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Fix sign error in documentation of Robin condition #968

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Oct 16, 2023
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Fix sign error in documentation of Robin condition #968

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34 changes: 17 additions & 17 deletions docs/source/USAGE.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -823,14 +823,14 @@ the value at the neighboring cell :math:`P` and the normal gradient at the bound
.. math::
:label: upwind1

\phi_f &\approx \phi_P - \left(\vec{d}_{fP}\cdot\nabla\phi\right)_f
\phi_f &\approx \phi_P + \left(\vec{d}_{Pf}\cdot\nabla\phi\right)_f
\\
&\approx \phi_P - \left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{fP}\cdot\hat{n}\right)_f
&\approx \phi_P + \left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{Pf}\cdot\hat{n}\right)_f

where :math:`\vec{d}_{fP}` is the distance vector from the face center to
the adjoining cell center. The approximation
:math:`\left(\vec{d}_{fP}\cdot\nabla\phi\right)_f \approx
\left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{fP}\cdot\hat{n}\right)_f`
where :math:`\vec{d}_{Pf}` is the distance vector to the center of the face
:math:`f` from the center of the adjoining cell :math:`P`. The
approximation :math:`\left(\vec{d}_{Pf}\cdot\nabla\phi\right)_f \approx
\left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{Pf}\cdot\hat{n}\right)_f`
is most valid when the mesh is orthogonal.

Substituting this expression into the Robin condition:
Expand All @@ -840,11 +840,11 @@ Substituting this expression into the Robin condition:

\hat{n}\cdot\left(\vec{a} \phi + b \nabla\phi\right)_f &= g \\
\hat{n}\cdot\left[\vec{a} \phi_P
- \vec{a} \left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{fP}\cdot\hat{n}\right)_f
+ \vec{a} \left(\hat{n}\cdot\nabla\phi\right)_f\left(\vec{d}_{Pf}\cdot\hat{n}\right)_f
+ b \nabla\phi\right]_f &\approx g \\
\left(\hat{n}\cdot\nabla\phi\right)_f
&\approx \frac{g_f - \left(\hat{n}\cdot\vec{a}\right)_f \phi_P}
{-\left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f}
{\left(\vec{d}_{Pf}\cdot\vec{a}\right)_f + b_f}

we obtain an expression for the gradient at the boundary face in terms of
its neighboring cell. We can, in turn, substitute this back into
Expand All @@ -854,11 +854,11 @@ its neighboring cell. We can, in turn, substitute this back into
:label: upwind2

\phi_f &\approx \phi_P
- \frac{g_f - \left(\hat{n}\cdot\vec{a}\right)_f \phi_P}
{-\left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f}
\left(\vec{d}_{fP}\cdot\hat{n}\right)_f \\
&\approx \frac{-g_f \left(\hat{n}\cdot\vec{d}_{fP}\right)_f + b_f\phi_P}
{- \left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f}
+ \frac{g_f - \left(\hat{n}\cdot\vec{a}\right)_f \phi_P}
{\left(\vec{d}_{Pf}\cdot\vec{a}\right)_f + b_f}
\left(\vec{d}_{Pf}\cdot\hat{n}\right)_f \\
&\approx \frac{g_f \left(\hat{n}\cdot\vec{d}_{Pf}\right)_f + b_f\phi_P}
{\left(\vec{d}_{Pf}\cdot\vec{a}\right)_f + b_f}

to obtain the value on the boundary face in terms of the neighboring cell.

Expand All @@ -874,7 +874,7 @@ Substituting :eq:`Robin_facegrad` into the discretization of the
+ \sum_{f \in S_R} \Gamma_f \left(\hat{n}\cdot\nabla\phi\right)_f A_f \\
&\approx \sum_{f \notin S_R} \Gamma_f \left(\hat{n}\cdot\nabla\phi\right)_f A_f
+ \sum_{f \in S_R} \Gamma_f \frac{g_f - \left(\hat{n}\cdot\vec{a}\right)_f \phi_P}
{-\left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f} A_f
{\left(\vec{d}_{Pf}\cdot\vec{a}\right)_f + b_f} A_f

An equation of the form

Expand All @@ -891,7 +891,7 @@ can be constrained to have a Robin condition at faces identified by
>>> a = FaceVariable(mesh=mesh, value=..., rank=1)
>>> b = FaceVariable(mesh=mesh, value=..., rank=0)
>>> g = FaceVariable(mesh=mesh, value=..., rank=0)
>>> RobinCoeff = (mask * Gamma0 * n / (-dPf.dot(a) + b)
>>> RobinCoeff = (mask * Gamma0 * n / (dPf.dot(a) + b)
>>> eqn = (TransientTerm() == DiffusionTerm(coeff=Gamma) + (RobinCoeff * g).divergence
... - ImplicitSourceTerm(coeff=(RobinCoeff * n.dot(a)).divergence)

Expand All @@ -905,8 +905,8 @@ substitute :eq:`upwind2`:
&\approx \sum_f \left(\hat{n}\cdot\vec{u}\right)_f \phi_f A_f \\
&= \sum_{f \notin S_R} \left(\hat{n}\cdot\vec{u}\right)_f \phi_f A_f
+ \sum_{f \in S_R} \left(\hat{n}\cdot\vec{u}\right)_f
\frac{-g_f \left(\hat{n}\cdot\vec{d}_{fP}\right)_f + b_f\phi_P}
{- \left(\vec{d}_{fP}\cdot\vec{a}\right)_f + b_f} A_f
\frac{g_f \left(\hat{n}\cdot\vec{d}_{Pf}\right)_f + b_f\phi_P}
{\left(\vec{d}_{Pf}\cdot\vec{a}\right)_f + b_f} A_f

.. note:: An expression like the heat flux convection boundary condition
:math:`-k\nabla T\cdot\hat{n} = h(T - T_\infty)` can be put in the form of the
Expand Down