} \right\|}

and in the case of buckling

e=∥Ku+λKgu∥∥Ku∥e = \frac{\left\| \mathbf{Ku} + \lambda\mathbf{K}_{g}\mathbf{u} \right\|}{\left\| \mathbf{Ku} \right\|}