Hi,
I have the following set of coupled equations,
\begin{align}
\partial a / \partial t = \Delta a - \Delta b + f(a) \\
\partial b / \partial t = \Delta b + g(a,b) \\
\partial c / \partial t = \nabla_S ( D_c \nabla_S c) + h(b,c)
\end{align}
where
$\nabla_S c$ is the surface gradient defined as $\nabla c - \hat{n} (
\hat{n} \cdot \nabla c)$ on the boundary. $a$ and $b$ are defined on
the whole domain; or mesh. Whereas, $c$ is only defined on the
boundary and only allowed to diffuse on the boundary surface.
Question: How do you implement $\nabla_S$?
I'm aware that the oriented surface normals are available via
mesh._orientedFaceNormals and presumably the necessary ones can be
pulled out using mesh.exteriorFaces.
Alternatively, should one solve for $a$ and $b$, grab their boundary
values and then solve for $c$? $c$ perhaps being defined on its own
mesh?
Of course, what I'm really after is a Laplace-Beltrami operator to
describe diffusion on a surface.
Question: Is there a way to define a diffusion tensor so that I can
use the usual DiffusionTerm to model tangential diffusion?
Any help or suggestions will be greatly appreciated.
Thanks,
Lafras
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