Convergence Criteria

As discussed in here, four different schemes are available in SeismoStruct for checking the convergence of a solution at the end of every iteration.

Displacement/Rotation based
Verification, at each individual degree-of-freedom of the structure, that the current iterative displacement/rotation is less or equal than a user-specified tolerance, provides the user with direct control over the degree of precision or, inversely, approximation, adopted in the solution of the problem. In addition, and for the large majority of analyses, such local precision check is also sufficient to guarantee the overall accuracy of the solution obtained. Therefore, this convergence check criterion is the default option in SeismoStruct for the majority of the predefined settings schemes with the default values for displacement and rotation tolerance varying between different settings schemes. For the default settings scheme, which lead to precise and stable solutions in the majority of cases, the values for the displacement and the rotation tolerances are 0.0001 m and 0.0001 rad, respectively.

Force/Moment based
There are occasions where the use of a displacement/rotation convergence check criterion is not sufficient to guarantee a numerically stable and/or accurate solution, due to the fact that displacement/rotation equilibrium does not guarantee, in such special cases, force/moment balance. This is the typical behaviour, for instance, of simple structural systems (e.g. vertical cantilever), where displacement/rotation convergence is obtained in a few iterations, such is the simplicity of the system and its deformed shape, which however may not be sufficient for the internal forces of the elements to be adequately balanced. Particularly, when a RC wall section is used, the stress-strain distribution across the section may assume very complex patterns, by virtue of its large width, thus requiring a much higher number of iterations to be fully equilibrated. In such cases, if a force/moment convergence check is not enforced, the response of the structure will result very irregular, with unrealistically abrupt variations of force/moment quantities (e.g. wiggly force-displacement response curve in pushover analysis). As described in here, a non-dimensional global tolerance is employed in this case, with a default value of 0.001.

Note: Force-based criteria will cause numerical problems and convergence difficulties when used with penalty functions. In these cases Lagrange multipliers should be employed.

Displacement/Rotation AND Force/Moment based
Taking into account the discussion made above, it results clear that maximum accuracy and solution control should be obtained when combining the displacement/rotation and force/moment convergence check criteria. This option, however, is not the default since the force/moment based criterion does, on occasions, create difficulties in models where infinitely stiff/rigid connections are modelled with rigid links, as discussed in here. Still, it is undoubtedly the most stringent convergence and accuracy control criterion available in SeismoStruct, and experienced users are advised to take advantage of it whenever accuracy is paramount.

Displacement/Rotation OR Force/Moment based
This last convergence criterion provides users with maximum flexibility as far as analysis stability is concerned, since converge is achieved when one of the two criteria is checked. This option is highly recommended when arriving at a particular final structural solution is the primary objective of the analysis, and accuracy assumes, at least momentarily, a secondary role.

General
Users may select if the convergence difficulties that might arise during the analysis will be visible in the Post-Processor. The default option is to show the convergence difficulties in the Post-Processor of pushover analysis.

Automatic Adaptation of the Convergence Norms
If this option is selected, in particular steps of the analysis, where convergence is difficult to achieve, the program may smartly increase the defined convergence norms, in order to enable convergence and to allow the program to move to the next step of the analysis. In order not to allow for infinite increase in the value of the convergence norms, a limit is set by the Largest Acceptable Increase of Norms combo box. The default option is to allow for the automatic adaptation of the convergence norm.

 

Note: Convergence difficulties in force-based elements are often caused by the employment of a large number of integration sections (e.g. 5) together with element discretisation (typically in beams, where the reinforcement details change). In such cases, users should decrease the number of integration sections to 3 (minimum accepted).