Variational formulation
==========================




Mechanics
---------

We note :math:`{U}_{\mathrm{ad}}` the set of eligible travel fields, that is to say the elements of :math:`{({H}^{1}(\Omega ))}^{d}` verifying the boundary conditions while traveling on the part of :math:`\partial \Omega` supporting such conditions.

The optimum conditions for energy () give the following variational formulation:


.. math::
: label: eq-4

    \ {\ begin {array} {c}\ sigma =\ sigma\ text {'} =\ sigma\ text {'} + {\ sigma} _ {p} I\\\ tau =\ tau\ text {'} -pn\\ {\ int} _ {\ omega\ setminus\ int} _ {\ int} _ {\ int} _ {\ omega\ setminus\ Gamma\ Gamma\ Gamma}\ Gamma}\ Gamma}\ gamma}\ gamma}\ gamma}\ gamma}\ gamma}\ gamma}\ gamma}\ gamma}\ tau\ sigma:\ epsilon (\ stackrel {} {u}) mathrm {.} \ stackrel {} {u} d\ Gamma = {W} _ {\ mathrm {ext}} (\ stackrel {} {u})\ forall\ stackrel {} {u} {u} {u}\ in {U} _ {\ mathrm {ad}} {u})\ forall\ stackrel {} {} {u} {u})\ forall\ stackrel {} {u} {u} {u})\ forall\ stackrel {} {u} {u} {u}\ forall\ stackrel {}




Hydraulics
-----------

We denote :math:`{P}_{\mathrm{ad}}^{\text{+}}` (resp. :math:`{P}_{\mathrm{ad}}^{\text{-}}`) the set of admissible pressure fields on :math:`{\Omega }^{\text{+}}` (resp. :math:`{\Omega }^{\text{-}}`), that is to say the elements of :math:`{H}^{1}({\Omega }^{\text{+}})` (resp. :math:`{H}^{1}({\Omega }^{\text{-}})`) verifying the pressure boundary conditions on :math:`\partial {\Omega }_{P}^{\text{+}}`, the part of :math:`\partial {\Omega }^{\text{+}}` supporting pressure limit conditions, (resp. :math:`\partial {\Omega }_{P}^{\text{-}}`). And we note :math:`{P}_{\mathrm{ad}}` the set of admissible pressure fields on :math:`\Gamma`, that is to say the elements of :math:`{H}^{1}(\Gamma )` verifying the pressure boundary conditions on :math:`\partial {\Gamma }_{P}`.


.. math::
: label: eq-4

    - {\ int} _ {{\ omega} ^ {\ text {+}}}}\ frac {\ partial m} {\ partial t} {\ stackrel {} {p}} ^ {\ text {+}} ^ {\ text {+}}} d\ omega + {+}} d\ omega + {\ int}} _ {\ int} _ {{\ omega}} ^ {\ text {+}}} ^ {\ text {+}}} ^ {\ text {+}}} ^ {\ text {+}}} ^ {\ text {+}}} ^ {\ text {+}}} d\ omega + {\ int}} d\ omega + {\ int}} _ {\ int} {\nabla\ stackrel {} {p}}} ^ {\ text {+}}} ^ {\ text {+}} d\ omega = {\ int} _ {F} ^ {\ text {+}}}} F\ mathrm {.}}}} F\ mathrm {.}}}} F\ mathrm {.}}}} F\ mathrm {.}}}} F\ mathrm {.}}} F\ mathrm {.}}} F\ mathrm {.}}} ^ {\ text {+}}}} F\ mathrm {.}}} F\ mathrm {.}}} F\ mathrm {.}}} f\ mathrm {.}}} f\ mathrm {.}} n {\ q} ^ {\ text {+}} {\ stackrel {} {p}}} ^ {\ text {+}} d\ Gamma\ forall {\ stackrel {} {p}}} ^ {\ text {+}}} ^ {\ text {+}}} ^ {\ text {+}}} ^ {\ text {+}}} ^ {\ text {+}}} ^ {\ text {+}}


.. math::
: label: eq-4

    - {\ int} _ {{\ omega} ^ {\ text {-}}}}\ frac {\ partial m} {\ partial t} {\ stackrel {} {p}} ^ {\ text {-}} ^ {\ text {-}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}} d\ omega + {\ int}} d\ omega + {\ int} _ {\ int} _ {\ omega} ^ {\ text {-}}} {\nabla\ stackrel {} {p}}} ^ {\ text {-}}} ^ {\ text {-}} d\ omega = {\ int} _ {F} ^ {\ text {-}}}} F\ mathrm {.}}}} F\ mathrm {.}}}} F\ mathrm {.}}}} F\ mathrm {.}}}} F\ mathrm {.}}}} F\ mathrm {.}}} f\ mathrm {.}}} f\ mathrm {.}}} f\ mathrm {.}} {.} n {\ stackrel {-}} n {\ stackrel {-}} d\ mathrm {.}}} F\ mathrm {.}} q} ^ {\ text {-}} {\ stackrel {} {p}}} ^ {\ text {-}} d\ Gamma\ forall {\ stackrel {} {p}}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}} ^ {\ text {-}}


.. math::
: label: eq-4

    - {\ int} _ {\ Gamma}\ frac {\ partial w} {\ partial w} {\ partial w} {\ partial w} {\ partial w} {\ partial w} {\ partial w} {\ partial w}\ frac {\ partial w}} {frac} {\ partial w} {\ partial w} {\ partial w}} {\ partial w} {\ partial w}} {\ partial w} {\ partial w} {\ partial w} {\ partial w} {\ partial w}} {\ partial t} {p}\ stackrel {} {p} {p} d\ Gamma + {\ int}} _ {\ gamma} ({q}} ^ {\ text {+}}} W {\nabla} _ {\ text {+}}} W {\nabla} ^ {\ text {-}})\ stackrel {} {p} {p} d\ Gamma = {\ int} _ {\ partial {\ Gamma} _ {F}} F\ mathrm {.} n\ stackrel {} {p} {p} {p} {p} {p}\ in {P}} _ {\ mathrm {ad}}


.. math::
: label: eq-4

    {\ int} _ {\ Gamma} ({p} ^ {\ text {+}} -p) {\ stackrel {} {q}}} ^ {\ text {+}} d\ Gamma =0\ forall {\ forall {\ stackrel {} {q}}} ^ {\ text {+}} ^ {-1}} d\ Gamma =0\ forall {\ stackrel {} {q}}} ^ {\ text {+}} d\ Gamma =0\ forall {\ stackrel {} {q}}} ^ {-1} (\ Gamma)


.. math::
: label: eq-4

    {\ int} _ {\ Gamma} ({p} ^ {\ text {-}} -p) {\ stackrel {-}}} ^ {\ text {-}} d\ Gamma =0\ forall {\ stackrel {\ stackrel {} {q}}} ^ {\ text {-}} ^ {-}} ^ {-1}} d\ Gamma =0\ forall {\ stackrel {} {q}}} ^ {\ text {-}} d\ Gamma =0\ forall {\ stackrel {} {q}}} ^ {\ text {-}} d\ Gamma =0\ forall {\ stackrel {} {q}}} ^




Temporal discretization
-------------------------

We adopt discretization in implicit time. The notations indexed by :math:`n` are the quantities at the start of the time step and those indexed by :math:`n+1` are the quantities at the end of the time step.

The time step is noted :math:`\Delta t={t}_{n+1}-{t}_{n}`.

Subsequently, in the absence of precision, the non-indexed notations will designate the quantities at the end of the time step.