10. Description of document versions#

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 \(\mathit{NB2}\) knots \({\left[{t}_{1}\mathrm{:}{t}_{2}\mathrm{:}n\right]}_{I}\)

Loop over the points of normal numerical integration of Gauss

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

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

formation of the symmetric tensor \(3\mathrm{\times }3\) of local constraints \(\mathrm{[}\tilde{S}\mathrm{]}\)
construction of the transformation matrix \(\mathrm{P}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\mathrm{=}\left[\begin{array}{c}{\mathrm{t}}_{1}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\\ {\mathrm{t}}_{2}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\\ {\mathrm{t}}_{3}^{\mathrm{T}}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\end{array}\right]\) where \({\mathrm{t}}_{3}({\xi }_{\mathrm{1,}}{\xi }_{\mathrm{2,}}{\xi }_{3})\mathrm{=}{n}_{1}({\xi }_{2})\)
calculation of the symmetric tensor \(3\mathrm{\times }3\) of global constraints \(\mathrm{[}S\mathrm{]}\mathrm{=}{P}^{T}\mathrm{[}\tilde{S}\mathrm{]}P\)
for the non-classical term, calculation of \(\text{HQ}\mathrm{=}\left[\frac{\mathrm{[}\stackrel{3\mathrm{\times }9}{\text{HSFM}}\mathrm{]}}{\mathrm{[}\stackrel{2\mathrm{\times }9}{\text{HSS}}\mathrm{]}}\right]\)

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

calculation of the inverse Jacobian matrix \({J}^{\mathrm{-}1}\) and the determinant \(\text{det}J\)
calculation of \({\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{3}\) with:

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

calculation of the third deformation operator \({\tilde{B}}_{3}\mathrm{=}\mathit{HQ}{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{3}\)
calculation and numerical integration \({Z}_{I}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{\zeta }}{\tilde{B}}_{3}^{T}\tilde{S}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}\)

calculation of the generalized tensor of**global* stresses

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

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

\(\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

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

End loop on integration points

Loop on all Lagrange knots with distinction from the super knot

calculation of \(\left[{z}_{I}\mathrm{\times }\right]\) knowing that \({Z}_{I}=\left\{\begin{array}{c}\text{.}\\ \text{.}\\ \text{.}\\ {z}_{I}\\ \text{.}\\ \text{.}\\ \text{.}\\ I=\mathrm{1,}\text{NB}2\end{array}\right\}\)
calculation of the normal vector \({n}_{I}\) and its anti-symmetric matrix \(\left[{n}_{I}\mathrm{\times }\right]\)
calculation of the non-classical geometric rigidity matrix

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

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

End loop on the knots

Storage of the upper triangular part of \({K}_{\sigma }^{e}\)

FIN

: Flow chart of geometric nonlinear calculation

Local landmarks at NB2 nodes \({\left[{t}_{1}:{t}_{2}:n\right]}_{I}\)

Start JN loop on the NB2 knots

IF JN :math:`` NB1

retrieving the total displacement vector already updated by MAJOUR:

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

retrieving the total rotation vector already updated by MAJOUR

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

ELSE JN

retrieving the total rotation vector

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

END IF N

calculation of the rotation matrix \({\Lambda }_{I}\mathrm{=}\text{exp}\left[{\theta }_{I}\mathrm{\times }\right]\)
transformed from normal \({n}_{I}^{\varphi }\mathrm{=}{\Lambda }_{I}{n}_{I}\)

End Loop on \(\mathit{NB2}\) knots

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

Start Loop INTSR on normal reduced Gauss integration points

construction of some of the operators \({\tilde{B}}_{1},{\tilde{B}}_{2}\)

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:

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

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

calculation of the inverse Jacobian matrix \({J}^{-1}\) and the determinant \(\text{det}J\)
calculation of \({\tilde{J}}^{-1}=\left[\begin{array}{ccc}{J}^{-1}& 0& 0\\ 0& {J}^{-1}& 0\\ 0& 0& {J}^{-1}\end{array}\right]\)
calculation of the second matrix of derivatives of form functions \({\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{1}\)

calculation of \(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\mathrm{=}{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{1}{p}^{e}\)
calculation of \(A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\mathrm{=}\left[\begin{array}{ccccccccc}{u}_{,x}& 0& 0& {v}_{,x}& 0& 0& {w}_{,x}& 0& 0\\ 0& {u}_{,y}& 0& 0& {v}_{,y}& 0& 0& {w}_{,y}& 0\\ 0& 0& {u}_{,z}& 0& 0& {v}_{,z}& 0& 0& {w}_{,z}\\ {u}_{,y}& {u}_{,x}& 0& {v}_{,y}& {v}_{,x}& 0& {w}_{,y}& {w}_{,x}& 0\\ {u}_{,z}& 0& {u}_{,x}& {v}_{,z}& 0& {v}_{,x}& {w}_{,z}& 0& {w}_{,x}\\ 0& {u}_{,z}& {u}_{,y}& 0& {v}_{,z}& {v}_{,y}& 0& {w}_{,z}& {w}_{,y}\end{array}\right]\)
calculation of \(H\left[Q+\frac{1}{2}A(\frac{\mathrm{\partial }u}{\mathrm{\partial }x})\right]\)

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

calculation of the first deformation operator

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

calculation of the local deformation vector \(\tilde{E}\mathrm{=}{\tilde{B}}_{1}{p}^{e}\)
calculation of the second matrix of derivatives of form functions \({\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{2}\)

\(\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

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

calculation and numerical integration \({r}^{e}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{\zeta }}{\tilde{B}}_{2}^{T}\tilde{S}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}\)
calculation of the behavior matrix \(D\)
calculation and numerical integration \({K}_{m}^{e}\mathrm{=}{\mathrm{\int }}_{{\Omega }_{\zeta }}{\tilde{B}}_{2}^{T}D{\tilde{B}}_{2}\text{det}Jd{\xi }_{1}d{\xi }_{2}d{\xi }_{3}\)
construction of the symmetric tensor \(3\mathrm{\times }3\) of local constraints \(\mathrm{[}\tilde{S}\mathrm{]}\)
calculation of the symmetric tensor \(3\mathrm{\times }3\) of global constraints \(\mathrm{[}S\mathrm{]}\mathrm{=}{P}^{T}\mathrm{[}\tilde{S}\mathrm{]}P\)
calculation of \({\tilde{B}}_{3}^{T}\mathrm{=}H\left[S\right]{\tilde{J}}^{\mathrm{-}1}{\left[\frac{\mathrm{\partial }N}{\mathrm{\partial }\xi }\right]}_{3}\)

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

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

calculation and numerical integration of classical stiffness

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

End loop INTSN on integration points

Start N loop on \(\mathit{NB2}\) knots

calculation of \(\left[{z}_{I}\mathrm{\times }\right]\) knowing that \({Z}_{I}=(\begin{array}{c}\text{.}\\ \text{.}\\ \text{.}\\ {z}_{I}\\ \text{.}\\ \text{.}\\ \text{.}\\ I=\mathrm{1,}\text{NB}2\end{array})\)
calculation of \(\left[{n}_{I}\mathrm{\times }\right]\)
calculation of the non-symmetric matrix \({\stackrel{3\mathrm{\times }3}{{K}_{g}^{e}}}_{\text{non classique}}(I,I)\mathrm{=}\left[{z}_{I}\mathrm{\times }\right]\left[{n}_{I}\mathrm{\times }\right]\)

IF JN \(\mathrm{\le }\) NB1

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

ELSE N

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

END IF N

Storing the entire non-symmetric matrix \({K}^{{e}_{T}}\)