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Appendix 15: Definition of the convergence norms
Convergence for the displacements (NORME3.F)
A step is converged for the displacements if $CONVE=\frac{UNOR}{DNOR} \leq PRECD$
with:
- $UNOR$ = displacement between current and previous iterations
- $DNOR$ = displacement between current iteration and beginning of the step (or end of the previous step)
The definition of the norm $CONVE$ may vary, depending on the type of norm $NTNOR$ and the absolute values for convergence COMPA(1…3) = STRAT (18…20) chosen in the execution file, as will be shown below. The algorithm for computing the norm can be decomposed in several steps.
Step 1: Compute UNORMi and DNORMi for every DOF
For every degree of freedom (DOF), two quantities can be computed: $UNORM_i$ and $DNORM_i$ as shown in the table below.
| NTNOR | $UNORM_i$ | $DNORM_i$ |
|---|---|---|
| 0 | $\displaystyle\sum_{\text{dof along i}} [\delta U (\text{current w.r.t. prev. iter})]²$ | $\displaystyle\sum_{\text{dof along i}} [\Delta U (\text{current w.r.t. prev. step})]²$ |
| 1 | $\displaystyle\sum_{\text{dof along i}} abs(\delta U (\text{current w.r.t. prev. iter}))$ | $\displaystyle\sum_{\text{dof along i}} abs(\Delta U (\text{current w.r.t. prev. step}))$ |
| 2 | $\displaystyle\max_{\text{dof along i}} abs(\delta U (\text{current w.r.t. prev. iter}))$ | $\displaystyle\max_{\text{dof along i}} abs(\Delta U (\text{current w.r.t. prev. step}))$ |
Step 2: Regroup $UNORM_i$ and $DNORM_i$ by dimension (i.e. translation, temperature and rotation) depending on the type of analysis
For example, for 3D analysis (NTANA = ±2), we only have one type of dimension (translation): \[\left \{\begin{array}{r c l} UNOR(1) & = & \displaystyle\sum_{i=1}^3 UNORM_i \\ DNOR(1) & = & \displaystyle\sum_{i=1}^3 DNORM_i \\ \end{array} \right .\]
For 3D shell analysis (NTANA = ±8), we have two types (translation and rotation): \[\left \{\begin{array}{r c l} UNOR(1) & = & \displaystyle\sum_{i=1}^3 UNORM_i \\ DNOR(1) & = & \displaystyle\sum_{i=1}^3 DNORM_i \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l} UNOR(3) & = & \displaystyle\sum_{i=4}^6 UNORM_i \\ DNOR(3) & = & \displaystyle\sum_{i=4}^6 DNORM_i \\ \end{array} \right .\]
For 3D thermo-mechanical analysis (NTANA=6), we also have two types (translation and temperature): \[\left \{\begin{array}{r c l} UNOR(1) & = & \displaystyle\sum_{i=1}^3 UNORM_i \\ DNOR(1) & = & \displaystyle\sum_{i=1}^3 DNORM_i \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l} UNOR(2) & = & UNORM_4 \\ DNOR(2) & = & DNORM_4 \\ \end{array} \right .\]
Step 3: Compute the norm of convergence CONVE
Depending on the type of norm NTNOR and the absolute values for convergence COMPA(1…3) = STRAT (18…20), the computation of the norm CONVE vary. The norm is
- relative if COMPA = 0
- absolute if COMPA > 0
- either absolute or relative if COMPA < 0
The computation is performed as shown in the table below:
| NTNOR | COMPA(i) ≤ 0 | COMPA(i) > 0 |
|---|---|---|
| 0 | \[\sqrt{\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{UNOR(i)}{\max(DNOR(i),COMPA(i)²)}} \] | \[\sqrt{\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{UNOR(i)}{COMPA(i)²}} \] |
| 1 or 2 | \[\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{UNOR(i)}{\max(DNOR(i),-COMPA(i))} \] | \[\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{UNOR(i)}{COMPA(i)} \] |
Remark: the norm can be absolute for some dimensions and relative for others, in which case the formula need to be adapted. This will be clearer with some examples…
For 3D analysis (NTANA = ±2) with NTNOR=0 and COMPA(1)=0:
\[CONVE=\sqrt{\frac{UNOR(1)}{DNOR(1)}}\]
For 3D shell analysis (NTANA = ±8) with NTNOR=0, COMPA(1)=0:
Convergence for the forces (NORME2.F)
