Optimal DesignProblem Formulation

Rudy J. Eggert, Professor Emeritus

http://coen.boisestate.edu/reggerthttp://highpeakpress.com/eggert/

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Today’s lecture

• Homework• Review

– Design, Form from Function– Optimal design

• Opt. Des. Problem Formulation• Examples

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Review

• Design-decision making activities– Form from Function– Phases:

• Formulation• Concept• Configuration• Parametric• Detail

• Opt. Design-systematic parametric design

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Design

Design

controlholdmoveprotectstore 

decision making processes

shapeconfigurationsizematerialsmanufacturing processes

Function

Form

 

 

Set of decision making processes and activities to determine: the form of an object, given the customer’s desired function.

Systematic Parametric Design

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Opt. Design Problem Formulation

Develop a mathematical model

Include mathematical relations for :1. A performance criterion or “cost function” (measures “goodness” of the candidate’s design)2. Necessary behaviors (must do or have) (obey laws of man, or nature, i.e. safety, physics, chemistry etc)

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Standard Design Optimzation Model

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n1=i x x x

m1= 0 )(g

p1= 0 =)(h

) (

: ToSubject

thatsuch*Find

) (Uii

) (Li

i

j

i

j

f :MINIMIZE

x

x

x

x

Design Problem Formulation (Arora)

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Step 1. Project/problem descriptionStep 2. Data and information collectionStep 3. Definition of design variablesStep 4. Optimization criterionStep 5. Formulation of constraints

Let’s reword these as actions to perform…

Opt. Des. Problem Formulation(Eggert)

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Step 1. Describe problemStep 2. Collect infoStep 3. Define DVsStep 4. Determine objective functionStep 5. Formulate constraints

Design of a can

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Step 1. Describe problem (restate w/bullets)•Must hold at least 400 ml•Min mfg cost which is proportional to surface area•Diameter no more than 8 cm•Diameter no less than 3.5•Height no more than 18 cm•Height no less than 8 cm

Design of a can

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Step 1. Describe problemStep 2. Collect infoStep 3. Define DVsStep 4. Determine objective functionStep 5. Formulate constraints

Design of a can

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Step 2. Collect infoDraw diagramRelation for volumeRelation for surface areaOther?

Volume = Area x height = (πD2/4)H

Surface area = top + bottom + side Area top, bottom = πD2 /4

Area side = πDH Total area = πD2 /4 + πD2/4 + πDH (cm3)

Design of a can

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Step 3. Define DVsDiameter, D, (cm)Height, H, (cm)x=[x1, x2] = [D, H]

Note: volume and area are functions of the DVs

Design of a can

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Step 4. Determine objective function

 Min f(x) = πD2 /2 + πDH (cm2)

Design of a can

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Step 5. Formulate constraints

Volume ≥ 400 ml (cm3), or (πD2/4)H ≥ 400 (cm3)

3.5 ≤ D ≤ 88 ≤ H ≤ 18

3.5 ≤ D

8 ≤ H D ≤ 8

H ≤ 18

Size limits

Design of a can - Summary

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(πD2/4)H ≥ 400 (cm3)

3.5 ≤ D

8 ≤ H D ≤ 8

H ≤ 18

 Min f(D,H) = πD2 /2 + πDH (cm2)

Subject to:

More on design variables (DVs)

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Parameters that:1.can be arbitrarily selected by the design engineer, AND that 2.influence the behavior of the product (or process) to be designed

For discrete variables... determine set of permissible values

DV Name heightSymbol HUnits (cm)Upper bound 18 cmLower bound 8 cm

Likely DV’s – think FORM

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SizesL, W, H, D, t

Shapessquare, circular, cylindrical, slender, short

Configurationsleft-handed, on-top, behind, over

Materialsmetals, polymers, ceramics

Manufacturing processesmachined, stamped, molded

More on Constraints

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Mechanical failure modes

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Tensile/compressive failure, plastic, brittleBuckleCorrosionExcess deflectionExcessive frictionThermal (melts, combusts…)WearVibrationUnsatisfactory motion (i.e. 4-bar, x,v,a) Other?

Electrical failure modes

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Short circuitOpen circuitExcessive power, heatPoor filteringEM interferenceOther?

Legal failure modes

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Violates codes/standardsCauses unforeseen property damageCauses unforeseen injuryInfringes existing patentOther?

Aids to math. modeling

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DiagramsClass notebooksHandbooksTextbooksTest resultsCAD

Summary

• Spend time on formulating• 5 Step process• Resulting in math. model

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