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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,160 @@ | ||
| import numpy as np | ||
| from scipy.interpolate import PchipInterpolator | ||
|
|
||
| def inverse(o, samples): | ||
| r""" | ||
| Extract `D` from samples of a solution. | ||
| Given a function :math:`\theta` of `r` and `t`, and scalars | ||
| :math:`\theta_i`, :math:`\theta_b` and :math:`o_b`, finds a positive | ||
| function `D` of the values of :math:`\theta` such that: | ||
| .. math:: \begin{cases} \dfrac{\partial\theta}{\partial t} = | ||
| \dfrac{\partial}{\partial r}\left(D(\theta)\dfrac{\partial\theta} | ||
| {\partial r}\right) & r>r_b(t),t>0\\ | ||
| \theta(r, 0) = \theta_i & r>0 \\ | ||
| \theta(r_b(t), t) = \theta_b & t>0 \\ | ||
| r_b(t) = o_b\sqrt t | ||
| \end{cases} | ||
| :math:`\theta` is taken as its values on a discrete set of points expressed | ||
| in terms of the Boltzmann variable. Problems in radial coordinates are not | ||
| supported. | ||
| Parameters | ||
| ---------- | ||
| o : numpy.array_like, shape (n,) | ||
| Points where :math:`\theta` is known, expressed in terms of the | ||
| Boltzmann variable. Must be strictly increasing. | ||
| samples : numpy.array_like, shape (n,) | ||
| Values of :math:`\theta` at `o`. Must be monotonic (either | ||
| non-increasing or non-decreasing) and ``samples[-1]`` must be the | ||
| initial value :math:`\theta_i`. | ||
| Returns | ||
| ------- | ||
| D : callable | ||
| Function to evaluate :math:`D` and its derivatives: | ||
| * ``D(theta)`` evaluates and returns :math:`D` at ``theta`` | ||
| * ``D(theta, 1)`` returns both the value of :math:`D` and its | ||
| first derivative at ``theta`` | ||
| * ``D(theta, 2)`` returns the value of :math:`D`, its first | ||
| derivative, and its second derivative at ``theta`` | ||
| In all cases, the argument ``theta`` may be a single float or a NumPy | ||
| array. | ||
| :math:`D` is guaranteed to be continuous; however, its derivatives are | ||
| not. | ||
| See also | ||
| -------- | ||
| o | ||
| Notes | ||
| ----- | ||
| An `o` function of :math:`\theta` is constructed by interpolating the input | ||
| data with a PCHIP monotonic cubic spline. The returned `D` uses the spline | ||
| to evaluate the expressions that result from solving the | ||
| Boltzmann-transformed equation for :math:`D`. | ||
| References | ||
| ---------- | ||
| [1] GERLERO, G. S.; BERLI, C. L. A.; KLER, P. A. Open-source | ||
| high-performance software packages for direct and inverse solving of | ||
| horizontal capillary flow. Capillarity, 2023, vol. 6, no. 2, pp. 31-40. | ||
| [2] BRUCE, R. R.; KLUTE, A. The measurement of soil moisture diffusivity. | ||
| Soil Science Society of America Journal, 1956, vol. 20, no. 4, pp. 458-462. | ||
| """ | ||
|
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| o = np.asarray(o) | ||
|
|
||
| if not np.all(np.diff(o) > 0): | ||
| raise ValueError("o must be strictly increasing") | ||
|
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| samples = np.asarray(samples) | ||
|
|
||
| dsamples = np.diff(samples) | ||
| if not(np.all(dsamples >= -1e-12) or np.all(dsamples <= 1e-12)): | ||
| raise ValueError("samples must be monotonic") | ||
|
|
||
| i = samples[-1] | ||
|
|
||
| samples, indices = np.unique(samples, return_index=True) | ||
| o = o[indices] | ||
|
|
||
| o_func = PchipInterpolator(x=samples, y=o, extrapolate=False) | ||
|
|
||
| o_antiderivative_func = o_func.antiderivative() | ||
| o_antiderivative_i = o_antiderivative_func(i) | ||
|
|
||
| o_funcs = [o_func.derivative(n) for n in range(4)] | ||
|
|
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| def D(theta, derivatives=0): | ||
|
|
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| Iodtheta = o_antiderivative_func(theta) - o_antiderivative_i | ||
|
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| do_dtheta = o_funcs[1](theta) | ||
|
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| D = -(do_dtheta*Iodtheta)/2 | ||
|
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| if derivatives == 0: return D | ||
|
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| o = o_funcs[0](theta) | ||
| d2o_dtheta2 = o_funcs[2](theta) | ||
|
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| dD_dtheta = -(d2o_dtheta2*Iodtheta + do_dtheta*o)/2 | ||
|
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| if derivatives == 1: return D, dD_dtheta | ||
|
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| d3o_dtheta3 = o_funcs[3](theta) | ||
|
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| d2D_dtheta2 = -(d3o_dtheta3*Iodtheta + 2*d2o_dtheta2*o + do_dtheta**2)/2 | ||
|
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| if derivatives == 2: return D, dD_dtheta, d2D_dtheta2 | ||
|
|
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| raise ValueError("derivatives must be 0, 1, or 2") | ||
|
|
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| return D | ||
|
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||
|
|
||
| def sorptivity(o, samples, *, i=None, b=None, ob=0): | ||
| r""" | ||
| Extract the sorptivity from samples of a solution. | ||
| Parameters | ||
| ---------- | ||
| o : numpy.array_like, shape (n,) | ||
| Points where :math:`\theta` is known, expressed in terms of the | ||
| Boltzmann variable. | ||
| samples : numpy.array_like, shape (n,) | ||
| Values of :math:`\theta` at `o`. | ||
| i : None or float, optional | ||
| Initial value :math:`\theta_i`. If not given, it is taken as | ||
| ``samples[-1]``. | ||
| b : None or float, optional | ||
| Boundary value :math:`\theta_b`. If not given, it is taken as | ||
| ``samples[0]``. | ||
| Returns | ||
| ------- | ||
| S : float | ||
| Sorptivity. | ||
| References | ||
| ---------- | ||
| [1] PHILIP, J. R. The theory of infiltration: 4. Sorptivity and | ||
| algebraic infiltration equations. Soil Science, 1957, vol. 84, no. 3, | ||
| pp. 257-264. | ||
| """ | ||
| o = np.insert(o, 0, ob) | ||
| if b is not None: | ||
| samples = np.insert(samples, 0, b) | ||
|
|
||
| if i is None: | ||
| i = samples[-1] | ||
|
|
||
| return np.trapz(samples - i, o) |
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