Source code for gemseo.problems.dataset.burgers

# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
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#    INITIAL AUTHORS - initial API and implementation and/or initial
#                           documentation
#        :author: Syver Doving Agdestein
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r"""
Burgers dataset
===============

This :class:`.Dataset` contains solutions to the Burgers' equation with
periodic boundary conditions on the interval :math:`[0, 2\pi]` for different
time steps:

.. math::

   u_t + u u_x = \nu u_{xx},

An analytical expression can be obtained for the solution, using the Cole-Hopf
transform:

.. math::

   u(t, x) = - 2 \nu \frac{\phi'}{\phi},

where :math:`\phi` is solution to the heat equation
:math:`\phi_t = \nu \phi_{xx}`.

This :class:`.Dataset` is based on a full-factorial
design of experiments. Each sample corresponds to a given time step :math:`t`,
while each feature corresponds to a given spatial point :math:`x`.

`More information about Burgers' equation
<https://en.wikipedia.org/wiki/Burgers%27_equation>`_
"""
from __future__ import annotations

from numpy import exp
from numpy import hstack
from numpy import linspace
from numpy import pi
from numpy import square

from gemseo.core.dataset import Dataset
from gemseo.core.discipline import MDODiscipline


[docs]class BurgersDiscipline(MDODiscipline): def __init__(self): super().__init__() self.input_grammar.initialize_from_data_names(["x", "z"]) self.output_grammar.initialize_from_data_names(["f", "g"])
[docs]class BurgersDataset(Dataset): """Burgers dataset parametrization.""" def __init__( self, name: str = "Burgers", by_group: bool = True, n_samples: int = 30, n_x: int = 501, fluid_viscosity: float = 0.1, categorize: bool = True, ) -> None: """ Args: name: The name of the dataset. by_group: Whether to store the data by group. Otherwise, store them by variables. n_samples: The number of samples. n_x: The number of spatial points. fluid_viscosity: The fluid viscosity. categorize: Whether to distinguish between the different groups of variables. """ super().__init__(name, by_group) time = linspace(0, 2, n_samples)[:, None] space = linspace(0, 2 * pi, n_x)[None, :] visc = fluid_viscosity alpha = space - 4 * time alpha_2 = square(alpha) beta = 4 * visc * (time + 1) gamma = space - 4 * time - 2 * pi gamma_2 = square(gamma) phi = exp(-alpha_2 / beta) + exp(-gamma_2 / beta) phi_deriv = -2 * alpha / beta * exp(-alpha_2 / beta) phi_deriv -= 2 * gamma / beta * exp(-gamma_2 / beta) u_t = -2 * visc / phi * phi_deriv if categorize: groups = {"t": Dataset.INPUT_GROUP, "u_t": Dataset.OUTPUT_GROUP} else: groups = None data = hstack([time, u_t]) self.set_from_array(data, ["t", "u_t"], {"t": 1, "u_t": n_x}, groups=groups) self.set_metadata("x", [[node] for node in space[0]]) self.set_metadata("nu", visc)