A step is converged for the displacements if $CONVE=\frac{UNOR}{DNOR} \leq PRECD$
with:
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.
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 | $\displaystyle\max_{\text{dof along i}} abs(\delta U (\text{current w.r.t. prev. iter}))$ | $\displaystyle\max_{\text{dof along i}} abs(\Delta U (\text{current w.r.t. prev. step}))$ |
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): \[\left \{\begin{array}{r c l} UNOR(1) & = & \displaystyle\sum_{i=1}^3 UNORM_i \\ DNOR(1) & = & \displaystyle\sum_{i=1}^3 DNORM_i \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l} UNOR(2) & = & UNORM_4 \\ DNOR(2) & = & DNORM_4 \\ \end{array} \right .\]
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
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 and COMPA(3)>0:
\[\sqrt{\frac{1}{2} \left(\frac{UNOR(1)}{\max(DNOR(1),COMPA(1)²)} +\frac{UNOR(3)}{COMPA(3)²}\right)} \]
The step is converged for the displacements if $CONVE \leq PRECD$
For example, for 3D analysis (NTANA = ±2) with NTNOR=0 and COMPA(1)=-10-1 and PRECD=10-2, the step is converged if:
\[\sqrt{\frac{UNOR(1)}{\max(DNOR(1),COMPA(1)²)}} \leq PRECD \\ \Leftrightarrow \sqrt{\frac{UNOR(1)}{\max(DNOR(1),10^{-2})}} \leq 10^{-2} \\ \Leftrightarrow UNOR(1) \leq 10^{-4}*\max(DNOR(1),10^{-2})\]
If $DNOR(1)=10^{-8}$, the step is converged if $UNOR(1) \leq 10^{-4}*10^{-2}=10^{-6}$
If $DNOR(1)=10^{-1}$, the step is converged if $UNOR(1) \leq 10^{-4}*DNOR(1)=10^{-5}$
A step is converged for the displacements if $CONVE=\frac{FNOR}{RNOR} \leq PRECF$
with:
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 (15…17) chosen in the execution file, as will be shown below. The algorithm for computing the norm can be decomposed in several steps.
For every degree of freedom (DOF), two quantities can be computed: $FNORM_i$ and $RNORM_i$ as shown in the table below.
| NTNOR | $FNORM_i$ | $RNORM_i$ |
|---|---|---|
| 0 | $\displaystyle\sum_{\text{dof along i}} [\text{out-of-balance forces}]²$ | $\displaystyle\sum_{\text{dof along i}} [\text{reactions}]²$ |
| 1 | $\displaystyle\sum_{\text{dof along i}} abs(\text{out-of-balance forces})$ | $\displaystyle\sum_{\text{dof along i}} abs(\text{reactions})$ |
| 2 | $\displaystyle\max_{\text{dof along i}} abs(\text{out-of-balance forces})$ | $\displaystyle\max_{\text{dof along i}} abs(\text{reactions})$ |
In addition, two quantities are also computed: $NFOR_i$ and $NREA_i$ which contains respectively the number of out-of-balance forces (positive equation number in ID matrix) and reactions (negative equation number in ID matrix) in every direction.
For example, for 3D analysis (NTANA = ±2), we only have one type of dimension (force): \[\left \{\begin{array}{r c l} FNOR(1) & = & \displaystyle\sum_{i=1}^3 FNORM_i \\ RNOR(1) & = & \displaystyle\sum_{i=1}^3 RNORM_i \\ \end{array} \right . \text{and} \left \{\begin{array}{r c l} NFO(1) & = & \displaystyle\sum_{i=1}^3 NFOR_i \\ NRE(1) & = & \displaystyle\sum_{i=1}^3 NREA_i \\ \end{array} \right .\]
For 3D shell analysis (NTANA = ±8), we have two types (translation and rotation): \[\left \{\begin{array}{r c l} FNOR(1) & = & \displaystyle\sum_{i=1}^3 FNORM_i \\ RNOR(1) & = & \displaystyle\sum_{i=1}^3 RNORM_i \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l} FNOR(3) & = & \displaystyle\sum_{i=4}^6 FNORM_i \\ RNOR(3) & = & \displaystyle\sum_{i=4}^6 RNORM_i \\ \end{array} \right .\] \[\left \{\begin{array}{r c l} NFO(1) & = & \displaystyle\sum_{i=1}^3 NFOR_i \\ NRE(1) & = & \displaystyle\sum_{i=1}^3 NREA_i \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l} NFO(3) & = & \displaystyle\sum_{i=4}^6 NFOR_i \\ NRE(3) & = & \displaystyle\sum_{i=4}^6 NREA_i \\ \end{array} \right .\]
For 3D thermo-mechanical analysis (NTANA=6), we also have two types (translation and temperature): \[\left \{\begin{array}{r c l} FNOR(1) & = & \displaystyle\sum_{i=1}^3 FNORM_i \\ RNOR(1) & = & \displaystyle\sum_{i=1}^3 RNORM_i \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l} FNOR(2) & = & FNORM_4 \\ RNOR(2) & = & RNORM_4 \\ \end{array} \right .\] \[\left \{\begin{array}{r c l} NFO(1) & = & \displaystyle\sum_{i=1}^3 NFOR_i \\ NRE(1) & = & \displaystyle\sum_{i=1}^3 NREA_i \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l} NFO(2) & = & NFOR_4 \\ NRE(2) & = & NREA_4 \\ \end{array} \right .\]
Depending on the type of norm NTNOR and the absolute values for convergence COMPA(1…3) = STRAT (15…17), the computation of the norm CONVE vary. The norm is
The computation is performed as shown in the table below:
| NTNOR | COMPA(i) ≤ 0 | COMPA(i) > 0 |
|---|---|---|
| 0 | \[\sqrt{\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{\frac{FNOR(i)}{NFO(i)}}{\max\left(\frac{RNOR(i)}{NRE(i)},\frac{COMPA(i)²}{NRE(i)}\right)}} \] | \[\sqrt{\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{\frac{FNOR(i)}{NFO(i)}}{COMPA(i)²}} \] |
| 1 or 2 | \[\sqrt{\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{\frac{FNOR(i)}{NFO(i)}}{\max\left(\frac{RNOR(i)}{NRE(i)},\frac{-COMPA(i)}{NRE(i)}\right)}} \] | \[\sqrt{\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{\frac{FNOR(i)}{NFO(i)}}{COMPA(i)}} \] |
Remark: the norm can be absolute for some dimensions and relative for others, in which case the formula needs to be adapted. This will be clearer with some examples…
For 3D analysis (NTANA = ±2) with NTNOR=0 and COMPA(1)=0:
\[CONVE=\sqrt{\frac{\frac{FNOR(i)}{NFO(i)}}{\frac{RNOR(i)}{NRE(i)}}}\]
For 3D shell analysis (NTANA = ±8) with NTNOR=0, COMPA(1)<0 and COMPA(3)>0:
\[ CONVE=\sqrt{\frac{1}{2} \left( \frac{\frac{FNOR(1)}{NFO(1)}}{\max\left(\frac{RNOR(1)}{NRE(1)},\frac{COMPA(1)²}{NRE(1)}\right)} +\frac{\frac{FNOR(3)}{NFO(3)}}{COMPA(3)²}\right)} \]
The step is converged for the displacements if $CONVE \leq PRECF$
For example, for 3D analysis (NTANA = ±2) with NTNOR=0 and COMPA(1)=-10-1 and PRECF=10-2, the step is converged if:
\[\sqrt{\frac{\frac{FNOR(1)}{NFO(1)}}{\frac{\max(RNOR(1),COMPA(1)²)}{NRE(1)}}} \leq PRECF \\ \Leftrightarrow \sqrt{\frac{\frac{FNOR(1)}{NFO(1)}}{\frac{\max(RNOR(1),10^{-2})}{NRE(1)}}} \leq 10^{-2} \\ \Leftrightarrow \frac{FNOR(1)}{NFO(1)} \leq 10^{-4}*\frac{\max(RNOR(1),10^{-2})}{NRE(1)}\]
If $RNOR(1)=10^{-8}$, the step is converged if $\frac{FNOR(1)}{NFO(1)} \leq 10^{-4}*\frac{10^{-2}}{NRE(1)}=\frac{10^{-6}}{NRE(1)}$
If $RNOR(1)=10^{-1}$, the step is converged if $\frac{FNOR(1)}{NFO(1)} \leq 10^{-4}*\frac{RNOR(1)}{NRE(1)}=\frac{10^{-5}}{NRE(1)}$