Assignment 1 MEEN 683
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MEEN 683 Multidisciplinary System Design Optimization (MSADO)
Spring 2024
Assignment 1
Part (a) Motivation:
Summarize in 5-10 sentences why you decided to take this class. What do you
expect to learn? How does this knowledge fit in with your career or research plans?
I am into mechanical design and manufacturing. I have been in design all along
and I had decided to take this course as I wanted to learn more on design
optimization for a product and be efficient in product development. Combined with
my knowledge and the course setup, I wish to learn to solve problems on a
mathematical approach and be able to arrive with a solution that would work best
if I were to work in projects in the future.
Chapter 1 — Principles of Optimal Design: In Sections 1.1 and 1.2 of Principles of Optimal Design (POD), the authors discuss
hierarchical levels in system definition and hierarchical system decomposition. To
answer the questions below, consider one of the following engineering systems: a
wind turbine power system, a cable stayed bridge, a plug-in hybrid electric car, or a
submarine. (P1) Describe the system boundary that you would choose in setting up a model for
your system. What are the inputs and outputs that cross this boundary and
characterize your system?
Solution: Wind Turbine Power System
System Boundary: The tower and base sustaining the turbine construction,
the wind turbine blades catching wind energy, and the link to the power grid
enabling the conversion and distribution of electrical power are all involved
in the interactions inside the system boundary.
Inputs: Temperature, Air density, Atmospheric pressure, Direction and
Speed of wind.
Outputs: As the rotor blades turn the rotor shaft, the energy is generated.
Noise produced during this process.
(P2) Propose a component decomposition for your system (use a similar level of
detail to that shown in Fig. 1.10). Solution:
(P3) Propose an aspect decomposition for your system (use a similar level of detail
to that shown in Fig. 1.12).
Solution:
Unconstrained Optimization/Math Review and Getting Coding (P4) (P4) Use software of your choosing (usually Matlab or Python) and create a
contour plot and a surface plot (3D) of the function below. Describe the features of
the function (based on your plots) in words. Note regarding the plotting tasks: This
is meant to get you working with some code. I recommend searching the internet
for assistance with whatever software you choose regarding surface and contour
plots. There are plenty of examples out there and being capable of figuring out on
your own how to do this (and tasks like these) is an important aspect of learning
and self-study. Learning by example from online resources you have found when
coding is often much faster than any tutorial. That being said, I can help with either
Matlab or Python if you’ve tried on your own and are still totally lost. Solution:
Fig 1. Contour Plot of the given function f(x).
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Fig 2. 3D Surface Plot for the given function f(x).
(P5) Compute the gradient and the Hessian of the function. Determine the nature of
any stationary points (maximum, minimum, saddle) and compare this with your
discussion from P4. Solution:
Stationary Points obtained from Hessian Matrix are:
x = [ - 0.1350 , 0.1350 ]
x = [ 0.5540 , - 0.5540 ]
x = [ - 0.4190 , 0.4190 ]
The stationary points are obtained in Python by equating gradient to zero and with
the above mentioned Hessian equations.
Python Code:
from scipy.optimize import minimize
import numpy as np
def objective_function
(
x
):
return (x[
1
] - x[
0
])**
4 + 8
*x[
0
]*x[
1
] - x[
0
] + x[
1
] + 3
result_saddle = minimize(objective_function, [
0
, 0
], method=
'BFGS'
)
result_local_min = minimize(objective_function, [
-0.5
, 0.5
], method=
'BFGS'
)
result_global_min = minimize(objective_function, [
0.5
, -0.5
], method=
'BFGS'
)
x_saddle, x_local_min, x_global_min = result_saddle.x, result_local_min.x, result_global_min.x
hessian_saddle = np.array(result_saddle.hess_inv)
hessian_local_min = np.array(result_local_min.hess_inv)
hessian_global_min = np.array(result_global_min.hess_inv)
print
(
"Stationary point (saddle) is at:"
, x_saddle, "with function value"
, objective_function(x_saddle))
print
(
"Stationary point (local minimum) is at:"
, x_local_min, "with function value"
, objective_function(x_local_min))
print
(
"Stationary point (global minimum) is at:"
, x_global_min, "with function value"
, objective_function(x_global_min))
print
(
"Hessian matrix at the saddle point is:\n"
, hessian_saddle)
print
(
"Hessian matrix at the local minimum point is:\n"
, hessian_local_min)
print
(
"Hessian matrix at the global minimum point is:\n"
, hessian_global_min)
Results:
Stationary point (saddle) is at: [ 0.55357989 -0.5535795 ] with function value
0.9438271147570854 Stationary point (local minimum) is at: [ 0.55358027 -0.5535796 ] with function
value 0.9438271147564343 Stationary point (global minimum) is at: [ 0.55357997 -0.55358 ] with function
value 0.9438271147555968 Hessian matrix at the saddle point is: [[0.52344293 0.47656422] [0.47656422
0.52342862]] Hessian matrix at the local minimum point is: [[0.52333255 0.47663737]
[0.47663737 0.52339272]] Hessian matrix at the global minimum point is: [[0.56213473 0.46965638]
[0.46965638 0.47495067]]
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(P6) Redo the P4 contour plot but with the following constraint added. Discuss any
impact this constraint may have on the stationary points you found for the
unconstrained problem.
Solution:
Fig 3. Contour Plot of P4 with the given constraint g(x) with stationary points.
The code has been modified accordingly in order to obtain the stationary points for
the function g(x) and plot obtained is provided above. The constraint is g(x)>=0.
The stationary point point1, g[-0.4190,0.4190] = 1.7280
The stationary point point2, g[-0.1350,0.1350] = 0.3018
The stationary point point3, g[0.5540,-0.5540]= -0.0590
For the stationary points “point1” and “point2”, the value of g(x) is greater than
zero. Hence, it satisfies the constraint.
For the stationary point “point3”, the value of g(x) is less than zero. Hence, it does
not satisfy the constraint. It has one minimum point and one saddle point.
(P7) Redo the P4 contour plot but with the following constraint added (and remove
the constraint from P6). Discuss any impact this constraint may have on the
stationary points you found for the unconstrained problem.
Solution:
Fig 4. Contour Plot of P4 with updated constraint with stationary points.
The code has been modified accordingly in order to obtain the stationary points for
the function g(x) and plot obtained is provided above. The constraint is g(x)>=0.
The stationary point s1, g[-0.4190,0.4190] = 1.338
The stationary point s2, g[-0.1350,0.1350] = 0.770
The stationary point s3, g[0.5540,-0.5540]= -0.608
For the stationary points “point1” and “point2”, the value of g(x) is greater than
zero. Hence, it satisfies the constraint.
For the stationary point “point3”, the value of g(x) is less than zero. Hence, it does
not satisfy the constraint. It has one minimum point and one saddle point.
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