MDO formulations and scalable models for a toy example in aerostructure#
This tutorial describes how to build and run a scalable problem by means of GEMSEO for the Sobieski SSBJ aerostructure problem.
Creation of the disciplines#
In this tutorial, we are creating the disciplines from their analytic formulation using the AnalyticDiscipline discipline class.
We also configure the GEMSEO logger with the configure_logger() function.
from typing import Iterable
from gemseo import configure_logger
from gemseo import create_discipline
from gemseo.core.discipline.discipline import Discipline
configure_logger()
def create_disciplines() -> Iterable[Discipline]:
"""Create the disciplines (aero, structure, OAD) with analytic formulas.
Returns:
The disciplines (aero, structure, OAD).
"""
aero_formulas = {"drag": "0.1*((sweep/360)**2 + 200 + thick_airfoils**2-thick_airfoils -4*displ)",
"forces": "10*sweep + 0.2*thick_airfoils-0.2*displ",
"lift": "(sweep + 0.2*thick_airfoils-2.*displ)/3000."}
aerodynamics = create_discipline("AnalyticDiscipline", name="Aerodynamics",
expressions=aero_formulas)
struc_formulas = {"mass": "4000*(sweep/360)**3 + 200000 + 100*thick_panels +200.0*forces",
"RF": "-3*sweep + -6*thick_panels+0.1*forces+55"
,
"displ": "2*sweep + 3*thick_panels-2.*forces"}
structure = create_discipline("AnalyticDiscipline", name="structure",
expressions=struc_formulas)
oad_formulas = {"range": "8e11*lift/(mass*drag)"}
oad = create_discipline("AnalyticDiscipline", name="OAD",
expressions=oad_formulas)
return [aerodynamics, structure, oad]
disciplines = create_disciplines()
Description of the MDO study#
The goal of this study is to maximize the range \(range\) with respect to the design parameters which are:
\(thick\_airfoils\)
\(thick\_panels\)
\(sweep\)
Different MDO formulations can be considered to address this study.
Monolithic formulations#
Monolithic formulations can be used to address this MDO study:
Multiple Discipline Feasible (
MDF): a Multiple Disciplinary Analysis is performed for each optimisation iteration to converge the coupling variables disciplines to their equilibrium valuesIndividual Discipline Feasible (
IDF): the coupling variables equalities between the disciplines are set as constraints of the optimizer.
MDF#
The MDF formulation for the current problem is as follows:
Optimization problem:
Minimize: -range(sweep, thick_airfoils, thick_panels)
With respect to:
sweep, thick_airfoils, thick_panels
Subject to constraints:
lift(sweep, thick_airfoils, thick_panels) = lift + -0.5 = 0
RF(sweep, thick_airfoils, thick_panels) = RF + -0.5 <= 0
The XDSM for the MDF formulation is displayed below:
IDF#
The IDF formulation for the current problem is as follows:
Optimization problem:
Minimize: -range(mass, drag, lift)
With respect to:
sweep, thick_airfoils, thick_panels, mass, drag, forces, displ, lift
Subject to constraints:
drag_lift_forces(sweep, thick_airfoils, displ) = drag(sweep, thick_airfoils, displ) - drag = 0
lift(sweep, thick_airfoils, displ) - lift = 0
forces(sweep, thick_airfoils, displ) - forces = 0
mass_displ(sweep, thick_panels, forces) = mass(sweep, thick_panels, forces) - mass = 0
displ(sweep, thick_panels, forces) - displ = 0
lift(sweep, thick_airfoils, displ) = lift + -0.5 = 0
RF(sweep, thick_panels, forces) = RF + -0.5 <= 0
As previously stressed, the coupling variables have become optimization variables in this MDO formulation. The XDSM for the IDF formulation is as follows:
BiLevel#
A BiLevel formulation can be considered for the following MDO problem. It consists of performing optimizations at two levels:
at the system level, an optimization is made with respect to the shared design variables
at the discipline level, for each discipline, an optimization is made with respect to the local discipline design variable.
Discipline optimizations are performed in parallel. MDAs are performed at the beginning and at the end of each system level optimization iteration in order to get converged coupling variables.
System level#
The optimization problem for the system level reads:
Optimization problem:
Minimize: -range(sweep)
With respect to:
sweep
Subject to constraints:
lift(sweep) = lift + -0.5 = 0
RF(sweep) = RF + -0.5 <= 0
Aerodynamics level#
The optimization problem for the aerodynamics discipline level reads:
Optimization sub-problem:
Minimize: -range(thick_airfoils)
With respect to:
thick_panels
Subject to constraints:
lift(thick_airfoils) = lift + -0.5 <= 0
Structure level#
The optimization problem for the structure discipline level reads:
Optimization sub-problem:
Minimize: -range(thick_panels)
With respect to:
thick_panels
Subject to constraints:
RF(thick_panels) = RF + -0.5 <= 0
The XDSM for the bi-level formulation#
The scalable problem#
In this section, we are going to build a scalable problem from the aerostructure problem. Further information on the scalable problem methodology can be found here.
Build the interpolation functions#
We build the interpolation functions from the disciplines:
from gemseo.problems.mdo.aerostructure.aerostructure_design_space import AerostructureDesignSpace
from gemseo.problems.mdo.scalable.data_driven.diagonal import ScalableDiagonalModel
def create_scalable_models(disciplines: Iterable[Discipline], n_samples: int = 20,
fill_factor: float = 0.8) -> Iterable[ScalableDiagonalModel]:
"""Create the scalable disciplines.
Args:
disciplines: The disciplines.
n_samples: The number of samples.
fill_factor: The fill factor.
Returns:
The scalable models.
"""
design_space = AerostructureDesignSpace()
design_space.set_current_value(design_space.get_current_value().real)
sizes = design_space.variable_sizes
scalable_models = []
for discipline in disciplines:
discipline.set_cache(cache_type=discipline.CacheType.MEMORY_FULL)
output = next(iter(discipline.output_grammar.names))
disc_design_space = deepcopy(design_space)
disc_design_space.filter(discipline.input_grammar.names)
scenario = create_scenario([discipline], "DisciplinaryOpt", output,
disc_design_space, scenario_type="DOE")
scenario.execute(algo_name="DiagonalDOE", n_samples=n_samples)
scalable_model = ScalableDiagonalModel(
discipline.cache.to_dataset(), sizes, fill_factor=fill_factor
)
scalable_models.append(scalable_model)
return scalable_models
scalable_models = create_scalable_models(disciplines)
Plot the interpolation function#
We can easily plot the interpolation functions:
for scalable_model in scalable_models:
scalable_model.plot_1d_interpolations(save=True)
We obtain the following plots for the three disciplines.
Aerodynamics#
|
|
|
Interpolation of drag(sweep, thickAirfoils, displ) |
Interpolation of lift(sweep, thickAirfoils, displ) |
Interpolation of forces(sweep, thickAirfoils, displ) |
Structure#
|
|
|
Interpolation of RF(sweep, thickPanels, forces) |
Interpolation of mass(sweep, thickAirfoils, forces) |
Interpolation of displ(sweep, thickAirfoils, forces) |
OAD#
Interpolation of range(lift, mass, drag)#