Reprinted with permission from the

Winter 2003 edition of

Driving Performance with Optimization

Leverage the

Power of

Linux Clusters

CAD/CAE

Integration:

Making the Grade

7

14

Wire Harness

Design Powers Up

18

Conducting

Computing

Performance

2

Page10

Ideas and Strategies in Product Development

Ideas and Strategies in Product Development

An Publication

WI N

TE

R 2

0 0

3

Altair Engineering, Inc.

DrivingPerformance

with Optimization

DrivingPerformance

with Optimization

12

Concept To Reality/ W

inter 2003

www.altair.co

m/c2

r

ing bodies, reducing the mo-

mentum

transferred from

one body to another. For ex-

ample, if the deform

ation

were perfect without energy

losses, a ball dropped on the

floor would bounce up to its

original height. If a ball of artist clay were dropped

on the floor, it would deform and stick to the floor.

In most impacts, bodies act somewhere between

these two extremes.

COR is the parameter used in rigid body dynamics to

distinguish between these extremes. For now, consider

the ball bouncing off the fixed floor. If a body bounces

to the same height from which it was dropped, it has a

COR of 1. A body that hits and sticks has a COR of 0.

The height a body bounces up is defined by the COR

between the two bodies (e.g., ball and floor). When the

floor is stationary, a collision having a 0.822 COR

value will cause the ball to bounce up 82.2 % of the

height from which it is dropped.

When both bodies are moving, the COR defini-

tion requires a more precise analysis. Remember, the

COR depends on the deformation in both bodies.Here is the full blown definition of COR: The COR is

the negative ratio of relative post-impact velocities to rela-

tive pre-impact velocities.

What exactly is meant by relative velocities? If

two balls move towards one another at 100 mph, the

relative velocity is 200 mph. In our simple case from

the previous paragraph, one velocity was zero.

Hence, the relative velocity is just that of the moving

body. Relative velocity reflects the vector nature of

velocities.To measure the COR for a club head and ball colli-

sion, the pre- and post-impact velocities for both the

club and the ball must be measured. COR is then a

simple calculation. As the ball approaches the club, it

passes through a pair of ballistic screens measuring the

time elapsed to travel a fixed distance. Thus, the in-

bound velocity is measured. As the ball rebounds, the

ballistic screens work the same way in reverse order.

The club head is at rest before impact, so the club

head’s post-impact velocity is all that’s needed to

compute the ball-club head COR. Use of physics’

conservation of momentum allows for the computa-

tion of the COR from the ball’s pre- and post-impact

velocities. This little trick makes for a more econom-

ical experiment, but it also confuses some because

the resulting formula has mass terms. Recall the def-

inition of COR is mass-independent.

Here is the bottom line on COR: Increasing COR

for a given club head velocity and ball will increase

the ball’s initial rebound velocity. Thus, the golf shot

should travel farther.

Getting to the COR with CAE

Designing golf clubs with a COR of less than 0.830

can be tricky. However, employing simulation soft-

ware enables manufacturers to evaluate a multitude

of designs without having to cast a single part.

One such simulation package is Altair Hyper-

Study, an open architecture optimization tool that

can be used in conjunction with any finite-element

solver. Using HyperStudy, with LS-Dyna3D soft-

ware, our objective was to obtain the maximum pos-

sible COR of the club head while maintaining a club

head mass of 200 g and keeping club head stress lev-

els below the material yield of 150 ksi.

The optimization problem is defined by the specifi-

cation of an objective function, constraints and design

variables. The model responses that are used for the ob-

jective and constraints are limited only to quantities

that can be obtained in the solver output. Through the

notation convention of HyperStudy, any value in the

input deck can be defined as a design variable. Thus,

the procedure involved is extremely general.

Figure 1. Definitions of regions on the golf club head.

Figure 2. Resulting

driver shapes for

maximum shape

variable values.

Hosel

FaceSmile

Sole

Toe

Crown

Skirt

Heel

Longer

Wider

Taller

OOn golf c

ourse

s arou

nd the w

orld,

plenty

of gre

at go

lf

has be

en pl

ayed

this s

ummer

at th

e prof

ession

al an

d am-

ateur

levels

. Off t

he links,

an on

going d

rama r

elatin

g to

golf c

lub pe

rform

ance

also

has be

en pl

aying o

ut.

At issu

e is t

he perf

orman

ce of

golf

club d

rivers

.

Specif

ically

, golf

’s ruli

ng bod

ies hav

e inde

pende

ntly

deter

mined a

rule t

hat set

s a un

iform

, worl

dwide

stan

-

dard

for “

sprin

g-like

” effe

ct in

drivi

ng club

s. The o

r-

ganiza

tions w

ill rig

orousl

y test

golf e

quipm

ent fo

r con

-

formity

to th

is and t

he oth

er Rule

s of G

olf.

What

this

means f

or eq

uipmen

t man

ufactu

rers i

s

that

they

must

deve

lop th

eir pr

oduc

ts to

meet t

he ap-

propri

ate st

anda

rds. W

hat’s m

ore, th

e club

s hav

e to b

e

pleasi

ng in th

e eye

s of th

e con

sumers

yet b

e trad

ition

al

and p

lain in

shap

e in th

e eye

s of g

olf’s r

uling b

odies

.

One way

equip

ment m

anufa

cturer

s can

achiev

e

thes

e goals

is th

rough

the u

se of c

omputatio

nal

analy

ses, in

cludin

g rec

ent a

dvan

cemen

ts in

optim

i-

zation

softw

are an

d meth

ods. A

rece

nt eva

luatio

n of

a golf

drive

r hea

d dem

onstr

ates t

hat usi

ng suc

h simu-

lation

tech

nology

can im

prov

e the d

esign

, resu

lting

in dr

ivers

that

are st

ructu

rally

optim

ized t

o the p

er-

forman

ce lim

it all

owed

by th

e Rule

s of G

olf.

Equipped for E

xcellence

Golf heri

tage h

as pr

oduc

ed tw

o ruli

ng bod

ies: t

he

Royal a

nd Anci

ent G

olf Clu

b of S

t. A

ndrews

(R&

A) and th

e Unite

d Sta

tes G

olf Asso

ciatio

n

(USGA).

The USGA is

, in a

way, t

he you

nger o

ff-

sprin

g of t

he R&

A. For

med on

Dec

embe

r 22,

1894

,

the U

SGA is a n

onpro

fit, m

embe

rship-

drive

n orga

n-

izatio

n that

gove

rns t

he gam

e of g

olf in

Nort

h Amer-

ica. E

ven th

ough

Can

ada h

as its

own go

lf ass

ocia-

tion, t

he Roy

al Can

adian

Golf

Asso

ciatio

n, it de

fers

to th

e USGA on

man

y matt

ers. F

or th

e mos

t part

,

the r

est of

the w

orld i

s gov

erned

by th

e R&A.

In te

rms o

f fac

ilitie

s, th

e USGA h

as tak

en su

b-

stantia

l step

s to s

uppo

rt its

equip

ment s

tanda

rds r

e-

by Tom M

ase

Michigan State

Universi

ty

Ea

op

et

Golf D

river

s

Golf D

river

sGo t

he D

is

10

Conce

pt To

Realit

y / W

inter 20

03

TH

E A

RT

OF

I NN

OV

AT

I ON

Enginee

ring en

thusia

sts

and s

oftwar

e tee-o

ff to

optimize

club d

river

perfo

rman

cewhil

e

ensu

ring co

nform

ance

to

he R

ules o

f Golf

.

stan

ce

spon

sibili

ties.

A state

-of-t

he-art,

14,00

0-sq

-ft te

st

center

was

open

ed in

1984

and i

s staf

fed by

a ca

dre of

engin

eers

and s

cientis

ts. T

he test

center

stud

ies th

e

whole ga

mbit of

golf e

quipm

ent a

nd dev

ices.

Behind t

he buil

ding i

s a le

vel, i

nstrum

ented

range

onto

which go

lf rob

ots hit

shots

. In th

e past

, most

of

the t

ime w

as sp

ent o

n testi

ng ball

confor

mation

to

the r

ules a

nd pub

lishin

g a lis

t of a

ccep

table

balls

. In

the m

id-19

70s,

the l

ist fi

t on on

e typ

ewrit

ten pa

ge;

now it

is a

book

of th

ousan

ds of

balls

.

The USGA W

eb si

te indic

ates t

hat ap

proxim

ately

20,00

0 ball

s are

tested

each

year.

Biom

echan

ics sc

i-

ence

s are

emplo

yed t

o unde

rstan

d how

the h

uman

as-

pect

of th

e gam

e is c

hangin

g. Rec

ent y

ears

have s

hown

an in

crease

in dr

iver c

lub hea

d test

ing and t

he form

a-

tion of

assoc

iated

regu

lation

s. Tod

ay, b

all an

d equ

ip-

ment m

anufac

ture

rs su

bmit th

eir pro

ducts t

o the

USGA to va

lidate

that

the e

quipm

ent c

onfor

ms to t

he

Rules o

f Golf

.

Why d

oes th

e USGA m

ake e

quipm

ent ru

les? U

SGA

Preside

nt Ree

d Mac

kenzie

says

for th

ree re

asons1 :

1.The o

verw

helmin

g majo

rity o

f golf

ers be

lieve

there

shou

ld be

rules

abou

t equ

ipmen

t.

2.Equ

ipmen

t rule

s hav

e bee

n the r

espon

sibilit

y of

the U

SGA for o

ver a

centur

y.

3.As a

n inde

pende

nt ruli

ng bod

y, th

e USGA has

no finan

cial in

terest

and c

an lo

ok ou

t for t

he goo

d of

the g

ame.

Dynamics R

ule

The Rule

s of G

olf is

a con

cise a

nd eloq

uent se

t of 3

4

rules.

Rule

4 is

titled

“Club

s,” fo

r the o

bviou

s rea

son.

Muc

h of th

is rule

has to

do w

ith th

e play

ing of a

roun

d.

For insta

nce, h

ow man

y clu

bs may

a play

er ca

rry?

Four

teen. F

urth

er, it

state

s, in

genera

l, these

club

s

shall

confor

m to A

ppen

dix II

of th

e Rule

s.

Appen

dix II

spell

s out

the m

eaning o

f USGA co

n-

forming c

lubs r

egard

ing shaft

strai

ghtn

ess, o

ffset,

grip,

groov

es (re

membe

r that

one?)

and a

newer

criter

ion

know

n as “c

oeffic

ient o

f resti

tution

” (COR).

This last

criter

ion is

in App

endix

II, 4

-1e, a

nd is c

ommon

ly re-

ferred

to as

“4-1

e” or

“COR.”

Curren

tly, t

he USGA

limits

golf c

lubs t

o hav

e a C

OR less

than

0.83

0.

The conce

pt of C

OR rests

in th

e engin

eerin

g

realm

of ri

gid bo

dy dy

namics

, the s

tudy

of how

rigid

bodies m

ove and ac

celer

ate u

nder th

e act

ions o

f

forc

es. T

he oper

ativ

e word

her

e is “

rigid

.” In

the

real w

orld,

defor

mation

take

s plac

e betw

een co

llid-

Conce

pt To

Realit

y / W

inter 20

03

OOn golf courses around the world, plenty of great golfhas been played this summer at the professional and am-ateur levels. Off the links, an ongoing drama relating togolf club performance also has been playing out.

At issue is the performance of golf club drivers.Specifically, golf’s ruling bodies have independentlydetermined a rule that sets a uniform, worldwide stan-dard for “spring-like” effect in driving clubs. The or-ganizations will rigorously test golf equipment for con-formity to this and the other Rules of Golf.

What this means for equipment manufacturers isthat they must develop their products to meet the ap-propriate standards. What’s more, the clubs have to bepleasing in the eyes of the consumers yet be traditionaland plain in shape in the eyes of golf’s ruling bodies.

One way equipment manufacturers can achievethese goals is through the use of computationalanalyses, including recent advancements in optimi-zation software and methods. A recent evaluation ofa golf driver head demonstrates that using such simu-lation technology can improve the design, resultingin drivers that are structurally optimized to the per-formance limit allowed by the Rules of Golf.

Equipped for ExcellenceGolf heritage has produced two ruling bodies: the

Royal and Ancient Golf Club of St. Andrews(R&A) and the United States Golf Association(USGA). The USGA is, in a way, the younger off-spring of the R&A. Formed on December 22, 1894,

the USGA is a nonprofit, membership-driven organ-ization that governs the game of golf in North Amer-ica. Even though Canada has its own golf associa-tion, the Royal Canadian Golf Association, it defersto the USGA on many matters. For the most part,the rest of the world is governed by the R&A.

In terms of facilities, the USGA has taken sub-stantial steps to support its equipment standards re-

by Tom MaseMichigan StateUniversity

EaopetGolf DriversGolf Drivers

Go the Dis

10Concept To Reality / Winter 2003

Reprinted with permission from the

Winter 2003 edition of

T H E A R T O F I N N O V A T I O N

Engineering enthusiastsand software tee-off tooptimize club driverperformance whileensuring conformance tohe Rules of Golf.

stance

sponsibilities. A state-of-the-art, 14,000-sq-ft testcenter was opened in 1984 and is staffed by a cadre ofengineers and scientists. The test center studies thewhole gambit of golf equipment and devices.

Behind the building is a level, instrumented rangeonto which golf robots hit shots. In the past, most ofthe time was spent on testing ball conformation tothe rules and publishing a list of acceptable balls. Inthe mid-1970s, the list fit on one typewritten page;now it is a book of thousands of balls.

The USGA Web site indicates that approximately20,000 balls are tested each year. Biomechanics sci-ences are employed to understand how the human as-pect of the game is changing. Recent years have shownan increase in driver club head testing and the forma-tion of associated regulations. Today, ball and equip-ment manufacturers submit their products to theUSGA to validate that the equipment conforms to theRules of Golf.

Why does the USGA make equipment rules? USGAPresident Reed Mackenzie says for three reasons1:

1. The overwhelming majority of golfers believethere should be rules about equipment.

2. Equipment rules have been the responsibility ofthe USGA for over a century.

3. As an independent ruling body, the USGA hasno financial interest and can look out for the good ofthe game.

Dynamics RuleThe Rules of Golf is a concise and eloquent set of 34

rules. Rule 4 is titled “Clubs,” for the obvious reason.Much of this rule has to do with the playing of a round.For instance, how many clubs may a player carry?Fourteen. Further, it states, in general, these clubsshall conform to Appendix II of the Rules.

Appendix II spells out the meaning of USGA con-forming clubs regarding shaft straightness, offset, grip,grooves (remember that one?) and a newer criterionknown as “coefficient of restitution” (COR). This lastcriterion is in Appendix II, 4-1e, and is commonly re-ferred to as “4-1e” or “COR.” Currently, the USGAlimits golf clubs to have a COR less than 0.830.

The concept of COR rests in the engineeringrealm of rigid body dynamics, the study of how rigidbodies move and accelerate under the actions offorces. The operative word here is “rigid.” In thereal world, deformation takes place between collid-

Concept To Reality / Winter 2003

12Concept To Reality / Winter 2003 www.altair.com/c2r

ing bodies, reducing the mo-mentum transferred fromone body to another. For ex-ample, if the deformationwere perfect without energylosses, a ball dropped on thefloor would bounce up to itsoriginal height. If a ball of artist clay were droppedon the floor, it would deform and stick to the floor.In most impacts, bodies act somewhere betweenthese two extremes.

COR is the parameter used in rigid body dynamics todistinguish between these extremes. For now, considerthe ball bouncing off the fixed floor. If a body bouncesto the same height from which it was dropped, it has aCOR of 1. A body that hits and sticks has a COR of 0.The height a body bounces up is defined by the CORbetween the two bodies (e.g., ball and floor). When thefloor is stationary, a collision having a 0.822 CORvalue will cause the ball to bounce up 82.2 % of theheight from which it is dropped.

When both bodies are moving, the COR defini-tion requires a more precise analysis. Remember, theCOR depends on the deformation in both bodies.

Here is the full blown definition of COR: The COR isthe negative ratio of relative post-impact velocities to rela-tive pre-impact velocities.

What exactly is meant by relative velocities? Iftwo balls move towards one another at 100 mph, therelative velocity is 200 mph. In our simple case fromthe previous paragraph, one velocity was zero.Hence, the relative velocity is just that of the movingbody. Relative velocity reflects the vector nature ofvelocities.

To measure the COR for a club head and ball colli-

sion, the pre- and post-impact velocities for both theclub and the ball must be measured. COR is then asimple calculation. As the ball approaches the club, itpasses through a pair of ballistic screens measuring thetime elapsed to travel a fixed distance. Thus, the in-bound velocity is measured. As the ball rebounds, theballistic screens work the same way in reverse order.

The club head is at rest before impact, so the clubhead’s post-impact velocity is all that’s needed tocompute the ball-club head COR. Use of physics’conservation of momentum allows for the computa-tion of the COR from the ball’s pre- and post-impactvelocities. This little trick makes for a more econom-ical experiment, but it also confuses some becausethe resulting formula has mass terms. Recall the def-inition of COR is mass-independent.

Here is the bottom line on COR: Increasing CORfor a given club head velocity and ball will increasethe ball’s initial rebound velocity. Thus, the golf shotshould travel farther.

Getting to the COR with CAEDesigning golf clubs with a COR of less than 0.830

can be tricky. However, employing simulation soft-ware enables manufacturers to evaluate a multitudeof designs without having to cast a single part.

One such simulation package is Altair Hyper-Study, an open architecture optimization tool thatcan be used in conjunction with any finite-elementsolver. Using HyperStudy, with LS-Dyna3D soft-ware, our objective was to obtain the maximum pos-sible COR of the club head while maintaining a clubhead mass of 200 g and keeping club head stress lev-els below the material yield of 150 ksi.

The optimization problem is defined by the specifi-cation of an objective function, constraints and designvariables. The model responses that are used for the ob-jective and constraints are limited only to quantitiesthat can be obtained in the solver output. Through thenotation convention of HyperStudy, any value in theinput deck can be defined as a design variable. Thus,the procedure involved is extremely general.

Figure 1. Definitions of regions on the golf club head.

Figure 2. Resultingdriver shapes formaximum shapevariable values.

Hosel

Face

SmileSole

Toe

Crown

SkirtHeel

Longer Wider Taller

Reprinted with permission from the

Winter 2003 edition of

T H E A R T O F I N N O V A T I O N

Structural optimization problems are distinguishedby the type of design variables utilized. Generally, sizeoptimization considers the effect of thickness whileshape optimization refers to the modification of geo-metric shape. For a golf driver, which is typically a hol-low structure, the key thickness variables are in theface of the club, but the other wall thicknesses alsocontribute to performance. In optimizing a driverhead, both size and shape optimizations should be con-sidered simultaneously.

In our recent CAE study to evaluate the COR for agolf head, three different shape variables were defined:longer in the toe-to-heel sense, wider in the face-to-back sense and taller. (See Figure 1 for golf club headnomenclature.) As part of the evaluation process, thesoftware alters the model slightly without distorting itor affecting the accuracy of the computation. A linearinterpolation is used to define the shape variables of allinternal nodes lying within the domain. The shapevariable changes can be animated to provide visualverification of the vectors and nodes selected. Figure 2shows the altered shapes of the three shape variablesdefined in the present study.

In addition to the three shape variables, 10 size vari-ables were defined in the optimization problem. Theclub face was divided up into five regions in a bulls-eyepattern while the remainder of the club head was di-vided into five other regions. All regions were defined asoptimization variables. The skirt and hosel thicknessvalues were not considered in the optimization problem.

The power of the optimization process can be seenin looking at the objective function as the design vari-ables are perturbed. Figure 3 shows the design variableevolution and the corresponding objective functionand constraint evolutions. Here, the COR signifi-cantly increases once the response surface has beendefined by initial iterations.

One caveat about the simulated optimization re-sults shown here: The ball model used did not haverate-dependent rubber properties. Rate-dependentmaterials would exhibit damping, which could possi-bly reduce the COR predicted. To get predictive CORnumbers, it is essential to have representative, accu-rate, validated material properties. However, a majorequipment manufacturer recently told me it has meas-ured club heads having COR as high as 0.91.

One final optimization result to point out is thatthe optimized shape has not radically been alteredinto something that doesn’t look like a golf club

head. The combination of shape and size variableshas given the program enough design room to greatlyenhance COR—resulting in a golf ball that travelsfarther—while maintaining a good golf head shape.

With the maturity and sophistication of CAE, op-timization software can now drive the design processversus its historical role of design evaluation and re-finement. As this program demonstrates, robust opti-mization tools, like HyperStudy, can play a key rolein the product development cycle, enabling manu-facturers to virtually design, optimize and validatenumerous concepts as well as save time and money inthe downstream prototype build and test phases.What’s more, optimization software can be appliedto a range of engineering problems, from golf driversto space craft, making it a “must-have” tool in manu-facturing’s arsenal of technological solutions.

Tom Mase is a Visiting Associate Professor at MichiganState University and is affiliated with its CompositeMaterials and Structures Center.

Acknowledgement: The author would like to acknowledgethe help of Eric A. Nelson, Altair Engineering, in theHyperStudy work herein.

LS-Dyna3D solver technology was developed by and isowned by LSTC. For more information visit www.lstc.com.

1 Referencewww.usga.org/about/Perspective/march_april_2002.html

13www.altair.com/c2r Concept To Reality / Winter 2003

C.O.R. Club Mass Peak Stress

Coef

ficie

nt o

f Res

titut

ion

0.92

0.91

0.9

0.89

0.88

0.87

0.86

0.85

0.84

0.830 4 8 12 16 20

Iteration #

Mas

s (g

)

270

260

250

240

230

220

210

200

1900 4 8 12 16 20

Iteration #

Stre

ss (k

si)

200

190

180

170

160

150

140

130

110

120

0 4 8 12 16 20Iteration #

Problem Statement:Objective - Maximize C.O.R.UB Constraint - Peak Stress < 150 ksiUB Constraint - Mass < 200 g

Figure 3. Resultingdriver shapes formaximum shapevariable values.

To request a complimentary copy of the technicalpaper of this case study, visit www.altair.com/c2r or

check 3 on the reply card.

Reprinted with permission from the

Winter 2003 edition of

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