Mer439 – Prof Anderson 1

Mer439 - Design of Thermal Fluid Systems

Optimization Techniques

Professor AndersonSpring Term 2012

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Need Identified

Problem Definition

Concept Generation

Modeling/Simulation

Workable Design

Optimization/Optimal Design

Concept Selection

Optimization in Design

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x1

x2

*

Set of all “workable”or “functional designs”(Allowed by physics,orange border)

Optimal DesignU(x1,x2) = Umax

Optimization

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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) Find Uopt

Equality Constraints: Gi( x1, x2, x3…)=0 i=1,2,…,m

Inequality Constraints:Hj(x1, x2, x3…) < or > Cj j=1,2,…l

If n>m → An Optimization problem resultsIf n=m → A unique solution exists…just solve all

equations simultaneouslyIf n<m → The problem is “over-constrained” no

solution which satisfies all of the constraints is possible

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Acceptable Designs

x1

x2

*

Set of all “workable”or “functional designs”(Allowed by physics,orange border)

Set of all “acceptable”designs. (allowed by constraints, yellow border)

Optimal DesignU(x1,x2) = Umax

H2 : X2 < c2

H1 : X1> c1

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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

Lagrange Multipliers “Programming” methods

Linear Programming Geometric Programming

Search Methods Elimination Methods

Exhaustive Fibonacci golden section search

“Hill Climbing” techniques Lattice Search Steepest ascent

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Exhaustive SearchExhaustive Search

x1

x2

*

Optimal DesignU(x1,x2) = Umax

H2 : X2 < c2

H1 : X1> c1

Note: None of the searchpoints exactly hits the optimum. The spacebetween search points isknown as the “interval of uncertainty”

The aptly named

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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 SearchLattice Search1

2

3

4

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Lattice SearchLattice Search

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Univariate Search

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Example (Univariate Search)

Find the minimum value for y, where:

Use only integer values of x1 and x2 and start w/ x2 = 3

2

16 2

211

x

xxxy

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The ProjectThe 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/non-linear) 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