Description of document versions
====================================


.. csv-table::

    "**Version** **Aster**", "**Author (s)**
    **Organization (s)**", "**Description of changes**"
    "5", "P.Massin, Mr. AL MIKDAD
    EDF -R&D/ MMN ", "Initial text"
    "7.4", "X.Desroches", "Update: minor changes"



**: Flowchart of the linear buckling calculation**

**Local landmarks at** :math:`\mathit{NB2}` **knots** :math:`{\left[{t}_{1}\mathrm{:}{t}_{2}\mathrm{:}n\right]}_{I}`

**Loop over the points of normal numerical integration of Gauss**


* retrieving the local constraints vector :math:`\tilde{S}\mathrm{=}(\begin{array}{c}{\tilde{S}}_{{t}_{1}{t}_{1}}\\ {\tilde{S}}_{{t}_{2}{t}_{2}}\\ {\tilde{\tau }}_{{t}_{1}{t}_{2}}\\ \\ {\tilde{\tau }}_{{t}_{1}n}\\ {\tilde{\tau }}_{{t}_{2}n}\end{array})\mathrm{=}(\begin{array}{c}{\tilde{S}}_{{t}_{1}{t}_{1}}\\ {\tilde{S}}_{{t}_{2}{t}_{2}}\\ \sqrt{2}{\tilde{S}}_{{t}_{1}{t}_{2}}\\ \\ \sqrt{2}{\tilde{S}}_{{t}_{1}n}\\ \sqrt{2}{\tilde{S}}_{{t}_{2}n}\end{array})`

from the 6 tensor components stored in the PCONTRR :math:`(\begin{array}{c}{\tilde{S}}_{{t}_{1}{t}_{1}}\\ {\tilde{S}}_{{t}_{2}{t}_{2}}\\ 0\\ {\tilde{S}}_{{t}_{1}{t}_{2}}\\ {\tilde{S}}_{{t}_{1}n}\\ {\tilde{S}}_{{t}_{2}n}\end{array})` mode


* formation of the symmetric tensor :math:`3\mathrm{\times }3` of local constraints :math:`\mathrm{[}\tilde{S}\mathrm{]}`


* construction of the transformation matrix :math:`\mathrm{P}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\mathrm{=}\left[\begin{array}{c}{\mathrm{t}}_{1}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\\ {\mathrm{t}}_{2}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\\ {\mathrm{t}}_{3}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\end{array}\right]` where :math:`{\mathrm{t}}_{3}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\mathrm{=}{n}_{1}({\xi }_{2})`


* calculation of the symmetric tensor :math:`3\mathrm{\times }3` of global constraints :math:`\mathrm{[}S\mathrm{]}\mathrm{=}{P}^{T}\mathrm{[}\tilde{S}\mathrm{]}P`


* for the non-classical term, calculation of :math:`\text{HQ}\mathrm{=}\left[\frac{\mathrm{[}\stackrel{3\mathrm{\times }9}{\text{HSFM}}\mathrm{]}}{\mathrm{[}\stackrel{2\mathrm{\times }9}{\text{HSS}}\mathrm{]}}\right]`

:math:`\begin{array}{c}\text{HQ}\mathrm{=}\\ \left[\begin{array}{cccccc}{({t}_{1}(1))}^{2}& {({t}_{1}(2))}^{2}& {({t}_{1}(3))}^{2}& {t}_{1}(1){t}_{1}(2)& {t}_{1}(2){t}_{1}(3)& {t}_{1}(3){t}_{1}(1)\\ {({t}_{2}(1))}^{2}& {({t}_{2}(2))}^{2}& {({t}_{2}(3))}^{2}& {t}_{2}(1){t}_{2}(2)& {t}_{2}(2){t}_{2}(3)& {t}_{2}(3){t}_{2}(1)\\ {({t}_{3}(1))}^{2}& {({t}_{3}(2))}^{2}& {({t}_{3}(3))}^{2}& {t}_{3}(1){t}_{3}(2)& {t}_{3}(2){t}_{3}(3)& {t}_{3}(3){t}_{3}(1)\\ {\mathrm{2t}}_{2}(1){t}_{3}(1)& {\mathrm{2t}}_{2}(2){t}_{3}(2)& {\mathrm{2t}}_{2}(3){t}_{3}(3)& {t}_{2}(1){t}_{3}(2)+{t}_{3}(1){t}_{2}(2)& {t}_{2}(2){t}_{3}(3)+{t}_{3}(2){t}_{2}(3)& {t}_{2}(3){t}_{3}(1)+{t}_{3}(3){t}_{3}(1)\\ {\mathrm{2t}}_{3}(1){t}_{1}(1)& {\mathrm{2t}}_{3}(2){t}_{1}(2)& {\mathrm{2t}}_{3}(3){t}_{1}(3)& {t}_{3}(1){t}_{1}(2)+{t}_{1}(1){t}_{3}(2)& {t}_{3}(2){t}_{1}(3)+{t}_{1}(2){t}_{3}(3)& {t}_{3}(3){t}_{1}(1)+{t}_{1}(3){t}_{3}(1)\end{array}\right]\\ \left[\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 1& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 1& 0\end{array}\right]\end{array}`


* calculation of the inverse Jacobian matrix :math:`{J}^{\mathrm{-}1}` and the determinant :math:`\text{det}J`


* calculation of :math:`{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{3}` with:

:math:`{\tilde{J}}^{\mathrm{-}1}\mathrm{=}\left[\begin{array}{ccc}{J}^{\mathrm{-}1}& 0& 0\\ 0& {J}^{\mathrm{-}1}& 0\\ 0& 0& {J}^{\mathrm{-}1}\end{array}\right];{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{3}\mathrm{=}\left[\mathrm{\cdots }\frac{h}{2}\left[\begin{array}{ccc}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}& 0& 0\\ {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}& 0& 0\\ {N}_{I}^{(2)}& 0& 0\\ 0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}& 0\\ 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}& 0\\ 0& {N}_{I}^{(2)}& 0\\ 0& 0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}\\ 0& 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}\\ 0& 0& {N}_{I}^{(2)}\end{array}\right]\mathrm{\cdots }I\mathrm{=}\mathrm{1,}\text{NB2}\right]`


* calculation of the third deformation operator :math:`{\tilde{B}}_{3}\mathrm{=}\mathit{HQ}{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{3}`


* calculation and numerical integration :math:`{Z}_{I}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{\zeta }}{\tilde{B}}_{3}^{T}\tilde{S}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}`


*calculation of the generalized tensor of**global** stresses

:math:`\stackrel{9\mathrm{\times }9}{\stackrel{ˉ}{S}}\mathrm{=}\left[\begin{array}{ccc}\mathrm{[}\stackrel{3\mathrm{\times }3}{S}\mathrm{]}& 0& 0\\ 0& \left[S\right]& 0\\ 0& 0& \left[S\right]\end{array}\right]`


* calculation of :math:`{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\text{avec}`:

:math:`\begin{array}{c}\mathrm{\cdots }\left[\left[\begin{array}{ccc}{N}_{I,{\xi }_{1}}^{(1)}& 0& 0\\ {N}_{I,{\xi }_{2}}^{(1)}& 0& 0\\ 0& 0& 0\\ 0& {N}_{I,{\xi }_{1}}^{(1)}& 0\\ 0& {N}_{I,{\xi }_{2}}^{(1)}& 0\\ 0& 0& 0\\ 0& 0& {N}_{I,{\xi }_{1}}^{(1)}\\ 0& 0& {N}_{I,{\xi }_{2}}^{(1)}\\ 0& 0& 0\end{array}\right]\frac{h}{2}\left[\begin{array}{ccc}0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{z}& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{y}\\ 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{z}& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{y}\\ 0& {N}_{1}^{(2)}{n}_{z}& \mathrm{-}{N}_{1}^{(2)}{n}_{y}\\ \mathrm{-}{x}_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{z}& 0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{x}\\ \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{z}& 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{x}\\ \mathrm{-}{N}_{1}^{(2)}{n}_{z}& 0& {N}_{1}^{(2)}{n}_{x}\\ {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{y}& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{x}& 0\\ {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{y}& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{x}& 0\\ {N}_{1}^{(2)}{n}_{y}& \mathrm{-}{N}_{1}^{(2)}{n}_{x}& 0\end{array}\right]\right]\mathrm{\cdots }I\mathrm{=}\mathrm{1,}\text{NB}1\\ \mathrm{\mid }\frac{h}{2}\left[\begin{array}{ccc}0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{z}& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{y}\\ 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{z}& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{y}\\ 0& {N}_{\text{NB}2}^{(2)}{n}_{z}& \mathrm{-}{N}_{1}^{(2)}{n}_{y}\\ \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{z}& 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{x}\\ \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{z}& 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{x}\\ \mathrm{-}{N}_{\text{NB}2}^{(2)}{n}_{z}& 0& {N}_{\text{NB}2}^{(2)}{n}_{x}\\ {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{y}& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{x}& 0\\ {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{y}& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{x}& 0\\ {N}_{\text{NB}2}^{(2)}{n}_{y}& \mathrm{-}{N}_{\text{NB}2}^{(2)}{n}_{x}& 0\end{array}\right]\\ {\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\mathrm{=}\begin{array}{c}\left[\right]\left[\right]\end{array}\mathrm{[}\mathrm{]}\end{array}`


* calculation and numerical integration of the classical geometric rigidity matrix

:math:`{K}_{\sigma }^{{e}_{\text{classique}}}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{\zeta }}{\left[{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\right]}^{T}\stackrel{ˉ}{S}\left[{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\right]\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}`

**End loop on integration points**

**Loop on all Lagrange knots with distinction from the super knot**


* calculation of :math:`\left[{z}_{I}\mathrm{\times }\right]` knowing that :math:`{Z}_{I}=\left\{\begin{array}{c}\text{.}\\ \text{.}\\ \text{.}\\ {z}_{I}\\ \text{.}\\ \text{.}\\ \text{.}\\ I=\mathrm{1,}\text{NB}2\end{array}\right\}`


* calculation of the normal vector :math:`{n}_{I}` and its anti-symmetric matrix :math:`\left[{n}_{I}\mathrm{\times }\right]`


* calculation of the non-classical geometric rigidity matrix

:math:`{\stackrel{3\mathrm{\times }3}{{K}_{\sigma }^{e}}}_{\text{non classique}}(I,I)\mathrm{=}\left[{z}_{I}\mathrm{\times }\right]\left[{n}_{I}\mathrm{\times }\right]I\mathrm{=}\mathrm{1,}\text{NB2}`


* addition of :math:`{\stackrel{3\mathrm{\times }3}{{K}_{\sigma }^{e}}}_{\text{non classique}}(I,I)` with distinction of the super node

**End loop on the knots**

**Storage of the upper triangular part of** :math:`{K}_{\sigma }^{e}`

**FIN**


**: Flow chart of geometric nonlinear calculation**

**Local landmarks at NB2 nodes** :math:`{\left[{t}_{1}:{t}_{2}:n\right]}_{I}`

**Start** **JN loop on the NB2 knots**

**IF** **JN** :math:`` **NB1**


* retrieving the total displacement vector already updated by MAJOUR:

:math:`{u}_{I}(\text{ii})\mathrm{=}\text{ZR}(\text{IDEPLP}+\text{IDEPLM}\mathrm{-}1+6\mathrm{\times }(\text{JN}\mathrm{-}1)+\text{ii});\text{ii}\mathrm{=}\mathrm{1,3};(\text{JN}\mathrm{=}\mathrm{1,}\text{NB}1)`


* retrieving the total rotation vector already updated by MAJOUR

:math:`{\theta }_{I}(\text{ii})\mathrm{=}\text{ZR}(\text{IDEPLP}\mathrm{-}1+6\mathrm{\times }(\text{JN}\mathrm{-}1)+\text{ii}+3);\text{ii}\mathrm{=}\mathrm{1,3};(\text{JN}\mathrm{=}\mathrm{1,}\text{NB}1)`

ELSE JN


* retrieving the total rotation vector

:math:`{\theta }_{I}(\text{ii})\mathrm{=}\text{ZR}(\text{IDEPLP}\mathrm{-}1+6\mathrm{\times }\text{NB}1+\text{ii});\text{ii}\mathrm{=}\mathrm{1,3};(\text{JN}\mathrm{=}\text{NB}2)`

**END IF** **N**


* calculation of the rotation matrix :math:`{\Lambda }_{I}\mathrm{=}\text{exp}\left[{\theta }_{I}\mathrm{\times }\right]`


* transformed from normal :math:`{n}_{I}^{\varphi }\mathrm{=}{\Lambda }_{I}{n}_{I}`

**End** **Loop on** :math:`\mathit{NB2}` **knots**

Calculation of :math:`{p}^{e}\mathrm{=}\left\{\begin{array}{c}\mathrm{⋮}\\ {(\begin{array}{c}u\\ v\\ w\\ {n}_{x}^{\varphi }\mathrm{-}{n}_{x}\\ {n}_{y}^{\varphi }\mathrm{-}{n}_{y}\\ {n}_{z}^{\varphi }\mathrm{-}{n}_{z}\end{array})}_{I}\\ \mathrm{⋮}\\ I\mathrm{=}\mathrm{1,}\text{NB}1\\ \mathrm{⋮}\\ {(\begin{array}{c}{n}_{x}^{\varphi }\mathrm{-}{n}_{x}\\ {n}_{y}^{\varphi }\mathrm{-}{n}_{y}\\ {n}_{z}^{\varphi }\mathrm{-}{n}_{z}\end{array})}_{\text{NB}2}\end{array}\right\}`

**Start** **Loop INTSR on normal reduced Gauss integration points**


* construction of some of the operators :math:`{\tilde{B}}_{1},{\tilde{B}}_{2}`

to J = 1, INTSR integration points to be able to extrapolate them

**End** **Loop INTSR on normal reduced Gauss integration points**

**Start** **Loop INTSN on Gauss normal numerical integration points**


* construction of the transformation matrix:

:math:`\mathrm{P}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\mathrm{=}\left[\begin{array}{c}{\mathrm{t}}_{1}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\\ {\mathrm{t}}_{2}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\\ {\mathrm{t}}_{3}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\end{array}\right]`

Where :math:`{t}_{3}({\xi }_{1},{\xi }_{2},{\xi }_{3})\mathrm{=}n({\xi }_{1},{\xi }_{2})`


* calculation of the inverse Jacobian matrix :math:`{J}^{-1}` and the determinant :math:`\text{det}J`


* calculation of :math:`{\tilde{J}}^{-1}=\left[\begin{array}{ccc}{J}^{-1}& 0& 0\\ 0& {J}^{-1}& 0\\ 0& 0& {J}^{-1}\end{array}\right]`


* calculation of the second matrix of derivatives of form functions :math:`{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{1}`

:math:`\begin{array}{c}\mathrm{\cdots }\left[\left[\begin{array}{ccc}{N}_{I,{\xi }_{1}}^{(1)}& 0& 0\\ {N}_{I,{\xi }_{2}}^{(1)}& 0& 0\\ 0& 0& 0\\ 0& {N}_{I,{\xi }_{1}}^{(1)}& 0\\ 0& {N}_{I,{\xi }_{2}}^{(1)}& 0\\ 0& 0& 0\\ 0& 0& {N}_{I,{\xi }_{1}}^{(1)}\\ 0& 0& {N}_{I,{\xi }_{2}}^{(1)}\\ 0& 0& 0\end{array}\right]\frac{h}{2}\left[\begin{array}{ccc}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}& 0& 0\\ {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}& 0& 0\\ {N}_{I}^{(2)}& 0& 0\\ 0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}& 0\\ 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}& 0\\ 0& {N}_{I}^{(2)}& 0\\ 0& 0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}\\ 0& 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}\\ 0& 0& {N}_{I}^{(2)}\end{array}\right]\right]\mathrm{...}I\mathrm{=}\mathrm{1,}\text{NB}1\\ \mathrm{\mid }\frac{h}{2}\left[\begin{array}{ccc}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}& 0& 0\\ {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}& 0& 0\\ {N}_{\text{NB}2}^{(2)}& 0& 0\\ 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}& 0\\ 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}& 0\\ 0& {N}_{\text{NB}2}^{(2)}& 0\\ 0& 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}\\ 0& 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}\\ 0& 0& {N}_{\text{NB}2}^{(2)}\end{array}\right]\\ {\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{1}\mathrm{=}\begin{array}{c}\left[\right]\left[\right]\end{array}\mathrm{[}\mathrm{]}\end{array}`


* calculation of :math:`\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\mathrm{=}{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{1}{p}^{e}`


* calculation of :math:`A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\mathrm{=}\left[\begin{array}{ccccccccc}{u}_{,x}& 0& 0& {v}_{,x}& 0& 0& {w}_{,x}& 0& 0\\ 0& {u}_{,y}& 0& 0& {v}_{,y}& 0& 0& {w}_{,y}& 0\\ 0& 0& {u}_{,z}& 0& 0& {v}_{,z}& 0& 0& {w}_{,z}\\ {u}_{,y}& {u}_{,x}& 0& {v}_{,y}& {v}_{,x}& 0& {w}_{,y}& {w}_{,x}& 0\\ {u}_{,z}& 0& {u}_{,x}& {v}_{,z}& 0& {v}_{,x}& {w}_{,z}& 0& {w}_{,x}\\ 0& {u}_{,z}& {u}_{,y}& 0& {v}_{,z}& {v}_{,y}& 0& {w}_{,z}& {w}_{,y}\end{array}\right]`


* calculation of :math:`H\left[Q+\frac{1}{2}A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\right]`

:math:`\begin{array}{c}H\mathrm{=}\\ \left[\begin{array}{cccccc}{({t}_{1}(1))}^{2}& {({t}_{1}(2))}^{2}& {({t}_{1}(3))}^{2}& {t}_{1}(1){t}_{1}(2)& {t}_{1}(2){t}_{1}(3)& {t}_{1}(3){t}_{1}(1)\\ {({t}_{2}(1))}^{2}& {({t}_{2}(2))}^{2}& {({t}_{2}(3))}^{2}& {t}_{2}(1){t}_{2}(2)& {t}_{2}(2){t}_{2}(3)& {t}_{2}(3){t}_{2}(1)\\ {({t}_{3}(1))}^{2}& {({t}_{3}(2))}^{2}& {({t}_{3}(3))}^{2}& {t}_{3}(1){t}_{3}(2)& {t}_{3}(2){t}_{3}(3)& {t}_{3}(3){t}_{3}(1)\\ {\mathrm{2t}}_{2}(1){t}_{3}(1)& {\mathrm{2t}}_{2}(2){t}_{3}(2)& {\mathrm{2t}}_{2}(3){t}_{3}(3)& {t}_{2}(1){t}_{3}(2)+{t}_{3}(1){t}_{2}(2)& {t}_{2}(2){t}_{3}(3)+{t}_{3}(2){t}_{2}(3)& {t}_{2}(3){t}_{3}(1)+{t}_{3}(3){t}_{3}(1)\\ {\mathrm{2t}}_{3}(1){t}_{1}(1)& {\mathrm{2t}}_{3}(2){t}_{1}(2)& {\mathrm{2t}}_{3}(3){t}_{1}(3)& {t}_{3}(1){t}_{1}(2)+{t}_{1}(1){t}_{3}(2)& {t}_{3}(2){t}_{1}(3)+{t}_{1}(2){t}_{3}(3)& {t}_{3}(3){t}_{1}(1)+{t}_{1}(3){t}_{3}(1)\end{array}\right]\end{array}`


* calculation of the first deformation operator

:math:`{\tilde{B}}_{1}\mathrm{=}H\left[Q+\frac{1}{2}A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\right]{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{1}`


* calculation of the local deformation vector :math:`\tilde{E}\mathrm{=}{\tilde{B}}_{1}{p}^{e}`


* calculation of the second matrix of derivatives of form functions :math:`{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}`

:math:`\begin{array}{c}\mathrm{\cdots }\left[\left[\begin{array}{ccc}{N}_{I,{\xi }_{1}}^{(1)}& 0& 0\\ {N}_{I,{\xi }_{2}}^{(1)}& 0& 0\\ 0& 0& 0\\ 0& {N}_{I,{\xi }_{1}}^{(1)}& 0\\ 0& {N}_{I,{\xi }_{2}}^{(1)}& 0\\ 0& 0& 0\\ 0& 0& {N}_{I,{\xi }_{1}}^{(1)}\\ 0& 0& {N}_{I,{\xi }_{2}}^{(1)}\\ 0& 0& 0\end{array}\right]\frac{h}{2}\left[\begin{array}{ccc}0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{z}^{\varphi }& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{y}^{\varphi }\\ 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{z}^{\varphi }& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{y}^{\varphi }\\ 0& {N}_{1}^{(2)}{n}_{z}^{\varphi }& \mathrm{-}{N}_{1}^{(2)}{n}_{y}^{\varphi }\\ \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{z}^{\varphi }& 0& {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{x}^{\varphi }\\ \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{z}^{\varphi }& 0& {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{x}^{\varphi }\\ \mathrm{-}{N}_{1}^{(2)}{n}_{z}^{\varphi }& 0& {N}_{1}^{(2)}{n}_{x}^{\varphi }\\ {\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{y}^{\varphi }& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{1}}^{(2)}{n}_{x}^{\varphi }& 0\\ {\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{y}^{\varphi }& \mathrm{-}{\xi }_{3}{N}_{I,{\xi }_{2}}^{(2)}{n}_{x}^{\varphi }& 0\\ {N}_{1}^{(2)}{n}_{y}^{\varphi }& \mathrm{-}{N}_{1}^{(2)}{n}_{x}^{\varphi }& 0\end{array}\right]\right]\mathit{LI}\mathrm{=}\mathrm{1,}\text{NB}1\\ \mathrm{\mid }\frac{h}{2}\left[\begin{array}{ccc}0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{z}^{\varphi }& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{y}^{\varphi }\\ 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{z}^{\varphi }& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{y}^{\varphi }\\ 0& {N}_{\text{NB}2}^{(2)}{n}_{z}^{\varphi }& \mathrm{-}{N}_{\text{NB}2}^{(2)}{n}_{y}^{\varphi }\\ \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{z}^{\varphi }& 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{x}^{\varphi }\\ \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{z}^{\varphi }& 0& {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{x}^{\varphi }\\ \mathrm{-}{N}_{\text{NB}2}^{(2)}{n}_{z}^{\varphi }& 0& {N}_{\text{NB}2}^{(2)}{n}_{x}^{\varphi }\\ {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{y}^{\varphi }& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{1}}^{(2)}{n}_{x}^{\varphi }& 0\\ {\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{y}^{\varphi }& \mathrm{-}{\xi }_{3}{N}_{\text{NB}\mathrm{2,}{\xi }_{2}}^{(2)}{n}_{x}^{\varphi }& 0\\ {N}_{\text{NB}2}^{(2)}{n}_{y}^{\varphi }& \mathrm{-}{N}_{\text{NB}2}^{(2)}{n}_{x}^{\varphi }& 0\end{array}\right]\\ {\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\mathrm{=}\begin{array}{c}\left[\right]\left[\right]\end{array}\mathrm{[}\mathrm{]}\end{array}`


* calculation of the second deformation operator

:math:`{\tilde{B}}_{2}\mathrm{=}H(Q+A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})){\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}`


* calculation and numerical integration :math:`{r}^{e}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{\zeta }}{\tilde{B}}_{2}^{T}\tilde{S}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}`


* calculation of the behavior matrix :math:`D`


* calculation and numerical integration :math:`{K}_{m}^{e}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{\zeta }}{\tilde{B}}_{2}^{T}D{\tilde{B}}_{2}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}`


* construction of the symmetric tensor :math:`3\mathrm{\times }3` of local constraints :math:`\mathrm{[}\tilde{S}\mathrm{]}`


* calculation of the symmetric tensor :math:`3\mathrm{\times }3` of global constraints :math:`\mathrm{[}S\mathrm{]}\mathrm{=}{P}^{T}\mathrm{[}\tilde{S}\mathrm{]}P`


* calculation of :math:`{\tilde{B}}_{3}^{T}\mathrm{=}H\left[S\right]{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{3}`

**calculation and numerical integration** :math:`{Z}_{I}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{z}}{\tilde{B}}_{3}^{T}\tilde{S}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}`


*calculation of the generalized tensor of**global** constraints :math:`\stackrel{9\mathrm{\times }9}{\stackrel{ˉ}{S}}\mathrm{=}\left[\begin{array}{ccc}\mathrm{[}\stackrel{3\mathrm{\times }3}{S}\mathrm{]}& 0& 0\\ 0& \left[S\right]& 0\\ 0& 0& \left[S\right]\end{array}\right]`


* calculation and numerical integration of classical stiffness

:math:`{K}_{g}^{{e}_{\text{classique}}}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{z}}{\left[{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\right]}^{T}\stackrel{ˉ}{S}\left[{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\right]\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}`

**End loop INTSN on integration points**

**Start** **N loop on** :math:`\mathit{NB2}` **knots**


* calculation of :math:`\left[{z}_{I}\mathrm{\times }\right]` knowing that :math:`{Z}_{I}=(\begin{array}{c}\text{.}\\ \text{.}\\ \text{.}\\ {z}_{I}\\ \text{.}\\ \text{.}\\ \text{.}\\ I=\mathrm{1,}\text{NB}2\end{array})`


* calculation of :math:`\left[{n}_{I}\mathrm{\times }\right]`


* calculation of the non-symmetric matrix :math:`{\stackrel{3\mathrm{\times }3}{{K}_{g}^{e}}}_{\text{non classique}}(I,I)\mathrm{=}\left[{z}_{I}\mathrm{\times }\right]\left[{n}_{I}\mathrm{\times }\right]`

**IF** **JN** :math:`\mathrm{\le }` **NB1**


* addition of :math:`{\stackrel{3\mathrm{\times }3}{{K}_{g}^{e}}}_{\text{non classique}}(I,I)` with distinction of extra-node

**ELSE** **N**


* assignment of :math:`{\stackrel{3\mathrm{\times }3}{{K}_{g}^{e}}}_{\text{non classique}}(I,I)` with distinction from extra-node

**END IF** **N**

**Storing the entire non-symmetric matrix** :math:`{K}^{{e}_{T}}`