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OFFSETVERSION_specification3.lyx
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OFFSETVERSION_specification3.lyx
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#LyX 2.0 created this file. For more info see http://www.lyx.org/
\lyxformat 413
\begin_document
\begin_header
\textclass article
\use_default_options true
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\language english
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\index Index
\shortcut idx
\color #008000
\end_index
\leftmargin 1.5cm
\topmargin 1cm
\rightmargin 1.5cm
\bottommargin 2cm
\secnumdepth 3
\tocdepth 3
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\end_header
\begin_body
\begin_layout Subsection*
Data conventions / boundary rules
\end_layout
\begin_layout Subsubsection
Frills
\end_layout
\begin_layout Standard
// OKay.
ASK // Is there instead a way to get rid of frills.
// We still want a constant number of tris per tri tile, # verts per vert
tile, // we still want a geometric correspondence between tri and vert
tiles.
// Frills then seems inevitable.? If we knock 1 off the number of tris per
tri tile, // that will bring down the number of tris by # tri tiles, which
is awkward.
\end_layout
\begin_layout Standard
// ** So yes, we need frills.
**
\end_layout
\begin_layout Standard
Frills are going to be along both the outermost and innermost boundaries.
\end_layout
\begin_layout Itemize
// Frill has vertex 0,1 as the two different vertices.
\end_layout
\begin_layout Itemize
// Frill has all tri neighbours set to the 1 neighbour.
\end_layout
\begin_layout Itemize
// Frill has area = 0.
\end_layout
\begin_layout Itemize
\begin_inset Quotes eld
\end_inset
centroid
\begin_inset Quotes erd
\end_inset
is on boundary ? NO:
\end_layout
\begin_deeper
\begin_layout Itemize
// So how are we going to get the corners of central cell?
\end_layout
\begin_layout Itemize
// Do we (i) go 1/3 along tri edge or (ii) make them the average of 2 tri
centroids and the vertex?
\end_layout
\begin_layout Itemize
Way (ii) allows us not to have to access extra information than triangles
in order to calculate Lap A on central cells.
\end_layout
\begin_layout Itemize
However, if we do this then we distort central cells near the insulator.
\end_layout
\begin_layout Itemize
And we cannot then sensibly be putting the frill centroid on the boundary.
That will spoil things.
\end_layout
\begin_layout Itemize
It is better if we do not do that then.
// Put the frill centroid the other side of the boundary.
\end_layout
\end_deeper
\begin_layout Itemize
Try to avoid traffic of
\begin_inset Formula $nv$
\end_inset
into frills or (for now) into insulator triangles -- whether by advection
or diffusion.
\end_layout
\begin_deeper
\begin_layout Itemize
Therefore we HAVE to load the flag of the triangle that we are looking into.
\end_layout
\end_deeper
\begin_layout Itemize
When we set
\begin_inset Formula $\nabla^{2}A$
\end_inset
we first set
\begin_inset Formula $A_{frill}$
\end_inset
in a preliminary routine so that we do not need to detect frill neighbours
of domain triangles.
\end_layout
\begin_layout Itemize
Ensure that in frills,
\begin_inset Formula $v=0$
\end_inset
.
\end_layout
\begin_layout Subsubsection
Outermost vertex-centered cells
\end_layout
\begin_layout Itemize
Outermost cells DO allow ingress of species and DO allow
\begin_inset Formula $\phi$
\end_inset
to pump electrons back to the domain: this is via triangle minors
\begin_inset Formula $v$
\end_inset
of course.
There is no traffic at the outer edge of the outermost cell.
\end_layout
\begin_layout Itemize
If we choose outermost
\begin_inset Formula $v=0$
\end_inset
-- which we might as well since neither
\begin_inset Formula $v_{r}$
\end_inset
nor
\begin_inset Formula $v_{z}$
\end_inset
has any sense in being nonzero - then we can still evolve
\begin_inset Formula $\partial A/\partial t$
\end_inset
all the same.
\end_layout
\begin_layout Itemize
When we set
\begin_inset Formula $\nabla^{2}A$
\end_inset
,
\family roman
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\uuline off
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\color none
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\begin_inset Formula $\nabla A$
\end_inset
or
\begin_inset Formula $\nabla\phi$
\end_inset
for outermost central, we can apply boundary conditions looking outwards.
Likewise setting
\begin_inset Formula $\nabla^{2}\phi$
\end_inset
for outermost major.
We can probably use
\begin_inset Formula $A_{frill}$
\end_inset
to create
\begin_inset Formula $\nabla^{2}A$
\end_inset
.
\end_layout
\begin_layout Itemize
\begin_inset Graphics
filename Edges.png
lyxscale 50
scale 50
\end_inset
\end_layout
\begin_layout Subsubsection
Insulator-crossing triangles
\end_layout
\begin_layout Standard
Parts of triangle-centered minor cells with flag CROSSING_INS will be outside
the insulator and parts will be inside it.
\end_layout
\begin_layout Standard
Terminology:
\end_layout
\begin_layout Standard
The green cells are Major Cells which store density and temperature.
\end_layout
\begin_layout Standard
The blue cells are Central Minors which store velocity.
\end_layout
\begin_layout Standard
The remainder of a triangle is a Tri Minor which store velocity.
\end_layout
\begin_layout Standard
Note that mass, momentum and heat are conserved where appropriate, but we
work in
\begin_inset Formula $v$
\end_inset
and
\begin_inset Formula $T$
\end_inset
because they are subject to spatial smoothing.
If we worked in
\begin_inset Formula $nT$
\end_inset
we would need to do floating-point divisions every time we populate a set
of shared data to get the gradient of
\begin_inset Formula $T$
\end_inset
, so even purely from a computer efficiency perspective there was a strong
argument.
\end_layout
\begin_layout Standard
The mesh continues into the insulator so that we can model
\begin_inset Formula $A$
\end_inset
there, basically.
\end_layout
\begin_layout Standard
\begin_inset Graphics
filename inscross2.png
\end_inset
\end_layout
\begin_layout Standard
How to handle:
\end_layout
\begin_layout Itemize
Insulator-crossing triangles have a
\begin_inset Formula $v$
\end_inset
that is 0 for now.
\end_layout
\begin_deeper
\begin_layout Itemize
We do want
\begin_inset Formula $\phi$
\end_inset
changing at vertices to be able to push electrons out.
In some insulator-crossing tris this means azimuthally pushing out.
The azimuthal gradient of
\begin_inset Formula $\phi$
\end_inset
can be relevant, leaving
\begin_inset Formula $a_{r}=0$
\end_inset
.
We would get the azimuthal electric and pressure gradients from where
\begin_inset Formula $n,\phi$
\end_inset
are stored at the vertices.
\end_layout
\begin_layout Itemize
When we come to do device-azimuthal advection of
\begin_inset Formula $N,NT$
\end_inset
, we can try to do this using the triangle above -- though adding this branching
is not ideal for CUDA so we want to avoid it.
\end_layout
\end_deeper
\begin_layout Itemize
It IS logical to have v cells crossing a boundary so that we can decide
how much traffic there is at the boundary of an electrode.
But do we want this to be legislated
\begin_inset Formula $v$
\end_inset
or evolving
\begin_inset Formula $v$
\end_inset
?
\end_layout
\begin_layout Itemize
Insulator-crossing triangles perceive
\begin_inset Formula $\partial\phi/\partial r$
\end_inset
and
\begin_inset Formula $\frac{\partial}{\partial r}\left(nT\right)$
\end_inset
as zero, if
\begin_inset Formula $v$
\end_inset
ever is allowed to evolve there.
Allowing nonzero
\begin_inset Formula $v$
\end_inset
means we have to create domain intersection area, which can be a little
bit fiddly.
\end_layout
\begin_layout Itemize
It was nice when
\begin_inset Formula $nv$
\end_inset
was found in major cells that abutted the insulator, for sure.
\end_layout
\begin_layout Itemize
Inner triangles should also have
\begin_inset Formula $v,n=0$
\end_inset
so that we can do evolution of
\begin_inset Formula $A$
\end_inset
without branching.
\end_layout
\begin_layout Itemize
Phi is defined at the vertices within the domain.
Within insulator we would in 3D be able to define it, and there is a big
influence from beneath.
\end_layout
\begin_deeper
\begin_layout Itemize
However, the important thing for now is that when there is charge at the
nearby vertices, they can affect
\begin_inset Formula $v_{e}$
\end_inset
nearby and eject the charge.
\end_layout
\end_deeper
\begin_layout Subsubsection
Summary
\end_layout
\begin_layout Itemize
There should be now 5 possible tri flags: OUTER_FRILL, DOMAIN_TRIANGLE,
INNER_FRILL, CROSSING_INS, INNER_TRIANGLE.
\end_layout
\begin_layout Itemize
whereas vertices can be OUTERMOST, DOMAIN_VERTEX, INNER_VERTEX, INNERMOST.
\end_layout
\begin_layout Itemize
whereas centrals can be OUTERMOST_CENTRAL = OUTERMOST, INNERMOST_CENTRAL
= INNERMOST, DOMAIN_VERTEX, INNER_VERTEX
\end_layout
\begin_layout Itemize
Do we make different flags for OUTERMOST_CENTRAL vs such as OUTER_FRILL
???
\end_layout
\begin_layout Standard
What happens for each of these, summarized for each of the 3 things: treatment
of A's derivs, treatment of flows, and treatment of grad phi, nT, etc:
\end_layout
\begin_layout Itemize
In this table,
\begin_inset Quotes eld
\end_inset
standard
\begin_inset Quotes erd
\end_inset
means there is nothing to watch out for.
\end_layout
\begin_layout Standard
\begin_inset Tabular
<lyxtabular version="3" rows="12" columns="4">
<features tabularvalignment="middle">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Treatment of
\begin_inset Formula $A$
\end_inset
derivs
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Momentum flows:
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Treatment of
\begin_inset Formula $\phi$
\end_inset
,pressure
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
OUTER_FRILL
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $A_{frill}$
\end_inset
set per BC
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
n/a,
\begin_inset Formula $nv=0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
n/a
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
DOMAIN_TRIANGLE
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
standard
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
no
\begin_inset Formula $nv$
\end_inset
into outer frill, ins tri
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
standard
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
INNER_FRILL
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $A_{frill}$
\end_inset
set per BC
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
n/a,
\begin_inset Formula $nv=0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
n/a
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
INNER_TRIANGLE
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
standard
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
n/a,
\begin_inset Formula $nv=0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
n/a
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
CROSSING_INS
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
standard
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
FOR NOW
\begin_inset Formula $nv=0$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
FOR NOW
\begin_inset Formula $v=a=0$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Mass, heat flows:
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
OUTERMOST
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
Careful use of
\begin_inset Formula $A_{frill}$
\end_inset
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Traffic with domain, not outwards
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
\begin_inset Formula $\phi$
\end_inset
exists;
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $\nabla^{2}\phi$
\end_inset
careful;
\begin_inset Formula $\nabla\left(nT\right)=0$
\end_inset
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
DOMAIN_VERTEX
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
standard
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
std:
\begin_inset Formula $v_{r}=0$
\end_inset
prevents flow thru ins
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
Prevent Inner vertex neighs: how?
\end_layout
\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text
\begin_layout Plain Layout
INNER_VERTEX
\end_layout
\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text