7 design optimization-sm
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n ro uc on o es gn
Optimization
Minimum Weight
under Allowable
Various DesignObjectives
Stress)
A PEM Fuel Cell Stackwith Even Compression
over Active Area
(Minimum StressDifference)
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A PEM Fuel Cell
Stack Multi-
Ideal Stiffness to
Accommodate
Thermal- and Hydro-
Expansions
(Minimum Difference
between Ideal
Stiffness and
Calculated Stiffness):
Find a panel design with
the ideal stiffness.
Minimum MaximumStress in the Structure
Optimized Groove Dimension
to Avoid Stress Concentration
or ea en ng o e ruc ure
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Engineering Applications of Optimization
Design - determining design parameters that lead to the bestperformance of a mechanical structure, device, or system.
Core of engineering design, or the systematic approach to design(Arora, 89)
Planning
production planning - minimizing manufacturing costs
management of financial resources - obtaining maximum profits
task planning (robot, traffic flow) - achieving best performances
Control and Manufacturing - identifying the optimal controlparameters for the best performance (machining, trajectory, etc.)
Mathematical Modeling - curve and surface fitting of given datawith minimum error
Commonly used tool: OPT function in FEA; MATLAB Optimization Toolbox
What are common for an optimizationproblem?
There are multiple solutions to the problem; and the
op ma so u on s o e en e .
There exist one or more objectives to accomplish and a
measure of how well these objectives are accomplished
(measurable performance).
Constraints of different forms (hard, soft) are imposed.
There are several key influencing variables. The change
of their values will influence (either improve or worsen)the measurable performance and the degree of
violation of the constraints.
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Optimization can provide either a closed-form solution, or
a numerical solution.
Solution Methods
umer ca opt m zat on systemat ca y an e c ent y a uststhe influencing variables to find the solution that has the bestperformance, satisfying given constraints.
Frequently, the design objective, or cost function cannot beexpressed in the form of simple algebra. Computer programshave to be used to carryout the evaluation on the designobjective or costs. For a given design variable, , the value ofthe objective function, f(), can only be obtained using a
. ,carried out numerically.
e.g. Minimize the maximum stress in a tents/tension structures using FEA.
Computer Program(no simple algebra)
f()
Definition of Design Optimization
An optimization problem is a problem in which
certain parameters (design variables) needed to be
determined to achieve the best measurable
performance (objective function) undergiven
.
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Type of design variables
optimization ofcontinuous variables
Classification of the Optimization Problems
n eger programm ng scre e var a es (examples)
mixed variables
Relations among design variables
nonlinearprogramming
linearprogramming
Type of optimization problems
1
2. . ( )x
e g f X Ae Bx= +
1 1 2 2. . ( ) n ne g f X c x c x c x= + + +K
uncons ra ne op m za on
constrained optimization (examples)
Capability of the search algorithm
search for a local minimum
global optimization; multiple objectives; etc.
Automation and Integration
Formulation of the optimization problems
specifying design objective(s)
specifying design constraints
identifying design variables
Solution of the optimization problems
selecting appropriate search algorithm
determining start point, step size, stopping criteria
interpreting/verifying optimization results
Integration with mechanical design and analysis black box analysis functions serve as objective and
constraint functions (e.g. FEA, CFD models)
incorporating optimization results into design
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An Example Optimization Problem
Design of a thin wall tray:
, , , .
The design problem is to select the length, l, and width, w,
of the tray.
Given
A workable design:
lwh V = h H=
V
Pick eitherlorw and solve for others
wH
=
lw
h
An Optimal Design
The design is to minimize material volume, (orweight),where T is an acceptable small value for wall thickness.
Minimize
subject to
V w l h T wl lh whmbottom sides
( , , ) ( )= + +2 2
lwh V
h H
=
=
constraints (functions)
Design variables: w, l, and h.
w 0
lw
h
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Standard Mathematical Form
- objective functionmin ( )T wl lh wh+ +2 2
Subject to - equality constraints
- inequality constrains
- variable bounds
. . . , ,
lwh V
h H
=
=
0
0
l 0
- design vector
[ ]r
x l w h T= , ,
- for use of any available optimization routines
Analytical (Closed Form) Solution Eliminate the equality constrains, convert the original problem into a
single variable problem, then solve it.
from h = H & l w H = V; solve forl: lV
Hw=
from
wminT( )
V
Hww
V
HwH wH + +2 2
wminT( ) ( )
V
H
V
wwH f w+ + =2 2
2 *( )0, ,
df w V V we have w then the design optimum w
dw H H = = =
- a stationary point
Discard the negative value, since the inequality constraint is violated. The optimal value forl:
lV
Hw
V
Hw*
*
*= = =
V TV
Hhw
V
wM
* *
*( )= + +2
2= +T
V
HVH( )4
lwh
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Graphical
Solution
lw
h
lw
lw=W
h
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Procedures for Solving an Eng. Optimization Problem
Formulation of the Optimization Problem
Simplifying the physical problem identifying the major factor(s) that determine the performance or outcome
of the physical system, such as costs, weight, power output, etc. objective
- the design variables
Modeling the relations between design variables and the identified majorfactor - objective function
Identifying any constraints imposed on the design variables and modelingtheir relationship constraint functions
Selecting the most suitable optimization technique or algorithm tosolve the formulated optimization problem.
- requiring an in-depth know-how of various optimization techniques.
Determining search control parameters
- determining the initial points, step size, and stopping criteria of the
numerical optimization Analyzing, interpreting, and validating the calculated results
An optimization program does not guarantee a correct answer, one needs to
prove the result mathematically.
verify the result using check points.
Standard Form for Using Software Tools forOptimization (e.g. MatLab Optimization Tool Box)
Use of MATLAB
Optimization Toolbox
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Notes
One Dimensional Search Methods
The 1-D Search Problem - Basis for ND Optimization Search Techniques
Computer Program
(no simple algebra) f()
(maximum stress less than allowed value)
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1-D Optimization
We dont know the curve. Given , we can calculate f().
By inspecting some points, we try to find the approximated
shape of the curve, and to find the minimum, *numerically, using as few function evaluation as possible.
The procedure can be divided into two parts:
a) Finding the range or region known to contain * .
b) Calculating the value of
* as accurately as designed or aspossible within the range narrowing down the range.
Search Methods
Typical approaches include:
Quadratic Interpolation (Interpolation Based)
Cubic Interpolation
Newton-Raphson Scheme (Derivative Based)
Fibonacei Search (Pattern Search Based)
Golden Section Search
Iterative Optimization Process:
o
k New start point k+1
Repeat this process until the stopping rules are satisfied,
then * =n .
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Iterative Process for Locating the Range
Picking up a start point, o, and a range;
Shrinking the range;
Doubling the range;
Periodically changing the sign.
Typical Stooping Rules:
* * * *
**
( ) ( )
( )
f new f last new last
or newf new
< 0 -- Variables
Find:D*, H*, and T* -- Optimum
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Lowest Cost Design Cheapest Coffee Mug
Formulation ofDifferent Design
Optimization Problem
Minimize Mfg. Cost of the Mug -- Objective Function
Subject to
Mug Volume Constant 1 -- Inequality Constraint
Mug Mass Constant 2
trengt onstant
H/D = 1.65 -- Equality Constraint
D, H, T, Material, Tolerances, etc. -- Variables
Find:D*, H*, and T* etc. -- Optimum
Are you ready for the first quiz?
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