d9.05.03 Architecture THM. Intégration des équations d’équilibre#

Résumé:

Cette note présente les arguments et variables informatiques utilisés dans les routines THM. Cette note commence par une présentation sommaire des équations, qui ne se substitue pas à la doc.R, seule référence dans le domaine.

Lois de comportement#

Mécanique#

Ecriture générale#

\(\lbrace \begin{array}{c}{\sigma}^{+}={\sigma}^{+}({\epsilon}^{+},{p}_{1}^{+},{p}_{2}^{+},{T}^{+};{\epsilon}^{-},{p}_{1}^{-},{p}_{2}^{-},{T}^{-},{\sigma}^{-},{\chi }^{-})\\ {\chi }^{+}={\chi }^{+}({\epsilon}^{+},{p}_{1}^{+},{p}_{2}^{+},{T}^{+};{\epsilon}^{-},{p}_{1}^{-},{p}_{2}^{-},{T}^{-},{\sigma}^{-},{\chi }^{-})\end{array}\) éq 2.1.1-1

Cas des contraintes effectives#

Dans le cas de l’hypothèse des contraintes effectives, cette fonction se décomposera sous la forme:

\(\begin{array}{c}\sigma =\sigma '+{\sigma}_{p}I\\ \sigma '\text{est le tenseur des contraintes effectives:}\\ {\sigma}_{p}\text{est un scalaire}\end{array}\)

\(\lbrace \begin{array}{c}\sigma {'}^{+}=\sigma {'}^{+}({\epsilon}^{+},{T}^{+};{\epsilon}^{-},{T}^{-},\sigma {'}^{-},{\chi }_{\sigma}^{-})\\ {\chi }_{\sigma}^{+}={\chi }_{\sigma}^{+}({\epsilon}^{+},{T}^{+};{\epsilon}^{-},{T}^{-},\sigma {'}^{-},{\chi }_{\sigma}^{-})\end{array}\) éq 2.1.2-1

\(\lbrace \begin{array}{c}{\sigma}_{p}^{+}={\sigma}_{p}^{+}({p}_{1}^{+},{p}_{2}^{+};{p}_{1}^{-},{p}_{2}^{-},{\chi }_{H}^{-})\\ {\chi }_{H}^{+}={\chi }_{H}^{+}({p}_{1}^{+},{p}_{2}^{+};{p}_{1}^{-},{p}_{2}^{-},{\chi }_{H}^{-})\end{array}\) éq 2.1.2-2

On remarque que dans cette décomposition:

  1. la dépendance par rapport à la thermique a été laissée dans les contraintes effectives; typiquement, on pense que les lois sur les contraintes effectives s’écrivent comme en thermo mécanique classique:

\(\sigma {'}^{+}=\sigma {'}^{+}({\epsilon}^{+}-{\alpha}^{+}{T}^{+};{\epsilon}^{-}-{\alpha}^{-}{T}^{-},\sigma {'}^{-},{\chi }_{\sigma}^{-})\)

  1. on a distingué les variables internes relatives à la loi de comportement sur les contraintes effectives, que l’on a écrites \({\chi }_{\sigma}\) , les variables internes d’origine hydraulique que l’on a écrites \({\chi }_{H}\) et les variables internes d’origine thermique que l’on a écrites \({\chi }_{T}\) (voir paragraphes suivants).

Choix des contraintes#

Du fait de l’utilisation assez fréquente de l’hypothèse des contraintes effectives, on décide que le vecteur des contraintes pour la partie mécanique contient dans tous les cas le tenseur des contraintes effectives \(\sigma '\) et le scalaire \({\sigma}_{p}\) . Dans le cas général où l’hypothèse des contraintes effectives n’est pas retenue, on aura simplement: . \({\sigma}_{p}=0\)

C’est donc à la charge du module d’intégration des équations d’équilibre de faire la somme: \(\sigma =\sigma '+{\sigma}_{p}I\) .

Hydraulique#

La loi de comportement hydraulique fournira les relations suivantes:

\(\lbrace \begin{array}{c}{m}_{c}^{p+}={m}_{c}^{p+}({\epsilon}^{+},{p}_{1}^{+},{p}_{2}^{+},{T}^{+};{\epsilon}^{-},{p}_{1}^{-},{p}_{2}^{-},{T}^{-},{m}_{d}^{q-},{\text{M}}_{d}^{q-},{\mathrm{\chi }}_{H}^{-})\\ {\text{M}}_{c}^{p+}={\text{M}}_{c}^{p+}\left(\begin{array}{c}{\epsilon}^{+},{p}_{1}^{+},\nabla {p}_{1}^{+},{p}_{2}^{+},\nabla {p}_{2}^{+},{T}^{+},\nabla {T}^{+};\\ {\epsilon}^{-},{p}_{1}^{-},\nabla {p}_{1}^{-},{p}_{2}^{-},\nabla {p}_{2}^{-},{T}^{-},\nabla {T}^{-},{\text{M}}_{d}^{q-},{\mathrm{\chi }}_{H}^{-}:{\text{F}}^{m+}\end{array}\right)\\ {\mathrm{\chi }}_{H}^{+}={\mathrm{\chi }}_{H}^{+}({\epsilon}^{+},{p}_{1}^{+},{p}_{2}^{+},{T}^{+};{\epsilon}^{-},{p}_{1}^{-},{p}_{2}^{-},{T}^{-},{m}_{1}^{-},{m}_{2}^{-},{\mathrm{\chi }}_{H}^{-})\end{array}\begin{array}{c}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\end{array}\rbrace \forall c\mathit{et}\forall p\phantom{\rule{2em}{0ex}}\mathit{de}1à{\mathit{np}}_{c}\) éq 2.2-1

On remarque que le champ de gravité est une donnée de la loi de comportement hydraulique par ce que l’évolution du vecteur de flux suit des relations du type: \(\text{M}={\lambda}_{H}{\rho}^{\mathit{fl}}[-\nabla P+{\rho}^{\mathit{fl}}{\text{F}}^{m}]\) .

Thermique#

Les lois de comportement fourniront:

\(\begin{array}{c}\lbrace \begin{array}{c}Q{'}^{+}=Q{'}^{+}({\epsilon}^{+},{p}_{1}^{+},{p}_{2}^{+},{T}^{+};{\epsilon}^{-},{p}_{1}^{-},{p}_{2}^{-},{T}^{-},S{'}^{-})\\ {h}_{cm}^{p+}={h}_{cm}^{p+}({\epsilon}^{+},{p}_{1}^{+},{p}_{2}^{+},{T}^{+};{\epsilon}^{-},{p}_{1}^{-},{p}_{2}^{-},{T}^{-},{s}_{\mathit{dm}}^{q-})\forall c\mathit{et}\forall p\mathit{de}1à{\mathit{np}}_{c}\\ {\text{q}}^{+}={\text{q}}^{+}({\epsilon}^{+},{p}_{1}^{+},{p}_{2}^{+},{T}^{+},\nabla {T}^{+};{\epsilon}^{-},{p}_{1}^{-},{p}_{2}^{-},{T}^{-},\nabla {T}^{-},{\text{q}}^{-})\\ {\chi }_{T}^{+}={\chi }_{T}^{+}({\epsilon}^{+},{p}_{1}^{+},{p}_{2}^{+},{T}^{+},\nabla {T}^{+};{\epsilon}^{-},{p}_{1}^{-},{p}_{2}^{-},{T}^{-},\nabla {T}^{-},{\chi }_{T}^{-})\end{array}\\ \text{Avec}{h}_{\mathit{dm}}^{q-}=({h}_{\mathrm{1m}}^{1-},{h}_{\mathrm{1m}}^{2-},{h}_{\mathrm{2m}}^{1-},{h}_{\mathrm{2m}}^{2-})\end{array}\) éq 2.3-1

Masse volumique homogénéisée#

\({r}^{+}={r}_{0}+{m}_{1}^{1+}+{m}_{1}^{2+}+{m}_{2}^{1+}+{m}_{2}^{2+}\) éq 2.4-1

Efforts généralisés#

Il ressort de ce qui est écrit plus haut que les contraintes généralisées sont:

\(\begin{array}{c}\underline{\underline{\sigma '}},{\sigma}_{p};\\ {m}_{1}^{1,}{\text{M}}_{1}^{1,}{h}_{\mathrm{1m}}^{1};{m}_{1}^{2,}{\text{M}}_{1}^{2,}{h}_{\mathrm{1m}}^{2};\\ {m}_{2}^{1,}{\text{M}}_{2}^{1,}{h}_{\mathrm{2m}}^{1};{m}_{2}^{2,}{\text{M}}_{2}^{2,}{h}_{\mathrm{2m}}^{2};\\ Q',\text{q}\end{array}\)

Les déformations généralisées associées sont:

\(\text{u},\underline{\underline{\epsilon}}(\text{u}):{p}_{1,}\nabla {p}_{1}:{p}_{2,}\nabla {p}_{2};T,\nabla T\)

Remarque:

Dans le cadre de la modélisation HM permanente saturée, les contraintes généralisées ne contiennent pas le terme d’apport massique.

Algorithme de résolution#

Algorithme non linéaire de résolution des équations d’équilibre#

Dans le cas général de la modélisation (coefficients variables, désaturation, convection) le problème variationnel présenté ci-dessus est non linéaire par rapport aux champs de déplacement, pression et température. Après discrétisation par éléments finis, on obtient un système matriciel non linéaire. La matrice de résolution contient de plus un terme non symétrique et est traité comme tel (pas de symétrisation de cette matrice pour utiliser des méthodes de minimum). On utilise dans tous les cas de modélisation le solveur non linéaire du Code_Aster STAT_NON_LINE reposant sur une méthode de Newton-Raphson, décrite en [R5.03.01]. Son principe est le suivant (les équations correspondant au traitement par dualisation des conditions aux limites ne sont pas indiquées explicitement ici).

L’équation d’équilibre thermo-poro-mécanique à l’instant \({t}^{+}\) , connaissant à l’instant précédent \(({\text{u}}_{-},{P}_{-},{T}_{-})\) , ainsi que les éventuelles variables interness’écrit :

\({F}_{i}({\text{u}}_{+},{P}_{+},{T}_{+})={L}_{e}({t}^{+})-G({\text{u}}_{-},{P}_{-},{T}_{-})\) ,

Pour trouver la solution de cette équation non linéaire, on construit une suite:

  • initialisée par une prédiction qui donne \(({\text{u}}_{0},{P}_{0},{T}_{0})=({\text{u}}_{-},{P}_{-},{T}_{-})+(\Delta {\text{u}}_{0},\Delta {P}_{0},\Delta {T}_{0})\) :

\({\mathit{DF}}_{i({\text{u}}_{-},{P}_{-},{T}_{-})}\circ (\Delta {\text{u}}_{0},\Delta {P}_{0},\Delta {T}_{0})={L}_{e}({t}^{+})-{L}_{e}({t}^{-})\)

  • corrigée par récurrence donnant:

\(({\text{u}}_{n+1},{P}_{n+1},{T}_{n+1})=({\text{u}}_{n},{P}_{n},{T}_{n})+(\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\)

\({\mathit{DF}}_{i}\circ (\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})=-{F}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})+{L}_{e}({t}^{+})-G({\text{u}}_{-},{P}_{-},{T}_{-})\)

Les notations suivantes ont été adoptées:

  • \({F}_{i}(\text{u},P,T)\) contient le travail de déformation, les contributions à l’instant actuel des termes de dissipation hydraulique et thermique exprimés au sein de la \(\theta\) ‑méthode, et des variations d’apport de masse fluide et d’entropie;

  • \({\mathit{DF}}_{i}\) désigne l’opérateur tangent, qui peut ne pas être actualisé à chaque itération en \(({\text{u}}_{n},{P}_{n},{T}_{n})\) , selon un compromis coût de calcul-vitesse de convergence; la convergence est vérifiée par un test sur la norme relative de la différence des itérés successifs (via le mot-clé INCO_GLOB_RELA);

  • \(G({\text{u}}_{-},{P}_{-},{T}_{-})\) contient les contributions à l’instant précédent des termes de dissipation hydraulique et thermique exprimées au sein de la \(\theta\) - méthode, et des variations d’apport de masse fluide et d’entropie;

  • \({L}_{e}(t)\) désigne le travail virtuel des forces «mortes» extérieures et d’apports extérieurs hydrauliques et de chaleur exprimés par la \(\theta\) -méthode.

à convergence à l’itération \(n+1\) , on opère une actualisation des champs . \(({\text{u}}_{+},{P}_{+},{T}_{+})=({\text{u}}_{n+1},{P}_{n+1},{T}_{n+1})\)

Dans la version présente de l’algorithme THM, nous avons décidé de regrouper tous les termes y compris ceux dus aux forces suiveuses et ceux du temps moins:

En posant:

\(-{R}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})=-{F}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})-G({\text{u}}_{-},{P}_{-},{T}_{-})\)

,

donc \({\mathit{DF}}_{i}={\mathit{DR}}_{i}\)

on a finalement:

\({\mathit{DF}}_{i}\circ (\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})=-{R}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})+{L}_{e}({t}^{+})\)

L’algorithme général d’équilibre s’écrira alors, pour un pas de temps:

Initialisations:

Calcul de \({L}_{e}({t}^{+})\) (option CHAR_MECA)

Calcul de \({\mathit{DF}}_{i({\text{u}}_{-},{P}_{-},{T}_{-})}\) (option RIGI_MECA-TANG)

Calcul de \((\Delta {\text{u}}_{0},\Delta {P}_{0},\Delta {T}_{0})\) par: \({\mathit{DF}}_{i({\text{u}}_{-},{P}_{-},{T}_{-})}\circ (\Delta {\text{u}}_{0},\Delta {P}_{0},\Delta {T}_{0})={L}_{e}({t}^{+})-{L}_{e}({t}^{-})\)

Itérations d’équilibre de Newton n

Si option FULL_MECA :

Calcul de \({\mathit{DF}}_{i({\text{u}}^{+},{P}^{+},{T}^{+})}\) et \(-{R}_{i}({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})\) :

Mise à jour matrice tangente: \({\mathit{DF}}_{i}={\mathit{DF}}_{i({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})}\)

Si option RAPH_MECA

Calcul de

../../../../_images/Object_383.svg

Calcul de \((\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\) par:

\({\mathit{DF}}_{i}\circ (\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})=-{R}_{i}({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})+{L}_{e}({t}^{+})\)

Actualisation :

\(({\text{u}}_{n+1}^{+},{P}_{n+1}^{+},{T}_{n+1}^{+})=({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})+(\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\)

SI test convergence OK

fin Newton: pas de temps suivant

Sinon

n = n+1

Boucle sur les éléments, les points de Gauss#

Comme dans tous les codes d’éléments finis, les termes sont calculés par boucle sur les éléments et boucle sur les points de Gauss:

\(\begin{array}{c}{R}_{i}({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})=\sum_{\mathit{el}}\sum_{g}{w}_{g}^{\mathit{el}}{R}_{gi}^{\mathit{el}}({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})\\ {\mathit{DF}}_{i({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})}=\sum_{\mathit{el}}\sum_{g}{w}_{g}^{\mathit{el}}{\mathit{DF}}_{gi({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})}^{\mathit{el}}\end{array}\)

Notons: \(\lbrace {X}^{\mathit{el}}\rbrace\) le vecteur des inconnues nodales, sur un élément fini el

par exemple \(\lbrace {X}^{\mathit{el}}\rbrace =\begin{array}{c}u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\\ u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\\ u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\end{array}\begin{array}{c}\begin{array}{c}\\ \\ \\ \\ \\ \end{array}\rbrace \text{noeud 1}\\ \begin{array}{c}\\ \\ \\ \\ \\ \end{array}\rbrace \text{noeud 2}\\ \begin{array}{c}\\ \\ \\ \\ \\ \end{array}\rbrace \text{noeud 3}\end{array}\)

Dans le présent paragraphe, pour simplifier la présentation, nous supposons que nous traitons d’un élément fini supportant des ddl de déplacement, deux ddl de pression et un ddl de température.

Notons \(\lbrace {\text{E}}_{g}^{\mathit{el}}\rbrace\) le vecteur des déformations généralisées au point de Gauss g de l’élément el

Par exemple:

\(\lbrace {\text{E}}_{g}^{\mathit{el}}\rbrace =\left\lbrace \begin{array}{c}\text{u}\\ \epsilon (\text{u})\\ {p}_{1}\\ \nabla {p}_{1}\\ {p}_{2}\\ \nabla {p}_{2}\\ T\\ \nabla T\end{array}\right\rbrace\)

Nous notons \(\lbrace {\Sigma}_{g}^{\mathit{el}}\rbrace\) le vecteur de contraintes généralisées pour le point de Gauss g de l’élément el

Par exemple, et toujours dans le cas le plus complet:

\(\lbrace {\Sigma}_{g}^{\mathit{el}}\rbrace =\left\lbrace \begin{array}{c}\underline{\underline{\sigma '}}\\ {\sigma}_{p}\\ {m}_{1}^{1}\\ {\text{M}}_{1}^{1}\\ {h}_{\mathrm{1m}}^{1}\\ {m}_{1}^{2}\\ {\text{M}}_{1}^{2}\\ {h}_{\mathrm{1m}}^{2}\\ {m}_{2}^{1}\\ {\text{M}}_{2}^{1}\\ {h}_{\mathrm{2m}}^{1}\\ {m}_{2}^{2}\\ {\text{M}}_{2}^{2}\\ {h}_{\mathrm{2m}}^{2}\\ Q'\\ \text{q}\end{array}\right\rbrace\)

Les routines éléments finis calculent la matrice: \({[B]}_{g}^{\mathit{el}}\) définie par:

\(\lbrace {E}_{g}^{\mathit{el}}\rbrace ={[B]}_{g}^{\mathit{el}}\lbrace X\rbrace\)

L’algorithme deviendra alors:

Initialisations:

Calcul de \({L}_{e}({t}^{+})\) (option CHAR_MECA)

Calcul de \({\mathit{DF}}_{i({\text{u}}_{-},{P}_{-},{T}_{-})}\) (option RIGI_MECA-TANG)

Calcul de \((\Delta {\text{u}}_{0},\Delta {P}_{0},\Delta {T}_{0})\) par: \({\mathit{DF}}_{i({\text{u}}_{-},{P}_{-},{T}_{-})}\circ (\Delta {\text{u}}_{0},\Delta {P}_{0},\Delta {T}_{0})={L}_{e}({t}^{+})-{L}_{e}({t}^{-})\)

Itérations d’équilibre de Newton n

Boucle éléments el

Boucle points de gauss g

Calcul \({[B]}_{g}^{\mathit{el}}\)

Calcul \(\left\lbrace {E}_{g}^{\mathit{el}-}\right\rbrace ={\left[B\right]}_{g}^{\mathit{el}}\left\lbrace {X}^{-}\right\rbrace\) et \(\left\lbrace {E}_{\mathit{gn}}^{\mathit{el}+}\right\rbrace ={\left[B\right]}_{g}^{\mathit{el}}\left\lbrace {X}_{n}^{+}\right\rbrace\)

Calcul \(\left\lbrace {\Sigma}_{\mathit{gn}}^{\mathit{el}+}\right\rbrace\) , \(-{R}_{\mathit{ig}}^{\mathit{el}}({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})\) et \({\mathit{DF}}_{gi({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})}^{\mathit{el}}\) (selon options) à partir de:

\(\left\lbrace {E}_{g}^{\mathit{el}-}\right\rbrace ,\left\lbrace {E}_{g}^{\mathit{el}+}\right\rbrace ,\left\lbrace {\Sigma}_{g}^{\mathit{el}-}\right\rbrace ,\left\lbrace {E}_{g}^{\mathit{el}+}\right\rbrace ,{\left[B\right]}_{g}^{\mathit{el}}\)

Calcul de \((\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\) par:

\({\mathit{DF}}_{i}\circ (\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})=-{R}_{i}({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})+{L}_{e}({t}^{+})\)

Actualisation :

\(({\text{u}}_{n+1}^{+},{P}_{n+1}^{+},{T}_{n+1}^{+})=({\text{u}}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})+(\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\)

SI test convergence OK

fin Newton: pas de temps suivant

Sinon

n = n+1

Vecteurs et matrices selon les options : routine EQUTHM#

La partie centrale encadrée de l’algorithme présenté ci dessus est réalisée par une routine générique EQUTHM. Nous donnons en annexe une représentation graphique de l’appel de cette routine.

Cette routine est paramétrée en fonction des équations présentes (mécanique, hydraulique avec 1 ou 2 pressions, thermique). Le travail effectué par cette routine est paramétré par l’option.

Le terme \(-{R}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})\) sera calculé par les options RAPH_MECA et FULL_MECA. Ce terme inclut les forces de volume suiveuses: on considérera que les forces suiveuses seront intégrées aux options RAPH_MECA, FULL_MECA et RIGI_MECA_TANG. Dans le cas où les données utilisateurs ne comportent pas de forces de volume, le vecteur \({\text{F}}^{{m}^{+}}\) sera simplement nul.

Les présentations faites dans les deux paragraphes suivants sont faites dans le cas le plus général où on a une équation de mécanique, deux équations d’hydraulique et une équation de thermique. La routine EQUTHM calculera ou non les différents termes selon la description qu’on lui fera des équations présentes.

Les indices g et el sont désormais omis, mais il est clair que ce qui est décrit s’applique à chaque point de Gauss de chaque élément.

Remarque:

Dans le cadre de la modélisation HM permanente saturée, une routine similaire à la routine EQUTHM a été implantée (la routine EQUTHP), qui tient compte des spécificités des équations de la modélisation permanente (pas d’apport massique).

Résidu ou force nodale : options RAPH_MECA et FULL_MECA#

On répartira les termes de la formulation variationnelle selon le principe suivant:

Si \({\text{E}}_{g}^{\text{*}\mathit{el}}\) désigne un champ de déformation virtuel, \({\text{E}}_{g}^{\text{*}\mathit{elT}}=(\text{v},\epsilon (\text{v}),{\pi}_{1,}\nabla {\pi}_{1,}{\pi}_{2,}\nabla {\pi}_{2,}\tau ,\nabla \tau )\) calculé à partir d’un vecteur de déplacement nodaux virtuels: \(\left\lbrace {X}^{\text{*}\mathit{el}}\right\rbrace\)

\({\text{E}}_{g}^{\text{*}\mathit{elT}}\cdot {R}_{\mathit{ig}}^{\mathit{el}}({\text{u}}_{+},{P}_{+},{T}_{+})={R}_{1}\text{v}+{R}_{2}\epsilon (\text{v})+{R}_{3}{\pi}_{1}+{R}_{4}\nabla {\pi}_{1}+{R}_{5}{\pi}_{2}+{R}_{6}\nabla {\pi}_{2}+{R}_{7}\tau +{R}_{8}\nabla \tau\)

On a alors:

Indice

R

associé à

1

\(-\left({m}_{1}^{1+}+{m}_{1}^{2+}+{m}_{2}^{1+}+{m}_{2}^{2+}\right){\text{F}}^{{m}^{+}}\)

\(\text{v}\)

2

\(\sigma {'}^{+}+{\sigma}_{p}^{+}I\)

\(\epsilon (\text{v})\)

3

\(-{m}_{1}^{1+}-{m}_{1}^{2+}+{m}_{1}^{1-}+{m}_{1}^{2-}\)

\({\pi}_{1}\)

4

\(\theta \Delta t\left({M}_{1}^{1+}+{M}_{1}^{2+}\right)+(1-\theta )\Delta t\left({M}_{1}^{1-}+{M}_{1}^{2-}\right)\)

\(\nabla {\pi}_{1}\)

5

\(-{m}_{2}^{1+}-{m}_{2}^{2+}+{m}_{2}^{1-}+{m}_{2}^{2-}\)

\({\pi}_{2}\)

6

\(\theta \Delta t\left({M}_{2}^{1+}+{M}_{2}^{2+}\right)+(1-\theta )\Delta t\left({M}_{2}^{1-}+{M}_{2}^{2-}\right)\)

\(\nabla {\pi}_{2}\)

7

\(\begin{array}{c}Q{'}^{+}-Q{'}^{-}\\ (\theta {h}_{\mathrm{1m}}^{1+}+(1-\theta ){h}_{\mathrm{1m}}^{1-})({m}_{1}^{1+}-{m}_{1}^{1-})+(\theta {h}_{\mathrm{1m}}^{2+}+(1-\theta ){h}_{\mathrm{1m}}^{2-})({m}_{1}^{2+}-{m}_{1}^{2-})\\ (\theta {h}_{\mathrm{2m}}^{1+}+(1-\theta ){h}_{\mathrm{2m}}^{1-})({m}_{2}^{1+}-{m}_{2}^{1-})+(\theta {h}_{\mathrm{2m}}^{2+}+(1-\theta ){h}_{\mathrm{2m}}^{2-})({m}_{2}^{2+}-{m}_{2}^{2-})\\ -\Delta t\theta ({\text{M}}_{1}^{1+}+{\text{M}}_{1}^{2+}+{\text{M}}_{2}^{1+}+{\text{M}}_{2}^{2+})\cdot {\text{F}}^{m}-\Delta t(1-\theta )({\text{M}}_{1}^{1-}+{\text{M}}_{1}^{2-}+{\text{M}}_{2}^{1-}+{\text{M}}_{2}^{2-})\cdot {\text{F}}^{m}\end{array}\)

\(\tau\)

8

\(\begin{array}{c}-\theta \Delta t\left({h}_{\mathrm{1m}}^{1+}{M}_{1}^{1+}+{h}_{\mathrm{1m}}^{2+}{M}_{1}^{2+}+{h}_{\mathrm{2m}}^{1+}{M}_{2}^{1+}+{h}_{\mathrm{2m}}^{2+}{M}_{2}^{2+}+{\text{q}}^{+}\right)\\ -(1-\theta )\Delta t\left({h}_{\mathrm{1m}}^{1+}{M}_{1}^{1+}+{h}_{\mathrm{1m}}^{2+}{M}_{1}^{2+}+{h}_{\mathrm{2m}}^{1+}{M}_{2}^{1+}+{h}_{\mathrm{2m}}^{2+}{M}_{2}^{2+}+{\text{q}}^{+}\right)\end{array}\)

\(\nabla \tau\)

A partir de là on définira le vecteur résidu nodal \(\left\lbrace {V}_{g}^{\mathit{el}}\right\rbrace\) tel que:

\({\left\lbrace {X}^{\text{*}\mathit{el}}\right\rbrace }^{T}\cdot \left\lbrace {V}_{g}^{\mathit{el}}\right\rbrace ={E}_{g}^{\text{*}{\mathit{el}}^{T}}\cdot {R}_{\mathit{ig}}^{\mathit{el}}({\text{u}}_{+},{P}_{+},{T}_{+})\)

\(\left\lbrace {V}_{g}^{\mathit{el}}\right\rbrace\) se calculera par:

\(\left\lbrace {V}_{g}^{\mathit{el}}\right\rbrace ={\left[{B}_{g}^{\mathit{el}}\right]}^{T}\cdot \lbrace R\rbrace\)

Remarque:

Dans le cadre de la modélisation HM permanente saturée, la routine EQUTHP n’assemble jamais les termes R3 et R5.

Chargement : options CHAR_MECA#

Ce chapitre n’est ici que pour mémoire car la routine EQUTHM ne s’occupera pas de ces termes.

On répartira les termes de la formulation variationnelle selon le principe suivant:

\({\text{E}}_{g}^{\text{*}{\mathit{el}}^{T}}\cdot {L}_{\mathit{eg}}^{\mathit{el}}(t+)={L}_{1}\text{v}+{L}_{2}\epsilon (\text{v})+{L}_{3}{\pi}_{1}+{L}_{4}\nabla {\pi}_{1}+{L}_{5}{\pi}_{2}+{L}_{6}\nabla {\pi}_{2}+{L}_{7}\tau +{L}_{8}\nabla \tau\)

Indice

L

type élément

associé à

1

\({\text{f}}^{{\mathit{ext}}^{+}}\)

bord

\(\text{v}\)

3

\(\Delta t\left({M}_{\mathrm{1ext}}^{1\theta }+{M}_{\mathrm{1ext}}^{2\theta }\right)\)

bord

\({\pi}_{1}\)

5

\(\Delta t\left({M}_{\mathrm{2ext}}^{1\theta }+{M}_{\mathrm{2ext}}^{2\theta }\right)\)

bord

\({\pi}_{2}\)

7

\(\begin{array}{c}\Delta t{R}^{\theta}\\ -\Delta t\left({\text{q}}_{\mathit{ext}}^{\theta}+\left({h}_{\mathrm{1m}}^{1\theta }{\text{M}}_{\mathrm{1ext}}^{1\theta }+{h}_{\mathrm{1m}}^{2\theta }{\text{M}}_{\mathrm{1ext}}^{2\theta }\right)\right)\\ -\Delta t\left({h}_{\mathrm{2m}}^{1\theta }{\text{M}}_{\mathrm{2ext}}^{1\theta }+{h}_{\mathrm{2m}}^{2\theta }{\text{M}}_{\mathrm{2ext}}^{2\theta }\right)\\ =-\Delta t\stackrel{̃}{\text{q}}{}_{\mathit{ext}}^{\theta}\end{array}\)

volume bord

\(\tau\)

Opérateur tangent : options FULL_MECA, RIGI_MECA_TANG#

Remarque sur les notations matricielles:

Dans ce qui suit, si \(X\) désigne un vecteur de composantes \({X}^{i}\) et \(Y\) un vecteur de composantes \({Y}^{i}\) , \(\left[\frac{\partial X}{\partial Y}\right]\) désignera une matrice dont l’élément \((\mathit{ligne}:i,\mathit{colonne}:j)\) est \(\frac{\partial {X}^{i}}{\partial {Y}^{j}}\) .

Pour calculer l’opérateur tangent \({\mathit{DF}}_{i}\) , on calculera les quantités suivantes:

\(\left[\text{DRDE}\right]\) =

DR1U

DR1E

DR1P1

DR1GP1

DR1P2

DR1GP2

DR1T

DR1GT

DR2U

DR2E

DR2P1

DR2GP1

DR2P2

DR2GP2

DR2T

DR2GT

DR3U

DR3E

DR3P1

DR3GP1

DR3P2

DR3GP2

DR3T

DR3GT

DR4U

DR4E

DR4P1

DR4GP1

DR4P2

DR4GP2

DR4T

DR4GT

DR5U

DR5E

DR5P1

DR5GP1

DR5P2

DR5GP2

DR5T

DR5GT

DR6U

DR6E

DR6P1

DR6GP1

DR6P2

DR6GP2

DR6T

DR6GT

DR7U

DR7E

DR7P1

DR7GP1

DR7P2

DR7GP2

DR7T

DR7GT

DR8U

DR8E

DR8P1

DR8GP1

DR8P2

DR8GP2

DR8T

DR8GT

Où on a noté:

\(\begin{array}{c}\mathit{DRiU}=\underline{\frac{\partial {F}_{i}}{\partial u}}\\ \mathit{DRiE}=\underline{\underline{\frac{\partial {F}_{i}}{\partial \epsilon }}}\\ \mathit{DRiP1}=\frac{\partial {F}_{i}}{\partial {p}_{1}}\\ \mathit{DRiP2}=\frac{\partial {F}_{i}}{\partial {p}_{2}}\\ \mathit{DRiGP1}=\underline{\frac{\partial {F}_{i}}{\partial \nabla {p}_{1}}}\\ \mathit{DRiGP2}=\underline{\frac{\partial {F}_{i}}{\partial \nabla {p}_{2}}}\\ \mathit{DRiT}=\frac{\partial {F}_{i}}{\partial T}\\ \mathit{DRiGT}=\underline{\frac{\partial {F}_{i}}{\partial \nabla T}}\end{array}\)

Pour faire ces calculs on considère que les lois de comportement fourniront, pour les options correspondantes, toutes les dérivées suivantes:

\(\left[\text{DSDE}\right]=\left[\begin{array}{cccccccc}\frac{\partial \sigma '}{\partial \text{u}}& \frac{\partial \sigma '}{\partial \epsilon }& \frac{\partial \sigma '}{\partial {p}_{1}}& \frac{\partial \sigma '}{\partial \nabla {p}_{1}}& \frac{\partial \sigma '}{\partial {p}_{2}}& \frac{\partial \sigma '}{\partial \nabla {p}_{2}}& \frac{\partial \sigma '}{\partial T}& \frac{\partial \sigma '}{\partial \nabla T}\\ \frac{\partial {\sigma}_{p}}{\partial \text{u}}& \frac{\partial {\sigma}_{p}}{\partial \epsilon }& \frac{\partial {\sigma}_{p}}{\partial {p}_{1}}& \frac{\partial {\sigma}_{p}}{\partial \nabla {p}_{1}}& \frac{\partial {\sigma}_{p}}{\partial {p}_{2}}& \frac{\partial {\sigma}_{p}}{\partial \nabla {p}_{2}}& \frac{\partial {\sigma}_{p}}{\partial T}& \frac{\partial {\sigma}_{p}}{\partial \nabla T}\\ \frac{\partial {m}_{1}^{1}}{\partial \text{u}}& \frac{\partial {m}_{1}^{1}}{\partial \epsilon }& \frac{\partial {m}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial T}& \frac{\partial {m}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial \text{u}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \epsilon }& \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial T}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \text{u}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \epsilon }& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla T}\\ \frac{\partial {m}_{1}^{2}}{\partial \text{u}}& \frac{\partial {m}_{1}^{2}}{\partial \epsilon }& \frac{\partial {m}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial T}& \frac{\partial {m}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial \text{u}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \epsilon }& \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial T}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \text{u}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \epsilon }& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla T}\\ \frac{\partial {m}_{2}^{1}}{\partial \text{u}}& \frac{\partial {m}_{2}^{1}}{\partial \epsilon }& \frac{\partial {m}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial T}& \frac{\partial {m}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial \text{u}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \epsilon }& \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial T}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \text{u}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \epsilon }& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla T}\\ \frac{\partial {m}_{2}^{2}}{\partial \text{u}}& \frac{\partial {m}_{2}^{2}}{\partial \epsilon }& \frac{\partial {m}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial T}& \frac{\partial {m}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial \text{u}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \epsilon }& \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial T}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \text{u}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \epsilon }& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla T}\\ \frac{\partial Q'}{\partial \text{u}}& \frac{\partial Q'}{\partial \epsilon }& \frac{\partial Q'}{\partial {p}_{1}}& \frac{\partial Q'}{\partial \nabla {p}_{1}}& \frac{\partial Q'}{\partial {p}_{2}}& \frac{\partial Q'}{\partial \nabla {p}_{2}}& \frac{\partial Q'}{\partial T}& \frac{\partial Q'}{\partial \nabla T}\\ \frac{\partial \text{q}}{\partial \text{u}}& \frac{\partial \text{q}}{\partial \epsilon }& \frac{\partial \text{q}}{\partial {p}_{1}}& \frac{\partial \text{q}}{\partial \nabla {p}_{1}}& \frac{\partial \text{q}}{\partial {p}_{2}}& \frac{\partial \text{q}}{\partial \nabla {p}_{2}}& \frac{\partial \text{q}}{\partial T}& \frac{\partial \text{q}}{\partial \nabla T}\end{array}\right]\)

En fait, dans ces expressions, les dérivées par rapport à u sont toutes nulles, mais nous gardons l’écriture compte tenu de la définition des matrices \({[B]}_{g}^{\mathit{el}}\) que nous avons adoptée.

L’appel aux lois de comportement fournira les morceaux de la matrice \(\left[\text{DSDE}\right]\) selon les équations présentes:

\(\left[\text{DMECDE}\right]=\left[\begin{array}{c}\frac{\partial \sigma '}{\partial \epsilon }\\ \frac{\partial {\sigma}_{p}}{\partial \epsilon }\end{array}\right];\left[\text{DMECP1}\right]=\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial {p}_{1}}& \frac{\partial \sigma '}{\partial \nabla {p}_{1}}\\ \frac{\partial {\sigma}_{p}}{\partial {p}_{1}}& \frac{\partial {\sigma}_{p}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DMECP2}\right]=\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial {p}_{2}}& \frac{\partial \sigma '}{\partial \nabla {p}_{2}}\\ \frac{\partial {\sigma}_{p}}{\partial {p}_{2}}& \frac{\partial {\sigma}_{p}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DMECDT}\right]=\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial T}& \frac{\partial \sigma '}{\partial \nabla T}\\ \frac{\partial {\sigma}_{p}}{\partial T}& \frac{\partial {\sigma}_{p}}{\partial \nabla T}\end{array}\right]\)

\(\left[\text{DP11DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{1}^{1}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \epsilon }\end{array}\right];\left[\text{DP11P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP11P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DP11DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial T}& \frac{\partial {m}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial T}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla T}\end{array}\right]\)

\(\left[\text{DP12DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{1}^{2}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \epsilon }\end{array}\right];\left[\text{DP12P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP12P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DP12DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial T}& \frac{\partial {m}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial T}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla T}\end{array}\right]\)

\(\left[\text{DP21DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{2}^{1}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \epsilon }\end{array}\right];\left[\text{DP21P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP21P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DP21DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial T}& \frac{\partial {m}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial T}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla T}\end{array}\right]\)

\(\left[\text{DP22DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{2}^{2}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \epsilon }\end{array}\right];\left[\text{DP22P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP22P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DP22DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial T}& \frac{\partial {m}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial T}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla T}\end{array}\right]\)

\(\left[\text{DTDE}\right]=\left[\begin{array}{c}\frac{\partial Q'}{\partial \epsilon }\\ \frac{\partial \text{q}}{\partial \epsilon }\end{array}\right];\left[\text{DTDP1}\right]=\left[\begin{array}{cc}\frac{\partial Q'}{\partial {p}_{1}}& \frac{\partial Q'}{\partial \nabla {p}_{1}}\\ \frac{\partial \text{q}}{\partial {p}_{1}}& \frac{\partial \text{q}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DTDP2}\right]=\left[\begin{array}{cc}\frac{\partial Q'}{\partial {p}_{2}}& \frac{\partial Q'}{\partial \nabla {p}_{2}}\\ \frac{\partial \text{q}}{\partial {p}_{2}}& \frac{\partial \text{q}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DTDT}\right]=\left[\begin{array}{cc}\frac{\partial Q'}{\partial T}& \frac{\partial Q'}{\partial \nabla T}\\ \frac{\partial \text{q}}{\partial T}& \frac{\partial \text{q}}{\partial \nabla T}\end{array}\right]\)

Par ailleurs, en dérivant l’expression du résidu par rapport aux contraintes, on définit:

\(\left[\text{DRDS}\right]=\left[\begin{array}{cccccccccccccccc}\frac{\partial {R}_{1}}{\partial \sigma '}& \frac{\partial {R}_{1}}{\partial {\sigma}_{p}}& \frac{\partial {R}_{1}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{1}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{1}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{1}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{1}}{\partial Q'}& \frac{\partial {R}_{1}}{\partial \text{q}}\\ \frac{\partial {R}_{2}}{\partial \sigma '}& \frac{\partial {R}_{2}}{\partial {\sigma}_{p}}& \frac{\partial {R}_{2}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{2}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{2}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{2}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{2}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{2}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{2}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{2}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{2}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{2}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{2}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{2}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{2}}{\partial Q'}& \frac{\partial {R}_{2}}{\partial \text{q}}\\ \frac{\partial {R}_{3}}{\partial \sigma '}& \frac{\partial {R}_{3}}{\partial {\sigma}_{p}}& \frac{\partial {R}_{3}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{3}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{3}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{3}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{3}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{3}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{3}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{3}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{3}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{3}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{3}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{3}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{3}}{\partial Q'}& \frac{\partial {R}_{3}}{\partial \text{q}}\\ \frac{\partial {R}_{4}}{\partial \sigma '}& \frac{\partial {R}_{4}}{\partial {\sigma}_{p}}& \frac{\partial {R}_{4}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{4}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{4}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{4}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{4}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{4}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{4}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{4}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{4}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{4}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{4}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{4}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{4}}{\partial Q'}& \frac{\partial {R}_{4}}{\partial \text{q}}\\ \frac{\partial {R}_{5}}{\partial \sigma '}& \frac{\partial {R}_{5}}{\partial {\sigma}_{p}}& \frac{\partial {R}_{5}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{5}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{5}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{5}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{5}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{5}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{5}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{5}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{5}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{5}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{5}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{5}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{5}}{\partial Q'}& \frac{\partial {R}_{5}}{\partial \text{q}}\\ \frac{\partial {R}_{6}}{\partial \sigma '}& \frac{\partial {R}_{6}}{\partial {\sigma}_{p}}& \frac{\partial {R}_{6}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{6}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{6}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{6}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{6}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{6}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{6}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{6}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{6}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{6}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{6}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{6}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{6}}{\partial Q'}& \frac{\partial {R}_{6}}{\partial \text{q}}\\ \frac{\partial {R}_{7}}{\partial \sigma '}& \frac{\partial {R}_{7}}{\partial {\sigma}_{p}}& \frac{\partial {R}_{7}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{7}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{7}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{7}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{7}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{7}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{7}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{7}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{7}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{7}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{7}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{7}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{7}}{\partial Q'}& \frac{\partial {R}_{7}}{\partial \text{q}}\\ \frac{\partial {R}_{8}}{\partial \sigma '}& \frac{\partial {R}_{8}}{\partial {\sigma}_{p}}& \frac{\partial {R}_{8}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{8}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{8}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{8}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{8}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{8}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{8}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{8}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{8}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{8}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{8}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{8}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{8}}{\partial Q'}& \frac{\partial {R}_{8}}{\partial \text{q}}\end{array}\right]\)

Toutes ces quantités n’étant pas forcément calculées, on notera:

\(\left[\text{DR1DS}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {\sigma}^{'+}}& \frac{\partial {R}_{1}}{\partial {\sigma}_{p}^{+}}\end{array}\right];\left[\text{DR1P11}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{1+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{1+}}\end{array}\right]\mathit{ou}\left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{1+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{1+}}& \frac{\partial {R}_{1}}{\partial {\sigma}_{\mathrm{1m}}^{1+}}\end{array}\right]\)

\(\left[\text{DR1P12}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{2+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{2+}}\end{array}\right]\mathit{ou}\left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{2+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{2+}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{1m}}^{2+}}\end{array}\right]\)

\(\left[\text{DR1P21}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{1+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{1+}}\end{array}\right]\mathit{ou}\left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{1+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{1+}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{1+}}\end{array}\right]\)

\(\left[\text{DR1P22}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{2+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{2+}}\end{array}\right]\mathit{ou}\left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{2+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{2+}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{2+}}\end{array}\right]\)

\(\left[\text{DR1DT}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {Q}^{'+}}& \frac{\partial {R}_{1}}{\partial {\text{q}}^{+}}\end{array}\right]\)

De même:

\(\left[\text{DR8DS}\right],\left[\text{DR8P11}\right],\left[\text{DR8P12}\right],\left[\text{DR8P21}\right],\left[\text{DR8P22}\right],\left[\text{DR8DT}\right]\)

Il est alors clair que:

\(\left[\text{DRDE}\right]=\left[\text{DRDS}\right]\cdot \left[\text{DSDE}\right]\)

Et la contribution du point de Gauss à la matrice tangente \({{\text{DF}}_{g}^{\mathit{el}}}_{i({u}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})}\) s’obtient par:

\(\left[{{\text{DF}}_{g}^{\mathit{el}}}_{i({u}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})}\right]={\left[{\text{B}}_{g}^{\mathit{el}}\right]}^{T}\cdot \left[\text{DRDE}\right]\cdot \left[{\text{B}}_{g}^{\mathit{el}}\right]\)

Schéma général#

../../../../_images/Shape131.gif

Spécifications du sous programme générique EQUTHM#

Arguments de la routine#

ARGUMENTS D’ENTREE: IN

COMPOR

Description du comportement

OPTION

Option à calculer

NDIM

dimension espace

2 ou 3

NDDL

Nombre total de degrés de liberté de l’élément appelant

DIMDEF

dimension du tableau des déformations généralisées au point de Gauss

DIMCON

dimension du tableau des contraintes généralisées au point de Gauss

NVIMEC

Nombre de variables internes «mécaniques»

ADVIME

Adresse des variables internes mécaniques dans le tableau des variables internes au point de Gauss

NVIHY

Nombre de variables internes «hydrauliques»

ADVIHY

Adresse des variables internes hydrauliques dans le tableau des variables internes au point de Gauss

NVITM

Nombre de variables internes «thermiques»

ADVITM

Adresse des variables internes thermiques dans le tableau des variables internes au point de Gauss

B(1:dimdef,1:nddl)

Matrice \({\left[B\right]}_{g}^{\mathit{el}}\)

DEFGEP(1:dimdef)

Valeurs de déformations généralisées au point de Gauss temps plus

DEFGEM(1:dimdef)

Valeurs de déformations généralisées au point de Gauss temps moins

CONGEM(1:dimcon)

Valeurs de contraintes généralisées au point de Gauss temps moins

VINTM(1:nvimec+nvihy+nvitm)

Valeurs des variables internes au point de Gauss temps moins

MECA(1:5)

YAMEC = MECA(1)

logique si 1 il y a une équation de mécanique

ADDEME = MECA(2)

Adresse dans les tableaux des déformations au point de Gauss DEFGEPet DEFGEMdes déformations correspondant à la mécanique

ADCOME = MECA(3)

Adresse dans les tableaux des contraintes au point de Gauss CONGEPet CONGEMdes contraintes correspondant à l’équation ieq

NDEFME = MECA(4)

Nombre de déformations mécaniques

NCONME = MECA(5)

Nombre de contraintes mécaniques

PRESS1(1:5)

YAP1 = PRESS1(1)

logique si 1 il y a une équation constituant 1

NBPHA1 = PRESS1(2)

nombre de phases pour le constituant 1

ADDEP1 = PRESS1(3)

Adresse dans les tableaux des déformations au point de Gauss DEFGEPet DEFGEMdes déformations correspondant à la première pression

ADCP11 = PRESS1(4)

Adresse dans les tableaux des contraintes au point de Gauss CONGEPet CONGEMdes contraintes correspondant à la première phase du premier constituant

ADCP12 = PRESS1(5)

Adresse dans les tableaux des contraintes au point de Gauss CONGEPet CONGEMdes contraintes correspondant à la deuxième phase du premier constituant

NDEFP1 = PRESS1(6)

Nombre de déformations pression 1

NCONP1 = PRESS1(7)

Nombre de contraintes pour chaque phase du constituant 1

PRESS2(1:5)

YAP2 = PRESS2(1)

logique si 1 il y a une équation constituant 2

NBPHA2 = PRESS2(2)

nombre de phases pour le constituant 2

ADDEP2 = PRESS2(3)

Adresse dans les tableaux des déformations au point de Gauss DEFGEPet DEFGEMdes déformations correspondant à PRE2

ADCP21 = PRESS2(4)

Adresse dans les tableaux des contraintes au point de Gauss CONGEPet CONGEMdes contraintes correspondant à la première phase du deuxième constituant

ADCP22 = PRESS2(5)

Adresse dans les tableaux des contraintes au point de Gauss CONGEPet CONGEMdes contraintes correspondant à la deuxième phase du deuxième constituant

NDEFP2 = PRESS2(6)

Nombre de déformations pression 2

NCONP2 = PRESS2(7)

Nombre de contraintes pour chaque phase du constituant 2

TEMPE(1:5)

YATE = TEMPE(1)

logique si 1 il y a une équation de thermique

ADDETE = TEMPE(2)

Adresse dans les tableaux des déformations au point de Gauss DEFGEPet DEFGEMdes déformations correspondant à la thermique

ADCOTE = TEMPE(3)

Adresse dans les tableaux des contraintes au point de Gauss CONGEPet première CONGEMdes contraintes correspondant à la thermique

NDEFT = TEMPE(4)

Nombre de déformations thermique

NCONT = TEMPE(5)

Nombre de contraintes thermique

ARGUMENTS DE SORTIE: OUT

CONGEP(1:dimcon)

Valeurs de contraintes généralisées au point de Gauss temps plus

VINTP(1:nvimec+nvihy+nvitm)

Valeurs des variables internes au point de Gauss temps plus

V(1:nddl)

\(\left\lbrace {\text{V}}_{g}^{\mathit{el}}\right\rbrace ={\left[{\text{B}}_{g}^{\mathit{el}}\right]}^{T}\left\lbrace R\right\rbrace\)

MAT(1:nddl,1:nddl)

\(\left[{{\text{DF}}_{g}^{\mathit{el}}}_{i({u}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})}\right]={\left[{\text{B}}_{g}^{\mathit{el}}\right]}^{T}\cdot \left[\text{DRDE}\right]\cdot \left[{\text{B}}_{g}^{\mathit{el}}\right]\)

TABLEAUX DE TRAVAIL

R(1:dimdef)

DRDS (1:dimdef,1:dimcon)

DSDE (1:dimcon,1:dimdef)

Adressage dans les tableaux de déformation et contrainte#

Adressage dans les déformations#

Déformations temps moins#

Partie (nom local dans routine COMTHM )

Signification

Adresse dans DEFGEM

DEMECM

\(\text{u},\underline{\underline{\epsilon}}(\text{u})\)

ADDEME

DEP1M

\({p}_{1,}\nabla {p}_{1}\)

ADDEP1

DEP2M

\({p}_{2,}\nabla {p}_{2}\)

ADDEP2

DETM

\(T,\nabla T\)

ADDETE
Déformations temps plus#

Partie (nom local dans routine COMTHM )

Signification

Adresse dans DEFGEP

DEMECP

\(\text{u},\underline{\underline{\epsilon}}(\text{u})\)

ADDEME

DEP1P

\({p}_{1,}\nabla {p}_{1}\)

ADDEP1

DEP2P

\({p}_{2,}\nabla {p}_{2}\)

ADDEP2

DETP

\(T,\nabla T\)

ADDETE

Adressage dans les contraintes#

Contraintes temps moins#

Partie (nom local dans routine COMTHM )

Signification

Adresse dans CONGEM

COMECM

\(\underline{\underline{\sigma '}},{\sigma}_{p}\)

ADCOME

CP11M

\({m}_{1}^{1,}{\text{M}}_{1}^{1}\text{ou}{m}_{1}^{1,}{\text{M}}_{1}^{1,}{h}_{\mathrm{1m}}^{1}\)

ADCP11

CP12M

\({m}_{1}^{2,}{\text{M}}_{1}^{2}\text{ou}{m}_{1}^{2,}{\text{M}}_{1}^{2,}{h}_{\mathrm{1m}}^{2}\)

ADCP12

CP21M

\({m}_{2}^{1,}{\text{M}}_{2}^{1}\text{ou}{m}_{2}^{1,}{\text{M}}_{2}^{1,}{h}_{\mathrm{2m}}^{1}\)

ADCP21

CP22M

\({m}_{2}^{2,}{\text{M}}_{2}^{2}\text{ou}{m}_{2}^{2,}{\text{M}}_{2}^{2,}{h}_{\mathrm{2m}}^{2}\)

ADCP22

COTM

\(Q',\text{q}\)

ADCOTE
Contraintes temps plus#

Partie (nom local dans routine COMTHM )

Signification

Adresse dans CONGEP

COMECP

\(\underline{\underline{\sigma '}},{\sigma}_{p}\)

ADCOME

CP11P

\({m}_{1}^{1,}{\text{M}}_{1}^{1}\text{ou}{m}_{1}^{1,}{\text{M}}_{1}^{1,}{h}_{\mathrm{1m}}^{1}\)

ADCP11

CP12P

\({m}_{1}^{2,}{\text{M}}_{1}^{2}\text{ou}{m}_{1}^{2,}{\text{M}}_{1}^{2,}{h}_{\mathrm{1m}}^{2}\)

ADCP12

CP21P

\({m}_{2}^{1,}{\text{M}}_{2}^{1}\text{ou}{m}_{2}^{1,}{\text{M}}_{2}^{1,}{h}_{\mathrm{2m}}^{1}\)

ADCP21

CP22P

\({m}_{2}^{2,}{\text{M}}_{2}^{2}\text{ou}{m}_{2}^{2,}{\text{M}}_{2}^{2,}{h}_{\mathrm{2m}}^{2}\)

ADCP22

COTP

\(Q',\text{q}\)

ADCOTE

Adressage dans les variables internes (exemple)#

Variables internes au temps moins#

Partie (nom local dans routine COMTHM )

Signification

Adresse dans VINTM

VIMEM

\(\varphi\)

ADVIME

VIHYM

\({S}_{\mathit{lq}},{P}_{\mathit{vq}},{P}_{\mathit{lq}}\)

ADVIHY
Variables internes au temps plus#

Partie (nom local dans routine COMTHM )

Signification

Adresse dans VINTP

VIMEP

\(\varphi\)

ADVIME

VIHYP

\({S}_{\mathit{lq}},{P}_{\mathit{vq}},{P}_{\mathit{lq}}\)

ADVIHY

Adressage R, DRDS, DSDE#

Adressage dans R#

Sous partie de R

Associé à

Adresse dans R

R1

\(\text{v}\)

ADDEME

R2

\(\epsilon (\text{v})\)

ADDEME+NDIM

R3

\({\pi}_{1}\)

ADDEP1

R4

\(\nabla {\pi}_{1}\)

ADDEP1+1

R5

\({\pi}_{2}\)

ADDEP2

R6

\(\nabla {\pi}_{2}\)

ADDEP2+1

R7

\(\tau\)

ADDETE

R8

\(\nabla \tau\)

ADDETE+1

Adressage dans DRDS#

Partie du tableau DRDS

Signification

Adresse dans DRDS

DR1DS

\(\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {\sigma}^{'+}}& \frac{\partial {R}_{1}}{\partial {\sigma}_{p}^{+}}\end{array}\right]\)

ADDEME, ADCOME

DR2DS

ADDEME+NDIM-1, ADCOME

DR1P11

\(\begin{array}{c}\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{1+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{1+}}\end{array}\right]\mathit{ou}\\ \left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{1+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{1+}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{1m}}^{1+}}\end{array}\right]\end{array}\)

ADDEME, ADCP11

DR2P11

ADDEME+NDIM-1, ADCP11

DR1P12

\(\begin{array}{c}\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{2+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{2+}}\end{array}\right]\mathit{ou}\\ \left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{2+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{2+}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{1m}}^{2+}}\end{array}\right]\end{array}\)

ADDEME, ADCP12

DR2P12

ADDEME+NDIM-1, ADCP12

DR1P21

\(\begin{array}{c}\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{1+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{1+}}\end{array}\right]\mathit{ou}\\ \left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{1+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{1+}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{1+}}\end{array}\right]\end{array}\)

ADDEME, ADCP21

DR2P21

ADDEME+NDIM-1, ADCP21

DR1P22

\(\begin{array}{c}\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{2+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{2+}}\end{array}\right]\mathit{ou}\\ \left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{2+}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{2+}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{2+}}\end{array}\right]\end{array}\)

ADDEME, ADCP22

DR2P22

ADDEME+NDIM-1, ADCP22

DR1DT

\(\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {Q}^{'+}}& \frac{\partial {R}_{1}}{\partial {\text{q}}^{+}}\end{array}\right]\)

ADDEME, ADCOTE

DR2DT

ADDEME+NDIM-1, ADCOTE

DR3DS

ADDEP1, ADCOME

DR4DS

ADDEP1+1, ADCOME

DR3P11

ADDEP1, ADCP11

DR4P11

ADDEP1+1, ADCP11

DR3P21

ADDEP1, ADCP21

DR4P21

ADDEP1+ 1, ADCP21

DR3DT

ADDEP1, ADCOTE

DR4DT

ADDEP1+ 1, ADCOTE

DR5DS

ADDEP2, ADCOME

DR6DS

ADDEP2+ 1, ADCOME

DR5P11

ADDEP2, ADCP11

DR6P11

ADDEP2+ 1, ADCP11

DR5P21

ADDEP2, ADCP21

DR6P21

ADDEP2+1, ADCP21

DR5DT

ADDEP2, ADCOTE

DR6DT

ADDEP2+ 1, ADCOTE

DR7DS

ADDETE, ADCOME

DR8DS

ADDETE+ 1, ADCOME

DR7P11

ADDETE, ADCP11

DR8P11

ADDETE+ 1, ADCP11

DR7P21

ADDETE, ADCP21

DR8P21

ADDETE+ 1, ADCP21

DR7DT

ADDETE, ADCOTE

DR8DT

ADDETE+1, ADCOTE

Adressage dans DSDE#

Partie (nom local à COMTHM )

Signification

Adresse dans DSDE

DMECDE

\(\left[\begin{array}{c}\frac{\partial \sigma '}{\partial \epsilon }\\ \frac{\partial {\sigma}_{p}}{\partial \epsilon }\end{array}\right]\)

ADCOME, ADDEME

DMECP1

\(\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial {p}_{1}}& \frac{\partial \sigma '}{\partial \nabla {p}_{1}}\\ \frac{\partial {\sigma}_{p}}{\partial {p}_{1}}& \frac{\partial {\sigma}_{p}}{\partial \nabla {p}_{1}}\end{array}\right]\)

ADCOME, ADDEP1

DMECP2

\(\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial {p}_{2}}& \frac{\partial \sigma '}{\partial \nabla {p}_{2}}\\ \frac{\partial {\sigma}_{p}}{\partial {p}_{2}}& \frac{\partial {\sigma}_{p}}{\partial \nabla {p}_{2}}\end{array}\right]\)

ADCOME, ADDEP2

DMECDT

\(\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial T}& \frac{\partial \sigma '}{\partial \nabla T}\\ \frac{\partial {\sigma}_{p}}{\partial T}& \frac{\partial {\sigma}_{p}}{\partial \nabla T}\end{array}\right]\)

ADCOME, ADDETE

DP11DE

\(\left[\begin{array}{c}\frac{\partial {m}_{1}^{1}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \epsilon }\end{array}\right]\)

ADCP11, ADDEME

DP11P1

\(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{1}}\end{array}\right]\)

ADCP11, ADDEP1

DP11P2

\(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{2}}\end{array}\right]\)

ADCP11, ADDEP2

DP11DT

\(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial T}& \frac{\partial {m}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial T}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla T}\end{array}\right]\)

ADCP11, ADDETE

DP12DE

\(\left[\begin{array}{c}\frac{\partial {m}_{1}^{2}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \epsilon }\end{array}\right]\)

ADCP12, ADDEME

DP12P1

\(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{1}}\end{array}\right]\)

ADCP12, ADDEP1

DP12P2

\(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{2}}\end{array}\right]\)

ADCP12, ADDEP2

DP12DT

\(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial T}& \frac{\partial {m}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial T}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla T}\end{array}\right]\)

ADCP12, ADDETE

DP21DE

\(\left[\begin{array}{c}\frac{\partial {m}_{2}^{1}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \epsilon }\end{array}\right]\)

ADCP21, ADDEME

DP21P1

\(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{1}}\end{array}\right]\)

ADCP21, ADDEP1

DP21P2

\(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{2}}\end{array}\right]\)

ADCP21, ADDEP2

DP21DT

\(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial T}& \frac{\partial {m}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial T}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla T}\end{array}\right]\)

ADCP21, ADDETE

DP22DE

\(\left[\begin{array}{c}\frac{\partial {m}_{2}^{2}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \epsilon }\end{array}\right]\)

ADCP22, ADDEME

DP22P1

\(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{1}}\end{array}\right]\)

ADCP22, ADDEP1

DP22P2

\(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{2}}\end{array}\right]\)

ADCP22, ADDEP2

DP22DT

\(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial T}& \frac{\partial {m}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial T}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla T}\end{array}\right]\)

ADCP22, ADDETE

DTDE

\(\left[\begin{array}{c}\frac{\partial Q'}{\partial \epsilon }\\ \frac{\partial \text{q}}{\partial \epsilon }\end{array}\right]\)

ADCOTE, ADDEME

DTDP1

\(\left[\begin{array}{cc}\frac{\partial Q'}{\partial {p}_{1}}& \frac{\partial Q'}{\partial \nabla {p}_{1}}\\ \frac{\partial \text{q}}{\partial {p}_{1}}& \frac{\partial \text{q}}{\partial \nabla {p}_{1}}\end{array}\right]\)

ADCOTE, ADDEP1

DTDP2

\(\left[\begin{array}{cc}\frac{\partial Q'}{\partial {p}_{2}}& \frac{\partial Q'}{\partial \nabla {p}_{2}}\\ \frac{\partial \text{q}}{\partial {p}_{2}}& \frac{\partial \text{q}}{\partial \nabla {p}_{2}}\end{array}\right]\)

ADCOTE, ADDEP2

DTDT

\(\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial T}& \frac{\partial \sigma '}{\partial \nabla T}\\ \frac{\partial \text{q}}{\partial T}& \frac{\partial \text{q}}{\partial \nabla T}\end{array}\right]\)

ADCOTE, ADDETE

Algorithme routine EQUTHM#

YAMEC = MECA(1)

ADDEME = MECA(2)

ADCOME = MECA(3)

NDEFME = MECA(4)

NCONME = MECA(5)

YAP1 = PRESS1(1)

NBPHA1 = PRESS1(2)

ADDEP1 = PRESS1(3)

ADCP11 = PRESS1(4)

ADCP12 = PRESS1(5)

NDEFP1 = PRESS1(6)

NCONP1 = PRESS1(7)

YAP2 = PRESS2(1)

NBPHA2 = PRESS2(2)

ADDEP2 = PRESS2(3)

ADCP21 = PRESS2(4)

ADCP22 = PRESS2(5)

NDEFP2 = PRESS2(6)

NCONP2 = PRESS2(7)

YATE = TEMPE(1)

ADDETE = TEMPE(2)

ADCOTE = TEMPE(3)

NDEFT = TEMPE(4)

NCONT = TEMPE(5)

CALL COMTHM(

COMPOR

OPTION

NDIM

NDDL

DIMDEF

DIMCON

NVIMEC

NVIHY , NVITM

NDEFME

NDEFP1

NDEFP2

NDEFT

NCONME

NCONP1

NCONP2

NCONT

YAP1

NBPHA1

YAP2

NBPHA2

DEFGEM(ADDEME)

DEFGEM(ADDEP1)

DEFGEM(ADDEP2)

DEFGEM(ADDETE)

DEFGEP(ADDEME)

DEFGEP(ADDEP1)

DEFGEP(ADDEP2)

DEFGEP(ADDETE)

CONGEM(ADCOME)

CONGEM(ADCOTE)

CONGEM(ADCP11)

CONGEM(ADCP12)

CONGEM(ADCP21)

CONGEM(ADCP21)

VINTM(ADVIME)

VINTM(ADVIHY)

VINTM (ADVITM)

CONGEP(ADCOME)

CONGEP(ADCP11)

CONGEP(ADCP21)

CONGEP(ADCOTE)

VINTP(ADVIME)

VINTP(ADVIHY)

VINTP (ADVITM)

DSDE (ADCOME,ADDEME)

DSDE (ADCOME,ADDEP1)

DSDE (ADCOME,ADDEP2)

DSDE (ADCOME,ADDETE)

DSDE (ADCP11,ADDEP1)

DSDE (ADCP11,ADDEME)

DSDE (ADCP11,ADDEP2)

DSDE (ADCP11,ADDETE)

DSDE (ADCP12,ADDEP1)

DSDE (ADCP12,ADDEME)

DSDE (ADCP12,ADDEP2)

DSDE (ADCP12,ADDETE)

DSDE (ADCP21,ADDEP2)

DSDE (ADCP21,ADDEME)

DSDE (ADCP21,ADDEP1)

DSDE (ADCP21,ADDETE)

DSDE (ADCP22,ADDEP2)

DSDE (ADCP22,ADDEME)

DSDE (ADCP22,ADDEP1)

DSDE (ADCP22,ADDETE)

DSDE (ADCOTE,ADDETE)

DSDE (ADCOTE,ADDEME)

DSDE (ADCOTE,ADDEP1)

DSDE (ADCOTE,ADDEP2)

)

Si FULL_MECA ou RAPH_MECA

Si YAMEC

Injection des termes \(\sigma {'}^{+}+{\sigma}_{p}^{+}I\) dans R(ADDEME+NDIM-1)

Injection des termes: \(-{r}_{0}{\text{F}}^{{m}^{+}}\) dans R(ADDEME)

Si YAP1

Injection des termes \(-{m}_{1}^{1+}+{m}_{1}^{1-}\text{ou}-{m}_{1}^{1+}-{m}_{1}^{2+}+{m}_{1}^{1-}+{m}_{1}^{2-}\) dans R(ADDEP1)

Injection des termes

\(\begin{array}{c}\Delta t\theta {\text{M}}_{1}^{1+}+(1-\theta )\Delta t{\text{M}}_{1}^{1-}\text{ou}\\ \theta \Delta t\left({\text{M}}_{1}^{1+}+{\text{M}}_{1}^{2+}\right)+(1-\theta )\Delta t\left({\text{M}}_{1}^{1-}+{\text{M}}_{1}^{2-}\right)\end{array}\)

dans R(ADDEP1+1)

SI YAMEC

Injection des termes:

\(-{m}_{1}^{1+}{\text{F}}^{{m}^{+}}\text{ou}-\left({m}_{1}^{1+}+{m}_{1}^{2+}\right){\text{F}}^{{m}^{+}}\) dans R(ADDEME)

Si YATE

Injection des termes:

\(\begin{array}{c}\Delta t\left(\theta {h}_{\mathrm{1m}}^{1+}+(1-\theta ){h}_{\mathrm{1m}}^{1-}\right)\left({m}_{1}^{1+}-{m}_{1}^{1-}\right)-\theta \Delta t{\text{M}}_{1}^{1+}{F}^{m}-(1-\theta )\Delta t{\text{M}}_{1}^{1-}{F}^{m}\\ \text{ou}\\ \Delta t\left(\theta {h}_{\mathrm{1m}}^{1+}+(1-\theta ){h}_{\mathrm{1m}}^{1-}\right)\left({m}_{1}^{1+}-{m}_{1}^{1-}\right)+\Delta t\left(\theta {h}_{\mathrm{1m}}^{2+}+(1-\theta ){h}_{\mathrm{1m}}^{2-}\right)\left({m}_{1}^{2+}-{m}_{1}^{2-}\right)\\ -\theta \Delta t{\text{M}}_{1}^{1+}{F}^{m}-(1-\theta )\Delta t{\text{M}}_{1}^{1-}{F}^{m}-\theta \Delta t{\text{M}}_{1}^{2+}{F}^{m}-(1-\theta )\Delta t{\text{M}}_{1}^{2-}{F}^{m}\end{array}\)

dans R(ADDETE)

Injection des termes

\(\begin{array}{c}-\theta \Delta t{h}_{\mathrm{1m}}^{1+}{\text{M}}_{1}^{1+}-(1-\theta )-\theta \Delta t{h}_{\mathrm{1m}}^{1-}{\text{M}}_{1}^{1-}\text{ou}\\ -\theta \Delta t\left({h}_{\mathrm{1m}}^{1+}{\text{M}}_{1}^{1+}+{h}_{\mathrm{1m}}^{2+}{\text{M}}_{1}^{2+}\right)-(1-\theta )\Delta t\left({h}_{\mathrm{1m}}^{1-}{\text{M}}_{1}^{1-}+{h}_{\mathrm{1m}}^{2-}{\text{M}}_{1}^{2-}\right)\end{array}\)

dans R(ADDETE+1)

Si YAP2

Injection des termes \(+{m}_{2}^{1+}-{m}_{2}^{1-}\text{ou}+{m}_{2}^{1+}+{m}_{2}^{2+}-{m}_{2}^{1-}-{m}_{2}^{2-}\) dans R(ADDEP2)

Injection des termes

\(\begin{array}{c}\Delta t\theta {\text{M}}_{2}^{1+}+(1-\theta )\Delta t{\text{M}}_{2}^{1-}\text{ou}\\ \theta \Delta t\left({\text{M}}_{2}^{1+}+{\text{M}}_{2}^{2+}\right)+(1-\theta )\Delta t\left({\text{M}}_{2}^{1-}+{\text{M}}_{2}^{2-}\right)\end{array}\)

dans R(ADDEP2+1)

SI YAMEC

Injection des termes:

\(-{m}_{2}^{1+}{\text{F}}^{{m}^{+}}\text{ou}-\left({m}_{2}^{1+}+{m}_{2}^{2+}\right){\text{F}}^{{m}^{+}}\) dans R(ADDEME)

Si YATE

Injection des termes:

\(\begin{array}{c}\Delta t\left(\theta {h}_{\mathrm{2m}}^{1+}+(1-\theta ){h}_{\mathrm{2m}}^{1-}\right)\left({m}_{2}^{1+}-{m}_{2}^{1-}\right)-\theta \Delta t{\text{M}}_{2}^{1+}{F}^{m}-(1-\theta )\Delta t{\text{M}}_{2}^{1-}{F}^{m}\\ \text{ou}\\ \Delta t\left(\theta {h}_{\mathrm{2m}}^{1+}+(1-\theta ){h}_{\mathrm{2m}}^{1-}\right)\left({m}_{2}^{1+}-{m}_{2}^{1-}\right)+\Delta t\left(\theta {h}_{\mathrm{2m}}^{2+}+(1-\theta ){h}_{\mathrm{2m}}^{2-}\right)\left({m}_{2}^{2+}-{m}_{2}^{2-}\right)\\ -\theta \Delta t{\text{M}}_{2}^{1+}{F}^{m}-(1-\theta )\Delta t{\text{M}}_{2}^{1-}{F}^{m}-\theta \Delta t{\text{M}}_{2}^{2+}{F}^{m}-(1-\theta )\Delta t{\text{M}}_{2}^{2-}{F}^{m}\end{array}\)

dans R(ADDETE)

Injection des termes

\(\begin{array}{c}-\theta \Delta t{h}_{\mathrm{2m}}^{1+}{\text{M}}_{2}^{1+}-(1-\theta )-\theta \Delta t{h}_{\mathrm{2m}}^{1-}{\text{M}}_{2}^{1-}\text{ou}\\ -\theta \Delta t\left({h}_{\mathrm{2m}}^{1+}{\text{M}}_{2}^{1+}+{h}_{\mathrm{2m}}^{2+}{\text{M}}_{2}^{2+}\right)-(1-\theta )\Delta t\left({h}_{\mathrm{2m}}^{1-}{\text{M}}_{2}^{1-}+{h}_{\mathrm{2m}}^{2-}{\text{M}}_{2}^{2-}\right)\end{array}\)

dans R(ADDETE+1)

Si YATE

Injection des termes: \(Q{'}^{+}-Q{'}^{-}\) dans R(ADDETE)

Injection des termes \(-\theta \Delta t{\text{q}}^{+}-(1-\theta )\Delta t{\text{q}}^{-}\) dans R(ADDETE+1)

Accumulation dans vecteur V:

\(\left\lbrace \text{V}\right\rbrace =\left\lbrace \text{V}\right\rbrace +{\left[{\text{B}}_{g}^{\mathit{el}}\right]}^{T}\left\lbrace R\right\rbrace\)

SI RAPH_MECA ou RIGI_MECA_TANG

SI YAMEC

calcul de DR1DS et injection en DRDS(ADDEME,ADCOME)

calcul de DR2DS et injection en DRDS(ADDEME+NDIM-1,ADCOME)

SI YAP1

calcul de DR1P11 et injection en DRDS(ADDEME,ADCP11)

calcul de DR2P11 et injection en DRDS(ADDEME+NDIM-1, ADCP11)

SI NBPHA1 > 1

calcul de DR1P12 et injection en DRDS(ADDEME,ADCP12)

calcul de DR2P12 et injection en DRDS(ADDEME+NDIM-1, ADCP12)

SI YAP2

calcul de DR1P21 et injection en DRDS(ADDEME,ADCP21)

calcul de DR2P21 et injection en DRDS(ADDEME+NDIM-1, ADCP21)

SI NBPHA2 > 1

calcul de DR1P22 et injection en DRDS(ADDEME,ADCP22)

calcul de DR2P22 et injection en DRDS(ADDEME+NDIM-1, ADCP22)

SI YATE

calcul de DR1DT et injection en DRDS(ADDEME,ADCOTE)

calcul de DR2DT et injection en DRDS(ADDEME+NDIM-1, ADCOTE)

SI YAP1

calcul de DR3P11 et injection en DRDS(ADDEP1,ADCP11)

calcul de DR4P11 et injection en DRDS(ADDEP1+1,ADCP11)

SI NBPHA1 > 1

calcul de DR3P12 et injection en DRDS(ADDEP1,ADCP12)

calcul de DR4P12 et injection en DRDS(ADDEP1+1,ADCP12)

SI YAMEC

calcul de DR3DS et injection en DRDS(ADDEP1,ADCOME)

calcul de DR4DS et injection en DRDS(ADDEP1+1, ADCOME)

SI YAP2

calcul de DR3P21 et injection en DRDS(ADDEP1,ADCP21)

calcul de DR4P21 et injection en DRDS(ADDEP1+ 1, ADCP21)

SI NBPHA2 > 1

calcul de DR3P22 et injection en DRDS(ADDEP1,ADCP22)

calcul de DR4P21 et injection en DRDS(ADDEP1+ 1, ADCP22)

SI YATE

calcul de DR3DT et injection en DRDS(ADDEP1,ADCOTE)

calcul de DR4DT et injection en DRDS(ADDEP1+ 1, ADCOTE)

SI YAP2

calcul de DR5P21 et injection en DRDS(ADDEP2,ADCP21)

calcul de DR6P21 et injection en DRDS(ADDEP2+1,ADCP21)

SI NBPHA2 > 1

calcul de DR5P22 et injection en DRDS(ADDEP2,ADCP22)

calcul de DR6P22 et injection en DRDS(ADDEP2+1,ADCP22)

SI YAMEC

calcul de DR5DS et injection en DRDS(ADDEP2,ADCOME)

calcul de DR6DS et injection en DRDS(ADDEP2+ 1, ADCOME)

YAP1 donc:

calcul de DR5P11 et injection en DRDS(ADDEP2,ADCP11)

calcul de DR6P11 et injection en DRDS(ADDEP2+ 1, ADCP11)

SI NBPHA1 > 1

calcul de DR5P12 et injection en DRDS(ADDEP2,ADCP12)

calcul de DR6P12 et injection en DRDS(ADDEP2+ 1, ADCP12)

SI YATE

calcul de DR5DT et injection en DRDS(ADDEP2,ADCOTE)

calcul de DR6DT et injection en DRDS(ADDEP2+ 1, ADCOTE)

SI YATE

calcul de DR7DT et injection en DRDS(ADDETE,ADCOTE)

calcul de DR8DT et injection en DRDS(ADDETE+1,ADCOTE)

SI YAMEC

calcul de DR7DS et injection en DRDS(ADDETE,ADCOME)

calcul de DR8DS et injection en DRDS(ADDETE+ 1, ADCOME)

SI YAP1

calcul de DR7P11 et injection en DRDS(ADDETE,ADCP11)

calcul de DR8P11 et injection en DRDS(ADDETE+ 1, ADCP11)

SI NBPHA1 > 1

calcul de DR7P12 et injection en DRDS(ADDETE,ADCP12)

calcul de DR8P12 et injection en DRDS(ADDETE+ 1, ADCP12)

SI YAP2

calcul de DR7P21 et injection en DRDS(ADDETE,ADCP21)

calcul de DR8P21 et injection en DRDS(ADDETE+ 1, ADCP21)

SI NBPHA1 > 1

calcul de DR7P22 et injection en DRDS(ADDETE,ADCP22)

calcul de DR8P22 et injection en DRDS(ADDETE+ 1, ADCP22)

\(\left[\text{DRDE}\right]=\left[\text{DRDS}\right]\cdot \left[\text{DSDE}\right]\)

\(\left[{{\text{DF}}_{g}^{\mathit{el}}}_{i({u}_{n}^{+},{P}_{n}^{+},{T}_{n}^{+})}\right]={\left[{\text{B}}_{g}^{\mathit{el}}\right]}^{T}\cdot \left[\text{DRDE}\right]\cdot \left[{\text{B}}_{g}^{\mathit{el}}\right]\) accumulé dans MAT

Arguments de la routine d’appel des lois de comportement#

SUBROUTINE COMTHM(

ARGUMENTS D’ENTREE: IN

COMPOR

OPTION

NDIM

NDDL

DIMDEF

DIMCON

NVIMEC

NVIHY , NVITM

NDEFME

NDEFP1

NDEFP2

NDEFT

NCONME

NCONP1

NCONP2

NCONT

YAP1

NBPHA1

YAP2

NBPHA2

DEMECM \(\text{u},\underline{\underline{\epsilon}}(\text{u})\) temps moins

DEP1M \({p}_{1,}\nabla {p}_{1}\) temps moins

DEP2M \({p}_{2,}\nabla {p}_{2}\) temps moins

DETM \(T,\nabla T\) temps moins

DEMECP \(\text{u},\underline{\underline{\epsilon}}(\text{u})\) temps plus

DEP1P \({p}_{1,}\nabla {p}_{1}\) temps plus

DEP2P \({p}_{2,}\nabla {p}_{2}\) temps plus

DETP \(T,\nabla T\) temps plus

COMECM \(\underline{\underline{\sigma '}},{\sigma}_{p}\) temps moins

COTM \(Q',\text{q}\) temps moins

CP11M \({m}_{1}^{1,}{\text{M}}_{1}^{1}\) ou \({m}_{1}^{1,}{\text{M}}_{1}^{1,}{h}_{\mathrm{1m}}^{1}\) 8 temps moins

CP12M \({m}_{1}^{2,}{\text{M}}_{1}^{2}\) ou \({m}_{1}^{2,}{\text{M}}_{1}^{2,}{h}_{\mathrm{1m}}^{2}\) temps moins

CP21M \({m}_{2}^{1,}{\text{M}}_{2}^{1}\) ou \({m}_{2}^{1,}{\text{M}}_{2}^{1,}{h}_{\mathrm{2m}}^{1}\) temps moins

CP21M \({m}_{2}^{2,}{\text{M}}_{2}^{2}\) ou \({m}_{2}^{1,}{\text{M}}_{2}^{1,}{h}_{\mathrm{2m}}^{1}\) temps moins

VIMEM variables internes méca temps moins

VIHYM variables internes hydro temps moins

VITMM variables internes therm temps moins

ARGUMENTS DE SORTIE: OUT

COMECP \(\underline{\underline{\sigma '}},{\sigma}_{p}\) temps plus

COTP \(Q',\text{q}\) temps plus

CP11P \({m}_{1}^{1,}{\text{M}}_{1}^{1}\) ou \({m}_{1}^{1,}{\text{M}}_{1}^{1,}{h}_{\mathrm{1m}}^{1}\) temps plus

CP12P \({m}_{1}^{2,}{\text{M}}_{1}^{2}\) ou \({m}_{1}^{2,}{\text{M}}_{1}^{2,}{h}_{\mathrm{1m}}^{2}\) temps plus

CP21P \({m}_{2}^{1,}{\text{M}}_{2}^{1}\) ou \({m}_{2}^{1,}{\text{M}}_{2}^{1,}{h}_{\mathrm{2m}}^{1}\) temps plus

CP21P \({m}_{2}^{2,}{\text{M}}_{2}^{2}\) ou \({m}_{2}^{1,}{\text{M}}_{2}^{1,}{h}_{\mathrm{2m}}^{1}\) temps plus

VIMEP variables internes méca temps plus

VIHYP variables internes hydro temps plus

VITMP variables internes therm temps plus

DMECDE \(\left[\begin{array}{c}\frac{\partial \sigma '}{\partial \epsilon }\\ \frac{\partial {\sigma}_{p}}{\partial \epsilon }\end{array}\right]\)

DMECP1 \(\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial {p}_{1}}& \frac{\partial \sigma '}{\partial \nabla {p}_{1}}\\ \frac{\partial {\sigma}_{p}}{\partial {p}_{1}}& \frac{\partial {\sigma}_{p}}{\partial \nabla {p}_{1}}\end{array}\right]\)

DMECP2 \(\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial {p}_{2}}& \frac{\partial \sigma '}{\partial \nabla {p}_{2}}\\ \frac{\partial {\sigma}_{p}}{\partial {p}_{2}}& \frac{\partial {\sigma}_{p}}{\partial \nabla {p}_{2}}\end{array}\right]\)

DMECDT \(\left[\begin{array}{cc}\frac{\partial \sigma '}{\partial T}& \frac{\partial \sigma '}{\partial \nabla T}\\ \frac{\partial {\sigma}_{p}}{\partial T}& \frac{\partial {\sigma}_{p}}{\partial \nabla T}\end{array}\right]\)

DP11DE \(\left[\begin{array}{c}\frac{\partial {m}_{1}^{1}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \epsilon }\end{array}\right]\)

DP11P1 \(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{1}}\end{array}\right]\)

DP11P2 \(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{2}}\end{array}\right]\)

DP11DT \(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial T}& \frac{\partial {m}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial T}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla T}\end{array}\right]\)

DP12DE \(\left[\begin{array}{c}\frac{\partial {m}_{1}^{2}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \epsilon }\end{array}\right]\)

DP12P1 \(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{1}}\end{array}\right]\)

DP12P2 \(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{2}}\end{array}\right]\)

DP12DT \(\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial T}& \frac{\partial {m}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial T}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla T}\end{array}\right]\)

DP21DE \(\left[\begin{array}{c}\frac{\partial {m}_{2}^{1}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \epsilon }\end{array}\right]\)

DP21P1 \(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{1}}\end{array}\right]\)

DP21P2 \(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{2}}\end{array}\right]\)

DP21DT \(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial T}& \frac{\partial {m}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial T}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla T}\end{array}\right]\)

DP22DE \(\left[\begin{array}{c}\frac{\partial {m}_{2}^{2}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \epsilon }\end{array}\right]\)

DP22P1 \(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{1}}\end{array}\right]\)

DP22P2 \(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{2}}\end{array}\right]\)

DP22DT \(\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial T}& \frac{\partial {m}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial T}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla T}\end{array}\right]\)

DTDE \(\left[\begin{array}{c}\frac{\partial Q'}{\partial \epsilon }\\ \frac{\partial \text{q}}{\partial \epsilon }\end{array}\right]\)

DTDP1 \(\left[\begin{array}{cc}\frac{\partial Q'}{\partial {p}_{1}}& \frac{\partial Q'}{\partial \nabla {p}_{1}}\\ \frac{\partial \text{q}}{\partial {p}_{1}}& \frac{\partial \text{q}}{\partial \nabla {p}_{1}}\end{array}\right]\)

DTDP2 \(\left[\begin{array}{cc}\frac{\partial Q'}{\partial {p}_{2}}& \frac{\partial Q'}{\partial \nabla {p}_{2}}\\ \frac{\partial \text{q}}{\partial {p}_{2}}& \frac{\partial \text{q}}{\partial \nabla {p}_{2}}\end{array}\right]\)

DTDT \(\left[\begin{array}{cc}\frac{\partial Q'}{\partial T}& \frac{\partial Q'}{\partial \nabla T}\\ \frac{\partial \text{q}}{\partial T}& \frac{\partial \text{q}}{\partial \nabla T}\end{array}\right]\)

 
)

REAL*8

DEMECM(NDEFME), DEP1M(NDEFP1), DEP2M(NDEFP2), DETM(NDEFT)

DEMECP(NDEFME), DEP1P(NDEFP1), DEP2P(NDEFP2), DETP(NDEFT)

COMECM(NCONME), CP11M(NCONP1), CP21M(NCONP2), COTM(NCONT)

VIMEM(NVIMEC), VIHYM(NVIHY) , VITMM (NVITM)

COMECP(NCONME), CP11P(NCONP1), CP21P(NCONP2), COTP(NCONT)

VIMEP(NVIMEC), VIHYP(NVIHY), VITMP (NVITM)

DMECDE(NCONME,NDEFME),DMECP1(NCONME,NDEFP1),

DMECP2(NCONME,NDEFP2),DMECDT(NCONME,NDEFT)

DP11DE(NCONP1,NDEFME),DP11P1(NCONP1,NDEFP1),

DP11P2(NCONP1,NDEFP2),DP11DT(NCONP1,NDEFT)

DP21DE(NCONP2,NDEFME),DP21P1(NCONP2,NDEFP1,

DP21P2(NCONP2,NDEFP2,DP21DT(NCONP2,NDEFT)

DP12DE(NCONP1,NDEFME),DP12P1(NCONP1,NDEFP1),

DP12P2(NCONP1,NDEFP2),DP12DT(NCONP1,NDEFT)

DP22DE(NCONP2,NDEFME),DP22P1(NCONP2,NDEFP1,

DP22P2(NCONP2,NDEFP2,DP22DT(NCONP2,NDEFT)

DTDE(NCONT2,NDEFME),DTDP1(NCONT2,NDEFP1),

DTDP2(NCONT2,NDEFP2),DTDT(NCONT2,NDEFT)

Éléments finis en THM#

Attributs dans les catalogues#

Pour identifier un élément fini de type THM dans le catalogue phenomenons_modelisation, on utilise les attributs suivants:

  • Attribut TYPMOD2=”THM” pour dire que cet élément permet le couplage THM;

  • Attribut THER = “OUI”/”NON” quand on de la thermique:

  • Attribut MECA = “OUI”/”NON” quand on de la mécanique:

  • Attribut HYDR1 = “0”, “1” ou “2” selon le nombre de phases du premier constituant;

  • Attribut HYDR2 = “0”, “1” ou “2” selon le nombre de phases du second constituant.

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