Welcome
ME 482/582 OPTIMAL DESIGN
Rudy J. Eggert, Professor Emeritus
Mechanical & Biomedical Engineering http://coen.boisestate.edu/reggert http://highpeakpress.com/eggert/
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Today’s lecture
• Optimization
• Design
• Analysis versus design
• Phases of design
• Parametric design
• Mathematics review
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OPTIMAL DESIGN
Definition:
The development and use of analytical and computer methods to provide an optimal design of a product or process with minimal computational effort.
That’s right…
.
The thing we design will be optimal AND the methods we use will be optimal.
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Product Realization Process
Realized
Product
Production Design
Engineering Design
Industrial Design
Sales /
Marketing
Disposal
Service
Distribution
Manufacturing
(Production)
Customer
Need

Design
Set of decision making processes and activities to determine: the form of an object, given the customer’s desired function .
Function
Design
Form control hold move protect store decision making processes shape configuration size materials manufacturing processes

Analysis is not Design
Which of the following is design and which is analysis?
A. Given that the customer wishes to fasten together two steel plates, select appropriate sizes for the bolt, nut and washer.
B. Given the cross-section geometry of a new airplane wing we determine the lift it produces by conducting wind tunnel experiments.
Problem Type
Design
Analysis
Solution
Form
(size, shape, matls,cnfg, mfg )
Predicted behavior
(performance)

System Evolution (Arora)
Figure 1.1
System evolution model.
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Preliminary
Design
Design Phases
Formulation
Concept
Configuration
Parametric
Embodiment
Design
Detail

From Customer Needs thru
Concept Design
Customer Needs
Formulation
Concept Design
Customer requirements
Importance weights
Eng. characteristics
House of Quality
Eng. Design Spec’s
Physical principles
Material
Geometry

Configuration Design
Abstract embodiment
Physical principles
Material
Geometry
Configuration
Design
Architecture
Special Purpose Parts:
Features
Arrangements
Relative dimensions
Attribute list (variables)
Standard Parts:
Type
Attribute list (variables)

Design Phases Cont’d
Design variable values e.g. Sizes, dimensions
Materials
Mfg. processes
Performance predictions
Overall satisfaction
Prototype test results
Parametric
Design
Detail
Design
Special Purpose Parts:
Features
Arrangements
Relative dimensions
Variable list
Standard Parts:
Type
Variable list
Product specifications
Production drawings
Performance Tests
Bills of materials
Mfg. specifications

Design Optimal Design
Figure 1.2
Comparison of (a) conventional design method and (b) optimum design method.
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Systematic Parametric Design
Formulate
Problem
Select Design Variables
Determine constraints
Re-Specify
Re-Design
Generate
Alternatives all alternatives
Select values for Design Variables
Analyze
Alternatives feasible alternatives
Predict Performance
Check Feasibility: Functional? Manufacturable ?
Evaluate
Alternatives best alternative
Determine best alternative
Refine
Optimize
Engineering Design
Eggert, 2010
, refined best alternative 13

Tools used in Optimal Design
• Algebra
• Calculus
• Vector and matrix aritmetic
• Excel (computation & graphing)
• Graphing (hand)
• Computer Programming (any language)
• Engineering principles
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Systematic Parametric Design
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Mathematical Notation
Recall from Calculus, a function of many variables: f (x, y, z)
We shall use vectors for multiple variables: f ( x ) note bold x
All vectors are columns x





 x x
 x n
1
2






 x
1
, x
2
The transpose is used to show a row
 x n

T
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Handwritten vectors
The book shows vectors as lower case bolded, for example: f ( x ) note bold x
For handwritten homework and tests… we will use lower case hand-printed with an underscore, for example: f ( x )
 note underscor e x

 x
1
, x
2
 x n

T
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Superscripts (1),(2) x
( 1 )  x
( 2 ) 


1


2
1 .
.
3
.
3


2









2
5
0
.
.
1
5
.
3




Points P, x (1) and x (2)
Figure 1.3
Vector representation of a point P that is in 3-dimensional space.
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Vector or point?
Is a “vector” a “point” in n-dimensional space denoted as R (n) ?
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Set of Points, S
S = {x|(x
1
– 4) 2 + (x
2
– 4) 2

9}.
Figure 1.4
Image of a geometrical representation for the set S = {x|(x
1
– 4) 2 + (x
2
– 4) 2

9}.
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Dot Product
From Engineering Statics: x
 y x
 y

 x y cos(

)
 i n

1 x i y i
 x
1 y
1
 x
2 y
2
 x
3 y
3
How do we know if two vectors are orthogonal (normal) ?
In optimal Design: x
 y
 x
T y x T y


 n
 x
1 i
,

1 x
2 x i
, y i x
3
 i n

1 x i y i
 x
1 y
1 y
1
, y
2
, y
3

T
 x
2 y
2
 x
3 y
3
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Vector or Scalar?
Is a dot product of two vectors a vector or scalar quantity?
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Norm of a vector
The length or magnitude of a vector is called the NORM.
x
 i n 

1 x i
2  x
 x
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Product of vector and matrix


1
 5
 2
1

3
1

1
1 


1 



 x
1 x
2 x
3
A
3 x 3 x
3




 x 1




( x
( 5
1 x
1
 ( 2 x
1


 x
2
3 x
2 x
2


 x
3 x
3 x
3
)
)
) 


 y
3 x 1
Is the product a scalar or vector?
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Triple Product c
 x
T
Ax

 x

Ax


3 2 1



1

5
 2

1
3
1

1
1



1 


3

2
 1





( 3 x 3 )( 3 x 1 )

3 2 1



( 1 )(

( 5 )(
 ( 2 )(
3 )
3 )
3 )



( 1 )(
(
( 1 )(
2

3 )(
2
)
)
2

)

(

(
1
1
)( 1 )
(

1 )( 1 )
)( 1 )





( 3 x 1 )

( i .
e .
3 rows x 1 column )

3 2 1



6
 8
 9





3 ( 6 )

2 ( 8 )

1 ( 9 )

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Rusty? …. Review appendix A, pgs 785-822
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Function continuity
Figure 1.5
Continuous and discontinuous functions: (a) and (b) continuous functions; (c) not a function;
(d) discontinuous function.
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First Partial Derivatives of a function
Gradient vector
 f ( x *)















 x f f
 x
2 f
 x n
1









 x *



 f
 x
1
 f
 x
2

 f
 x n


T x *
We’ll se a lot of these in chapter 4.
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Second Partial Deriivatives of a function…
Hessian Matrix
 2 f

H ( x *)










 2
 2
 2
 x
2
2
 x n
2 f f f
 x
1

2
2


 x
1
2
2 f
 x
1
2 f
 2 f
 x
1

2
2
 2
 x
2
2 f f
 x
2

2



 2
 x
1
2
 2 f f
 2
 x n
 2
2 f f
 x
2
2
 2
 f
 x n
2
 x n
2








 x *
What does the x* mean?
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Summary
• Design
• Optimal design
• Design Phases
• Systematic Parametric Design
• Vector, matrix review
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