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"""
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Basic discrete calculus for finite difference (fd) simulations.
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TODO: short description of functional vs operator form
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Discrete calculus
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=================
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This documentation and approach is roughly based on W.C. Chew's excellent
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"Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770),
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which covers a superset of this material with similar notation and more detail.
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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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where \\( f \\) is a function defined at discrete locations on the x-axis (labeled using \\( m \\)).
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The value at \\( m \\) occupies a length \\( \\Delta_{x, m} \\) along the x-axis. Note that \\( m \\)
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is an index along the x-axis, _not_ necessarily an x-coordinate, since each length
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\\( \\Delta_{x, m}, \\Delta_{x, m+1}, ...\\) is independently chosen.
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If we treat `f` as a 1D array of values, with the `i`-th value `f[i]` taking up a length `dx[i]`
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along the x-axis, the forward derivative is
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deriv_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
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Likewise, discrete reverse derivative is
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$$ [\\hat{\\partial}_x f ]_{m - \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m} - f_{m - 1}) $$
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or
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deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
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The derivatives' values are shifted by a half-cell relative to the original function, and
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will have different cell widths if all the `dx[i]` ( \\( \\Delta_{x, m} \\) ) are not
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identical:
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[figure: derivatives and cell sizes]
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dx0 dx1 dx2 dx3 cell sizes for function
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----- ----- ----------- -----
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______________________________
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| | | |
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f0 | f1 | f2 | f3 | function
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_____|_____|___________|_____|
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| | | |
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| Df0 | Df1 | Df2 | Df3 forward derivative (periodic boundary)
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__|_____|________|________|___
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dx'3] dx'0 dx'1 dx'2 [dx'3 cell sizes for forward derivative
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-- ----- -------- -------- ---
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dx'0] dx'1 dx'2 dx'3 [dx'0 cell sizes for reverse derivative
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______________________________
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| | | |
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| df1 | df2 | df3 | df0 reverse derivative (periodic boundary)
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__|_____|________|________|___
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Periodic boundaries are used here and elsewhere unless otherwise noted.
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In the above figure,
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`f0 =` \\(f_0\\), `f1 =` \\(f_1\\)
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`Df0 =` \\([\\tilde{\\partial}f]_{0 + \\frac{1}{2}}\\)
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`Df1 =` \\([\\tilde{\\partial}f]_{1 + \\frac{1}{2}}\\)
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`df0 =` \\([\\hat{\\partial}f]_{0 - \\frac{1}{2}}\\)
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etc.
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The fractional subscript \\( m + \\frac{1}{2} \\) is used to indicate values defined
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at shifted locations relative to the original \\( m \\), with corresponding lengths
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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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and generalization to three dimensions.
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Gradients and fore-vectors
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--------------------------
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Expanding to three dimensions, we can define two gradients
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$$ [\\tilde{\\nabla} f]_{m,n,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
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\\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
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\\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
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$$ [\\hat{\\nabla} f]_{m,n,p} = \\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
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\\vec{y} [\\hat{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
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\\vec{z} [\\hat{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
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or
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[code: gradients]
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grad_forward(f)[i,j,k] = [Dx_forward(f)[i, j, k],
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Dy_forward(f)[i, j, k],
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Dz_forward(f)[i, j, k]]
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= [(f[i + 1, j, k] - f[i, j, k]) / dx[i],
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(f[i, j + 1, k] - f[i, j, k]) / dy[i],
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(f[i, j, k + 1] - f[i, j, k]) / dz[i]]
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grad_back(f)[i,j,k] = [Dx_back(f)[i, j, k],
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Dy_back(f)[i, j, k],
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Dz_back(f)[i, j, k]]
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= [(f[i, j, k] - f[i - 1, j, k]) / dx[i],
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(f[i, j, k] - f[i, j - 1, k]) / dy[i],
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(f[i, j, k] - f[i, j, k - 1]) / dz[i]]
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The three derivatives in the gradient cause shifts in different
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directions, so the x/y/z components of the resulting "vector" are defined
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at different points: the x-component is shifted in the x-direction,
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y in y, and z in z.
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We call the resulting object a "fore-vector" or "back-vector", depending
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on the direction of the shift. We write it as
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$$ \\tilde{g}_{m,n,p} = \\vec{x} g^x_{m + \\frac{1}{2},n,p} +
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\\vec{y} g^y_{m,n + \\frac{1}{2},p} +
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\\vec{z} g^z_{m,n,p + \\frac{1}{2}} $$
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$$ \\hat{g}_{m,n,p} = \\vec{x} g^x_{m - \\frac{1}{2},n,p} +
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\\vec{y} g^y_{m,n - \\frac{1}{2},p} +
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\\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
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[figure: gradient / fore-vector]
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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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/ : / |
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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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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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Divergences
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-----------
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There are also two divergences,
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$$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p}
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= [\\tilde{\\partial}_x g^x]_{m,n,p} +
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[\\tilde{\\partial}_y g^y]_{m,n,p} +
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[\\tilde{\\partial}_z g^z]_{m,n,p} $$
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$$ d_{n,m,p} = [\\hat{\\nabla} \\cdot \\tilde{g}]_{n,m,p}
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= [\\hat{\\partial}_x g^x]_{m,n,p} +
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[\\hat{\\partial}_y g^y]_{m,n,p} +
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[\\hat{\\partial}_z g^z]_{m,n,p} $$
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or
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[code: divergences]
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div_forward(g)[i,j,k] = Dx_forward(gx)[i, j, k] +
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Dy_forward(gy)[i, j, k] +
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Dz_forward(gz)[i, j, k]
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= (gx[i + 1, j, k] - gx[i, j, k]) / dx[i] +
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(gy[i, j + 1, k] - gy[i, j, k]) / dy[i] +
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(gz[i, j, k + 1] - gz[i, j, k]) / dz[i]
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div_back(g)[i,j,k] = Dx_back(gx)[i, j, k] +
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Dy_back(gy)[i, j, k] +
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Dz_back(gz)[i, j, k]
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= (gx[i, j, k] - gx[i - 1, j, k]) / dx[i] +
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(gy[i, j, k] - gy[i, j - 1, k]) / dy[i] +
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(gz[i, j, k] - gz[i, j, k - 1]) / dz[i]
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where `g = [gx, gy, gz]` is a fore- or back-vector field.
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Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
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is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not at the locations of its components
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\\( (m \\pm \\frac{1}{2},n,p) \\) etc.
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[figure: divergence]
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^^
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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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/ : || // / | The divergence at (m, n, p) (the center
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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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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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Curls
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-----
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The two curls are then
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$$ \\begin{align*}
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\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
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[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
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\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
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&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
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&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
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\\end{align*} $$
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and
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$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
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[\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
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where \\( \\hat{g} \\) and \\( \\tilde{g} \\) are located at \\((m,n,p)\\)
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with components at \\( (m \\pm \\frac{1}{2},n,p) \\) etc.,
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while \\( \\hat{h} \\) and \\( \\tilde{h} \\) are located at \\((m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\)
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with components at \\((m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\) etc.
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[code: curls]
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curl_forward(g)[i,j,k] = [Dy_forward(gz)[i, j, k] - Dz_forward(gy)[i, j, k],
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Dz_forward(gx)[i, j, k] - Dx_forward(gz)[i, j, k],
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Dx_forward(gy)[i, j, k] - Dy_forward(gx)[i, j, k]]
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curl_back(g)[i,j,k] = [Dy_back(gz)[i, j, k] - Dz_back(gy)[i, j, k],
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Dz_back(gx)[i, j, k] - Dx_back(gz)[i, j, k],
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Dx_back(gy)[i, j, k] - Dy_back(gx)[i, j, k]]
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For example, consider the forward curl, at (m, n, p), of a back-vector field `g`, defined
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on a grid containing (m + 1/2, n + 1/2, p + 1/2).
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The curl will be a fore-vector, so its z-component will be defined at (m, n, p + 1/2).
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Take the nearest x- and y-components of `g` in the xy plane where the curl's z-component
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is located; these are
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[curl components]
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(m, n + 1/2, p + 1/2) : x-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
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(m + 1, n + 1/2, p + 1/2) : x-component of back-vector at (m + 3/2, n + 1/2, p + 1/2)
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(m + 1/2, n , p + 1/2) : y-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
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(m + 1/2, n + 1 , p + 1/2) : y-component of back-vector at (m + 1/2, n + 3/2, p + 1/2)
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These four xy-components can be used to form a loop around the curl's z-component; its magnitude and sign
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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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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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Maxwell's Equations
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|
===================
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If we discretize both space (m,n,p) and time (l), Maxwell's equations become
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$$ \\begin{align*}
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\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
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+ \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times \\hat{H}_{l,\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
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+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
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\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
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\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
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\\end{align*} $$
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with
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$$ \\begin{align*}
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\\hat{B}_\\vec{r} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\tilde{D}_\\vec{r} &= \\epsilon_\\vec{r} \\cdot \\tilde{E}_\\vec{r}
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\\end{align*} $$
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where the spatial subscripts are abbreviated as \\( \\vec{r} = (m, n, p) \\) and
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\\( \\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}) \\),
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\\( \\tilde{E} \\) and \\( \\hat{H} \\) are the electric and magnetic fields,
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\\( \\tilde{J} \\) and \\( \\hat{M} \\) are the electric and magnetic current distributions,
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and \\( \\epsilon \\) and \\( \\mu \\) are the dielectric permittivity and magnetic permeability.
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The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
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The time derivatives can be expanded to form the update equations:
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[code: Maxwell's equations]
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H[i, j, k] -= (curl_forward(E[t])[i, j, k] - M[t, i, j, k]) / mu[i, j, k]
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E[i, j, k] += (curl_back( H[t])[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
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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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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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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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/ 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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| / | /
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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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and combining them with charge continuity,
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$$ \\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho = 0 $$
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implying that the discrete Maxwell's equations do not produce spurious charges.
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Wave equation
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-------------
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Taking the backward curl of the \\( \\tilde{\\nabla} \\times \\tilde{E} \\) equation and
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replacing the resulting \\( \\hat{\\nabla} \\times \\hat{H} \\) term using its respective equation,
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and setting \\( \\hat{M} \\) to zero, we can form the discrete wave equation:
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$$
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\\begin{align*}
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\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
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-\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
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+ \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
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\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
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-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
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\\hat{\\nabla} \\times (-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}) \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
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-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
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-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
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\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}})
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+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_\\vec{r} \\cdot \\tilde{E}_{l, \\vec{r}}
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&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
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\\end{align*}
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$$
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Grid description
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================
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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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"""
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|
|
from .types import fdfield_t, vfdfield_t, dx_lists_t, fdfield_updater_t
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|
|
from .vectorization import vec, unvec
|
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|
|
from . import operators, functional, types, vectorization
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|
