6. Solving the coupled problem#
6.1. Linearization of the coupled problem#
6.1.1. Linearization using the Newton-Raphson method#
Since the coupled problem is non-linear (the nonlinearity of the problem is due to the mass terms of the variational formulations of the mass conservation equations for the mass and for the interface and to the cohesive terms for mechanics) we proceed with its linearization using the Newton-Raphson method.
Let \(F\) be the nonlinear system associated with the variational formulations of the mass conservation equations (for the mass and for the interface), the mechanical equilibrium equation and the pressure continuity condition \({p}_{f}\) at the interface level. Let \({x}^{k}\) be the vector of nodal unknowns at the Newton iteration \(k\) such that:
\({x}^{k}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\{{u}^{k}\phantom{\rule{1em}{0ex}}{p}^{k}\phantom{\rule{1em}{0ex}}{p}_{f}^{k}\phantom{\rule{1em}{0ex}}{q}_{1}^{k}\phantom{\rule{1em}{0ex}}{q}_{2}^{k}\phantom{\rule{1em}{0ex}}\mathrm{\lambda }\phantom{\rule{1em}{0ex}}\mathrm{\mu }\phantom{\rule{1em}{0ex}}w\}}^{T}\)
In iteration \(k+1\) (the vector of the nodal unknowns \({x}^{k+1}\) is not known) we set:
\(F({x}^{k+1})=0\)
In order to be able to determine \({x}^{k+1}\), we use a Taylor expansion of \(F\) (which is a vector function that is assumed to be continuous and differentiable) in the vicinity of \({x}^{k}\) (then known at iteration \(k+1\)). So the linear system at iteration \(k+1\) is written:
\(-F({x}^{k})=\frac{\partial F({x}^{k})}{\partial {x}^{k}}\cdot \delta {x}^{k}\)
with \(\delta {x}^{k}={x}^{k+1}-{x}^{k}\) the increment of the values of the nodal unknowns between two successive iterations (which is an unknown at iteration \(k+1\)), \(\frac{\partial F({x}^{k})}{\partial {x}^{k}}\) the tangent matrix and \(F({x}^{k})\) the second member. These last two terms are known in iteration \(k+1\) and are functions of \({x}^{k}\).
6.1.2. Integral writing of the linearized problem#
In the following, we consider that the unknowns are noted with a \(\delta\) and the test fields with a \(\text{*}\) as an exponent.
The linear system at Newton’s iteration \(k+1\) is written (for a time step):
mechanical balance equation
\(\forall {u}^{\text{*}}\in {U}_{0}\):
\(\begin{array}{c}{\int }_{\mathrm{\Omega }}{\mathrm{\sigma }}^{\text{' +}}(\mathrm{\delta }u)\mathrm{:}\mathrm{\epsilon }({u}^{\text{*}})d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}b\mathrm{\delta }p(1\mathrm{:}\mathrm{\epsilon }({u}^{\text{*}}))d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}b{F}^{m\text{+}}\mathit{Tr}(\nabla (\mathrm{\delta }u)){u}^{\text{*}}d\mathrm{\Omega }\hfill \\ \phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}\left(\frac{{\mathrm{\rho }}^{\text{+}}(b-{\mathrm{\varphi }}^{\text{+}})}{{K}_{s}^{}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{{\mathrm{\rho }}^{\text{+}}{\mathrm{\varphi }}^{\text{+}}}{{K}_{w}^{}}\right)\mathrm{\delta }p{F}^{m\text{+}}{u}^{\text{*}}d\mathrm{\Omega }+{\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\delta }\mathrm{\mu }\cdot ⟦{u}^{\text{*}}⟧d{\mathrm{\Gamma }}_{c}\hfill \\ \phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{\mathrm{\Omega }}{\mathrm{\sigma }}^{\text{' +}}(u)\mathrm{:}\mathrm{\epsilon }({u}^{\text{*}})d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}b{p}^{\text{+}}(1\mathrm{:}\mathrm{\epsilon }({u}^{\text{*}}))d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}{r}^{\text{+}}{F}^{m\text{+}}{u}^{\text{*}}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{t}}{t}^{\text{+}}{u}^{\text{*}}d{\mathrm{\Gamma }}_{t}\hfill \\ \phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\mu }\cdot ⟦{u}^{\text{*}}⟧d{\mathrm{\Gamma }}_{c}\hfill \end{array}\)
projection of the displacement jump
\(\forall {\mathrm{\mu }}^{\text{*}}\in {L}_{0}\):
\({\int }_{{\mathrm{\Gamma }}_{c}}\left(⟦\mathrm{\delta }u⟧-\mathrm{\delta }w\right)\cdot {\mathrm{\mu }}^{\text{*}}d{\mathrm{\Gamma }}_{c}=-{\int }_{{\mathrm{\Gamma }}_{c}}\left(⟦u⟧-w\right)\cdot {\mathrm{\mu }}^{\text{*}}d{\mathrm{\Gamma }}_{c}\)
cohesive constraints
\(\forall {w}^{\text{*}}\in {L}_{0}\):
\(\begin{array}{c}-{\int }_{{\mathrm{\Gamma }}_{c}}\left[\mathrm{\delta }\mathrm{\mu }-\frac{\partial {t\text{'}}_{c}}{\partial (\mathrm{\lambda }+rw)}\cdot \left(\mathrm{\delta }\mathrm{\lambda }+r\mathrm{\delta }w\right)\right]\cdot {w}^{\text{*}}d{\mathrm{\Gamma }}_{c}-{\int }_{{\mathrm{\Gamma }}_{c}}\mathrm{\delta }{p}_{f}{n}_{c}\cdot {w}^{\text{*}}d{\mathrm{\Gamma }}_{c}\hfill \\ ={\int }_{{\mathrm{\Gamma }}_{c}}\left[\mathrm{\mu }-{t\text{'}}_{c}(\mathrm{\lambda }+rw)+{p}_{f}{n}_{c}\right]\cdot {w}^{\text{*}}d{\mathrm{\Gamma }}_{c}\hfill \end{array}\)
interface law
\(\forall {\mathrm{\lambda }}^{\text{*}}\in {L}_{0}\):
\(\begin{array}{c}-{\int }_{{\mathrm{\Gamma }}_{c}}\left[\frac{\mathrm{\delta }\mathrm{\lambda }}{r}-\frac{\partial {t\text{'}}_{c}}{\partial (\mathrm{\lambda }+rw)}\cdot \left(\frac{\mathrm{\delta }\mathrm{\lambda }}{r}+\mathrm{\delta }w\right)\right]\cdot {\mathrm{\lambda }}^{\text{*}}d{\mathrm{\Gamma }}_{c}\hfill \\ ={\int }_{{\mathrm{\Gamma }}_{c}}\frac{\left[\mathrm{\lambda }-{t\text{'}}_{c}(\mathrm{\lambda }+rw)\right]}{r}\cdot {\mathrm{\lambda }}^{\text{*}}d{\mathrm{\Gamma }}_{c}\hfill \end{array}\)
mass conservation equation (case of the massif)
So \(\forall {p}^{\text{*}}\in {P}_{0}\):
\(\begin{array}{c}-{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}b\mathit{Tr}(\nabla (\mathrm{\delta }u)){p}^{\text{*}}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}\left(\left(\frac{{\mathrm{\rho }}^{\text{+}}(b-{\mathrm{\varphi }}^{\text{+}})}{{K}_{s}^{}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{{\mathrm{\rho }}^{\text{+}}{\mathrm{\varphi }}^{\text{+}}}{{K}_{w}^{}}\right)\mathrm{\delta }p\right){p}^{\text{*}}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0ex}{0ex}}\hfill \\ \mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{\mathrm{\Omega }}\left({\mathrm{\lambda }}^{\text{+}}\left(-\nabla {p}^{\text{+}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\mathrm{\rho }}^{\text{+}}{F}^{m\text{+}}\right)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\mathrm{\lambda }}^{\text{+}}{\mathrm{\rho }}^{\text{+}}{F}^{m\text{+}}\right)\frac{{\mathrm{\rho }}^{\text{+}}}{{K}_{w}^{}}\mathrm{\delta }p\nabla {p}^{\text{*}}d\mathrm{\Omega }\right]\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0ex}{0ex}}\hfill \\ \mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}\left(-\nabla {p}^{\text{+}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\mathrm{\rho }}^{\text{+}}{F}^{m\text{+}}\right)\frac{\partial {\mathrm{\lambda }}_{}^{\text{+}}}{\partial {p}^{\text{+}}}\mathrm{\delta }p\nabla {p}^{\text{*}}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}{\mathrm{\lambda }}^{\text{+}}\nabla (\mathrm{\delta }p)\nabla {p}^{\text{*}}d\mathrm{\Omega }\right]\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0ex}{0ex}}\\ \mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}\left(-\nabla {p}^{\text{+}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\mathrm{\rho }}^{\text{+}}{F}^{m\text{+}}\right)\frac{\partial {\mathrm{\lambda }}_{}^{\text{+}}}{\partial {\mathrm{\epsilon }}_{\text{v}}^{\text{+}}}\mathit{Tr}(\nabla (\mathrm{\delta }u))\nabla {p}^{\text{*}}d\mathrm{\Omega }\right]\hfill \\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{1}}\mathrm{\delta }{q}_{1}{p}^{\text{*}}d{\mathrm{\Gamma }}_{1}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{2}}\mathrm{\delta }{q}_{2}{p}^{\text{*}}d{\mathrm{\Gamma }}_{2}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}\left({m}_{w}^{\text{+}}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{m}_{w}^{\text{-}}\right){p}^{\text{*}}d\mathrm{\Omega }\hfill \\ \phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }{\int }_{\mathrm{\Omega }}{M}^{\text{+}}\nabla {p}^{\text{*}}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t(1\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }){\int }_{\mathrm{\Omega }}{M}^{\text{-}}\nabla {p}^{\text{*}}d\mathrm{\Omega }\hfill \\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{F}}{M}_{\text{ext}}^{\text{+}}{p}^{\text{*}}d{\mathrm{\Gamma }}_{F}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t(1\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{F}}{M}_{\text{ext}}^{\text{-}}{p}^{\text{*}}d{\mathrm{\Gamma }}_{F}\hfill \\ \phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{1}}{q}_{1}^{\text{+}}{p}^{\text{*}}d{\mathrm{\Gamma }}_{1}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t(1\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{1}}{q}_{1}^{\text{-}}{p}^{\text{*}}d{\mathrm{\Gamma }}_{1}\hfill \\ \phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{2}}{q}_{2}^{\text{+}}{p}^{\text{*}}d{\mathrm{\Gamma }}_{2}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t(1\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{2}}{q}_{2}^{\text{-}}{p}^{\text{*}}d{\mathrm{\Gamma }}_{2}\hfill \end{array}\)
mass conservation equation (interface case)
So \(\forall {p}_{f}^{\text{*}}\in {F}_{0}\):
\(\begin{array}{c}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}{\mathrm{\rho }}^{\text{+}}\mathrm{\delta }⟦u⟧\cdot {n}_{c}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}\left(\frac{{\mathrm{\rho }}^{\text{+}}⟦{u}^{\text{+}}⟧\cdot {n}_{c}}{{K}_{w}}\mathrm{\delta }{p}_{f}\right){p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\hfill \\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{{\mathrm{\Gamma }}_{c}}-\phantom{\rule{0.5em}{0ex}}\frac{{\mathrm{\rho }}^{\text{+}}\nabla {p}_{f}^{\text{+}}}{12\mathrm{\mu }}\left(3{\left(⟦{u}^{\text{+}}⟧\cdot {n}_{c}\right)}^{2}\right)\mathrm{\delta }⟦u⟧\cdot {n}_{c}\nabla {p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\right]\hfill \\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{{\mathrm{\Gamma }}_{c}}-\phantom{\rule{0.5em}{0ex}}\frac{{\left(⟦{u}^{\text{+}}⟧\cdot {n}_{c}\right)}^{3}\nabla {p}_{f}^{\text{+}}}{12\mathrm{\mu }}\frac{{\mathrm{\rho }}^{\text{+}}}{{K}_{w}}\mathrm{\delta }{p}_{f}\nabla {p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\right]\hfill \\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{{\mathrm{\Gamma }}_{c}}-\phantom{\rule{0.5em}{0ex}}\frac{{\mathrm{\rho }}^{\text{+}}{\left(⟦{u}^{\text{+}}⟧\cdot {n}_{c}\right)}^{3}}{12\mathrm{\mu }}\nabla \left(\mathrm{\delta }{p}_{f}\right)\nabla {p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\right]\hfill \\ \phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{{\mathrm{\Gamma }}_{1}}\mathrm{\delta }{q}_{1}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{1}\right]\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{{\mathrm{\Gamma }}_{2}}\mathrm{\delta }{q}_{2}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{2}\right]\hfill \\ \phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}({w}^{\text{+}}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{w}^{\text{-}}){p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{c}}{W}^{\text{+}}\nabla {p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{c}}{W}^{\text{-}}\nabla {p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{c}\hfill \\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{{\mathrm{\Gamma }}_{1}}{q}_{1}^{\text{+}}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{1}\right]\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t(1-\mathrm{\theta })\left[{\int }_{{\mathrm{\Gamma }}_{1}}{q}_{1}^{\text{-}}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{1}\right]\hfill \\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }\left[{\int }_{{\mathrm{\Gamma }}_{2}}{q}_{2}^{\text{+}}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{2}\right]\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t(1-\mathrm{\theta })\left[{\int }_{{\mathrm{\Gamma }}_{2}}{q}_{2}^{\text{-}}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{2}\right]\hfill \\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{f}}{W}_{\mathit{ext}}^{\text{+}}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{f}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\mathrm{\Delta }t(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{f}}{W}_{\mathit{ext}}^{\text{-}}{p}_{f}^{\text{*}}d{\mathrm{\Gamma }}_{f}\hfill \end{array}\)
pressure continuity equation \({p}_{f}\) at the interface
So \(\forall {q}_{1}^{\text{*}}\in {Q}_{1}\) and \(\forall {q}_{2}^{\text{*}}\in {Q}_{2}\)
\({\int }_{{\Gamma }_{1}}\delta {p}^{\mathit{inf}}{q}_{1}^{\text{*}}d{\Gamma }_{1}-{\int }_{{\Gamma }_{1}}\delta {p}_{f}{q}_{1}^{\text{*}}d{\Gamma }_{1}=-{\int }_{{\Gamma }_{1}}\left({p}^{\mathit{inf}}-{p}_{f}\right){q}_{1}^{\text{*}}d{\Gamma }_{1}\) out of \({\Gamma }_{1}\)
\({\int }_{{\Gamma }_{2}}\delta {p}^{\text{sup}}{q}_{2}^{\text{*}}d{\Gamma }_{2}-{\int }_{{\Gamma }_{2}}\delta {p}_{f}{q}_{2}^{\text{*}}d{\Gamma }_{2}=-{\int }_{{\Gamma }_{2}}\left({p}^{\text{sup}}-{p}_{f}\right){q}_{2}^{\text{*}}d{\Gamma }_{2}\) out of \({\Gamma }_{2}\)
6.2. Writing basic terms with XFEM#
6.2.1. Writing the coupled problem in matrix form#
The system of equations previously discretized at the Newton iteration \(k+1\) can be put in the form (where \(\delta u\), \(\delta p\), \(\delta {p}_{f}\), \(\delta {q}_{1}\), \(\delta {q}_{2}\), \(\mathrm{\delta }\mathrm{\lambda }\), \(\mathrm{\delta }\mathrm{\mu }\) and \(\mathrm{\delta }w\) are the unknowns of the problem to be solved):
Mechanical balance |
\(\begin{array}{c}\{{u}^{\text{*}}\}\left[{K}_{\mathit{meca}}^{1}\right](\mathrm{\delta }u)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{u}^{\text{*}}\}\left[{K}_{\mathit{meca}}^{2}\right](\mathrm{\delta }p)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{u}^{\text{*}}\}\left[A\right](\mathrm{\delta }u)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0ex}{0ex}}\\ \{{u}^{\text{*}}\}\left[B\right](\mathrm{\delta }p)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{⟦u⟧}^{\text{*}}\}\left[{C}^{1}\right]\{\mathrm{\delta }\mathrm{\mu }\}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0ex}{0ex}}\hfill \\ \{{u}^{\text{*}}\}({L}_{\mathit{meca}})\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{⟦u⟧}^{\text{*}}\}({L}^{1})\hfill \end{array}\) |
Move jump projection |
\(\begin{array}{c}\{{\mathrm{\mu }}^{\text{*}}\}\left[{K}^{\mathrm{\mu }u}\right](\mathrm{\delta }u)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{\mathrm{\mu }}^{\text{*}}\}{\left[-{K}^{w\mathrm{\mu }}\right]}^{T}(\mathrm{\delta }w)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0ex}{0ex}}\\ \{{\mathrm{\mu }}^{\text{*}}\}({L}_{u})\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}\{{\mathrm{\mu }}^{\text{*}}\}({L}_{w})\hfill \end{array}\) |
Cohesive constraint |
\(\begin{array}{c}\{{w}^{\text{*}}\}\left[-{K}^{w\mathrm{\mu }}\right](\mathrm{\delta }\mathrm{\mu })\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{w}^{\text{*}}\}\left[{D}^{ww}\right](\mathrm{\delta }w)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{w}^{\text{*}}\}\left[{K}^{wp}\right](\mathrm{\delta }{p}_{f})\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0ex}{0ex}}\\ \{{w}^{\text{*}}\}{\left[{D}^{\mathrm{\lambda }w}\right]}^{T}(\mathrm{\delta }\mathrm{\lambda })\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-\{{w}^{\text{*}}\}({L}_{\mathrm{\mu }}^{2})\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{w}^{\text{*}}\}({L}_{\mathit{cohe}}^{1})\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{w}^{\text{*}}\}({L}_{p})\end{array}\) |
Interface law |
\(\begin{array}{c}\{{\mathrm{\lambda }}^{\text{*}}\}\left[{D}^{\mathrm{\lambda }w}\right](\mathrm{\delta }w)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{\mathrm{\lambda }}^{\text{*}}\}\left[{D}^{\mathrm{\lambda }\mathrm{\lambda }}\right](\mathrm{\delta }\mathrm{\lambda })\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0ex}{0ex}}\\ -\{{\mathrm{\lambda }}^{\text{*}}\}({L}_{\mathrm{\mu }}^{1})\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\{{\mathrm{\lambda }}^{\text{*}}\}({L}_{\mathit{cohe}}^{2})\hfill \end{array}\) |
Conservation of Mass (case of the massif) |
\(\begin{array}{c}\{{p}^{\text{*}}\}[{M}_{\mathit{hydro}}^{1}](\delta u)+\{{p}^{\text{*}}\}[{M}_{\mathit{hydro}}^{2}](\delta p)+\Delta t\theta \{{p}^{\text{*}}\}[{K}_{\mathit{hydro}}^{1}](\delta p)+\\ \Delta t\theta \{{p}^{\text{*}}\}[{K}_{\mathit{hydro}}^{2}](\delta p)+\Delta t\theta \{{p}^{\text{*}}\}[{K}_{\mathit{hydro}}^{3}](\delta p)+\\ \Delta t\theta \{{p}^{\text{*}}\}[{K}_{\mathit{hydro}}^{4}](\delta u)+\Delta t\theta \{{p}^{\text{*}}\}[{E}^{1}](\delta {q}_{1})+\Delta t\theta \{{p}^{\text{*}}\}[{E}^{2}](\delta {q}_{2})\\ =\{{p}^{\text{*}}\}({L}_{\mathit{hydro}}^{1})+\Delta t\{{p}^{\text{*}}\}{({L}_{\mathit{hydro}}^{2})}_{\theta }+\Delta t\{{p}^{\text{*}}\}{({L}_{\mathit{hydro}}^{3})}_{\theta }+\\ \Delta t\{{p}^{\text{*}}\}{({L}_{\mathit{hydro}}^{4})}_{\theta }+\Delta t\{{p}^{\text{*}}\}{({L}_{\mathit{hydro}}^{5})}_{\theta }\end{array}\) |
Conservation of mass (interface case) |
\(\begin{array}{c}\{{p}_{f}^{\text{*}}\}[{W}_{\mathit{hydro}}^{1}](\delta ⟦u⟧)+\{{p}_{f}^{\text{*}}\}[{W}_{\mathit{hydro}}^{2}](\delta {p}_{f})+\Delta t\theta \{{p}_{f}^{\text{*}}\}[{H}_{\mathit{hydro}}^{1}](\delta ⟦u⟧)+\\ \Delta t\theta \{{p}_{f}^{\text{*}}\}[{H}_{\mathit{hydro}}^{2}](\delta {p}_{f})+\Delta t\theta \{{p}_{f}^{\text{*}}\}[{H}_{\mathit{hydro}}^{3}](\delta {p}_{f})+\\ \Delta t\theta \{{p}_{f}^{\text{*}}\}[{D}^{1}](\delta {q}_{1})+\Delta t\theta \{{p}_{f}^{\text{*}}\}[{D}^{2}](\delta ⟦u⟧)\\ \Delta t\theta \{{p}_{f}^{\text{*}}\}[{D}^{3}](\delta {q}_{2})+\Delta t\theta \{{p}_{f}^{\text{*}}\}[{D}^{4}](\delta ⟦u⟧)=\\ \{{p}_{f}^{\text{*}}\}({L}_{\mathit{hydro}}^{6})+\Delta t\{{p}_{f}^{\text{*}}\}{({L}_{\mathit{hydro}}^{7})}_{\theta }+\Delta t\{{p}_{f}^{\text{*}}\}{({L}_{\mathit{hydro}}^{8})}_{\theta }+\\ \Delta t\{{p}_{f}^{\text{*}}\}{({L}_{\mathit{hydro}}^{9})}_{\theta }+\Delta t\{{p}_{f}^{\text{*}}\}{({L}_{\mathit{hydro}}^{10})}_{\theta }\end{array}\) |
Continuity of pressure |
\(\{{q}_{1}^{\text{*}}\}[{D}^{1}](\delta p)+\{{q}_{1}^{\text{*}}\}[{D}^{2}](\delta {p}_{f})=\{{q}_{1}^{\text{*}}\}({J}_{\mathit{cont}}^{1})\) \(\{{q}_{2}^{\text{*}}\}[{D}^{3}](\delta p)+\{{q}_{2}^{\text{*}}\}[{D}^{4}](\delta {p}_{f})=\{{q}_{2}^{\text{*}}\}({J}_{\mathit{cont}}^{2})\) |
\({K}_{\mathit{meca}}^{1}\) is the elementary mechanical stiffness matrix classically encountered in mechanics,
\({K}_{\mathit{meca}}^{2}\) is due to the decomposition of the stress tensor of the massif (hypothesis of effective constraints),
\(A\) and \(B\) are due to the consideration of mass inputs in the expression of the homogenized density involved in the expression of the density forces on \(\Omega\),
\({C}^{1}\) is the elementary stiffness matrix for the interface,
\({K}^{\mu u}\) and \({K}^{w\mu }\) are matrices discretizing « mortar » operators, the latter also managing the base change.
\({K}^{wp}\) is an elementary stiffness matrix for the interface,
The matrices \(D\) are all block diagonals: for \(I\) and \(J\) two distinct DDL Lagrange numbers, they verify \({D}_{\mathit{IJ}}=0\),
\({M}_{\mathit{hydro}}^{1}\) and \({M}_{\mathit{hydro}}^{2}\) are the elementary mass matrices in the case of the massif for hydrodynamics,
\({W}_{\mathit{hydro}}^{1}\) and \({W}_{\mathit{hydro}}^{2}\) are the elementary mass matrices in the case of the hydrodynamic interface,
\({K}_{\mathit{hydro}}^{1}\), \({K}_{\mathit{hydro}}^{2}\), \({K}_{\mathit{hydro}}^{3}\) and \({K}_{\mathit{hydro}}^{4}\) are the elementary stiffness matrices for hydrodynamics in the case of the massif,
\({H}_{\mathit{hydro}}^{1}\), \({H}_{\mathit{hydro}}^{2}\) and \({H}_{\mathit{hydro}}^{3}\) are the elementary stiffness matrices for hydrodynamics in the case of the interface,
\({E}^{1}\) and \({E}^{2}\) are the exchange matrices in the case of the massif,
\({D}^{1}\) and \({D}^{2}\) are the exchange matrices in the case of the interface,
\({F}^{1}\), \({F}^{2}\), \({F}^{3}\) and \({F}^{4}\) are the pressure continuity matrices at the interface level,
\({L}_{\mathit{meca}}\) is the second member of the volume and surface forces applied to the domain and its border,
\({L}^{1}\) is the second member for the interface,
\({L}_{u}\), \({L}_{w}\) and \({L}_{p}\) are second members for the projection of the jump of movements,
\({L}_{\mathrm{\mu }}^{2}\), and \({L}_{\mathit{cohe}}^{1}\) are second members for the cohesive constraint,
\({L}_{\mathrm{\mu }}^{1}\) and \({L}_{\mathit{cohe}}^{2}\) are second members for the interface law,
\({L}_{\mathit{hydro}}^{i}\) with \(i\in ⟦\mathrm{1,5}⟧\) the second members due to mass inputs and flows in the case of the massif for hydrodynamics,
\({L}_{\mathit{hydro}}^{i}\) with \(i\in ⟦\mathrm{6,10}⟧\) the second members due to mass inputs and flows in the case of the hydrodynamic interface,
\({J}_{\mathit{hydro}}^{1}\) and \({J}_{\mathit{hydro}}^{2}\) the second members due to exchanges on \({\Gamma }_{1}\) and \({\Gamma }_{2}\).
Note:
As we can see in the expressions for the elementary matrices defined below, the quantities \({\rho }^{\text{+}}\) , * , \({\varphi }^{\text{+}}\) (functions of displacement and pressure), \({p}^{\text{+}}\) and \({u}^{\text{+}}(\mathit{ou}⟦{u}^{\text{+}}⟧)\) are left in the state (not discretized), because they are quantities obtained during the previous Newton iteration (for the current time step \(\text{+}\) ). They are therefore a prima facie known. In the expression of elementary matrices (for the mechanical and hydrodynamic case) we will not deliberately indicate the Newton iteration number on these quantities to avoid overloading the expressions.
6.2.2. Expression of elementary matrices for mechanics#
The elementary mechanical stiffness matrix at the Newton iteration \(k+1\) is written as:
\(\{{u}^{\text{*}}\}\left[{K}_{\mathit{meca}}^{1}\right](\mathrm{\delta }u)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}({a}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{b}_{i}^{\text{*}})\nabla {\mathrm{\phi }}_{i}{C}_{ij}\nabla {\mathrm{\phi }}_{j}({\mathrm{\delta }a}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{\mathrm{\delta }b}_{j})d\mathrm{\Omega }\)
The elementary matrices due to the decomposition of the stress tensor of the massif (hypothesis of effective constraints) are written in the Newton iteration \(k+1\):
\(\{{u}^{\text{*}}\}\left[{K}_{\mathit{meca}}^{2}\right](\mathrm{\delta }p)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{\mathrm{\Omega }}b({a}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{b}_{i}^{\text{*}})\nabla {\mathrm{\phi }}_{i}[\mathit{Id}]{\mathrm{\psi }}_{j}({\mathrm{\delta }c}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{\mathrm{\delta }d}_{j})d\mathrm{\Omega }\)
The matrix \(A\) at Newton’s iteration \(k+1\) is written as:
\(\{{u}^{\text{*}}\}\left[A\right](\mathrm{\delta }u)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}b({a}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{b}_{i}^{\text{*}}){\mathrm{\phi }}_{i}[\mathit{Id}]\nabla {\mathrm{\phi }}_{j}({\mathrm{\delta }a}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{\mathrm{\delta }b}_{j}){F}^{m\text{+}}d\mathrm{\Omega }\)
The matrix \(B\) at Newton’s iteration \(k+1\) is written as:
\(\{{u}^{\text{*}}\}\left[B\right](\mathrm{\delta }p)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{\mathrm{\Omega }}\left(\left(\frac{{\mathrm{\rho }}^{\text{+}}(b-{\mathrm{\varphi }}^{\text{+}})}{{K}_{s}^{}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{{\mathrm{\rho }}^{\text{+}}{\mathrm{\varphi }}^{\text{+}}}{{K}_{w}^{}}\right)\right)({a}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{b}_{i}^{\text{*}}){\mathrm{\phi }}_{i}{\mathrm{\psi }}_{j}(\mathrm{\delta }{c}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}\mathrm{\delta }{d}_{j}){F}^{m\text{+}}d\mathrm{\Omega }\)
The matrix associated with the projection of the jump in displacements at the Newton iteration \(k+1\) is written:
\(\{{⟦u⟧}^{\text{*}}\}\left[{C}^{1}\right](\mathrm{\delta }\mathrm{\mu })\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}\left(2\stackrel{~}{H}{b}_{i}^{\text{*}}\right){\mathrm{\phi }}_{i}\stackrel{~}{{\mathrm{\psi }}_{j}}\left(\mathrm{\delta }{\mathrm{\mu }}_{j}\right)d{\mathrm{\Gamma }}_{c}\)
The components of the unknowns \(u\) and \(\mathrm{\mu }\) are defined in a fixed base \(\left({e}_{X},{e}_{Y},{e}_{Z}\right)\), while the components of \(w\) and \(\mathrm{\lambda }\) are defined in the local base \(\left(n,{\mathrm{\tau }}_{1},{\mathrm{\tau }}_{2}\right)\) at the cracked surface \({\mathrm{\Gamma }}_{c}\) at each point \(x\in {\mathrm{\Gamma }}_{c}\), so that :
\(w(x)=\sum _{i=1}^{{N}_{\mathrm{\lambda }}}{\mathrm{\psi }}_{I}(x)\left({w}_{I,n}n(x)+{w}_{I,\mathrm{\tau }1}{\mathrm{\tau }}_{1}(x)+{w}_{I,\mathrm{\tau }2}{\mathrm{\tau }}_{2}(x)\right)\)
A similar definition applies to \(\mathrm{\lambda }\). For a degree of freedom \(I\) of the reduced space (see § 5.2.2), it is possible to determine the \(t{\text{'}}_{c,n}^{I},t{\text{'}}_{t,\mathrm{\tau }1}^{I},t{\text{'}}_{c,\mathrm{\tau }2}^{I}\) components of the cohesive force from \(\left({w}_{I,n},{w}_{I,\mathrm{\tau }1},{w}_{I,\mathrm{\tau }2}\right)\), \(\left({\mathrm{\lambda }}_{I,n},{\mathrm{\lambda }}_{I,\mathrm{\tau }1},{\mathrm{\lambda }}_{I,\mathrm{\tau }2}\right)\) and the cohesive law. These components are not intended to be associated with a particular direction \(I\) of the degree of freedom, but intended to be linked in a weak sense to the global constraint \(\mathrm{\mu }\) written in a fixed base (confer [R7.02.19]).
\({\{u\}}^{\text{*}}\left[{K}^{\mathrm{\mu }u}\right](\mathrm{\delta }\mathrm{\mu })={\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{j}}{\mathrm{\phi }}_{i}2\stackrel{~}{H}{b}_{i}^{\text{*}}\cdot (\mathrm{\delta }{\mathrm{\mu }}_{j})d{\mathrm{\Gamma }}_{c}\)
\({\{w\}}^{\text{*}}\left[{K}^{w\mathrm{\mu }}\right](\mathrm{\delta }\mathrm{\mu })={\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{i}}\stackrel{~}{{\mathrm{\psi }}_{j}}{{w}_{j}}^{\text{*}}\cdot Q\cdot (\mathrm{\delta }{\mathrm{\mu }}_{J})d{\mathrm{\Gamma }}_{c}\) with \(Q\) the orthonormal base change matrix defined as before.
\({\{w\}}^{\text{*}}\left[{K}^{wp}\right](\mathrm{\delta }{p}_{f})={\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{j}}\stackrel{~}{{\mathrm{\psi }}_{i}}{w}_{i}^{\text{*}}\cdot {(\mathrm{\delta }{p}_{f})}_{j}{n}_{c}d{\mathrm{\Gamma }}_{c}\)
\({\{w\}}^{\text{*}}\left[{D}^{ww}\right](\mathrm{\delta }w)={\int }_{{\mathrm{\Gamma }}_{c}}{w}_{i}^{\text{*}}(\mathrm{\delta }{w}_{i})r\frac{\partial {t\text{'}}_{c}}{\partial (\mathrm{\lambda }+rw)}({\mathrm{\lambda }}_{i}+r{w}_{i})\stackrel{~}{{\mathrm{\psi }}_{i}^{2}}d{\mathrm{\Gamma }}_{c}\) \({\{\mathrm{\lambda }\}}^{\text{*}}\left[{D}^{\mathrm{\lambda }w}\right](\mathrm{\delta }w)={\int }_{{\mathrm{\Gamma }}_{c}}{\mathrm{\lambda }}_{i}^{\text{*}}(\mathrm{\delta }{w}_{i})\frac{\partial {t\text{'}}_{c}}{\partial (\mathrm{\lambda }+rw)}({\mathrm{\lambda }}_{i}+r{w}_{i})\stackrel{~}{{\mathrm{\psi }}_{i}^{2}}d{\mathrm{\Gamma }}_{c}\)
\({\{\mathrm{\lambda }\}}^{\text{*}}\left[{D}^{\mathrm{\lambda }\mathrm{\lambda }}\right](\mathrm{\delta }\mathrm{\lambda })={\int }_{{\mathrm{\Gamma }}_{c}}{\mathrm{\lambda }}_{i}^{\text{*}}(\mathrm{\delta }{\mathrm{\lambda }}_{i})\frac{1}{r}\left(\frac{\partial {t\text{'}}_{c}}{\partial (\mathrm{\lambda }+rw)}({\mathrm{\lambda }}_{i}+r{w}_{i})-1\right)\stackrel{~}{{\mathrm{\psi }}_{i}^{2}}d{\mathrm{\Gamma }}_{c}\)
6.2.3. Expression of the second members for mechanics#
In the expressions for the second members presented here we indicate the number of the previous Newton iteration \(k\).
\(\begin{array}{c}\{{u}^{\text{*}}\}({L}_{\mathit{meca}})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{\mathrm{\Omega }}({a}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{b}_{i}^{\text{*}})\nabla {\mathrm{\phi }}_{i}\phantom{\rule{1em}{0ex}}{({\mathrm{\sigma }}^{\text{'}\text{+}}(u))}^{k}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}b({a}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{b}_{i}^{\text{*}})\nabla {\mathrm{\phi }}_{i}[\mathit{Id}]{({p}^{\text{+}})}^{k}d\mathrm{\Omega }\\ \phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}({a}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{b}_{i}^{\text{*}}){\mathrm{\phi }}_{i}({r}_{0}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{({m}_{w}^{\text{+}})}^{k}){F}^{m\text{+}}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{t}}({a}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{b}_{i}^{\text{*}}){\mathrm{\phi }}_{i}\phantom{\rule{0.5em}{0ex}}{({t}^{\text{+}})}^{k}d{\mathrm{\Gamma }}_{t}\end{array}\)
\(\{{⟦u⟧}^{\text{*}}\}({L}^{1})\phantom{\rule{0.5em}{0ex}}=-\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}\left(2\stackrel{~}{H}{b}_{i}^{\text{*}}\right){\mathrm{\phi }}_{i}\stackrel{~}{{\mathrm{\psi }}_{j}}\text{}{\mathrm{\mu }}_{j}^{k}d{\mathrm{\Gamma }}_{c}\)
\({\{u\}}_{i}^{\text{*}}{\left({L}_{\mathrm{\mu }}^{1}\right)}_{i}=-{b}_{i}^{\text{*}}\cdot {\int }_{{\mathrm{\Gamma }}_{c}}2{\mathrm{\phi }}_{i}\mathrm{\mu }d{\mathrm{\Gamma }}_{c}\)
\({\{\mathrm{\mu }\}}^{\text{*}}\left({L}_{u}\right)=-{\mathrm{\mu }}_{i}^{\text{*}}\cdot {\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{i}}⟦u⟧d{\mathrm{\Gamma }}_{c}\)
\({\{w\}}^{\text{*}}\left({L}_{p}\right)=-{w}_{i}^{\text{*}}\cdot {\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{i}}{p}_{f}{n}_{c}d{\mathrm{\Gamma }}_{c}\)
\({\{\mathrm{\mu }\}}^{\text{*}}\left({L}_{w}\right)=-{\mathrm{\mu }}_{i}^{\text{*}}\cdot {\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{i}}{Q}^{T}\cdot wd{\mathrm{\Gamma }}_{c}\)
\({\{w\}}^{\text{*}}\left({L}_{\mathrm{\mu }}^{2}\right)={w}_{i}^{\text{*}}\cdot {\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{i}}Q\cdot \mathrm{\mu }d{\mathrm{\Gamma }}_{c}\)
\({\{w\}}^{\text{*}}\left({L}_{\mathit{coh}}^{1}\right)={w}_{i}^{\text{*}}\cdot {t\text{'}}_{c}({\mathrm{\lambda }}_{i}+r{w}_{i}){\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{i}}d{\mathrm{\Gamma }}_{c}\)
\({\{w\}}^{\text{*}}{\left({L}_{\mathit{coh}}^{2}\right)}_{I}=\frac{{w}_{i}^{\text{*}}}{r}{t\text{'}}_{c}({\mathrm{\lambda }}_{i}+r{w}_{i}){\int }_{{\mathrm{\Gamma }}_{c}}\stackrel{~}{{\mathrm{\psi }}_{I}}d{\mathrm{\Gamma }}_{c}\)
6.2.4. Expression of elementary matrices for hydrodynamics#
6.2.4.1. Case of the massif#
The elementary mass matrices at the Newton iteration \(k+1\) are written as:
\(\{{p}^{\text{*}}\}\left[{M}_{\mathit{hydro}}^{1}\right](\mathrm{\delta }u)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{\mathrm{\Omega }}b{\mathrm{\rho }}^{\text{+}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}[\mathit{Id}]\nabla {\mathrm{\phi }}_{j}({\mathrm{\delta }a}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{\mathrm{\delta }b}_{j})d\mathrm{\Omega }\)
\(\{{p}^{\text{*}}\}\left[{M}_{\mathit{hydro}}^{2}\right](\mathrm{\delta }p)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{\mathrm{\Omega }}\left(\left(\frac{{\mathrm{\rho }}^{\text{+}}(b-{\mathrm{\varphi }}^{\text{+}})}{{K}_{s}^{}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\frac{{\mathrm{\rho }}^{\text{+}}{\mathrm{\varphi }}^{\text{+}}}{{K}_{w}^{}}\right)\right)({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}{\mathrm{\psi }}_{j}(\mathrm{\delta }{c}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}\mathrm{\delta }{d}_{j})d\mathrm{\Omega }\)
The elementary stiffness matrices at the Newton \(k+1\) iteration are written as:
\(\{{p}^{\text{*}}\}[{K}_{\mathit{hydro}}^{1}](\mathrm{\delta }p)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}\left({\mathrm{\lambda }}^{\text{+}}\left(-\nabla {p}^{\text{+}}+{\mathrm{\rho }}^{\text{+}}{F}^{m\text{+}}\right)\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}{\mathrm{\rho }}^{\text{+}}{\mathrm{\lambda }}^{\text{+}}{F}^{m\text{+}}\right)\frac{{\mathrm{\rho }}^{\text{+}}}{{K}_{w}^{\text{}}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}})\nabla {\mathrm{\psi }}_{i}{\mathrm{\psi }}_{j}(\mathrm{\delta }{c}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}\mathrm{\delta }{d}_{j})d\mathrm{\Omega }\)
\(\{{p}^{\text{*}}\}[{K}_{\mathit{hydro}}^{2}](\mathrm{\delta }p)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}\left(-\nabla {p}^{\text{+}}+{\mathrm{\rho }}^{\text{+}}{F}^{m\text{+}}\right)\frac{\partial {\mathrm{\lambda }}_{}^{\text{+}}}{\partial {p}^{\text{+}}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}})\nabla {\mathrm{\psi }}_{i}{\mathrm{\psi }}_{j}(\mathrm{\delta }{c}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}\mathrm{\delta }{d}_{j})d\mathrm{\Omega }\)
\(\{{p}^{\text{*}}\}[{K}_{\mathit{hydro}}^{3}](\mathrm{\delta }p)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}{\mathrm{\lambda }}^{\text{+}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}})\nabla {\mathrm{\psi }}_{i}\nabla {\mathrm{\psi }}_{j}(\mathrm{\delta }{c}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}\mathrm{\delta }{d}_{j})d\mathrm{\Omega }\)
\(\{{p}^{\text{*}}\}[{K}_{\mathit{hydro}}^{4}](\mathrm{\delta }u)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}{\mathrm{\rho }}^{\text{+}}\left(-\nabla {p}^{\text{+}}+{\mathrm{\rho }}^{\text{+}}{F}^{m\text{+}}\right)\frac{\partial {\mathrm{\lambda }}_{}^{\text{+}}}{\partial {\mathrm{\epsilon }}_{v}^{\text{+}}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}})\nabla {\mathrm{\psi }}_{i}[\mathit{Id}]\nabla {\mathrm{\phi }}_{j}(\mathrm{\delta }{a}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}\mathrm{\delta }{b}_{j})d\mathrm{\Omega }\)
The elementary exchange matrices for the massive at the Newton iteration \(k+1\) are written:
\(\{{p}^{\text{*}}\}[{E}^{1}](\mathrm{\delta }{q}_{1})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{1}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}\stackrel{~}{{\mathrm{\psi }}_{j}}{(\mathrm{\delta }{q}_{1})}_{j}d{\mathrm{\Gamma }}_{1}\)
\(\{{p}^{\text{*}}\}[{E}^{2}](\mathrm{\delta }{q}_{2})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{2}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}\stackrel{~}{{\mathrm{\psi }}_{j}}{(\mathrm{\delta }{q}_{2})}_{j}d{\mathrm{\Gamma }}_{2}\)
6.2.4.2. Interface case#
The elementary mass matrices at the Newton iteration \(k+1\) are written as:
\(\{{p}_{f}^{\text{*}}\}[{W}_{\mathit{hydro}}^{1}](\mathrm{\delta }⟦u⟧)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{c}}{\mathrm{\rho }}^{\text{+}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{\mathrm{\phi }}_{j}\left(2\stackrel{~}{H}\mathrm{\delta }{b}_{j}\right){n}_{c}d{\mathrm{\Gamma }}_{c}\)
\(\{{p}_{f}^{\text{*}}\}[{W}_{\mathit{hydro}}^{2}](\mathrm{\delta }{p}_{f})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{c}}\frac{{\mathrm{\rho }}^{\text{+}}⟦u⟧\cdot {n}_{c}}{{K}_{w}^{}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}\stackrel{~}{{\mathrm{\psi }}_{j}}{\left(\mathrm{\delta }{p}_{f}\right)}_{j}d{\mathrm{\Gamma }}_{c}\)
The elementary stiffness matrices at the Newton \(k+1\) iteration are written as:
\(\{{p}_{f}^{\text{*}}\}[{H}_{\mathit{hydro}}^{1}](\mathrm{\delta }⟦u⟧)\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{c}}\frac{{\mathrm{\rho }}^{\text{+}}\left(3{\left(⟦u⟧\cdot {n}_{c}\right)}^{2}\right)\nabla {p}_{f}^{\text{+}}}{12\mathrm{\mu }}{({p}_{f}^{\text{*}})}_{i}\nabla \stackrel{~}{{\mathrm{\psi }}_{i}}{\mathrm{\phi }}_{j}\left(2\stackrel{~}{H}\mathrm{\delta }{b}_{j}\right){n}_{c}d{\mathrm{\Gamma }}_{c}\)
\(\{{p}_{f}^{\text{*}}\}[{H}_{\mathit{hydro}}^{2}](\mathrm{\delta }{p}_{f})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{c}}\frac{{\left(⟦u⟧\cdot {n}_{c}\right)}^{3}\nabla {p}_{f}^{\text{+}}}{12\mathrm{\mu }}\frac{{\mathrm{\rho }}^{\text{+}}}{{K}_{w}^{}}{({p}_{f}^{\text{*}})}_{i}\nabla \stackrel{~}{{\mathrm{\psi }}_{i}}\stackrel{~}{{\mathrm{\psi }}_{j}}{(\mathrm{\delta }{p}_{f})}_{j}d{\mathrm{\Gamma }}_{c}\)
\(\{{p}_{f}^{\text{*}}\}[{H}_{\mathit{hydro}}^{3}](\mathrm{\delta }{p}_{f})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{c}}\frac{{\mathrm{\rho }}^{\text{+}}{\left(⟦u⟧\cdot {n}_{c}\right)}^{3}}{12\mathrm{\mu }}{({p}_{f}^{\text{*}})}_{i}\nabla \stackrel{~}{{\mathrm{\psi }}_{i}}\nabla \stackrel{~}{{\mathrm{\psi }}_{j}}{(\mathrm{\delta }{p}_{f})}_{j}d{\mathrm{\Gamma }}_{c}\)
The elementary exchange matrices for the Newton \(k+1\) iteration interface are written as:
\(\{{p}_{f}^{\text{*}}\}[{D}^{1}](\mathrm{\delta }{q}_{1})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{1}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}\stackrel{~}{{\mathrm{\psi }}_{j}}{(\mathrm{\delta }{q}_{1})}_{j}d{\mathrm{\Gamma }}_{1}\)
\(\{{p}_{f}^{\text{*}}\}[{D}^{2}](\mathrm{\delta }{q}_{2})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{2}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}\stackrel{~}{{\mathrm{\psi }}_{j}}{(\mathrm{\delta }{q}_{2})}_{j}d{\mathrm{\Gamma }}_{2}\)
6.2.4.3. Continuity of pressure#
The elementary exchange matrices (for the pressure continuity equation at the interface level) at the Newton iteration \(k+1\) are written:
\(\{{q}_{1}^{\text{*}}\}[{F}^{1}](\mathrm{\delta }{p}^{\mathit{inf}})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{1}}{({q}_{1}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{\mathrm{\psi }}_{j}(\mathrm{\delta }{c}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}\mathrm{\delta }{d}_{j})d{\mathrm{\Gamma }}_{1}\)
\(\{{q}_{1}^{\text{*}}\}[{F}^{2}](\mathrm{\delta }{p}_{f})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{1}}{({q}_{1}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}\stackrel{~}{{\mathrm{\psi }}_{j}}{(\mathrm{\delta }{p}_{f})}_{j}d{\mathrm{\Gamma }}_{1}\)
\(\{{q}_{2}^{\text{*}}\}[{F}^{3}](\mathrm{\delta }{p}^{\text{sup}})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{2}}{({q}_{2}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{\mathrm{\psi }}_{j}(\mathrm{\delta }{c}_{j}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}\mathrm{\delta }{d}_{j})d{\mathrm{\Gamma }}_{2}\)
\(\{{q}_{2}^{\text{*}}\}[{F}^{4}](\mathrm{\delta }{p}_{f})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{2}}{({q}_{2}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}\stackrel{~}{{\mathrm{\psi }}_{j}}{(\mathrm{\delta }{p}_{f})}_{j}d{\mathrm{\Gamma }}_{2}\)
Note:
In the expression of the elementary matrices defined in this part, we can see the presence of two additional « unknowns », \(\delta {p}^{\mathit{inf}}\) and \(\delta {p}^{\text{sup}}\). They actually correspond to the unknown relative to the pressure fields \(\delta p\) defined on \({\Gamma }_{1}\) (i.e. \(\delta {p}^{\mathit{inf}}\) ) and on \({\Gamma }_{2}\) (i.e. \(\delta {p}^{\text{sup}}\) ) respectively (ie ) .
6.2.5. Expression of the second elementary members for hydrodynamics#
6.2.5.1. Case of the massif#
In the expressions for the second members presented here we indicate the number of the previous Newton iteration:
\(\{{p}^{\text{*}}\}({L}_{\mathit{hydro}}^{1})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{\mathrm{\Omega }}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}\left({({m}_{w}^{\text{+}})}^{k}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{({m}_{w}^{\text{-}})}^{k}\right)d\mathrm{\Omega }\)
\(\{{p}^{\text{*}}\}{({L}_{\mathit{hydro}}^{2})}_{\mathrm{\theta }}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-\mathrm{\theta }{\int }_{\mathrm{\Omega }}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}})\nabla {\mathrm{\psi }}_{i}{({M}^{\text{+}})}^{k}d\mathrm{\Omega }\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}(1-\mathrm{\theta }){\int }_{\mathrm{\Omega }}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}})\nabla {\mathrm{\psi }}_{i}{({M}^{\text{-}})}^{k}d\mathrm{\Omega }\)
\(\{{p}^{\text{*}}\}{({L}_{\mathit{hydro}}^{3})}_{\mathrm{\theta }}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{F}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}{({M}_{\mathit{ext}}^{\text{+}})}^{k}d{\mathrm{\Gamma }}_{F}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{F}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}{({M}_{\mathit{ext}}^{\text{-}})}^{k}d{\mathrm{\Gamma }}_{F}\)
\(\{{p}^{\text{*}}\}{({L}_{\mathit{hydro}}^{4})}_{\mathrm{\theta }}\phantom{\rule{0.5em}{0ex}}=-\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{1}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}{({q}_{1}^{\text{+}})}^{k}d{\mathrm{\Gamma }}_{1}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{1}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}{({q}_{1}^{\text{-}})}^{k}d{\mathrm{\Gamma }}_{1}\)
\(\{{p}^{\text{*}}\}{({L}_{\mathit{hydro}}^{5})}_{\mathrm{\theta }}\phantom{\rule{0.5em}{0ex}}=-\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{2}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}{({q}_{2}^{\text{+}})}^{k}d{\mathrm{\Gamma }}_{2}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{2}}({c}_{i}^{\text{*}}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}\stackrel{~}{H}{d}_{i}^{\text{*}}){\mathrm{\psi }}_{i}{({q}_{2}^{\text{-}})}^{k}d{\mathrm{\Gamma }}_{2}\)
6.2.5.2. Interface case#
In the expressions for the second members presented here we indicate the number of the previous Newton iteration.
\(\{{p}_{f}^{\text{*}}\}({L}_{\mathit{hydro}}^{6})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}{\int }_{{\mathrm{\Gamma }}_{c}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}\left({({w}^{\text{+}})}^{k}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{({w}^{\text{-}})}^{k}\right)d{\mathrm{\Gamma }}_{c}\)
\(\{{p}_{f}^{\text{*}}\}{({L}_{\mathit{hydro}}^{7})}_{\mathrm{\theta }}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{c}}{({p}_{f}^{\text{*}})}_{i}\nabla \stackrel{~}{{\mathrm{\psi }}_{i}}{({W}^{\text{+}})}^{k}d{\mathrm{\Gamma }}_{c}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{c}}{({p}_{f}^{\text{*}})}_{i}\nabla \stackrel{~}{{\mathrm{\psi }}_{i}}{({W}^{\text{-}})}^{k}d{\mathrm{\Gamma }}_{c}\)
\(\{{p}_{f}^{\text{*}}\}{({L}_{\mathit{hydro}}^{8})}_{\mathrm{\theta }}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{f}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{({W}_{\mathit{ext}}^{\text{+}})}^{k}d{\mathrm{\Gamma }}_{f}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{f}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{({W}_{\mathit{ext}}^{\text{-}})}^{k}d{\mathrm{\Gamma }}_{f}\)
\(\{{p}_{f}^{\text{*}}\}{({L}_{\mathit{hydro}}^{9})}_{\mathrm{\theta }}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{1}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{({q}_{1}^{\text{+}})}^{k}d{\mathrm{\Gamma }}_{1}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{1}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{({q}_{1}^{\text{-}})}^{k}d{\mathrm{\Gamma }}_{1}\)
\(\{{p}_{f}^{\text{*}}\}{({L}_{\mathit{hydro}}^{10})}_{\mathrm{\theta }}\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}\mathrm{\theta }{\int }_{{\mathrm{\Gamma }}_{2}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{({q}_{2}^{\text{+}})}^{k}d{\mathrm{\Gamma }}_{2}\phantom{\rule{0.5em}{0ex}}+\phantom{\rule{0.5em}{0ex}}(1-\mathrm{\theta }){\int }_{{\mathrm{\Gamma }}_{2}}{({p}_{f}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}{({q}_{2}^{\text{-}})}^{k}d{\mathrm{\Gamma }}_{2}\)
6.2.5.3. Continuity of pressure#
In the expressions for the second members presented here (for the pressure continuity equation at the interface level), we indicate the number of the previous Newton iteration.
\(\{{q}_{1}^{\text{*}}\}({J}_{\mathit{cont}}^{1})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{1}}{({q}_{1}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}\left({({p}^{\mathit{inf}})}^{k}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{({p}_{f}^{\text{+}})}^{k}\right)d{\mathrm{\Gamma }}_{1}\)
\(\{{q}_{2}^{\text{*}}\}({J}_{\mathit{cont}}^{2})\phantom{\rule{0.5em}{0ex}}=\phantom{\rule{0.5em}{0ex}}-{\int }_{{\mathrm{\Gamma }}_{1}}{({q}_{2}^{\text{*}})}_{i}\stackrel{~}{{\mathrm{\psi }}_{i}}\left({({p}^{\text{sup}})}^{k}\phantom{\rule{0.5em}{0ex}}-\phantom{\rule{0.5em}{0ex}}{({p}_{f}^{\text{+}})}^{k}\right)d{\mathrm{\Gamma }}_{2}\)