MECH4301 2008 L# 10 Conflicting Objectives 1/30 MECH4301 2008, Lecture 10 Objectives in Conflict:...
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Transcript of 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)
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
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)
MECH4301 2008 L# 10 Conflicting
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
Light Metric 1: Mass m Heavy
Cheap Metric 2: Cost C Expensive
Trade-offsurface
select
current material
MECH4301 2008 L# 10 Conflicting
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
Light Metric 1: Mass m Heavy
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: $200.
MECH4301 2008 L# 10 Conflicting
Objectives 7/30
Light Metric 1: Mass m Heavy
Cheap Metric 2: Cost C Expensive
Strategy 3: Penalty functions and exchange constants
Optimum solution,minimising Z
(lowers both m and c)
Z1
Z2Z3
Z4 Contours of constant Z
Decreasingvalues of Z
α
Seek material with smallest Z:• Either evaluate Z for each solution, and rank,
Or make trade-off plot
But what is the meaning of ?
• plot on it contours of Z
-- lines of constant Z have slope -
ZmC
• Read off solution with lowest Z
Define locally linearPenalty function Z
Cm Z
Z = y-intcpt (in this example)
MECH4301 2008 L# 10 Conflicting
Objectives 8/30
Light Metric 1: Mass m Heavy
Cheap Metric 2: Cost C Expensive
Z = penalty, value or utility function.
Z1
α
Along the line Z = cost + mass
= constant
cost
mass
Z is the combined “value” of (cost & mass)
MECH4301 2008 L# 10 Conflicting
Objectives 9/30
The exchange constant
The quantity is called an “exchange constant” -- it measures the value of performance, here the value of saving 1 kg of mass ($/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.
MECH4301 2008 L# 10 Conflicting
Objectives 10/30
Materials substitution and exchange constants
2P
1P2
P
Engineering
definition of
Cost of substituting D
for A ($/kg)
Cost of substituting B
for A
Upper bound to
C. H. Cáceres, "Economical and environmental factors in light alloys automotive applications", Metall. Mater. Trans. A, 2007, 38, 1649-1662.
MECH4301 2008 L# 10 Conflicting
Objectives 11/30
Family car (based on fuel saving)
Truck (based on payload)
Civil aircraft (based on payload)
Military aircraft (performance payload)
Bicycle frame (perceived value)
Space vehicle (based on payload)
Transport System: mass saving ($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.
MECH4301 2008 L# 10 Conflicting
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
-
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
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)
MECH4301 2008 L# 10 Conflicting
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 *
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
Objectives 19/30
Tute 5: E 7.4. Compressed air cylinders for trucksDesign goal: lighter, cheap air cylinders for trucks
Compressed air tank
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
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)
MECH4301 2008 L# 10 Conflicting
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 $): eg, Ti, Mg, GFRP or
CFRP
Al alloys; stronger
steels
For 2009: Explain how the normalising is done by reading the bubble’s
coordinates on CES, and then dividing the axes scales by those
values)
MECH4301 2008 L# 10 Conflicting
Objectives 24/30
The trade-off plot: Conclusions
Aluminium alloys and low alloy steels offer modest reductions in mass and material cost.
Need a strategy to explore the win-lose (trade-off) sectors as well:
Penalty functions and Exchange constants
Win-win sector: Safe options, but kind of boring.
MECH4301 2008 L# 10 Conflicting
Objectives 25/30
[Density] * [Price] / [Yield strength (elastic limit)] / rel to steel0.1 1 10 100
[D
ensit
y] /
[Yie
ld s
trength (
ela
stic
lim
it)] /
rel to s
teel
0.01
0.1
1
10
100
CFRP
Titanium alloys
Cast iron, gray
GFRP,
Magnesium alloys
Aluminum alloys
Low alloy steel
Low carbon steel
Cost relative to plain carbon steel, C/Co
Mas
s re
lativ
e to
pla
in c
arbo
n st
eel,
m/m
o
Penalty Functions and Exchange Constants
Z*=1 *-1 = 0.05 (trucks, = 20$/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
MECH4301 2008 L# 10 Conflicting
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)
MECH4301 2008 L# 10 Conflicting
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.
MECH4301 2008 L# 10 Conflicting
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
MECH4301 2008 L# 10 Conflicting
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.
MECH4301 2008 L# 10 Conflicting
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.