Variational writing of equilibrium equations
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Mechanics
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We note :math:`{U}_{ad}` the set of kinematically admissible displacement fields, that is to say the elements of :math:`{({H}^{1}(\Omega ))}^{3}` verifying the boundary conditions while moving on the part of :math:`\partial \Omega` supporting such conditions [:ref:`bib3 <bib3>`].

The variational form of [:ref:`éq 3.1-1 <éq 3.1-1>`] is:

:math:`\{\begin{array}{c}\mathrm{\sigma }=\mathrm{\sigma }\text{'}+{\sigma }_{p}\mathrm{I}\\ \int {}_{\mathrm{\Omega }}\text{}\mathrm{\sigma }\text{.}\mathrm{\epsilon }\left(\mathrm{v}\right)=\int {}_{\mathrm{\Omega }}\text{}r{\mathrm{F}}^{m}\text{.}\mathrm{v}d\mathrm{\Omega }+\int {}_{\partial \mathrm{\Omega }}\text{}{\mathrm{f}}^{\mathit{ext}}\text{.}\mathrm{v}d\mathrm{\Gamma }\text{}\forall \mathrm{v}\in {U}_{\mathit{ad}}\end{array}` **eq 4.1-1**




Hydraulics
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We denote :math:`{P}_{1\mathrm{ad}}` (resp. :math:`{P}_{2\mathrm{ad}}`) the set of admissible pressure fields :math:`{\pi }_{1}` (resp. :math:`{\pi }_{2}`), that is to say the elements of :math:`{H}^{1}(\Omega )` verifying the pressure boundary conditions :math:`{P}_{1}` (resp. :math:`{P}_{2}`) on the part of :math:`\partial \Omega` supporting such conditions [:ref:`bib3 <bib3>`]. The variational form of [:ref:`éq 3.2-1 <éq 3.2-1>`] is:

:math:`\begin{array}{}-{\int }_{\Omega }(\dot{{m}_{1}^{1}}+\dot{{m}_{1}^{2}})\text{.}{\pi }_{1}+{\int }_{\Omega }({M}_{1}^{1}+{M}_{1}^{2})\text{.}\nabla {\pi }_{1}d\Omega =\text{}\\ {\int }_{\partial \Omega }({M}_{1\mathrm{ext}}^{1}+{M}_{1\mathrm{ext}}^{2})\text{.}{\pi }_{1}d\Gamma \forall {\pi }_{1}\in {P}_{1\mathrm{ad}}\\ -{\int }_{\Omega }(\dot{{m}_{2}^{1}}+\dot{{m}_{2}^{2}})\text{.}{\pi }_{2}+{\int }_{\Omega }({M}_{2}^{1}+{M}_{2}^{2})\text{.}\nabla {\pi }_{2}d\Omega =\text{}\\ {\int }_{\partial \Omega }({M}_{2\mathrm{ext}}^{1}+{M}_{2\mathrm{ext}}^{2})\text{.}{\pi }_{2}d\Gamma \forall {\pi }_{2}\in {P}_{2\mathrm{ad}}\end{array}\}` **eq 4.2-1**

which shows scalar hydraulic flows :math:`{M}_{i\mathrm{ext}}^{j}` at the edges.




Thermal
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We note :math:`{T}_{\mathrm{ad}}` the set of admissible temperature fields :math:`\tau`, that is to say the elements of :math:`{H}^{1}(\Omega )` verifying the temperature boundary conditions on the part of :math:`\partial \Omega` supporting such conditions. [:ref:`bib3 <bib3>`]. The variational form of [:ref:`éq 3.3.3-3 <éq 3.3.3-3>`] is:

:math:`\begin{array}{}{\int }_{\Omega }\dot{Q}\text{'}\text{.}\tau d\Omega +\sum _{p,c}{\int }_{\Omega }{{h}^{m}}_{c}^{p}\dot{{m}_{c}^{p}}\text{.}\tau d\Omega -{\int }_{\Omega }(\sum _{p,c}{{h}^{m}}_{c}^{p}{M}_{c}^{p}+q)\text{.}\nabla \tau d\Omega =\text{}\\ {\int }_{\Omega }(\Theta +\sum _{p,c}{M}_{c}^{p}\text{.}{F}^{m})\text{.}\tau d\Omega -{\int }_{\partial \Omega }(\sum _{p,c}{{h}^{m}}_{c}^{p}{M}_{c\mathrm{ext}}^{p}+{q}_{\mathrm{ext}})\text{.}\tau d\Gamma \\ \forall \tau \in {T}_{\mathrm{ad}}\end{array}` **eq 4.3-1**

Note that, unlike other presentations, and in particular [:ref:`bib8 <bib8>`] we did not inject the mass conservation equations, and we integrated the transport term :math:`\sum _{p,c}\text{Div}({{h}^{m}}_{c}^{p}{M}_{c}^{p})` in part.

This last point has the advantage of not causing higher-order derivatives to appear, and, on the contrary, of naturally causing boundary conditions relating to heat input linked to hydraulic flows to appear: :math:`\sum _{p,c}{\int }_{\partial \Omega }{{h}^{m}}_{c}^{p}{M}_{c\mathrm{ext}}^{p}\text{.}\tau d\Gamma`.

It may in fact be considered that the heat flow conditions directly define:

:math:`\stackrel{~}{{q}_{\mathit{ext}}}={{h}^{m}}_{c}^{p}{M}_{c\mathit{ext}}^{p}+{q}_{\mathit{ext}}`