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Put H at vertices and label them +1/2

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Jan Petykiewicz 4 years ago
parent 8bccc69706
commit 808a8879ae
1 changed files with 146 additions and 97 deletions
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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
in distinct locations for all six E- and H-field components:
[figure: Yee cell]
(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)
[figure: Field components]
(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:
[figure: E-fields for adjacent cells]
________Ex(p+1, m+1)_____
z y /: /|
|/_x / : / |
/ : / |
Ey(p+1) Ey(m+1, p+1)
/ : / |
/ Ez(n+1) / Ez(m+1, n+1)
/__________Ex(p+1)_______/ |
| : | |
| : | | This figure shows which fore-vector
| : | | each e-field component belongs to.
| :.........Ex(n+1).|......| Indices are shortened; e.g. Ex(p+1)
| / | / means "Ex for the fore-vector located
Ez / Ez(m+1) at (m, n, p+1)".
| Ey | /
| / | Ey(m+1)
| / | /
|/ |/
r=(m, n, p)|___________Ex___________|
H1__________Hx0_________H0
z y /: /|
|/_x / : / | This figure shows H back-vector locations
/ : / | H0, H1, etc. and their associated components
Hy1 : Hy0 | H0 = (Hx0, Hy0, Hz0) etc.
/ : / |
/ Hz1 / Hz0
H2___________Hx3_________H3 | The equivalent drawing for E would have
| : | | fore-vectors located at the cube's
| : | | center (and the centers of adjacent cubes),
| : | | with components on the cube's faces.
| H5..........Hx4...|......H4
| / | /
Hz2 / Hz2 /
| / | /
| Hy6 | Hy4
| / | /
|/ |/
H6__________Hx7__________H7
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
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.
[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]
Base grid Shifted one half-cell right
(e.g. for 1D forward x derivative of all components)
Some lines are omitted for clarity.
z y : / : / :dz'[1]
Base grid Shifted one half-cell right (e.g. for 1D
forward x derivative of all components).
z y : / : / :dz'[1] Some lines are omitted for clarity.
|/_x :/ :/ :/
.......:..........:.......:...
| /: | /: | /:
@ -416,7 +416,7 @@ one or more axes.
|/ : |/ : |/ :dz'[0]
______________________________
/| :/ /| :/ /| :/dy'[1]
/.|...:..../.|...:./.|.. :....
/.|...:..../.|...:./.|...:....
| /: | /: | /:
| / : | / : | /dy'[0]
|/ : |/ : |/ :
@ -428,68 +428,117 @@ one or more axes.
All three dimensions shifted by one half-
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
[figure: Component centers]
___________________________________________
z y /: /: /|
|/_x / : / : / |
/ : / : / |
Ey...........Hz Ey.....Hz / |
/ : / / : / / |
/ : / / : / / |
/ : / / :/ / |
/___________Ex____________/______Ex________/ |
| : | : | |
| : | : | |
| Hx : | Hx : | Hx |
| /: :.................|../:...:........|../:...|
| / : / | / : / | / : /
|/ : / |/ : / |/ : /
Ez...........Hy Ez......Hy Ez :/
| Ey........:..Hz | Ey...:..Hz | Ey
| / : / | / : / | /
| / : / | / : / | /
|/ :/ |/ :/ |/
|___________Ex____________|_______Ex_______|
[H]__________Hx___________[H]______Hx______[H]
z y /: /: /: /: /| |
|/_x / : / : / : / : / | |
/ : / : / : / : / | |
Hy : Ez...........Hy : Ez......Hy | |
/: : : : /: : : : /| | |
/ : Hz : Ey....../.:..Hz : Ey./.|..Hz | dz[0]
/ : /: : / / : /: : / / | /| |
/_________________________/________________/ | / | |
| :/ : :/ | :/ : :/ | |/ | |
| Ex : [E].......|..Ex : [E]..|..Ex | |
| : | : | | |
| [H]..........Hx....|......[H].....Hx|......[H]
| / | / | / /
| / | / | / /
Hz / Hz / Hz / /
| Hy | Hy | Hy /
| / | / | / / dy[0]
| / | / | / /
|/ |/ |/ /
[H]__________Hx___________[H]______Hx______[H] /
------------------------- ----------------
dx[0] dx[1]
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:
m : m :
Ey...........m..:.........Ey......m..:.....Ey
m : m :
m : m :
_____m_____:______________m_____:________
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm
| / : | / :
| / : | / :
| / : | / :
wwww|w/wwww:wwwwwwwwwwwww|w/wwww:wwwwwwwwww
|/ w |/ w
____________|____________________|__________
Ey.......|...w............Ey..|...w........Ey
| w | w
| w | w
|w |w
wwwwwwwwwwww|wwwwwwwwwwwwwwwwwwww|wwwwwwwwww
<__________________________________________>
z y << /: / /: >> |
|/_x < < / : / / : > > |
< < / : / / : > > |
< < / : / / : > > |
<: < / : : / : >: > |
< : < / : : / : > : > | dz[0]
< : < / : : / : > : > |
<____________/_____________________________> : > |
< : < | : :| : > : > |
< Ex < | : Ex| : > Ex > |
< : < | : :| : > : > |
< : <....|.......:........:|.......:...>...:...>
< : < | / :| / / > : > /
< : < | / :| / / > : > /
< :< | / :|/ / > :> /
< < | / :| / > > /
< < | / | / > > / dy[0]
< < | / | / > > /
<< |/ |/ >> /
<____________|_________________|___________> /
The Ey values are positioned on the y-edges of the base
grid, but they represent the Ey field in a volume that
contains (but isn't necessarily centered on) the points
at which they are defined.
~------------ ----------------- -----------~
dx'[-1] dx'[0] dx'[1]
Here, the 'Ey' labels represent the same points as before;
the grid lines _|:/ are edges of the area represented
by each Ey value, and the lines drawn using m.w represent
areas where a cell's faces extend beyond the drawn area
(i.e. where the drawing is truncated in the z-direction).
The Ex values are positioned on the x-faces of the base
grid. They represent the Ex field in volumes shifted by
a half-cell in the x-dimension, as shown here. Only the
center cell is fully shown; the other two are truncated
(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

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