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more grid work

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Jan Petykiewicz 2019-12-09 21:28:26 -08:00
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meanas/fdmath/__init__.py
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@ -12,8 +12,8 @@ This documentation and approach is roughly based on W.C. Chew's excellent
which covers a superset of this material with similar notation and more detail. which covers a superset of this material with similar notation and more detail.
Derivatives and shifted values Scalar derivatives and cell shifts
------------------------------ ----------------------------------
Define the discrete forward derivative as Define the discrete forward derivative as
$$ [\\tilde{\\partial}_x f ]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$ $$ [\\tilde{\\partial}_x f ]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
@ -71,10 +71,14 @@ The fractional subscript \\( m + \\frac{1}{2} \\) is used to indicate values def
$$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$ $$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
Just as \\( m \\) is not itself an x-coordinate, neither is \\( m + \\frac{1}{2} \\); Just as \\( m \\) is not itself an x-coordinate, neither is \\( m + \\frac{1}{2} \\);
carefully note the positions of the various cells in the above figure vs their labels. carefully note the positions of the various cells in the above figure vs their labels.
If the positions labeled with \\( m \\) are considered the "base" or "original" grid,
the positions labeled with \\( m + \\frac{1}{2} \\) are said to lie on a "dual" or
"derived" grid.
For the remainder of the `Discrete calculus` section, all figures will show For the remainder of the `Discrete calculus` section, all figures will show
constant-length cells in order to focus on the vector derivatives themselves. constant-length cells in order to focus on the vector derivatives themselves.
See the `Grid description` section below for additional information on this topic. See the `Grid description` section below for additional information on this topic
and generalization to three dimensions.
Gradients and fore-vectors Gradients and fore-vectors
@ -128,10 +132,10 @@ on the direction of the shift. We write it as
(m, n, p+1)/_____________/ | The forward derivatives are defined (m, n, p+1)/_____________/ | The forward derivatives are defined
| : | | at the Dx, Dy, Dz points, | : | | at the Dx, Dy, Dz points,
| :.........|...| but the forward-gradient fore-vector | :.........|...| but the forward-gradient fore-vector
Dz / | / is the set of all three z y Dz / | / is the set of all three
| Dy | / and is said to be "located" at (m,n,p) |/_x | Dy | / and is said to be "located" at (m,n,p)
| / | / |/ |/
(m, n, p)|/_____Dx_____|/ (m+1, n, p) (m, n, p)|_____Dx______| (m+1, n, p)
@ -181,11 +185,11 @@ is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not
/ : // / | of this cube) of a fore-vector field / : // / | of this cube) of a fore-vector field
(m-1/2, n-1/2, p+1/2)/_____________/ | is the sum of the outward-pointing (m-1/2, n-1/2, p+1/2)/_____________/ | is the sum of the outward-pointing
| : | | fore-vector components, which are | : | | fore-vector components, which are
<==|== :.........|.====> located at the face centers. z y <==|== :.........|.====> located at the face centers.
| / | / |/_x | / | /
| / // | / Note that in a nonuniform grid, each | / // | / Note that in a nonuniform grid, each
| / // || | / dimension is normalized by the cell width. |/ // || |/ dimension is normalized by the cell width.
(m-1/2, n-1/2, p-1/2)|/___//_______|/ (m+1/2, n-1/2, p-1/2) (m-1/2, n-1/2, p-1/2)|____//_______| (m+1/2, n-1/2, p-1/2)
'' || '' ||
VV VV
@ -241,12 +245,12 @@ For example, consider the forward curl, at (m, n, p), of a back-vector field `g`
[figure: z-component of curl] [figure: z-component of curl]
: | : |
: ^^ | z y : ^^ |
:....||.<.....| (m, n+1, p+1/2) |/_x :....||.<.....| (m, n+1, p+1/2)
/ || / / || /
| v || | ^ | v || | ^
| / | / |/ |/
(m, n, p+1/2) |/_____>______|/ (m+1, n, p+1/2) (m, n, p+1/2) |_____>______| (m+1, n, p+1/2)
@ -290,8 +294,8 @@ in distinct locations for all six E- and H-field components:
[figure: Yee cell] [figure: Yee cell]
(m, n+1, p+1) _________________________ (m+1, n+1, p+1) (m, n+1, p+1) _________________________ (m+1, n+1, p+1)
/: /| /: /|
/ : / | z y / : / |
/ : / | Locations of the |/_x / : / | Locations of the
/ : / | E- and H-field components / : / | E- and H-field components
/ : / | for the E fore-vector at / : / | for the E fore-vector at
/ : / | r = (m, n, p) and its associated / : / | r = (m, n, p) and its associated
@ -305,16 +309,16 @@ in distinct locations for all six E- and H-field components:
| Ey.......:..Hz | / This is the Yee discretization | Ey.......:..Hz | / This is the Yee discretization
| / : / | / scheme ("Yee cell"). | / : / | / scheme ("Yee cell").
| / : / | / | / : / | /
|/ :/ | / |/ :/ |/
r=(m, n, p)|___________Ex___________|/ (m+1, n, p) r=(m, n, p)|___________Ex___________| (m+1, n, p)
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)_____ ________Ex(p+1, m+1)_____
/: /| z y /: /|
/ : / | |/_x / : / |
/ : / | / : / |
Ey(p+1) Ey(m+1, p+1) Ey(p+1) Ey(m+1, p+1)
/ : / | / : / |
@ -325,12 +329,12 @@ Each component forms its own grid, offset from the others:
| : | | each e-field component belongs to. | : | | each e-field component belongs to.
| :.........Ex(n+1).|......| Indices are shortened; e.g. Ex(p+1) | :.........Ex(n+1).|......| Indices are shortened; e.g. Ex(p+1)
| / | / means "Ex for the fore-vector located | / | / means "Ex for the fore-vector located
Ez / Ez(m+1)/ at (m, n, p+1)". Ez / Ez(m+1) at (m, n, p+1)".
| Ey | / | Ey | /
| / | Ey(m+1) | / | Ey(m+1)
| / | / | / | /
|/ | / |/ |/
r=(m, n, p)|___________Ex___________|/ r=(m, n, p)|___________Ex___________|
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
@ -370,7 +374,123 @@ $$
Grid description Grid description
================ ================
The 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
one or more axes.
[figure: 3D base and derived grids]
_____________________________ _____________________________
z y /: /: /: /| z y /: /: /:
|/_x / : / : / : / | |/_x / : / : / :
/ : / : / : / | / : / : / :
/___________________________/ | dz[1] ________________________/____
/ : / : / : /| | /: : / : /: : dz[1]
/: : / : / : / | | / : : / : / : :
/ : :..../......:/......:/..|...| / .:...:../......:/..:...:.....
/___________/_______/_______/ | /| ______/_________/_______/___: :
| : / : | | | | / | | : : | | : :
| : / : | | | |/ | | : : | | : :
| :/ : | | | | | dz[0] | : : | | : : dz[0]
| / : | | | /| | | : : | | : :
| /: :...|.......|.......|./ |...| | ..:...:.|.......|...:...:.....
|/ : / | /| /|/ | / | : / | /| : /
|___________|_______|_______| | / dy[1] ______|_________|_______|___: / dy[1]
| : / | / | / | |/ | :/ | / | :/
| :/.......|.../...|.../...|...| ..|...:.....|.../...|...:...
| / | / | / | / | / | / | / dy[0]
| / | / | / | / dy[0] | / | / | /
|/ |/ |/ |/ |/ |/ |/
|___________|_______|_______| ______|_________|_______|___
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]
|/_x :/ :/ :/
.......:..........:.......:...
| /: | /: | /:
| / : | / : | / :
|/ : |/ : |/ :dz'[0]
______________________________
/| :/ /| :/ /| :/dy'[1]
/.|...:..../.|...:./.|.. :....
| /: | /: | /:
| / : | / : | /dy'[0]
|/ : |/ : |/ :
_______________________________
/| /| /|
/ | / | / |
| | |
dx'[0] dx'[1] dx'[2]
All three dimensions shifted by one half-
cell. This is quite hard to visualize
(and probably not entirely to scale).
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_______|
Part of a nonuniform "base grid", with labels specifying
positions of the various field components.
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
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.
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).
TODO: explain dxes TODO: explain dxes