4. Variational formulation#
4.1. Mechanics#
We note \({U}_{\mathrm{ad}}\) the set of eligible travel fields, that is to say the elements of \({({H}^{1}(\Omega ))}^{d}\) verifying the boundary conditions while traveling on the part of \(\partial \Omega\) supporting such conditions.
The optimum conditions for energy () give the following variational formulation:
: label: eq-4
{begin {array} {c}sigma =sigmatext {”} =sigmatext {”} + {sigma} _ {p} I\tau =tautext {”} -pn\ {int} _ {omegasetminusint} _ {int} _ {int} _ {omegasetminusGammaGammaGamma}Gamma}Gamma}gamma}gamma}gamma}gamma}gamma}gamma}gamma}gamma}tausigma:epsilon (stackrel {} {u}) mathrm {.} stackrel {} {u} dGamma = {W} _ {mathrm {ext}} (stackrel {} {u})forallstackrel {} {u} {u} {u}in {U} _ {mathrm {ad}} {u})forallstackrel {} {} {u} {u})forallstackrel {} {u} {u} {u})forallstackrel {} {u} {u} {u}forallstackrel {}
4.2. Hydraulics#
We denote \({P}_{\mathrm{ad}}^{\text{+}}\) (resp. \({P}_{\mathrm{ad}}^{\text{-}}\)) the set of admissible pressure fields on \({\Omega }^{\text{+}}\) (resp. \({\Omega }^{\text{-}}\)), that is to say the elements of \({H}^{1}({\Omega }^{\text{+}})\) (resp. \({H}^{1}({\Omega }^{\text{-}})\)) verifying the pressure boundary conditions on \(\partial {\Omega }_{P}^{\text{+}}\), the part of \(\partial {\Omega }^{\text{+}}\) supporting pressure limit conditions, (resp. \(\partial {\Omega }_{P}^{\text{-}}\)). And we note \({P}_{\mathrm{ad}}\) the set of admissible pressure fields on \(\Gamma\), that is to say the elements of \({H}^{1}(\Gamma )\) verifying the pressure boundary conditions on \(\partial {\Gamma }_{P}\).
: label: eq-4
{int} _ {{omega} ^ {text {+}}}}frac {partial m} {partial t} {stackrel {} {p}} ^ {text {+}} ^ {text {+}}} domega + {+}} domega + {int}} _ {int} _ {{omega}} ^ {text {+}}} ^ {text {+}}} ^ {text {+}}} ^ {text {+}}} ^ {text {+}}} ^ {text {+}}} domega + {int}} domega + {int}} _ {int} {nablastackrel {} {p}}} ^ {text {+}}} ^ {text {+}} domega = {int} _ {F} ^ {text {+}}}} Fmathrm {.}}}} Fmathrm {.}}}} Fmathrm {.}}}} Fmathrm {.}}}} Fmathrm {.}}} Fmathrm {.}}} Fmathrm {.}}} ^ {text {+}}}} Fmathrm {.}}} Fmathrm {.}}} Fmathrm {.}}} fmathrm {.}}} fmathrm {.}} n {q} ^ {text {+}} {stackrel {} {p}}} ^ {text {+}} dGammaforall {stackrel {} {p}}} ^ {text {+}}} ^ {text {+}}} ^ {text {+}}} ^ {text {+}}} ^ {text {+}}} ^ {text {+}}
: label: eq-4
{int} _ {{omega} ^ {text {-}}}}frac {partial m} {partial t} {stackrel {} {p}} ^ {text {-}} ^ {text {-}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}} domega + {int}} domega + {int} _ {int} _ {omega} ^ {text {-}}} {nablastackrel {} {p}}} ^ {text {-}}} ^ {text {-}} domega = {int} _ {F} ^ {text {-}}}} Fmathrm {.}}}} Fmathrm {.}}}} Fmathrm {.}}}} Fmathrm {.}}}} Fmathrm {.}}}} Fmathrm {.}}} fmathrm {.}}} fmathrm {.}}} fmathrm {.}} {.} n {stackrel {-}} n {stackrel {-}} dmathrm {.}}} Fmathrm {.}} q} ^ {text {-}} {stackrel {} {p}}} ^ {text {-}} dGammaforall {stackrel {} {p}}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}} ^ {text {-}}
: 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} dGamma + {int}} _ {gamma} ({q}} ^ {text {+}}} W {nabla} _ {text {+}}} W {nabla} ^ {text {-}})stackrel {} {p} {p} dGamma = {int} _ {partial {Gamma} _ {F}} Fmathrm {.} nstackrel {} {p} {p} {p} {p} {p}in {P}} _ {mathrm {ad}}
: label: eq-4
{int} _ {Gamma} ({p} ^ {text {+}} -p) {stackrel {} {q}}} ^ {text {+}} dGamma =0forall {forall {stackrel {} {q}}} ^ {text {+}} ^ {-1}} dGamma =0forall {stackrel {} {q}}} ^ {text {+}} dGamma =0forall {stackrel {} {q}}} ^ {-1} (Gamma)
: label: eq-4
{int} _ {Gamma} ({p} ^ {text {-}} -p) {stackrel {-}}} ^ {text {-}} dGamma =0forall {stackrel {stackrel {} {q}}} ^ {text {-}} ^ {-}} ^ {-1}} dGamma =0forall {stackrel {} {q}}} ^ {text {-}} dGamma =0forall {stackrel {} {q}}} ^ {text {-}} dGamma =0forall {stackrel {} {q}}} ^
4.3. Temporal discretization#
We adopt discretization in implicit time. The notations indexed by \(n\) are the quantities at the start of the time step and those indexed by \(n+1\) are the quantities at the end of the time step.
The time step is noted \(\Delta t={t}_{n+1}-{t}_{n}\).
Subsequently, in the absence of precision, the non-indexed notations will designate the quantities at the end of the time step.