Source code for gemseo.problems.dataset.burgers

# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry,
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# modify it under the terms of the GNU Lesser General Public
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# Contributors:
#    INITIAL AUTHORS - initial API and implementation and/or initial
#                           documentation
#        :author: Syver Doving Agdestein
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

.. 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
from __future__ import division, unicode_literals

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

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

[docs]class BurgersDiscipline(MDODiscipline): def __init__(self): super(BurgersDiscipline, self).__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="Burgers", by_group=True, n_samples=30, n_x=501, fluid_viscosity=0.1, categorize=True, ): """Constructor. :param str name: name of the dataset. :param bool by_group: if True, store the data by group. Otherwise, store them by variables. Default: True. :param int n_samples: number of samples. Default: 30. :param int n_x: number of spatial points. Default: 501. :param float fluid_viscosity: fluid viscosity. Default: 0.1. :param bool categorize: distinguish between the different groups of variables. Default: True. :parma bool opt_naming: use an optimization naming. Default: True. """ super(BurgersDataset, self).__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)