Note: This page is currently under construction. Additional boundary conditions will be added in future updates.

Robin Boundary Conditions

Note: Robin boundary conditions are only available with the VEF Pnc/P0 discretization.

Robin boundary conditions consist of a linear combination between the flux term \(\boldsymbol{F}\) and the variable term. They are useful for fluid-structure interactions or domain decomposition. The implementation in TRUST decomposes the condition into normal and tangential parts.

Let \(\boldsymbol{n} = (n_x, n_y)\) be the outward normal vector. The flux term is:

\[\boldsymbol{F} = F_n\boldsymbol{n} + \boldsymbol{F_t} \]

For the Navier-Stokes equations:

\[\begin{aligned} F_n &= \nu\nabla_n\boldsymbol{u}\cdot\boldsymbol{n} + \chi(\boldsymbol{u}\cdot\boldsymbol{n})(\boldsymbol{u}\cdot\boldsymbol{n}) - p \\ \boldsymbol{F_t} &= \nu\nabla_n\boldsymbol{u}\times\boldsymbol{n} + \chi(\boldsymbol{u}\cdot\boldsymbol{n})(\boldsymbol{u}\times\boldsymbol{n}) \end{aligned} \]

with \(\nu\) the viscosity and \(\chi \in \{0,1\}\). In 2D, the cross product is replaced by a projection onto the tangential vector \(\boldsymbol{t} = (-n_y, n_x)\).

Two Robin parameters are defined: \(\alpha\) for the normal part and \(\beta\) for the tangential part, with Robin data:

a normal scalar function \(g_N\)
a tangential function \(\boldsymbol{g_T}\) (scalar in 2D, vector in 3D)

The Robin boundary conditions implemented in TRUST are:

\[\begin{aligned} \alpha F_n + \boldsymbol{u}\cdot\boldsymbol{n} &= g_N \\ \beta\boldsymbol{F_t} + \boldsymbol{u}\times\boldsymbol{n} &= \boldsymbol{g_T} \end{aligned} \]

Keyword: Robin_VEF

Parameters:

alpha, beta as defined above.
champ_front_normal_et_tangentiel followed by the field data (concatenation of \(g_N\) and \(\boldsymbol{n}\times\boldsymbol{g_T}\)).

2D example — for \(\boldsymbol{u}=(y,-x)\), \(p=0.5(x^2+y^2)-1/3\), \(\boldsymbol{n}=(1,0)\):

Robin_VEF {
    alpha 3
    beta 4
    champ_front_normal_et_tangentiel_robin champ_front_fonc_txyz 2
        -1.5*x^2-4.5*y^2+y+1.0
        4*x*y-x-4
}

The first function is \(g_N\) and the second is \(g_T\).

3D: the field champ_front_normal_et_tangentiel will have 4 components (one for \(g_N\) and three for \(\boldsymbol{n}\times\boldsymbol{g_T}\)).

Note: The notation \(\boldsymbol{n}\times\boldsymbol{g_T}\) is used because we want to write the real tangential component of \(\boldsymbol{u}\).