Put H at vertices and label them +1/2
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@ -291,50 +291,52 @@ The time derivatives can be expanded to form the update equations:
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Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting
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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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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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(m, n, p+1)/________________________/ | H back-vector at r + 1/2 =
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| : | | (m + 1/2, n + 1/2, p + 1/2)
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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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[figure: Field components]
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(m - 1/2,=> ____________Hx__________[H] <= r + 1/2 = (m + 1/2,
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n + 1/2, /: /: /| n + 1/2,
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z y p + 1/2) / : / : / | p + 1/2)
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|/_x / : / : / |
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/ : Ez__________Hy | Locations of the E- and
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/ : : : /| | H-field components for the
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(m - 1/2, / : : Ey...../.|..Hz [E] fore-vector at r = (m,n,p)
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n - 1/2, =>/________________________/ | /| (the large cube's center)
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p + 1/2) | : : / | | / | and [H] back-vector at r + 1/2
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| : :/ | |/ | (the top right corner)
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| : [E].......|.Ex |
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| :.................|......| <= (m + 1/2, n + 1/2, p + 1/2)
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| / | /
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| / | /
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| / | / This is the Yee discretization
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| / | / scheme ("Yee cell").
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r - 1/2 = | / | /
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(m - 1/2, |/ |/
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n - 1/2,=> |________________________| <= (m + 1/2, n - 1/2, p - 1/2)
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p - 1/2)
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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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H1__________Hx0_________H0
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z y /: /|
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|/_x / : / |
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|/_x / : / | This figure shows H back-vector locations
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/ : / | H0, H1, etc. and their associated components
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Hy1 : Hy0 | H0 = (Hx0, Hy0, Hz0) etc.
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/ : / |
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Ey(p+1) Ey(m+1, p+1)
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/ : / |
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/ Ez(n+1) / Ez(m+1, n+1)
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/__________Ex(p+1)_______/ |
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| : | |
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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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/ Hz1 / Hz0
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H2___________Hx3_________H3 | The equivalent drawing for E would have
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| : | | fore-vectors located at the cube's
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| : | | center (and the centers of adjacent cubes),
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| : | | with components on the cube's faces.
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| H5..........Hx4...|......H4
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| / | /
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Hz2 / Hz2 /
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| / | /
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| Hy6 | Hy4
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| / | /
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|/ |/
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r=(m, n, p)|___________Ex___________|
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H6__________Hx7__________H7
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The divergence equations can be derived by taking the divergence of the curl equations
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@ -376,7 +378,7 @@ Grid description
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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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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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@ -404,11 +406,9 @@ one or more axes.
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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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Base grid Shifted one half-cell right (e.g. for 1D
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forward x derivative of all components).
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z y : / : / :dz'[1] Some lines are omitted for clarity.
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|/_x :/ :/ :/
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.......:..........:.......:...
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| /: | /: | /:
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@ -416,7 +416,7 @@ one or more axes.
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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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| /: | /: | /:
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| / : | / : | /dy'[0]
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|/ : |/ : |/ :
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@ -428,68 +428,117 @@ one or more axes.
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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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(and probably not entirely to scale); see
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later figures for a better representation.
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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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[H]__________Hx___________[H]______Hx______[H]
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z y /: /: /: /: /| |
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|/_x / : / : / : / : / | |
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/ : / : / : / : / | |
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Hy : Ez...........Hy : Ez......Hy | |
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/: : : : /: : : : /| | |
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/ : Hz : Ey....../.:..Hz : Ey./.|..Hz | dz[0]
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/ : /: : / / : /: : / / | /| |
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/_________________________/________________/ | / | |
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| :/ : :/ | :/ : :/ | |/ | |
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| Ex : [E].......|..Ex : [E]..|..Ex | |
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| : | : | | |
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| [H]..........Hx....|......[H].....Hx|......[H]
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| / | / | / /
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| / | / | / /
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Hz / Hz / Hz / /
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| Hy | Hy | Hy /
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| / | / | / / dy[0]
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| / | / | / /
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|/ |/ |/ /
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[H]__________Hx___________[H]______Hx______[H] /
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------------------------- ----------------
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dx[0] dx[1]
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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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positions of the various field components. [E] fore-vectors
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are at the cell centers, and [H] back-vectors are at the
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vertices. H components along the near (-y) top (+z) edge
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have been omitted to make the insides of the cubes easier
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to visualize.
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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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<__________________________________________>
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z y << /: / /: >> |
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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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<____________/_____________________________> : > |
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< : < | : :| : > : > |
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< Ex < | : Ex| : > Ex > |
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< : < | : :| : > : > |
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< : <....|.......:........:|.......:...>...:...>
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< : < | / :| / / > : > /
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< : < | / :| / / > : > /
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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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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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~------------ ----------------- -----------~
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dx'[-1] dx'[0] dx'[1]
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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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The Ex values are positioned on the x-faces of the base
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grid. They represent the Ex field in volumes shifted by
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a half-cell in the x-dimension, as shown here. Only the
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center cell is fully shown; the other two are truncated
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(shown using >< markers).
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Note that the Ex positions are the in the same positions
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as the previous figure; only the cell boundaries have moved.
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Also note that the points at which Ex is defined are not
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necessarily centered in the volumes they represent; non-
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uniform cell sizes result in off-center volumes like the
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center cell here.
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z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm s
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|/_x << m: m: >> |
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< < m : m : > > | dz'[1]
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Hy............m...........Hy........m......Hy > |
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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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< < | / : | / :> > | dz'[0]
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< < | / : | / :> > |
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< wwwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwww s
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< < |/ w |/ w > > /
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_____________|_____________________|________ > /
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< Hy........|...w...........Hy....|...w...>..Hy /
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< < | w | w > > / dy[0]
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< < | w | w > > /
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<< |w |w >> /
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wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww
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~------------ --------------------- -------~
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dx'[-1] dx'[0] dx'[1]
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The Hy values are positioned on the y-edges of the base
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grid. Again here, the 'Hy' labels represent the same points
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as in the basic grid figure above; the edges have shifted
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by a half-cell along the x- and z-axes.
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The grid lines _|:/ are edges of the area represented by
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each Hy value, and the lines drawn using <m>.w represent
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edges where a cell's faces extend beyond the drawn area
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(i.e. where the drawing is truncated in the x- or z-
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directions).
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TODO: explain dxes
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