# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
# License version 3 as published by the Free Software Foundation.
#
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# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# Lesser General Public License for more details.
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# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
# Contributors:
# INITIAL AUTHORS - API and implementation and/or documentation
# :author: Charlie Vanaret
# Francois Gallard
# OTHER AUTHORS - MACROSCOPIC CHANGES
r"""The disciplines for the MDO problem proposed by Sellar et al. in
Sellar, R., Batill, S., & Renaud, J. (1996).
Response surface based, concurrent subspace optimization
for multidisciplinary system design.
In 34th aerospace sciences meeting and exhibit (p. 714).
The MDO problem is written as follows:
.. math::
\begin{aligned}
\text{minimize the objective function }&obj=x_{local}^2 + x_{shared,2}
+y_1^2+e^{-y_2} \\
\text{with respect to the design variables }&x_{shared},\,x_{local} \\
\text{subject to the general constraints }
& c_1 \leq 0\\
& c_2 \leq 0\\
\text{subject to the bound constraints }
& -10 \leq x_{shared,1} \leq 10\\
& 0 \leq x_{shared,2} \leq 10\\
& 0 \leq x_{local} \leq 10.
\end{aligned}
where the coupling variables are
.. math::
\text{Discipline 1: } y_1 = \sqrt{x_{shared,1}^2 + x_{shared,2} +
x_{local} - 0.2\,y_2},
and
.. math::
\text{Discipline 2: }y_2 = |y_1| + x_{shared,1} + x_{shared,2}.
and where the general constraints are
.. math::
c_1 = 3.16 - y_1^2
c_2 = y_2 - 24
This module implements three disciplines to compute the different coupling variables,
constraints and objective:
- :class:`.Sellar1`:
this :class:`.MDODiscipline` computes :math:`y_1`
from :math:`y_2`, :math:`x_{shared,1}`, :math:`x_{shared,2}` and :math:`x_{local}`.
- :class:`.Sellar2`:
this :class:`.MDODiscipline` computes :math:`y_2`
from :math:`y_1`, :math:`x_{shared,1}` and :math:`x_{shared,2}`.
- :class:`.SellarSystem`:
this :class:`.MDODiscipline` computes both objective and constraints
from :math:`y_1`, :math:`y_2`, :math:`x_{local}` and :math:`x_{shared,2}`.
"""
from __future__ import division, unicode_literals
from cmath import exp, sqrt
from typing import Dict, Iterable, List, Optional
from numpy import array, atleast_2d, complex128, ndarray, ones, zeros
from gemseo.core.discipline import MDODiscipline
Y_1 = "y_1"
Y_2 = "y_2"
X_SHARED = "x_shared"
X_LOCAL = "x_local"
OBJ = "obj"
C_1 = "c_1"
C_2 = "c_2"
R_1 = "r_1"
R_2 = "r_2"
[docs]class SellarSystem(MDODiscipline):
"""The discipline to compute the objective and constraints of the Sellar problem."""
def __init__(self): # type: (...) -> None
super(SellarSystem, self).__init__(auto_detect_grammar_files=True)
self.default_inputs = get_inputs()
self.re_exec_policy = self.RE_EXECUTE_DONE_POLICY
def _run(self): # type: (...) -> None
x_local, x_shared, y_1, y_2 = self.get_inputs_by_name(
[X_LOCAL, X_SHARED, Y_1, Y_2]
)
obj = array([self.compute_obj(x_local, x_shared, y_1, y_2)], dtype=complex128)
c_1 = array([self.compute_c_1(y_1)], dtype=complex128)
c_2 = array([self.compute_c_2(y_2)], dtype=complex128)
self.store_local_data(obj=obj, c_1=c_1, c_2=c_2)
[docs] @staticmethod
def compute_obj(
x_local, # type: ndarray
x_shared, # type: ndarray
y_1, # type: ndarray
y_2, # type: ndarray
): # type: (...) -> float
"""Evaluate the objective :math:`obj`.
Args:
x_local: The design variables local to the first discipline.
x_shared: The shared design variables.
y_1: The coupling variable coming from the first discipline.
y_2: The coupling variable coming from the second discipline.
Returns:
The value of the objective :math:`obj`.
"""
return x_local[0] ** 2 + x_shared[1] + y_1[0] ** 2 + exp(-y_2[0])
[docs] @staticmethod
def compute_c_1(
y_1, # type: ndarray
): # type: (...) -> float
"""Evaluate the constraint :math:`c_1`.
Args:
y_1: The coupling variable coming from the first discipline.
Returns:
The value of the constraint :math:`c_1`.
"""
return 3.16 - y_1[0] ** 2
[docs] @staticmethod
def compute_c_2(
y_2, # type: ndarray
): # type: (...) -> float
"""Evaluate the constraint :math:`c_2`.
Args:
y_2: The coupling variable coming from the second discipline.
Returns:
The value of the constraint :math:`c_2`.
"""
return y_2[0] - 24.0
def _compute_jacobian(
self,
inputs=None, # type: Optional[Iterable[str]]
outputs=None, # type: Optional[Iterable[str]]
): # type: (...) -> None
self._init_jacobian(inputs, outputs, with_zeros=True)
x_local, _, y_1, y_2 = self.get_inputs_by_name([X_LOCAL, X_SHARED, Y_1, Y_2])
self.jac[C_1][Y_1] = atleast_2d(array([-2.0 * y_1]))
self.jac[C_2][Y_2] = ones((1, 1))
self.jac[OBJ][X_LOCAL] = atleast_2d(array([2.0 * x_local[0]]))
self.jac[OBJ][X_SHARED] = atleast_2d(array([0.0, 1.0]))
self.jac[OBJ][Y_1] = atleast_2d(array([2.0 * y_1[0]]))
self.jac[OBJ][Y_2] = atleast_2d(array([-exp(-y_2[0])]))
[docs]class Sellar1(MDODiscipline):
"""The discipline to compute the coupling variable :math:`y_1`."""
def __init__(
self,
residual_form=False, # type: bool
): # type: (...) -> None
"""
Args:
residual_form: If True,
then only residuals are computed,
not the coupling variables.
"""
self.residual_form = residual_form
super(Sellar1, self).__init__(auto_detect_grammar_files=True)
self.re_exec_policy = self.RE_EXECUTE_DONE_POLICY
if residual_form:
self.residual_variables = [R_1]
self.output_grammar.remove_item(Y_1)
else:
self.output_grammar.remove_item(R_1)
self.input_grammar.remove_item(Y_1)
self.default_inputs = get_inputs(self.input_grammar.get_data_names())
[docs] def get_attributes_to_serialize(self): # type: (...) -> List[str]
base_d = super(Sellar1, self).get_attributes_to_serialize()
base_d.append("residual_form")
return base_d
def _run(self): # type: (...) -> None
x_local, x_shared, y_2 = self.get_inputs_by_name([X_LOCAL, X_SHARED, Y_2])
if self.residual_form:
y_1 = self.get_inputs_by_name(Y_1)
r_1_out = array(
[self.compute_r_1(x_local, x_shared, y_1, y_2)], dtype=complex128
)
self.store_local_data(r_1=r_1_out)
else:
# functional form
y_1_out = array(
[self.compute_y_1(x_local, x_shared, y_2)], dtype=complex128
)
self.store_local_data(y_1=y_1_out)
[docs] @staticmethod
def compute_y_1(
x_local, # type: ndarray
x_shared, # type: ndarray
y_2, # type: ndarray
): # type: (...) -> float
"""Evaluate the first coupling equation in functional form.
Args:
x_local: The design variables local to first discipline.
x_shared: The shared design variables.
y_2: The coupling variable coming from the second discipline.
Returns:
The value of the coupling variable :math:`y_1`.
"""
return sqrt(x_shared[0] ** 2 + x_shared[1] + x_local[0] - 0.2 * y_2[0])
[docs] @staticmethod
def compute_r_1(
x_local, # type: ndarray
x_shared, # type: ndarray
y_1, # type: ndarray
y_2, # type: ndarray
): # type: (...) -> float
"""Evaluate the first coupling equation in residual form.
Args:
x_local: The design variables local to the first discipline.
x_shared: The shared design variables.
y_1: The coupling variable coming from the first discipline.
y_2: The coupling variable coming from the second discipline.
Returns:
The value of the residues related to the first discipline.
"""
return Sellar1.compute_y_1(x_local, x_shared, y_2) - y_1[0]
def _compute_jacobian(
self,
inputs=None, # type: Optional[Iterable[str]]
outputs=None, # type: Optional[Iterable[str]]
): # type: (...) -> None
self._init_jacobian(inputs, outputs, with_zeros=True)
x_local, x_shared, y_2 = self.get_inputs_by_name([X_LOCAL, X_SHARED, Y_2])
inv_denom = 1.0 / (self.compute_y_1(x_local, x_shared, y_2))
self.jac[Y_1] = {}
self.jac[Y_1][X_LOCAL] = atleast_2d(array([0.5 * inv_denom]))
self.jac[Y_1][X_SHARED] = atleast_2d(
array([x_shared[0] * inv_denom, 0.5 * inv_denom])
)
self.jac[Y_1][Y_2] = atleast_2d(array([-0.1 * inv_denom]))
if self.residual_form:
self.jac[R_1][X_LOCAL] = atleast_2d(self.jac[Y_1][X_LOCAL])
self.jac[R_1][X_SHARED] = atleast_2d(self.jac[Y_1][X_SHARED])
self.jac[R_1][Y_2] = atleast_2d(self.jac[Y_1][Y_2])
self.jac[R_1][Y_1] = -ones((1, 1))
del self.jac[Y_1]
[docs]class Sellar2(MDODiscipline):
"""The discipline to compute the coupling variable :math:`y_2`."""
def __init__(
self,
residual_form=False, # type: bool
): # type: (...) -> None
"""
Args:
residual_form: If True,
then only residuals are computed,
not the coupling variables.
"""
self.residual_form = residual_form
super(Sellar2, self).__init__(auto_detect_grammar_files=True)
self.re_exec_policy = self.RE_EXECUTE_DONE_POLICY
if residual_form:
self.residual_variables = [R_2]
self.output_grammar.remove_item(Y_2)
else:
self.output_grammar.remove_item(R_2)
self.input_grammar.remove_item(Y_2)
self.default_inputs = get_inputs(self.input_grammar.get_data_names())
[docs] def get_attributes_to_serialize(self):
base_d = super(Sellar2, self).get_attributes_to_serialize()
base_d.append("residual_form")
return base_d
def _run(self): # type: (...) -> None
x_shared, y_1 = self.get_inputs_by_name([X_SHARED, Y_1])
if self.residual_form:
y_2 = self.get_inputs_by_name(Y_2)
r_2_out = array([self.compute_r_2(x_shared, y_1, y_2)], dtype=complex128)
self.store_local_data(r_2=r_2_out)
else:
# functional form
y_2_out = array([self.compute_y_2(x_shared, y_1)], dtype=complex128)
self.store_local_data(y_2=y_2_out)
def _compute_jacobian(
self,
inputs=None, # type: Optional[Iterable[str]]
outputs=None, # type: Optional[Iterable[str]]
): # type: (...) -> None
self._init_jacobian(inputs, outputs, with_zeros=True)
y_1 = self.get_inputs_by_name(Y_1)
self.jac[Y_2] = {}
self.jac[Y_2][X_LOCAL] = zeros((1, 1))
self.jac[Y_2][X_SHARED] = ones((1, 2))
if y_1[0] < 0.0:
self.jac[Y_2][Y_1] = -ones((1, 1))
elif y_1[0] == 0.0:
self.jac[Y_2][Y_1] = zeros((1, 1))
else:
self.jac[Y_2][Y_1] = ones((1, 1))
if self.residual_form:
self.jac[R_2][X_LOCAL] = self.jac[Y_2][X_LOCAL]
self.jac[R_2][X_SHARED] = self.jac[Y_2][X_SHARED]
self.jac[R_2][Y_1] = self.jac[Y_2][Y_1]
self.jac[R_2][Y_2] = -ones((1, 1))
del self.jac[Y_2]
[docs] @staticmethod
def compute_y_2(
x_shared, # type: ndarray
y_1, # type: ndarray
): # type: (...) -> float
"""Evaluate the second coupling equation in functional form.
Args:
x_shared: The shared design variables.
y_1: The coupling variable coming from the first discipline.
Returns:
The value of the coupling variable :math:`y_2`.
"""
out = x_shared[0] + x_shared[1]
if y_1[0].real == 0:
y_2 = out
elif y_1[0].real > 0:
y_2 = y_1[0] + out
else:
y_2 = -y_1[0] + out
return y_2
[docs] @staticmethod
def compute_r_2(
x_shared, # type: ndarray
y_1, # type: ndarray
y_2, # type: ndarray
): # type: (...) -> float
"""Evaluate the second coupling equation in residual form.
Args:
x_shared: The shared design variables.
y_1: The coupling variable coming from the first discipline.
y_2: The coupling variable coming from the second discipline.
Returns:
The value of the residues related to the second discipline.
"""
return Sellar2.compute_y_2(x_shared, y_1) - y_2[0]
