mer439 - design of thermal fluid systems optimization techniques professor anderson
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Mer439 - Design of Thermal Fluid Systems Optimization Techniques Professor Anderson Spring Term 2012. Modeling/Simulation. Need Identified. Problem Definition. Workable Design. Concept Generation. Optimization/ Optimal Design. Concept Selection. Optimization in Design. Optimization. - PowerPoint PPT PresentationTRANSCRIPT
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Mer439 – Prof Anderson 1
Mer439 - Design of Thermal Fluid Systems
Optimization Techniques
Professor AndersonSpring Term 2012
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Mer439 – Prof Anderson 2
Need Identified
Problem Definition
Concept Generation
Modeling/Simulation
Workable Design
Optimization/Optimal Design
Concept Selection
Optimization in Design
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Mer439 – Prof Anderson 3
x1
x2
*
Set of all “workable”or “functional designs”(Allowed by physics,orange border)
Optimal DesignU(x1,x2) = Umax
Optimization
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Mer439 – Prof Anderson 4
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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Mer439 – Prof Anderson 5
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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Mer439 – Prof Anderson 6
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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Mer439 – Prof Anderson 7
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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Mer439 – Prof Anderson 8
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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Mer439 – Prof Anderson 9
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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Mer439 – Prof Anderson 10
Search Methods
Types of Approaches Elimination Methods Hill Climbing Techniques Constrained Optimization
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Mer439 – Prof Anderson 11
Unconstrained Search with Multiple Variables.
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Mer439 – Prof Anderson 12
*
*
*
*
Lattice SearchLattice Search1
2
3
4
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Mer439 – Prof Anderson 13
Lattice SearchLattice Search
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Mer439 – Prof Anderson 14
Univariate Search
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Mer439 – Prof Anderson 15
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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Mer439 – Prof Anderson 16
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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Mer439 – Prof Anderson 17
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