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Put H at vertices and label them +1/2

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Jan Petykiewicz 2019-12-10 01:14:21 -08:00
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meanas/fdmath/__init__.py
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@ -291,50 +291,52 @@ The time derivatives can be expanded to form the update equations:
Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting
in distinct locations for all six E- and H-field components: in distinct locations for all six E- and H-field components:
[figure: Yee cell] [figure: Field components]
(m, n+1, p+1) _________________________ (m+1, n+1, p+1)
/: /|
z y / : / |
|/_x / : / | Locations of the
/ : / | E- and H-field components
/ : / | for the E fore-vector at
/ : / | r = (m, n, p) and its associated
(m, n, p+1)/________________________/ | H back-vector at r + 1/2 =
| : | | (m + 1/2, n + 1/2, p + 1/2)
| : | | (the large cube's center)
| Hx : | |
| /: :.................|......| (m+1, n+1, p)
|/ : / | /
Ez..........Hy | /
| Ey.......:..Hz | / This is the Yee discretization
| / : / | / scheme ("Yee cell").
| / : / | /
|/ :/ |/
r=(m, n, p)|___________Ex___________| (m+1, n, p)
(m - 1/2,=> ____________Hx__________[H] <= r + 1/2 = (m + 1/2,
n + 1/2, /: /: /| n + 1/2,
z y p + 1/2) / : / : / | p + 1/2)
|/_x / : / : / |
/ : Ez__________Hy | Locations of the E- and
/ : : : /| | H-field components for the
(m - 1/2, / : : Ey...../.|..Hz [E] fore-vector at r = (m,n,p)
n - 1/2, =>/________________________/ | /| (the large cube's center)
p + 1/2) | : : / | | / | and [H] back-vector at r + 1/2
| : :/ | |/ | (the top right corner)
| : [E].......|.Ex |
| :.................|......| <= (m + 1/2, n + 1/2, p + 1/2)
| / | /
| / | /
| / | / This is the Yee discretization
| / | / scheme ("Yee cell").
r - 1/2 = | / | /
(m - 1/2, |/ |/
n - 1/2,=> |________________________| <= (m + 1/2, n - 1/2, p - 1/2)
p - 1/2)
Each component forms its own grid, offset from the others: Each component forms its own grid, offset from the others:
[figure: E-fields for adjacent cells] [figure: E-fields for adjacent cells]
________Ex(p+1, m+1)_____
z y /: /| H1__________Hx0_________H0
|/_x / : / | z y /: /|
/ : / | |/_x / : / | This figure shows H back-vector locations
Ey(p+1) Ey(m+1, p+1) / : / | H0, H1, etc. and their associated components
/ : / | Hy1 : Hy0 | H0 = (Hx0, Hy0, Hz0) etc.
/ Ez(n+1) / Ez(m+1, n+1) / : / |
/__________Ex(p+1)_______/ | / Hz1 / Hz0
| : | | H2___________Hx3_________H3 | The equivalent drawing for E would have
| : | | This figure shows which fore-vector | : | | fore-vectors located at the cube's
| : | | each e-field component belongs to. | : | | center (and the centers of adjacent cubes),
| :.........Ex(n+1).|......| Indices are shortened; e.g. Ex(p+1) | : | | with components on the cube's faces.
| / | / means "Ex for the fore-vector located | H5..........Hx4...|......H4
Ez / Ez(m+1) at (m, n, p+1)". | / | /
| Ey | / Hz2 / Hz2 /
| / | Ey(m+1) | / | /
| / | / | Hy6 | Hy4
|/ |/ | / | /
r=(m, n, p)|___________Ex___________| |/ |/
H6__________Hx7__________H7
The divergence equations can be derived by taking the divergence of the curl equations The divergence equations can be derived by taking the divergence of the curl equations
@ -376,7 +378,7 @@ Grid description
As described in the section on scalar discrete derivatives above, cell widths along As described in the section on scalar discrete derivatives above, cell widths along
each axis can be arbitrary and independently defined. Moreover, all field components each axis can be arbitrary and independently defined. Moreover, all field components
are defined at "derived" or "dual" positions, in between the "base" grid points on are defined at "derived" or "dual" positions, in-between the "base" grid points on
one or more axes. one or more axes.
[figure: 3D base and derived grids] [figure: 3D base and derived grids]
@ -404,11 +406,9 @@ one or more axes.
|___________|_______|_______| ______|_________|_______|___ |___________|_______|_______| ______|_________|_______|___
dx[0] dx[1] dx[2] dx'[0] dx'[1] dx'[2] dx[0] dx[1] dx[2] dx'[0] dx'[1] dx'[2]
Base grid Shifted one half-cell right Base grid Shifted one half-cell right (e.g. for 1D
(e.g. for 1D forward x derivative of all components) forward x derivative of all components).
Some lines are omitted for clarity. z y : / : / :dz'[1] Some lines are omitted for clarity.
z y : / : / :dz'[1]
|/_x :/ :/ :/ |/_x :/ :/ :/
.......:..........:.......:... .......:..........:.......:...
| /: | /: | /: | /: | /: | /:
@ -416,7 +416,7 @@ one or more axes.
|/ : |/ : |/ :dz'[0] |/ : |/ : |/ :dz'[0]
______________________________ ______________________________
/| :/ /| :/ /| :/dy'[1] /| :/ /| :/ /| :/dy'[1]
/.|...:..../.|...:./.|.. :.... /.|...:..../.|...:./.|...:....
| /: | /: | /: | /: | /: | /:
| / : | / : | /dy'[0] | / : | / : | /dy'[0]
|/ : |/ : |/ : |/ : |/ : |/ :
@ -428,68 +428,117 @@ one or more axes.
All three dimensions shifted by one half- All three dimensions shifted by one half-
cell. This is quite hard to visualize cell. This is quite hard to visualize
(and probably not entirely to scale). (and probably not entirely to scale); see
later figures for a better representation.
Nevertheless, while the spacing Nevertheless, while the spacing
[figure: Component centers] [figure: Component centers]
___________________________________________ [H]__________Hx___________[H]______Hx______[H]
z y /: /: /| z y /: /: /: /: /| |
|/_x / : / : / | |/_x / : / : / : / : / | |
/ : / : / | / : / : / : / : / | |
Ey...........Hz Ey.....Hz / | Hy : Ez...........Hy : Ez......Hy | |
/ : / / : / / | /: : : : /: : : : /| | |
/ : / / : / / | / : Hz : Ey....../.:..Hz : Ey./.|..Hz | dz[0]
/ : / / :/ / | / : /: : / / : /: : / / | /| |
/___________Ex____________/______Ex________/ | /_________________________/________________/ | / | |
| : | : | | | :/ : :/ | :/ : :/ | |/ | |
| : | : | | | Ex : [E].......|..Ex : [E]..|..Ex | |
| Hx : | Hx : | Hx | | : | : | | |
| /: :.................|../:...:........|../:...| | [H]..........Hx....|......[H].....Hx|......[H]
| / : / | / : / | / : / | / | / | / /
|/ : / |/ : / |/ : / | / | / | / /
Ez...........Hy Ez......Hy Ez :/ Hz / Hz / Hz / /
| Ey........:..Hz | Ey...:..Hz | Ey | Hy | Hy | Hy /
| / : / | / : / | / | / | / | / / dy[0]
| / : / | / : / | / | / | / | / /
|/ :/ |/ :/ |/ |/ |/ |/ /
|___________Ex____________|_______Ex_______| [H]__________Hx___________[H]______Hx______[H] /
------------------------- ----------------
dx[0] dx[1]
Part of a nonuniform "base grid", with labels specifying Part of a nonuniform "base grid", with labels specifying
positions of the various field components. positions of the various field components. [E] fore-vectors
are at the cell centers, and [H] back-vectors are at the
vertices. H components along the near (-y) top (+z) edge
have been omitted to make the insides of the cubes easier
to visualize.
z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm <__________________________________________>
|/_x m: m: z y << /: / /: >> |
m : m : |/_x < < / : / / : > > |
Ey...........m..:.........Ey......m..:.....Ey < < / : / / : > > |
m : m : < < / : / / : > > |
m : m : <: < / : : / : >: > |
_____m_____:______________m_____:________ < : < / : : / : > : > | dz[0]
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm < : < / : : / : > : > |
| / : | / : <____________/_____________________________> : > |
| / : | / : < : < | : :| : > : > |
| / : | / : < Ex < | : Ex| : > Ex > |
wwww|w/wwww:wwwwwwwwwwwww|w/wwww:wwwwwwwwww < : < | : :| : > : > |
|/ w |/ w < : <....|.......:........:|.......:...>...:...>
____________|____________________|__________ < : < | / :| / / > : > /
Ey.......|...w............Ey..|...w........Ey < : < | / :| / / > : > /
| w | w < :< | / :|/ / > :> /
| w | w < < | / :| / > > /
|w |w < < | / | / > > / dy[0]
wwwwwwwwwwww|wwwwwwwwwwwwwwwwwwww|wwwwwwwwww < < | / | / > > /
<< |/ |/ >> /
<____________|_________________|___________> /
The Ey values are positioned on the y-edges of the base ~------------ ----------------- -----------~
grid, but they represent the Ey field in a volume that dx'[-1] dx'[0] dx'[1]
contains (but isn't necessarily centered on) the points
at which they are defined.
Here, the 'Ey' labels represent the same points as before; The Ex values are positioned on the x-faces of the base
the grid lines _|:/ are edges of the area represented grid. They represent the Ex field in volumes shifted by
by each Ey value, and the lines drawn using m.w represent a half-cell in the x-dimension, as shown here. Only the
areas where a cell's faces extend beyond the drawn area center cell is fully shown; the other two are truncated
(i.e. where the drawing is truncated in the z-direction). (shown using >< markers).
Note that the Ex positions are the in the same positions
as the previous figure; only the cell boundaries have moved.
Also note that the points at which Ex is defined are not
necessarily centered in the volumes they represent; non-
uniform cell sizes result in off-center volumes like the
center cell here.
z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm s
|/_x << m: m: >> |
< < m : m : > > | dz'[1]
Hy............m...........Hy........m......Hy > |
< < m : m : > > |
< < m : m : > > |
< _______m_____:_______________m_____:_>______
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm > |
< < | / : | / :> > |
< < | / : | / :> > | dz'[0]
< < | / : | / :> > |
< wwwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwww s
< < |/ w |/ w > > /
_____________|_____________________|________ > /
< Hy........|...w...........Hy....|...w...>..Hy /
< < | w | w > > / dy[0]
< < | w | w > > /
<< |w |w >> /
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww
~------------ --------------------- -------~
dx'[-1] dx'[0] dx'[1]
The Hy values are positioned on the y-edges of the base
grid. Again here, the 'Hy' labels represent the same points
as in the basic grid figure above; the edges have shifted
by a half-cell along the x- and z-axes.
The grid lines _|:/ are edges of the area represented by
each Hy value, and the lines drawn using <m>.w represent
edges where a cell's faces extend beyond the drawn area
(i.e. where the drawing is truncated in the x- or z-
directions).
TODO: explain dxes TODO: explain dxes