Source code for gemseo_umdo.use_cases.heat_equation.model

# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
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r"""The heat equation model.

This model solves the 1D transient equation, a.k.a. heat equation.
It describes the temperature evolution $u$ in a $L$-length rod
from the initial time 0 to the final time $T$
with a thermal diffusivity $\nu(\mathbf{X})$
depending on a random vector $\mathbf{X}$.
The heat equation is

$$\frac{\partial u(x,t;\mathbf{X})}{\partial t}
   - \nu(\mathbf{X})\frac{\partial^2 u(x,t;\mathbf{X})}{\partial x^2} = 0$$

with the boundary condition $u(0,t;\mathbf{X})=u(L,t;\mathbf{X})=0$
where $x\in\mathcal{D}=[0,L]$ and $t\in[0,T]$.

To obtain an analytical solution,
Geraci et al. (2015) chose $L=1$ and the uncertain initial condition:

$$u(x,0;\mathbf{X}) =
   \mathcal{G}(\mathbf{X})\mathcal{F}_1(x)
   +\mathcal{I}(\mathbf{X})\mathcal{F}_2(x)
$$

where

- $\mathcal{F}_1(x)=\sin(\pi x)$,
- $\mathcal{F}_2(x)=\sin(2\pi x)+\sin(3\pi x)
          +50\left(\sin(9\pi x)+\sin(21\pi x)\right)$,
- $\mathcal{I}(\mathbf{X})=3.5
         \left(\sin(X_1)+7\sin^2(X_2)+0.1X_3^4\sin(X_1)\right)$,
- $\mathcal{G}(\mathbf{X})=50\prod_{i=5}^7(4|X_i|-1)$.

This uncertainty on the initial condition is modelled
by the random variables $X_1,\ldots X_7$ that are independent and distributed as:

- $X_i\sim\mathcal{U}(-\pi,\pi)$, for $i\in\{1,2,3\}$,
- $\nu(\mathbf{X})=X_4\sim\mathcal{U}(\nu_{\min},\nu_{\max})$,
- $X_i\sim\mathcal{U}(-1,1)$, for $i\in\{5,6,7\}$.

Then,
Geraci et al. (2015) consider the integral of the temperature along the rod

$$\mathcal{M}(\mathbf{X}) = \int_{\mathcal{D}}u(x,T;\mathbf{X})dx$$

and are interested in the estimation
of its
[HeatEquationConfiguration][gemseo_umdo.use_cases.heat_equation.configuration.HeatEquationConfiguration.expectation]:

$$\mathbb{E}[\mathcal{M}(\mathbf{X})] = 50H_1+\frac{49}{4}(H_3+50H_9+50H_{21})$$

where
$H_k=\frac{2}{k^3\pi^3T}
\frac{\exp(-\nu_{\min}k^2\pi^2T)-\exp(-\nu_{\max}k^2\pi^2T)}{\nu_{\max}-\nu_{\min}}$.

The [HeatEquationModel][gemseo_umdo.use_cases.heat_equation.model.HeatEquationModel]
computes the temperature at final time
from instances of the random variables ``"X_1"``, ..., ``"X_7"``
defined over the
[HeatEquationUncertainSpace]
[gemseo_umdo.use_cases.heat_equation.uncertain_space.HeatEquationUncertainSpace].
The temperature ``"u_mesh"`` is computed at each mesh node
while the temperature ``"u"`` is an integral over the rod.

See Also:
    Geraci et al., A multifidelity control variate approach
    for the multilevel Monte Carlo technique, Center for Turbulence Research,
    Annual Research Briefs, 2015.
"""
from __future__ import annotations

import numpy as np
from numpy import abs
from numpy import array
from numpy import linspace
from numpy import meshgrid
from numpy import newaxis
from numpy import pi
from numpy import sin
from numpy import trapz
from numpy.typing import NDArray

from gemseo_umdo.use_cases.heat_equation.configuration import (
    HeatEquationConfiguration,
)


[docs]class HeatEquationModel: """The discipline computing the temperature averaged over the rod at final time. This discipline can also compute a first-order polynomial centered at the mean input value. """ configuration: HeatEquationConfiguration """The configuration of the heat equation problem.""" taylor_mean: float """The expectation of the output of the first-order Taylor polynomial.""" def __init__( self, mesh_size: int = 100, n_modes: int = 21, final_time: float = 0.5, nu_bounds: tuple[float, float] = (0.001, 0.009), rod_length: float = 1.0, ) -> None: """ Args: mesh_size: The number of equispaced spatial nodes. n_modes: The number of modes of the truncated Fourier expansion. final_time: The time of interest. nu_bounds: The bounds of the thermal diffusivity. rod_length: The length of the rod. """ # noqa: D205 D212 D415 self.configuration = HeatEquationConfiguration( mesh_size, n_modes, final_time, nu_bounds, rod_length ) self.__nu_delta = nu_bounds[1] - nu_bounds[0] self.__modes = linspace(1, n_modes, n_modes) xx, nn = meshgrid(self.configuration.mesh, self.__modes, copy=False) self.__sinus = np.sin(xx * nn * pi)[:, :, newaxis] self.__default_input_value = array([0.0, 0.0, 0.0, 0.005, 0.0, 0.0, 0.0]) pi_mesh = pi * self.configuration.mesh self.__F1 = sin(pi_mesh) # noqa: N806 self.__F2 = ( # noqa: N806 sin(2 * pi_mesh) + sin(3 * pi_mesh) + 50 * (sin(9 * pi_mesh) + sin(21 * pi_mesh)) ) self.__term1 = self.__term2 = self.__term3 = self.__f_at_mu_X = 0 self.__compute_taylor_materials() self.taylor_mean = self.__f_at_mu_X + 600 * self.__term1 def __compute_initial_temperature( self, X: NDArray[float] # noqa: N803 ) -> NDArray[float]: """Compute the initial temperature for each mesh nodes. From Geraci et al., 2015 (Equation 5.2). Args: X: The input samples shaped as ``(sample_size, input_dimension)``. Returns: The initial temperature for each mesh nodes. """ G = 50 * (4 * abs(X[:, 4:7]) - 1).T.prod(0) # noqa: N806 I = 3.5 * ( # noqa: N806, E741 sin(X[:, 0]) + 7 * sin(X[:, 1]) ** 2 + 0.1 * X[:, 2] ** 4 * sin(X[:, 0]) ) return ( self.__F1[:, newaxis] * G[newaxis, :] + self.__F2[:, newaxis] * I[newaxis, :] ) def __call__( self, input_samples: NDArray[float] | None = None, batch_size: int = 50000 ) -> tuple[NDArray[float] | float, NDArray[float]]: """Compute the temperature. Args: input_samples: The input samples shaped as ``(sample_size, input_dimension)`` or ``(input_dimension, )``. batch_size: The maximum number of samples per batch. Returns: - The integrated temperature shaped as ``(sample_size, )`` or ``()``. - The temperature at the different nodes shaped as ``(sample_size, n_nodes)`` or ``(n_nodes, )``. """ if input_samples is None: input_samples = self.__default_input_value if input_samples.ndim == 1: is_input_samples_1d = True input_samples = input_samples[newaxis, :] else: is_input_samples_1d = False n_samples = len(input_samples) if n_samples <= batch_size: u, u_mesh = self.__evaluate(input_samples) else: u = np.zeros(n_samples) u_mesh = np.zeros((n_samples, self.configuration.mesh_size)) i_start = 0 while n_samples > 0: n_samples_batch = min(batch_size, n_samples) indices = slice(i_start, i_start + n_samples_batch) u[indices], u_mesh[indices] = self.__evaluate(input_samples[indices]) i_start += n_samples_batch n_samples -= n_samples_batch if is_input_samples_1d: return u[0], u_mesh[0] else: return u, u_mesh def __evaluate( self, X: NDArray[float] # noqa: N803 ) -> tuple[NDArray[float], NDArray[float]]: """Compute the temperature. From Geraci et al., 2015 (Equation 5.4). Args: X: The input samples shaped as ``(sample_size, input_dimension)``. Returns: The integrated temperature shaped as ``(sample_size, )``, the temperature at the different nodes shaped as ``(sample_size, n_nodes)``. """ term = np.trapz( self.__sinus * self.__compute_initial_temperature(X)[newaxis, :, :], x=self.configuration.mesh, axis=1, ) * np.exp( -X[:, 3][newaxis, :] * (self.__modes[:, newaxis] * pi) ** 2 * self.configuration.final_time ) u_mesh = 2 * np.sum(self.__sinus * term[:, newaxis, :], axis=0) return trapz(u_mesh, x=self.configuration.mesh, axis=0), u_mesh.T def __compute_taylor_materials(self) -> None: """Compute the materials of the first-order Taylor polynomial.""" mu_X = self.__default_input_value # noqa: N806 x = self.configuration.mesh n = self.__modes sn = self.__sinus u0_at_mu_X = self.__compute_initial_temperature( # noqa: N806 mu_X[newaxis, :] ).reshape( -1 ) # -> (nx, 1) => (nx,) snu0_at_mu_X = sn[:, :, 0] * u0_at_mu_X[None, :] # -> (n_modes, nx)# noqa: N806 snF1 = sn[:, :, 0] * self.__F1[None, :] # noqa: N806 -> (n_modes, nx) snF2 = sn[:, :, 0] * self.__F2[None, :] # noqa: N806 -> (n_modes, nx) sn_quad = np.trapz(sn, x=x, axis=1).ravel() # -> (n_modes,) snF1_quad = np.trapz(snF1, x=x, axis=1) # noqa: N806 -> (n_modes,) snF2_quad = np.trapz(snF2, x=x, axis=1) # noqa: N806 -> (n_modes,) A_n_at_mu_X_quad = 2 * np.trapz(snu0_at_mu_X, x=x, axis=1) # noqa: N806 # -> (n_modes,) B_n_at_mu_X_quad = ( # noqa: N806 np.exp(-mu_X[3] * (n * np.pi) ** 2 * 0.5) * sn_quad ) # -> (n_modes,) self.__term1 = np.sum(B_n_at_mu_X_quad * snF1_quad) # (scalar) self.__term2 = np.sum(B_n_at_mu_X_quad * snF2_quad) # (scalar) self.__term3 = np.sum( # (scalar) A_n_at_mu_X_quad * sn_quad * n**2 * np.exp(-mu_X[3] * n**2 * np.pi**2 * 0.5) ) self.__f_at_mu_X = self(mu_X)[0] # -> (1,) => (scalar)
[docs] def compute_taylor(self, input_samples: NDArray[float]) -> NDArray[float]: """Evaluate the first-order Taylor polynomial. Args: input_samples: The input samples shaped as ``(sample_size, input_dimension)`` or ``(input_dimension, )``. Returns: The output samples of the first-order Taylor polynomial shaped as ``(sample_size, n_nodes)`` or ``(n_nodes, )``. """ X = input_samples # noqa: N806 mu_X = self.__default_input_value # noqa: N806 return self.__f_at_mu_X + ( 7 * X[..., [0]] * self.__term2 - (X[..., [3]] - mu_X[3]) * np.pi**2 * 0.5 * self.__term3 + 400 * self.__term1 * (np.abs(X[..., [4]]) + np.abs(X[..., [5]]) + np.abs(X[..., [6]])) )