mech4301 2008 l# 10 conflicting objectives 1/30 mech4301 2008, lecture 10 objectives in conflict:...

MECH4301 2008 L# 10 Conflicting

Objectives 1/30

MECH4301 2008, Lecture 10 Objectives in Conflict: Trade-off Methods and Penalty Functions

•Textbook Chapters 9 & 10•Tutorial 5 (2 exercises, two afternoons, due Oct 13)

Technical Papers: •P. Sirisalee, M. F. Ashby, G. T. Parks and P. J. Clarkson, "Multi-Criteria Material Selection in Engineering Design", Adv. Engng. Mater., 2004, 6, 84-92. (Simple, readable)C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38, 1649-1662. (Automotive applications)M. F. Ashby, "Multi-objective optimization in material design and selection", Acta Materialia, 2000, 48, 359-369. (Advanced reading)


Objectives 2/30

Examples of Conflicting Objectives in design

Some objectives may mass, m conflict with another cost, c We wish to minimize both (all constraints being met)

Common design objectives:

Minimising mass (sprint bike; satellite components)

Minimising volume (mobile phone; minidisk player)

Minimising environmental impact (packaging, cars)

Minimising cost (everything)

Objectives

Conflict : the choice that optimises one does not optimise the other.

Best choice is a compromise.

Each defines a performance metric


Objectives 3/30

Light Metric 1: Mass m Heavy

Cheap Metric 2: Cost C Expensive

Multi-objective optimisation: The terminology

• Trade-off surface: the surface on which the non-dominated solutions lie (also called the Pareto Front) (after Pareto, 1898)

• Solution: a viable choice, meeting constraints, but not necessarily optimum by either criterion.

Trade-offsurface

• Plot all viable solutions as function of performance metrics. (Convention: express objectives to be minimised)

• Dominated solution: one that is unambiguously non-optimal (as A) (there are better ones)

A Dominatedsolution

• Non-dominated solution: one that is optimal by one metric (as B: optimal by one criterion but not necessarily by both)

B Non-dominatedsolution


Objectives 4/30

Example of Conflicting Objectives in PushbikesPrice vs. mass of bicycles: a matter of perception?

Price $

Mass (kg)

The price we are prepared to pay for a light bike does not relate to the actual cost of the materials it is made of.

Then, how do we decide what is

the “best” material?

Three strategies for finding the best compromise (next 4 frames)


Objectives 5/30

Strategy 1: compromise by intuition and experience

• Make trade-off plot and Sketch trade-off surface

• Use intuition to select a solution on the trade-off surface

• “Solutions” on or near the surface offer the best compromise between mass and cost

•The choice depends on how highly you value a light weight, -- a question of relative values



Trade-offsurface

select

current material


Objectives 6/30

Finding a compromise: Strategy 2

• Reformulate all but one of the objectives as constraints, setting an upper limit for it

Optimum solutionminimising m


Cheap Metric 2: Cost C Expensive Trade-offsurface

Mass and price of bicycles:

• Good if you have budget limit

• Trade-off surface leads you to the best choice within budget

• But not a true optimisation -- mass has been treated as a constraint, not an objective.

Optimum solutionminimising c

Constraint: mass = 11 kg

Upper limit for cost: $200.


Objectives 7/30



Strategy 3: Penalty functions and exchange constants

Optimum solution,minimising Z

(lowers both m and c)

Z1

Z2Z3

Z4 Contours of constant Z

Decreasingvalues of Z

α

Seek material with smallest Z:• Either evaluate Z for each solution, and rank,

Or make trade-off plot

But what is the meaning of ?

• plot on it contours of Z

-- lines of constant Z have slope -

ZmC

• Read off solution with lowest Z

Define locally linearPenalty function Z

Cm Z

Z = y-intcpt (in this example)


Objectives 8/30



Z = penalty, value or utility function.

Z1

α

Along the line Z = cost + mass

= constant

cost

mass

Z is the combined “value” of (cost & mass)


Objectives 9/30

The exchange constant

The quantity is called an “exchange constant” -- it measures the value of performance, here the value of saving 1 kg of mass ($/kg).

Cm

ZCmZ

How get …? Effect of metric on Zmarket survey (perceived value)full life cost (engineering criteria)

= drop in Z per unit mass, at constant

cost

Metric P1: Mass m

Metric P2: Cost C

Exchange Constant: quantifies the effect of a material substitution on the total value, or the (value) penalty involved in the substitution.


Objectives 10/30

Materials substitution and exchange constants

2P

1P2

P

Engineering

definition of

Cost of substituting D

for A ($/kg)

Cost of substituting B

for A

Upper bound to

C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38, 1649-1662.


Objectives 11/30

Family car (based on fuel saving)

Truck (based on payload)

Civil aircraft (based on payload)

Military aircraft (performance payload)

Bicycle frame (perceived value)

Space vehicle (based on payload)

Transport System: mass saving ($US per kg)

0.5 ~ 6

5 to 20

100 to 500

500 to 1000

20-4000

3000 to 10000

(Upper bounds to) Exchange constants for mass saving in transport systems

Finding : engineering criteria.Example of upper bounds to exchange constants for transport systems

The is how much you can afford to expend in a material substitution. If the substitution costs you more than the upper bound, you won’t get your $ back.

Savings over 2x105km

C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38, 1649-1662. M. F. Ashby, "Multy-objective optimization in material design and selection", Acta Materialia, 2000, 48, 359-369.


Objectives 12/30

Penalty function on log scales

Log scales

Lighter mass, m Heavier

Cheap Cost, C Expensive

Decreasing values of Z

A linear relation, on log scales, plots as a curve ZmαC

CmαZ

Linear scales

Lighter mass, m Heavier

Cheap Cost, C Expensive

Decreasing values of Z

-


Objectives 13/30

Density/Sqrt Modulus50 100 200 500 1000 2000 5000

De

nsi

ty x

Pri

ce /S

qrt

Mo

du

lus

10

100

1000

10000

100000

1e6

MAGNESIUM alloys

GFRP

Epoxy/HS Carbon weave

ALUMINUM alloys

HSLA steels CAST IRONS

Zinc alloys

Lead alloys

Copper alloys

Tungsten alloys

BronzeCFRP epoxy laminate

Ti-alloys

Ni-based superalloys

Cobased superalloys

Penalty function in transport systems. Mass of a beam vs. cost for given stiffness

P2= Costfor givenstiffness

P1=Massfor givenstiffness

Exchange constant

= 1 $/kg

Exchange constant

= 50 $/kgExchange constant

= 5 $/kg

Exchange constant

= 500 $/kg

Trade-off surface

c/E1/2

/E1/2

Family car

Truck

Civil aircraft

Military aircraft

Bicycle frame

Space vehicle

System ($US per kg)

0.5~6

5 to 20

100 to 500

500 to 1000

20-4000

3000 to 10000

2P

1P2

P

Engineering

definition of

Penalty Function & Exchange Constants: Powerful and Unambiguous Strategy for Material Substitutions under Conflicting Objectives


Objectives 14/30

Case study: casing for electronic equipment

Electronic equipment -- portable computers, players, mobile phones, cameras – are miniaturised; many less than 12 mm thick

Minidisk player: An ABS or Polycarbonate casing has to be > 1mm thick to be stiff enough to protect; casing takes 20% of the volume

stiff, light, thin casing bending stiffness EI at least that of existing case

minimise casing thicknessminimise casing mass

choice of material casing thickness, t

Constraints

Objectives

Function

Free variables

The thinnest may not be the lightest … need to explore trade-off


Objectives 15/30

Performance metrics for the casing: t and m Function Stiff casing

tw

L

F

Metric 1 3/1

3/13

E

1wE4

LSt

Objective 2 Minimise mass m

Metric 23/13/1

23/12

EEL

CwS12

m

m = massw = widthL = length = densityt = thicknessS = required stiffnessI = second moment of areaE = Youngs Modulus

Objective 1 Minimise thickness t

3L

IE48S

Constraints

12tw

I3

Adequate toughness, G1c > 1kJ/m2

Stiffness, S

with

Unit 5, Frame 5.10

Materials Index to minimise the thickness

Materials Index to minimise the mass


Objectives 16/30

Relative performance metrics

The thickness of a casing made from an alternative material M, differs (for the same stiffness) from one made of Mo by the factor

3/1o

o EE

tt

The mass differs by the factor

o

3/1o

3/1o

E.

Emm

omm

Explore the trade-off between and ott

We are interested here in substitution. Suppose the casing is currently made of a material Mo, elastic modulus Eo, density o.

Define a relativepenalty function, Z* oo t

tmm ** αZ ( now dimensionless)

Relative mass = ratio of Materials Indices (mass)

Relative thickness = ratio of Materials Indices (t)


Objectives 17/30

Plotting the relative penalty function, Z*

Penalty lines for casing

Assume mass and thickness are equally important: * = 1

** Zoo tt

mm

Thickness relative to ABS0.1 1 10

Mas

s re

lativ

e to

AB

S

1

10

Low alloy steel

Al-alloysMg-alloys

GFRPCFRP

Al-SiC Composites

Ti-alloys

ABSNi-alloys

Thickness relative to ABS

Mas

s re

lati

ve t

o A

BS

Z*1Z*2

Z*3

Polymers are all dominated solutions

Materials on trade- off surface are

metals and high performance composites

Explains the use of Mg alloys in mobile phones

and laptop computer casings, cameras

Penalty functions

of gradient -* = -1

* = ???

Current casing

Decreasing values of Z* at constant *


Objectives 18/30

Thickness relative to ABS0.1 1 10

Ma

ss r

ela

tive

to A

BS

0.1

1

10

PTFE

PC

ABS

PMMA

PP

NylonPolyester

PE

Ionomer Ni-alloys

Cu-alloys

Steels

Al-alloys

Al-SiC Composite

Ti-alloys

Mg-alloys

CFRPGFRP

Lead

Polymer foams.

Elastomers

Thickness relative to ABS, t/to

Mas

s re

lativ

e to

AB

S,

m/m

o

Trade-offsurface

Conclusion: Four-sector trade-off plot for minidisk player

Q: Is material cost relevant? Not a lot -- the case only weighs a few grams. Volume and weight are much more valuable.

The four sectors of a trade-off plot for substitution

A. Better by both metrics

C. Lighter but thicker

D. Worse by both metrics

B. Thinner but heavier

win-win sector

win-lose sectors: worth exploring

win-lose sector: worth exploring

sometimes

Don’t bother

Current casing


Objectives 19/30

Tute 5: E 7.4. Compressed air cylinders for trucksDesign goal: lighter, cheap air cylinders for trucks

Compressed air tank


Objectives 20/30

Design requirements for the air cylinder

Pressure vessel

• Minimise mass• Minimise cost

• Dimensions L, R, pressure p, given• Must not corrode in water or oil• Working temperature -50 to +1000C• Safety: must not fail by yielding• Adequate toughness: K1c > 15 MPa.m1/2

• Wall thickness, t; • Choice of material

Specification

Function

Objectives

Constraints

Free variables

R = radiusL = length = densityp = pressuret = wall thickness

L

2R

Pre

ssur

e p

t


Objectives 21/30

Light and Cheap Air Cylinder

Met

ric

1: m

ass

Eliminate t to give:

L

2R

Pre

ssur

e p

t

Constraint (no yielding)

Objective 2

y

2 Sp)Q1(LR2m

mCC m

StRp y

Vol of material in cylinder wall

Asp

ect

ratio

Q

Objective 1 tR4tLR2m 2

LR2

1tLR2

Metric 2: cost

R = radiusL = length = densityp = pressuret = wall thickness = yield strengthS = safety factorQ = aspect ratio 2R/L

y

y

m2 CSp)Q1(LR2C

Materials Index to minimise the mass

Materials Index to minimise the cost


Objectives 22/30

Conflicting Objectives: Relative mass and cost This is a problem of material substitution. The tank is currently made of a plain carbon steel.

The mass m and cost C of a tank made from an alternative material M, differs (for the same strength) from one made of Mo by the factors

Explore the trade-off between and

o

o,y

yo.

mm

oo,m

o,y

y

m

o C.

C

CC

omm

oCC

Relative mass = ratio of

Materials Indices (mass)

Relative cost = ratio of Materials

Indices (cost)


Objectives 23/30

[Density] * [Price] / [Yield strength (elastic limit)] / rel to steel0.1 1 10 100

[Densit

y]

/ [

Yie

ld s

trength

(ela

sti

c lim

it)]

/re

l to

ste

el

0.01

0.1

1

10

100

CFRP

Titanium alloys

Cast iron, gray

Low carbon steel

GFRP,

Magnesium alloys

Aluminum alloys

Low alloy steel

Four sectors trade-off plot for air tank

Trade-offsurface

Additional constraints:

K1c >15 MPa.m1/2

Tmax > 373 K

Tmin < 223 K

Water: good +

Organics: good +

Anything in this corner is slightly better (cheaper

and lighter)

Current tank. Axes normalised to locate

current tank material at origin (1,1)

win-win sector

win-lose sector:

win-lose sector:

Anything in this corner is a trade-off (lighter but more $): eg, Ti, Mg, GFRP or

CFRP

Al alloys; stronger

steels

For 2009: Explain how the normalising is done by reading the bubble’s

coordinates on CES, and then dividing the axes scales by those

values)


Objectives 24/30

The trade-off plot: Conclusions

Aluminium alloys and low alloy steels offer modest reductions in mass and material cost.

Need a strategy to explore the win-lose (trade-off) sectors as well:

Penalty functions and Exchange constants

Win-win sector: Safe options, but kind of boring.


Objectives 25/30

[Density] * [Price] / [Yield strength (elastic limit)] / rel to steel0.1 1 10 100

[D

ensit

y] /

[Yie

ld s

trength (

ela

stic

lim

it)] /

rel to s

teel

0.01

0.1

1

10

100

CFRP

Titanium alloys

Cast iron, gray

GFRP,

Magnesium alloys

Aluminum alloys

Low alloy steel

Low carbon steel

Cost relative to plain carbon steel, C/Co

Mas

s re

lativ

e to

pla

in c

arbo

n st

eel,

m/m

o

Penalty Functions and Exchange Constants

Z*=1 *-1 = 0.05 (trucks, = 20$/kg)To the left: OK; to the right: too expensive

Z*=1 *-1 = 0.01 = 100$/kg Ti, expensive!

Z*= 2 = 1 (current,

cheap and heavy)

Z*= 0.6 = 1 (cheaper and lighter)

(safe bet, boring)

GFRP: border line for 20$/kg CFRP is a cheaper option


Objectives 26/30

The main points

Real design problems involve conflicting objectives -- often technical or environmental performance vs. economic performance (cost).

Trade-off plots reveal the options for material selection or material substitutions that solve the conflict, and (when combined with the other constraints of the design) frequently point to a sensible final choice.

If the relative value of the two metrics of performance (measured by an exchange constant) is known, a penalty function allows an unambiguous selection: the exchange constants allow exploring the chart's win-lose (trade-off) sectors as well as the win-win sector.

2P

1P2

PEngineeri

ng definition

of

P1, P2 = performance

metrics (mass, cost)


Objectives 27/30

Tute 5, E 7.5. Refrigerated truck: Solution 1: CES chart for and 1/E Use foamed materials data base (level 3) Grapher version

=-3*x+.7

=-0.001*x+.035

For 2009: Explain here that using a high alpha means that you value thermal properties more than stiffness. A low alpha puts stiffness ahead of thermal behaviour.


Objectives 28/30

1/ [Young's Modulus] 0.001 0.01 0.1 1 10 100 1000 10000 100000

Therm

al co

nduct

ivit

y (W

/m.K

)

0.01

0.1

1

10

100

Graphite (General Purpose Industrial)(perp. to plane)

Aluminum-SiC Foam (0.27)

Nitrile Rubber, Hydrogenated (HNBR, 25-40% carbon black)

Polyurethane Elastomeric Foam Open Cell (0.065)

Gray (Flake graphite) cast iron (BS grade 150)

Mullite (Al2O3-SiO2 alloys)

Epoxy (Mineral Filler)

Polyphtalamide (General Purpose)

Birch (Betula verrucosa) (l)

x y=-3*x+.7 y=-0.001*x+.035

0.001 0.697 0.034999

0.002 0.694 0.034998

0.003 0.691 0.034997

0.005 0.685 0.034995

0.01 0.67 0.03499

0.02 0.64 0.03498

0.03 0.61 0.03497

0.05 0.55 0.03495

0.1 0.4 0.0349

0.15 0.25 0.03485

0.17 0.19 0.03483

0.2 0.1 0.0348

0.21 0.07 0.03479

0.22 0.04 0.03478

0.23 0.01 0.03477

0.3 0.0347

0.5 0.0345

1 0.034

2 0.033

3 0.032

5 0.03

10 0.025

20 0.015

24 0.011

=-3*x+.7

=-0.001*x+.035

Tute 5, E 7.5. Refrigerated truck: Solution 1: CES chart for and 1/E Use foamed materials data base (level 3) Excel version


Objectives 29/30

Refrigerated Truck

•Penalty Function Lines=-3*x+0.7 makes stiffness very important. As Ceramic foams are very stiff, they are selected but the thermal losses may be high, and the toughness may be low.

=-0.001*x+0.035 Medium density polymeric foams (0.08-0.16) are good if thermal losses are more important than having a high stiffness.


Objectives 30/30

The End

For 2009: This lecture is too messy and complicated. The penalty functions Z are not well explained. Use the minidisk case as an illustration of how to reduce Z at constant alpha, and the truck tank as an example of changing alpha at constant Z. Cut the maths

a bit.

mech4301 2008 l# 10 conflicting objectives 1/30 mech4301 2008, lecture 10 objectives in conflict:...

Documents

cost mass

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tradeoff surface solutions

unit mass

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best compromise