Mer439 - Design of Thermal
Fluid Systems
Optimization Techniques
Professor Anderson
Spring Term 2012
Mer439 – Prof Anderson
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Optimization in Design
Need Identified
Modeling/Simulation
Problem Definition
Workable Design
Concept Generation
Optimization/
Optimal Design
Concept Selection
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Optimization
Set of all “workable”
or “functional designs”
(Allowed by physics,
orange border)
x2
*
Optimal Design
U(x1,x2) = Umax
x1
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Lingo



Objective Function: represents the
quantity (U) which is to be optimized
(the “objective”) as a function of one or
more independent variables (x1, x2, x3…)
Design Variables: The independent
variables (x1, x2, x3…) that the objective
function depends on.
Constraints: Relations which limit the
possible (physical limitations) or the
permissible (external constraints)
solutions to the objective function.
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Mathematical Formulation

Objective Function of n independent design
variables:
For

U( x1, x2, x3…xn)
Equality Constraints:
Gi( x1, x2, x3…)=0

Find Uopt
i=1,2,…,m
Inequality Constraints:
Hj(x1, x2, x3…) < or > Cj
j=1,2,…l
If n>m → An Optimization problem results
If n=m → A unique solution exists…just solve all
equations simultaneously
If n<m → The problem is “over-constrained” no
solution which satisfies all of the constraints is
possible
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Acceptable Designs
x2
Set of all “workable”
or “functional designs”
(Allowed by physics,
orange border)
H1 : X1> c1
*
Optimal Design
U(x1,x2) = Umax
H2 : X2 < c2
Set of all “acceptable”
designs. (allowed by
constraints, yellow
border)
x1
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Example

Set up a mathematical statement to optimize a
water chilling system. The requirement of the
system is that it cool 20 kg/s of water from 13
to 8 oC, rejecting the heat back to the atmosphere
though a cooling tower. We seek a system with a
minimum first cost to perform this duty.
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Classification of Optimization Techniques

Calculus based Techniques


“Programming” methods



Lagrange Multipliers
Linear Programming
Geometric Programming
Search Methods

Elimination Methods
 Exhaustive
Fibonacci
 golden section search
“Hill Climbing” techniques
 Lattice Search
 Steepest ascent


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The aptly named
Exhaustive Search
x2
H1 : X1> c1
H2 : X2 < c2
*
Optimal Design
U(x1,x2) = Umax
Note: None of the search
points exactly hits the
optimum. The space
between search points is
known as the “interval of
uncertainty”
x1
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Search Methods

Types of Approaches



Elimination Methods
Hill Climbing Techniques
Constrained Optimization
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Unconstrained Search with Multiple
Variables.
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Lattice Search
1
3
*
2
*
4
*
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Lattice Search
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Univariate Search
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Example (Univariate Search)

Find the minimum value for y, where:
x2
16
y  x1 

x1x2
2

Use only integer values of x1 and x2
and start w/ x2 = 3
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The Project




What exactly are you trying to optimize?
→ “What is your Objective Function?”
What is the Absolute maximum that one
would be willing to pay? → “Is there a
cost inequality constraint that we can
use to help limit our design domain?”
What are your “design variables” ?
What is the nature of your functions?
(continuous / discrete) (linear/nonlinear) etc.
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Presentation Must Include




A clear representation of your
Objective Function
Clear Representations of your
constraints
A description of your optimization
methods
Evidence of a Sanity Check on your
proposed solution
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