more grid work
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@ -12,8 +12,8 @@ This documentation and approach is roughly based on W.C. Chew's excellent
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which covers a superset of this material with similar notation and more detail.
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Derivatives and shifted values
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------------------------------
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Scalar derivatives and cell shifts
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----------------------------------
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Define the discrete forward derivative as
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$$ [\\tilde{\\partial}_x f ]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
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@ -71,10 +71,14 @@ The fractional subscript \\( m + \\frac{1}{2} \\) is used to indicate values def
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$$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
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Just as \\( m \\) is not itself an x-coordinate, neither is \\( m + \\frac{1}{2} \\);
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carefully note the positions of the various cells in the above figure vs their labels.
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If the positions labeled with \\( m \\) are considered the "base" or "original" grid,
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the positions labeled with \\( m + \\frac{1}{2} \\) are said to lie on a "dual" or
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"derived" grid.
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For the remainder of the `Discrete calculus` section, all figures will show
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constant-length cells in order to focus on the vector derivatives themselves.
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See the `Grid description` section below for additional information on this topic.
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See the `Grid description` section below for additional information on this topic
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and generalization to three dimensions.
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Gradients and fore-vectors
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@ -128,10 +132,10 @@ on the direction of the shift. We write it as
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(m, n, p+1)/_____________/ | The forward derivatives are defined
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| : | | at the Dx, Dy, Dz points,
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| :.........|...| but the forward-gradient fore-vector
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Dz / | / is the set of all three
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| Dy | / and is said to be "located" at (m,n,p)
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| / | /
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(m, n, p)|/_____Dx_____|/ (m+1, n, p)
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z y Dz / | / is the set of all three
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|/_x | Dy | / and is said to be "located" at (m,n,p)
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|/ |/
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(m, n, p)|_____Dx______| (m+1, n, p)
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@ -181,11 +185,11 @@ is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not
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/ : // / | of this cube) of a fore-vector field
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(m-1/2, n-1/2, p+1/2)/_____________/ | is the sum of the outward-pointing
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| : | | fore-vector components, which are
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<==|== :.........|.====> located at the face centers.
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| / | /
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| / // | / Note that in a nonuniform grid, each
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| / // || | / dimension is normalized by the cell width.
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(m-1/2, n-1/2, p-1/2)|/___//_______|/ (m+1/2, n-1/2, p-1/2)
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z y <==|== :.........|.====> located at the face centers.
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|/_x | / | /
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| / // | / Note that in a nonuniform grid, each
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|/ // || |/ dimension is normalized by the cell width.
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(m-1/2, n-1/2, p-1/2)|____//_______| (m+1/2, n-1/2, p-1/2)
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'' ||
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VV
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@ -240,13 +244,13 @@ For example, consider the forward curl, at (m, n, p), of a back-vector field `g`
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is set by their loop-oriented sum (i.e. two have their signs flipped to complete the loop).
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[figure: z-component of curl]
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: |
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: ^^ |
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:....||.<.....| (m, n+1, p+1/2)
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/ || /
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| v || | ^
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| / | /
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(m, n, p+1/2) |/_____>______|/ (m+1, n, p+1/2)
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: |
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z y : ^^ |
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|/_x :....||.<.....| (m, n+1, p+1/2)
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/ || /
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| v || | ^
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|/ |/
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(m, n, p+1/2) |_____>______| (m+1, n, p+1/2)
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@ -290,8 +294,8 @@ in distinct locations for all six E- and H-field components:
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[figure: Yee cell]
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(m, n+1, p+1) _________________________ (m+1, n+1, p+1)
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/: /|
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/ : / |
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/ : / | Locations of the
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z y / : / |
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|/_x / : / | Locations of the
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/ : / | E- and H-field components
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/ : / | for the E fore-vector at
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/ : / | r = (m, n, p) and its associated
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@ -300,21 +304,21 @@ in distinct locations for all six E- and H-field components:
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| : | | (the large cube's center)
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| Hx : | |
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| /: :.................|......| (m+1, n+1, p)
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|/ : / | /
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Ez..........Hy | /
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| Ey.......:..Hz | / This is the Yee discretization
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| / : / | / scheme ("Yee cell").
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| / : / | /
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|/ :/ | /
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r=(m, n, p)|___________Ex___________|/ (m+1, n, p)
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|/ : / | /
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Ez..........Hy | /
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| Ey.......:..Hz | / This is the Yee discretization
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| / : / | / scheme ("Yee cell").
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| / : / | /
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|/ :/ |/
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r=(m, n, p)|___________Ex___________| (m+1, n, p)
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Each component forms its own grid, offset from the others:
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[figure: E-fields for adjacent cells]
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________Ex(p+1, m+1)_____
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/: /|
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/ : / |
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z y /: /|
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|/_x / : / |
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/ : / |
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Ey(p+1) Ey(m+1, p+1)
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/ : / |
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@ -324,13 +328,13 @@ Each component forms its own grid, offset from the others:
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| : | | This figure shows which fore-vector
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| : | | each e-field component belongs to.
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| :.........Ex(n+1).|......| Indices are shortened; e.g. Ex(p+1)
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| / | / means "Ex for the fore-vector located
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Ez / Ez(m+1)/ at (m, n, p+1)".
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| Ey | /
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| / | Ey(m+1)
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| / | /
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|/ | /
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r=(m, n, p)|___________Ex___________|/
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| / | / means "Ex for the fore-vector located
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Ez / Ez(m+1) at (m, n, p+1)".
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| Ey | /
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| / | Ey(m+1)
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| / | /
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|/ |/
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r=(m, n, p)|___________Ex___________|
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The divergence equations can be derived by taking the divergence of the curl equations
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@ -370,7 +374,123 @@ $$
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Grid description
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================
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The
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As described in the section on scalar discrete derivatives above, cell widths along
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each axis can be arbitrary and independently defined. Moreover, all field components
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are defined at "derived" or "dual" positions, in between the "base" grid points on
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one or more axes.
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[figure: 3D base and derived grids]
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_____________________________ _____________________________
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z y /: /: /: /| z y /: /: /:
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|/_x / : / : / : / | |/_x / : / : / :
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/ : / : / : / | / : / : / :
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/___________________________/ | dz[1] ________________________/____
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/ : / : / : /| | /: : / : /: : dz[1]
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/: : / : / : / | | / : : / : / : :
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/ : :..../......:/......:/..|...| / .:...:../......:/..:...:.....
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/___________/_______/_______/ | /| ______/_________/_______/___: :
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| : / : | | | | / | | : : | | : :
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| : / : | | | |/ | | : : | | : :
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| :/ : | | | | | dz[0] | : : | | : : dz[0]
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| / : | | | /| | | : : | | : :
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| /: :...|.......|.......|./ |...| | ..:...:.|.......|...:...:.....
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|/ : / | /| /|/ | / | : / | /| : /
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|___________|_______|_______| | / dy[1] ______|_________|_______|___: / dy[1]
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| : / | / | / | |/ | :/ | / | :/
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| :/.......|.../...|.../...|...| ..|...:.....|.../...|...:...
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| / | / | / | / | / | / | / dy[0]
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| / | / | / | / dy[0] | / | / | /
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|/ |/ |/ |/ |/ |/ |/
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|___________|_______|_______| ______|_________|_______|___
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dx[0] dx[1] dx[2] dx'[0] dx'[1] dx'[2]
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Base grid Shifted one half-cell right
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(e.g. for 1D forward x derivative of all components)
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Some lines are omitted for clarity.
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z y : / : / :dz'[1]
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|/_x :/ :/ :/
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.......:..........:.......:...
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| /: | /: | /:
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| / : | / : | / :
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|/ : |/ : |/ :dz'[0]
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______________________________
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/| :/ /| :/ /| :/dy'[1]
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/.|...:..../.|...:./.|.. :....
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| /: | /: | /:
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| / : | / : | /dy'[0]
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|/ : |/ : |/ :
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_______________________________
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/| /| /|
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/ | / | / |
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| | |
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dx'[0] dx'[1] dx'[2]
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All three dimensions shifted by one half-
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cell. This is quite hard to visualize
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(and probably not entirely to scale).
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Nevertheless, while the spacing
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[figure: Component centers]
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___________________________________________
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z y /: /: /|
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|/_x / : / : / |
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/ : / : / |
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Ey...........Hz Ey.....Hz / |
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/ : / / : / / |
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/ : / / : / / |
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/ : / / :/ / |
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/___________Ex____________/______Ex________/ |
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| : | : | |
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| : | : | |
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| Hx : | Hx : | Hx |
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| /: :.................|../:...:........|../:...|
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| / : / | / : / | / : /
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|/ : / |/ : / |/ : /
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Ez...........Hy Ez......Hy Ez :/
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| Ey........:..Hz | Ey...:..Hz | Ey
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| / : / | / : / | /
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| / : / | / : / | /
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|/ :/ |/ :/ |/
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|___________Ex____________|_______Ex_______|
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Part of a nonuniform "base grid", with labels specifying
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positions of the various field components.
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z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm
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|/_x m: m:
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m : m :
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Ey...........m..:.........Ey......m..:.....Ey
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m : m :
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m : m :
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_____m_____:______________m_____:________
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mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm
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| / : | / :
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| / : | / :
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| / : | / :
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wwww|w/wwww:wwwwwwwwwwwww|w/wwww:wwwwwwwwww
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|/ w |/ w
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____________|____________________|__________
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Ey.......|...w............Ey..|...w........Ey
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| w | w
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| w | w
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|w |w
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wwwwwwwwwwww|wwwwwwwwwwwwwwwwwwww|wwwwwwwwww
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The Ey values are positioned on the y-edges of the base
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grid, but they represent the Ey field in a volume that
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contains (but isn't necessarily centered on) the points
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at which they are defined.
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Here, the 'Ey' labels represent the same points as before;
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the grid lines _|:/ are edges of the area represented
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by each Ey value, and the lines drawn using m.w represent
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areas where a cell's faces extend beyond the drawn area
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(i.e. where the drawing is truncated in the z-direction).
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TODO: explain dxes
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