Occurrence:
I have solved my analysis in GeoStudio, but when I turn on Node Convergence under View Preferences in the Results View window, I see numerous red X’s and/or grey O’s displayed on my model.
Resolution:
The red X’s and/or grey O’s are convergence indicators. The cause of each of these symbols are listed below.
Grey O’s (maximum difference criterion):
These indicators may be displayed at nodes when the difference between the independent variable (pressure head, temperature …) reported at two successive iterations is larger than a specified tolerance. As an example, let us consider a seepage analysis in which the maximum pressure head difference (or tolerance) has been set equal to 0.005 m. If the pressure head reported at a specific node is equal to 23.5645 m on a given iteration and 23.5585 m on the subsequent iteration, a grey O is displayed at the node given that the absolute difference in pressure head (0.006 m) is larger than the specified tolerance (0.005 m).
It must be noted that nodes that meet the maximum difference criterion are assumed to have converged whereas nodes that have not met the criterion are subjected to the significant digit criterion.
Red X’s (significant digit criterion):
Theses indicators may be displayed at nodes when the maximum difference criterion discussed above has not been met and the independent variable (pressure head, temperature …) reported at two successive iterations is smaller than the specified significant digit criterion. As an example, let us consider a seepage analysis in which the precision has been set equal to two (2) significant digits. If the pressure head reported at a specific node is equal to 23.5645 m on a given iteration and 23.5585 m on the subsequent iteration, these numbers written to a precision of 2 () significant figures can be rewritten as 24. In this case. the grey O will be displayed at the node. I the significant digits criteria was changed to five (5), then a red X would replace the grey O, as the values of pressure head are no longer the same within the specified precision (23.565 m and 23.559 m, respectively). It is important to note that the count of significant digits starts at the first non-zero digit. The desired (or required) precision will depend on the project at hand. In most cases, the precision can be set equal to two (2) significant digits.
It must be noted that nodes that do not meet the maximum difference and significant digit criteria are considered as unconverged and reported as unconverged nodes in convergence graph types.
Notes:
Repeated substitution and under-relaxation are commonly used numerical procedures for coping with material non-linearity that involve repeatedly solving the finite element equations and updating the material properties based on the solution at the previous iteration. Mathematically, the procedures can be expressed as follows:
where x is the independent variable (pressure head, temperature …), and Rate is the under-relaxation rate. In the case of repeated substitution, the under-relaxation rate is set equal to one (1), and the equation reduces to
This implies that the solution of the previous iteration is used as the starting point of the next iteration. In the case of under-relaxation, the under-relaxation rate is set equal to a value smaller than one (e.g. 0.1), so the starting value of the independent variable becomes equal to the independent variable of the previous iteration plus one-tenth of the difference in independent variable between both iterations. The independent variable (and computed properties) are thus much closer to that of the previous iteration, which favors convergence.
For numerically challenging problems, it may be difficult to reach a converged solution using the default convergence parameters. The under-relaxation parameters may be modified to an Initial Rate of less than 1 (e.g. 0.65) and a Minimum Rate of less than 0.1 (e.g. 0.01) to help solve highly non-linear material properties.