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749 lines
36 KiB
Python
749 lines
36 KiB
Python
"""
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Basic discrete calculus for finite difference (fd) simulations.
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Fields, Functions, and Operators
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================================
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Discrete fields are stored in one of two forms:
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- The `fdfield_t` form is a multidimensional `numpy.NDArray`
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+ For a scalar field, this is just `U[m, n, p]`, where `m`, `n`, and `p` are
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discrete indices referring to positions on the x, y, and z axes respectively.
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+ For a vector field, the first index specifies which vector component is accessed:
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`E[:, m, n, p] = [Ex[m, n, p], Ey[m, n, p], Ez[m, n, p]]`.
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- The `vfdfield_t` form is simply a vectorzied (i.e. 1D) version of the `fdfield_t`,
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as obtained by `meanas.fdmath.vectorization.vec` (effectively just `numpy.ravel`)
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Operators which act on fields also come in two forms:
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+ Python functions, created by the functions in `meanas.fdmath.functional`.
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The generated functions act on fields in the `fdfield_t` form.
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+ Linear operators, usually 2D sparse matrices using `scipy.sparse`, created
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by `meanas.fdmath.operators`. These operators act on vectorized fields in the
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`vfdfield_t` form.
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The operations performed should be equivalent: `functional.op(*args)(E)` should be
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equivalent to `unvec(operators.op(*args) @ vec(E), E.shape[1:])`.
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Generally speaking the `field_t` form is easier to work with, but can be harder or less
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efficient to compose (e.g. it is easy to generate a single matrix by multiplying a
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series of other matrices).
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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-vector's) 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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$$ \\begin{aligned}
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\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
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[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
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\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
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&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
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&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
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\\end{aligned} $$
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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+1, 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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$$ \\begin{aligned}
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\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
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- \\hat{M}_{l, \\vec{r} + \\frac{1}{2}} \\\\
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\\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
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+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
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\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
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\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
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\\end{aligned} $$
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with
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$$ \\begin{aligned}
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\\hat{B}_{\\vec{r}} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
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\\tilde{D}_{\\vec{r}} &= \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}}
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\\end{aligned} $$
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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 updates]
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H[i, j, k] -= dt * (curl_forward(E)[i, j, k] + M[t, i, j, k]) / mu[i, j, k]
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E[i, j, k] += dt * (curl_back( H)[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: 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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H1__________Hx0_________H0
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z y /: /|
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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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/ 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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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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and combining them with charge continuity,
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|
$$ \\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho = 0 $$
|
|
implying that the discrete Maxwell's equations do not produce spurious charges.
|
|
|
|
|
|
Wave equation
|
|
-------------
|
|
|
|
Taking the backward curl of the $\\tilde{\\nabla} \\times \\tilde{E}$ equation and
|
|
replacing the resulting $\\hat{\\nabla} \\times \\hat{H}$ term using its respective equation,
|
|
and setting $\\hat{M}$ to zero, we can form the discrete wave equation:
|
|
|
|
$$
|
|
\\begin{aligned}
|
|
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
|
|
-\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
|
|
- \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
|
|
\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
|
|
-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
|
|
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
|
|
\\hat{\\nabla} \\times (-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}) \\\\
|
|
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
|
|
-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
|
|
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
|
|
-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
|
|
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
|
|
+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}}
|
|
&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
|
|
\\end{aligned}
|
|
$$
|
|
|
|
|
|
Frequency domain
|
|
----------------
|
|
|
|
We can substitute in a time-harmonic fields
|
|
|
|
$$
|
|
\\begin{aligned}
|
|
\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
|
|
\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
|
|
\\end{aligned}
|
|
$$
|
|
|
|
resulting in
|
|
|
|
$$
|
|
\\begin{aligned}
|
|
\\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
|
|
\\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
|
|
\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
|
|
\\end{aligned}
|
|
$$
|
|
|
|
This gives the frequency-domain wave equation,
|
|
|
|
$$
|
|
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
|
|
-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = -\\imath \\Omega \\tilde{J}_{\\vec{r}} e^{\\imath \\omega \\Delta_t / 2} \\\\
|
|
$$
|
|
|
|
|
|
Plane waves and Dispersion relation
|
|
------------------------------------
|
|
|
|
With uniform material distribution and no sources
|
|
|
|
$$
|
|
\\begin{aligned}
|
|
\\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
|
|
\\epsilon_{\\vec{r}} &= \\epsilon \\\\
|
|
\\tilde{J}_{\\vec{r}} &= 0 \\\\
|
|
\\end{aligned}
|
|
$$
|
|
|
|
the frequency domain wave equation simplifies to
|
|
|
|
$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} - \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
|
|
|
|
Since $\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0$, we can simplify
|
|
|
|
$$
|
|
\\begin{aligned}
|
|
\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}
|
|
&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
|
|
&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
|
|
&= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}
|
|
\\end{aligned}
|
|
$$
|
|
|
|
and we get
|
|
|
|
$$ \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} + \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
|
|
|
|
We can convert this to three scalar-wave equations of the form
|
|
|
|
$$ (\\tilde{\\nabla}^2 + K^2) \\phi_{\\vec{r}} = 0 $$
|
|
|
|
with $K^2 = \\Omega^2 \\mu \\epsilon$. Now we let
|
|
|
|
$$ \\phi_{\\vec{r}} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)} $$
|
|
|
|
resulting in
|
|
|
|
$$
|
|
\\begin{aligned}
|
|
\\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\
|
|
\\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\
|
|
K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
|
|
\\end{aligned}
|
|
$$
|
|
|
|
with similar expressions for the y and z dimnsions (and $K_y, K_z$).
|
|
|
|
This implies
|
|
|
|
$$
|
|
\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\
|
|
K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
|
|
$$
|
|
|
|
where $c = \\sqrt{\\mu \\epsilon}$.
|
|
|
|
Assuming real $(k_x, k_y, k_z), \\omega$ will be real only if
|
|
|
|
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
|
|
|
|
If $\\Delta_x = \\Delta_y = \\Delta_z$, this simplifies to $c \\Delta_t < \\Delta_x / \\sqrt{3}$.
|
|
This last form can be interpreted as enforcing causality; the distance that light
|
|
travels in one timestep (i.e., $c \\Delta_t$) must be less than the diagonal
|
|
of the smallest cell ( $\\Delta_x / \\sqrt{3}$ when on a uniform cubic grid).
|
|
|
|
|
|
Grid description
|
|
================
|
|
|
|
As described in the section on scalar discrete derivatives above, cell widths
|
|
(`dx[i]`, `dy[j]`, `dz[k]`) along each axis can be arbitrary and independently
|
|
defined. Moreover, all field components are actually defined at "derived" or "dual"
|
|
positions, in-between the "base" grid points on one or more axes.
|
|
|
|
To get a better sense of how this works, let's start by drawing a grid with uniform
|
|
`dy` and `dz` and nonuniform `dx`. We will only draw one cell in the y and z dimensions
|
|
to make the illustration simpler; we need at least two cells in the x dimension to
|
|
demonstrate how nonuniform `dx` affects the various components.
|
|
|
|
Place the E fore-vectors at integer indices $r = (m, n, p)$ and the H back-vectors
|
|
at fractional indices $r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2},
|
|
p + \\frac{1}{2})$. Remember that these are indices and not coordinates; they can
|
|
correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths.
|
|
|
|
Draw lines to denote the planes on which the H components and back-vectors are defined.
|
|
For simplicity, don't draw the equivalent planes for the E components and fore-vectors,
|
|
except as necessary to show their locations -- it's easiest to just connect them to their
|
|
associated H-equivalents.
|
|
|
|
The result looks something like this:
|
|
|
|
[figure: Component centers]
|
|
p=
|
|
[H]__________Hx___________[H]_____Hx______[H] __ +1/2
|
|
z y /: /: /: /: /| | |
|
|
|/_x / : / : / : / : / | | |
|
|
/ : / : / : / : / | | |
|
|
Hy : Ez...........Hy : Ez......Hy | | |
|
|
/: : : : /: : : : /| | | |
|
|
/ : Hz : Ey....../.:..Hz : Ey./.|..Hz __ 0 | dz[0]
|
|
/ : /: : / / : /: : / / | /| | |
|
|
/_________________________/_______________/ | / | | |
|
|
| :/ : :/ | :/ : :/ | |/ | | |
|
|
| Ex : [E].......|..Ex : [E]..|..Ex | | |
|
|
| : | : | | | |
|
|
| [H]..........Hx....|......[H].....H|x.....[H] __ --------- (n=+1/2, p=-1/2)
|
|
| / | / | / / /
|
|
Hz / Hz / Hz / / /
|
|
| / | / | / / /
|
|
| Hy | Hy | Hy __ 0 / dy[0]
|
|
| / | / | / / /
|
|
| / | / | / / /
|
|
|/ |/ |/ / /
|
|
[H]__________Hx___________[H]_____Hx______[H] __ -1/2 /
|
|
=n
|
|
|------------|------------|-------|-------|
|
|
-1/2 0 +1/2 +1 +3/2 = m
|
|
|
|
------------------------- ----------------
|
|
dx[0] dx[1]
|
|
|
|
Part of a nonuniform "base grid", with labels specifying
|
|
positions of the various field components. [E] fore-vectors
|
|
are at the cell centers, and [H] back-vectors are at the
|
|
vertices. H components along the near (-y) top (+z) edge
|
|
have been omitted to make the insides of the cubes easier
|
|
to visualize.
|
|
|
|
The above figure shows where all the components are located; however, it is also useful to show
|
|
what volumes those components correspond to. Consider the Ex component at `m = +1/2`: it is
|
|
shifted in the x-direction by a half-cell from the E fore-vector at `m = 0` (labeled `[E]`
|
|
in the figure). It corresponds to a volume between `m = 0` and `m = +1` (the other
|
|
dimensions are not shifted, i.e. they are still bounded by `n, p = +-1/2`). (See figure
|
|
below). Since `m` is an index and not an x-coordinate, the Ex component is not necessarily
|
|
at the center of the volume it represents, and the x-length of its volume is the derived
|
|
quantity `dx'[0] = (dx[0] + dx[1]) / 2` rather than the base `dx`.
|
|
(See also `Scalar derivatives and cell shifts`).
|
|
|
|
[figure: Ex volumes]
|
|
p=
|
|
<_________________________________________> __ +1/2
|
|
z y << /: / /: >> | |
|
|
|/_x < < / : / / : > > | |
|
|
< < / : / / : > > | |
|
|
< < / : / / : > > | |
|
|
<: < / : : / : >: > | |
|
|
< : < / : : / : > : > __ 0 | dz[0]
|
|
< : < / : : / :> : > | |
|
|
<____________/____________________/_______> : > | |
|
|
< : < | : : | > : > | |
|
|
< Ex < | : Ex | > Ex > | |
|
|
< : < | : : | > : > | |
|
|
< : <....|.......:........:...|.......>...:...> __ --------- (n=+1/2, p=-1/2)
|
|
< : < | / : /| /> : > / /
|
|
< : < | / : / | / > : > / /
|
|
< :< | / :/ | / > :> / /
|
|
< < | / : | / > > _ 0 / dy[0]
|
|
< < | / | / > > / /
|
|
< < | / | / > > / /
|
|
<< |/ |/ >> / /
|
|
<____________|____________________|_______> __ -1/2 /
|
|
=n
|
|
|------------|------------|-------|-------|
|
|
-1/2 0 +1/2 +1 +3/2 = m
|
|
|
|
~------------ -------------------- -------~
|
|
dx'[-1] dx'[0] dx'[1]
|
|
|
|
The Ex values are positioned on the x-faces of the base
|
|
grid. They represent the Ex field in volumes shifted by
|
|
a half-cell in the x-dimension, as shown here. Only the
|
|
center cell (with width dx'[0]) is fully shown; the
|
|
other two are truncated (shown using >< markers).
|
|
|
|
Note that the Ex positions are the in the same positions
|
|
as the previous figure; only the cell boundaries have moved.
|
|
Also note that the points at which Ex is defined are not
|
|
necessarily centered in the volumes they represent; non-
|
|
uniform cell sizes result in off-center volumes like the
|
|
center cell here.
|
|
|
|
The next figure shows the volumes corresponding to the Hy components, which
|
|
are shifted in two dimensions (x and z) compared to the base grid.
|
|
|
|
[figure: Hy volumes]
|
|
p=
|
|
z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm __ +1/2 s
|
|
|/_x << m: m: >> | |
|
|
< < m : m : > > | | dz'[1]
|
|
< < m : m : > > | |
|
|
Hy........... m........Hy...........m......Hy > | |
|
|
< < m : m : > > | |
|
|
< ______ m_____:_______________m_____:_>______ __ 0
|
|
< < m /: m / > > | |
|
|
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm > | |
|
|
< < | / : | / > > | | dz'[0]
|
|
< < | / : | / > > | |
|
|
< < | / : | / > > | |
|
|
< wwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwwww __ s
|
|
< < |/ w |/ w> > / /
|
|
_____________|_____________________|________ > / /
|
|
< < | w | w > > / /
|
|
< Hy........|...w........Hy.......|...w...>..Hy _ 0 / dy[0]
|
|
< < | w | w > > / /
|
|
<< | w | w > > / /
|
|
< |w |w >> / /
|
|
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww __ -1/2 /
|
|
|
|
|------------|------------|--------|-------|
|
|
-1/2 0 +1/2 +1 +3/2 = m
|
|
|
|
~------------ --------------------- -------~
|
|
dx'[-1] dx'[0] dx'[1]
|
|
|
|
The Hy values are positioned on the y-edges of the base
|
|
grid. Again here, the 'Hy' labels represent the same points
|
|
as in the basic grid figure above; the edges have shifted
|
|
by a half-cell along the x- and z-axes.
|
|
|
|
The grid lines _|:/ are edges of the area represented by
|
|
each Hy value, and the lines drawn using <m>.w represent
|
|
edges where a cell's faces extend beyond the drawn area
|
|
(i.e. where the drawing is truncated in the x- or z-
|
|
directions).
|
|
|
|
|
|
Datastructure: dx_lists_t
|
|
-------------------
|
|
|
|
In this documentation, the E fore-vectors are placed on the base grid. An
|
|
equivalent formulation could place the H back-vectors on the base grid instead.
|
|
However, in the case of a non-uniform grid, the operation to get from the "base"
|
|
cell widths to "derived" ones is not its own inverse.
|
|
|
|
The base grid's cell sizes could be fully described by a list of three 1D arrays,
|
|
specifying the cell widths along all three axes:
|
|
|
|
[dx, dy, dz] = [[dx[0], dx[1], ...], [dy[0], ...], [dz[0], ...]]
|
|
|
|
Note that this is a list-of-arrays rather than a 2D array, as the simulation domain
|
|
may have a different number of cells along each axis.
|
|
|
|
Knowing the base grid's cell widths and the boundary conditions (periodic unless
|
|
otherwise noted) is enough information to calculate the cell widths `dx'`, `dy'`,
|
|
and `dz'` for the derived grids.
|
|
|
|
However, since most operations are trivially generalized to allow either E or H
|
|
to be defined on the base grid, they are written to take the a full set of base
|
|
and derived cell widths, distinguished by which field they apply to rather than
|
|
their "base" or "derived" status. This removes the need for each function to
|
|
generate the derived widths, and makes the "base" vs "derived" distinction
|
|
unnecessary in the code.
|
|
|
|
The resulting data structure containing all the cell widths takes the form of a
|
|
list-of-lists-of-arrays. The first list-of-arrays provides the cell widths for
|
|
the E-field fore-vectors, while the second list-of-arrays does the same for the
|
|
H-field back-vectors:
|
|
|
|
[[[dx_e[0], dx_e[1], ...], [dy_e[0], ...], [dz_e[0], ...]],
|
|
[[dx_h[0], dx_h[1], ...], [dy_h[0], ...], [dz_h[0], ...]]]
|
|
|
|
where `dx_e[0]` is the x-width of the `m=0` cells, as used when calculating dE/dx,
|
|
and `dy_h[0]` is the y-width of the `n=0` cells, as used when calculating dH/dy, etc.
|
|
|
|
|
|
Permittivity and Permeability
|
|
=============================
|
|
|
|
Since each vector component of E and H is defined in a different location and represents
|
|
a different volume, the value of the spatially-discrete `epsilon` and `mu` can also be
|
|
different for all three field components, even when representing a simple planar interface
|
|
between two isotropic materials.
|
|
|
|
As a result, `epsilon` and `mu` are taken to have the same dimensions as the field, and
|
|
composed of the three diagonal tensor components:
|
|
|
|
[equations: epsilon_and_mu]
|
|
epsilon = [epsilon_xx, epsilon_yy, epsilon_zz]
|
|
mu = [mu_xx, mu_yy, mu_zz]
|
|
|
|
or
|
|
|
|
$$
|
|
\\epsilon = \\begin{bmatrix} \\epsilon_{xx} & 0 & 0 \\\\
|
|
0 & \\epsilon_{yy} & 0 \\\\
|
|
0 & 0 & \\epsilon_{zz} \\end{bmatrix}
|
|
$$
|
|
$$
|
|
\\mu = \\begin{bmatrix} \\mu_{xx} & 0 & 0 \\\\
|
|
0 & \\mu_{yy} & 0 \\\\
|
|
0 & 0 & \\mu_{zz} \\end{bmatrix}
|
|
$$
|
|
|
|
where the off-diagonal terms (e.g. `epsilon_xy`) are assumed to be zero.
|
|
|
|
High-accuracy volumetric integration of shapes on multiple grids can be performed
|
|
by the [gridlock](https://mpxd.net/code/jan/gridlock) module.
|
|
|
|
The values of the vacuum permittivity and permability effectively become scaling
|
|
factors that appear in several locations (e.g. between the E and H fields). In
|
|
order to limit floating-point inaccuracy and simplify calculations, they are often
|
|
set to 1 and relative permittivities and permeabilities are used in their places;
|
|
the true values can be multiplied back in after the simulation is complete if non-
|
|
normalized results are needed.
|
|
"""
|
|
|
|
from .types import fdfield_t, vfdfield_t, cfdfield_t, vcfdfield_t, dx_lists_t, dx_lists_mut
|
|
from .types import fdfield_updater_t, cfdfield_updater_t
|
|
from .vectorization import vec, unvec
|
|
from . import operators, functional, types, vectorization
|
|
|