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--- appendices:a15 [2019/06/21 16:47]
helene created
+++ appendices:a15 [2020/08/25 15:46] (current)
@@ Line 41: / Line 41: @@
 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:
+For 3D shell analysis (NTANA = ±8) with NTNOR=0, COMPA(1)<0 and COMPA(3)>0:
+\[\sqrt{\frac{1}{2} \left(\frac{UNOR(1)}{\max(DNOR(1),COMPA(1)²)} +\frac{UNOR(3)}{COMPA(3)²}\right)} \]
+==== Step 4: Check the convergence of the step ====
+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<sup>-1</sup> and PRECD=10<sup>-2</sup>, the step is converged if:
+\[\sqrt{\frac{UNOR(1)}{\max(DNOR(1),COMPA(1)²)}} \leq PRECD \\ <=> \sqrt{\frac{UNOR(1)}{\max(DNOR(1),10^{-2})}} \leq 10^{-2} \\ <=> UNOR(1) \leq 10^{-4}*\max(DNOR(1),10^{-2})\]
+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 $UNOR(1) \leq 10^{-4}*DNOR(1)=10^{-5}$ \\
 ===== Convergence for the forces (NORME2.F) =====
+A step is converged for the displacements if $CONVE=\frac{FNOR}{RNOR} \leq PRECF$
+with:
+  * $FNOR$ = out-of-balance forces ($F_{ex}-F_{in}$)
+  * $RNOR$ = reaction forces (force on the fixed DOF)
+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.
+==== Step 1: Compute FNORMi and RNORMi for every DOF ====
+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.
+==== Step 2: Regroup $FNORM_i$ and $RNORM_i$ by dimension (i.e. force, heat flux and momentum) depending on the type of analysis ====
+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):
+\[\left \{\begin{array}{r c l}  FNOR(1)  & = & \displaystyle\sum_{i=1}^3 FNORM_i \\  RNOR(1)   & = & \displaystyle\sum_{i=1}^3 RNORM_i  \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l}  FNOR(2)  & = &  FNORM_4 \\  RNOR(2)   & = & RNORM_4  \\ \end{array} \right .\]
+\[\left \{\begin{array}{r c l}  NFO(1)  & = & \displaystyle\sum_{i=1}^3 NFOR_i \\  NRE(1)   & = & \displaystyle\sum_{i=1}^3 NREA_i  \\ \end{array} \right . \text{ and } \left \{\begin{array}{r c l}  NFO(2)  & = &  NFOR_4 \\  NRE(2)   & = & NREA_4  \\ \end{array} \right .\]
+==== 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 (15...17), 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  |\[\sqrt{\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{\frac{FNOR(i)}{NFO(i)}}{\max\left(\frac{RNOR(i)}{NRE(i)},\frac{COMPA(i)²}{NRE(i)}\right)}} \]|\[\sqrt{\frac{1}{NDIM} \displaystyle\sum_{i=1}^{NDIM} \frac{\frac{FNOR(i)}{NFO(i)}}{COMPA(i)²}} \]|
+|  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:
+\[ CONVE=\sqrt{\frac{1}{2} \left( \frac{\frac{FNOR(1)}{NFO(1)}}{\max\left(\frac{RNOR(1)}{NRE(1)},\frac{COMPA(1)²}{NRE(1)}\right)} +\frac{\frac{FNOR(3)}{NFO(3)}}{COMPA(3)²}\right)} \]
+==== Step 4: Check the convergence of the step ====
+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<sup>-1</sup> and PRECF=10<sup>-2</sup>, the step is converged if:
+\[\sqrt{\frac{\frac{FNOR(1)}{NFO(1)}}{\frac{\max(RNOR(1),COMPA(1)²)}{NRE(1)}}} \leq PRECF \\ <=> \sqrt{\frac{\frac{FNOR(1)}{NFO(1)}}{\frac{\max(RNOR(1),10^{-2})}{NRE(1)}}} \leq 10^{-2} \\ <=> \frac{FNOR(1)}{NFO(1)} \leq 10^{-4}*\frac{\max(RNOR(1),10^{-2})}{NRE(1)}\]
+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 $\frac{FNOR(1)}{NFO(1)} \leq 10^{-4}*\frac{RNOR(1)}{NRE(1)}=\frac{10^{-5}}{NRE(1)}$  \\

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