Both scalar and vector operations are described to demonstrate usefulness of supercomputers (computers with peak computing speeds exceeding 100 million operative per second) in solving tribological problems. A simple kernel of the film shape calculations in an elastohydrodynamic lubricated rectangular contact is presented and the relevant equations are described. Both scalar and vector versions of the film shape code are presented. The run times of the two types of code indicate that over a 50-to-1 speedup of scalar to vector computational time for vector lengths typically used in elastohydrodynamic lubrication analysis is obtained

Hamrock, B. J.

English

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FILM SHAPE CALCULATIONS ON SUPERCOMPUTERS
Bernard J. HamrW;
National Aeronautics and Space Administration
Lewes Research Center
Cleveland, Ohio 44135
Abstract
A brief description of supercomputers is presented along with a definition and
illustration of vectorization. To demonstrate the usefulness of thef;e super-
N	 computers in solving tribological problems, a simple kernel of the 11'Ml shape
► 	 calculations in an elastohydrodynamic lubricated rectangular contact is pre-
w sented. The relevant equations are briefly described. Both scalar and vector
versions of the film shape code are presented. The run times of the two types
of code indicate that over a 50-to-1 speedup of scalar to vector computational
time for vector lengths typically used in elastohydrodynamic lubrication anal-
ysis is obtained.
Introduction
The present day powerful computing machines known as suipercomputers have peak
computing speeds exceeding l-0 nlil?ion operative per second. This is to be
contrasted with the first commerical electronic computer delivered in 1951
which had a peak computing speed of around 600 operations per second. This
means that the speed of large scale scientific computers has doubled on the
average every 2 years. Although the current performance levels of such
machines owe much to the rapid advance of microelectronics, new concepts in
computer architecture have been equally important. The term "architecture"
refers to the logical organization of the computer as it is seen by the pro-
grammer. The recent architectural innovations of greatest significance are
those that enable the machine to carry out many similar operations in par-
allel, or concurrently. This is referred to as vector computation. The
latest supercomputers allow the programmer to specify that many different ele-
mentary steps be executed simultaneously, whereas earlier computers obliged
the programmer to break his computational program down into a sequence of ele-
mentary steps that would be executed one at a time. This is referred to as
scalar computation. The speed with which signals can be propagated from one
part of the computer to another is limited by the speed of light, which is
close to 0.3 m per nanosecond. To get any further improvement in computation
speed meant that innovative approaches to the basic architecture of the com-
puter were required. These circumstances brought about the development of the
latest supercomputers which utilize vector operations to achieve faster oper-
ational speeds. Levine (1982) gives a thorough and useful description of
supercomputers.
The latest supercomputers represent a quantum jump in the speed of operation
of large scientific computers, and their effect in solving tribological prob-
lems of the next decade should be significant. The present paper will attempt
to demonstrate the effect of using these computers in performing film shape
calculations in solving elastohydrodynamic lubrication problems. Comparisons
will be made between scalar and vector operations to demonstrate the utility
of new computers. The film shape calculations were chosen since in elastohy-
drodynamic lubrication calculations over 80 percent of the computation time is
spent in the film shape calculations, For simplistic considerations a rec-
tangular contract (sometimes referred to as a line contact) is studied rather
than an elliptical contact,
Film Shape Formulation
Only the film shape calculations of the elastohydrodynamic lubrication analy-
sis will be considered in this paper. The film shape for a rectangular con-
tact can be written as:
h- ho +SW +a(x)	 (1)
where;
ho - constant
SW - separation due to geometry of undeformed solids
a(x) - elastic deformation
The separation due to the geometry of the underformed rollers shown in figure
1(a) can be described by an equivalent cylindrical solid near 
	
plane as shown
in figure 1(b). The gaimetrical requirement is that the separation of the two
rollers in the initial and equivalent situations should be the same at equal
values of x. Therefore the separation due to the underformed geometry of the
two rollers can be written as:
S(x)	 x-
x
(2)
where:
1	 1	 1
llx
m r
ax	 ray (S)
Equation (2) makes use of the well-known parabolic approximation to the circu-
lar section of the solid and is valid as long as the separation is much small--
er than the radius cif,-durvature.
From Timoshenko and Goodier (1951), the elastic deformation at a point of a
semi-infinite solid subjected to a pressure p along the line x - xl, can be
written as:
a F'
t ' a(x) - —	 p	 ln(x -- x 1 ) 2 dx l (Q)
1 2
a
d k (x) _ 2 F Pi Dj (g)
i=1,2,...
(10)
where:
2	 (6)
E =1-v^^1-v^
as
Ea ^
v - Poisson's ratio
E R modulus of elasticity
Figure 2 shows a rectangular area of uniform pressure. If the pressure is
assumed to be uniform over the rectangular area the pressure can be put in
front of the integral in equation (4). Therefore the integration can be per-
formed, resulting in the following:
a(x)	
n 
P D	 (6)
where:
D = b [(X - B)ln(x - B) 2 - (X + B)ln(X + B)' + 4B(1 - In b)] 	 (7)
n^ci	 (R)P	 p^ G
b = semiwidth of Hertzian contact
X = x/b, dimensionless coordinate
B ° IS/ b " rn
n	 number of nodes within the semiwidth of the contact
0
i
h
is
rf
j
a i
	
j	 .
x,
S	 x
i
i y
}: f
j'.
In equation (6) the elastic deformation is found to be equal to a constant
multiplied by the dimensionless pressure multiplied by a distance influence
factor. Now the term s(x) in equation (6) represents the elastic deformation
at a point A due to a rectangular area of uniform pressure and width 2 76. If
the conjunction is divided into a number of equal rectangular areas, the total
deformation at a point A due to the contribution of the various rectangular
areas of uniform pressure in the conjunction can be evaluated numerically.
The total elastic deformation caused by the rectangular areas of uniform pres-
sure within a conjunction can be written as:
where:j= Ik	 +1
t
V
+ 1 X2(70) + ?
	 p 
D7xn
	 i
i•1,2,..^
(11)
h
Hk " 17 . Hax
R
Therefore substituting equations (2) and ( g ) into equation (1) while writing
the film shape in dimensionless form gives;
0
Scalar and Vector Programs
Having formulated the film shape, the next task was to program it both scalar-
ly and vectorially. Before proceeding it might be well to point out more spe-
cifically what is meant by scalar and vector coding. Scalar coding is conven—
tional coding that has been in existence since the first commerically avail-
able computer in 1951. Vectorization, or the process of writing programs in
vector code, consists of designing, organizing, and writing programs so the
maximum possible number of arithmetic and logical operations are executed as
vector instructions. Vectorization is most easily introduced and illustrated
by example. Consider arrays A and B, each consisting of 100 numbers. Assume
that.one wishes to compute array C, where Ci m Ai + Bi, i
	
1,	 100. The
traditional ""scalar" computer executes five assembly language instructions 100
times. There are two memory fetches (Ai and Bi), one addition, one store to
memory (for C i ), and an instruction that increments a counter, tests, and
branches back to load the next pair of input operands. Thus, 500 scalar inL
struct Ioi"s are executed to add arrays A aild u. A vector computer "s compiler
can generate "vector" object code, which executes very differently. The vec-
tor code for adding the 100 pairs of operands consists of a singe hardware
instruction. The vector instruction executes by continuously streaming oper—
ands from central memory into the central processor, where the addition takes
place, and continuously streaming answers back to memory. During execution of
the vectorized addition, some elements of A and 5 are being read from central
memory, some elements of A and B are undergoing addition, and some are being
written in memory.
Figures 3 and 4 show the computer code for the film shape of the scalar and
vector kernels, respectively. The only calculations given in these figures
and not f oomulated in the earlier portion of the paper are the calculations
of T where:
rz
(12)
^h
E
T - QH3/2 w (1 — e—GP)H3/2/G
and:
G - aE", dimensionless materials parameter
a - pressure viscosity coefficient of lubricant
In the vector film shape code (figure 4) the expressions are written in vector
form throughout and the IF Statement in figure 3 is completely eliminated.
Vectorization requires the elimination of IF, CALL, and other interrupts with-
in  Da loops.
4
Results
Table I shows the scalar and vector computation times expressed in nanoseconds
for varying vector length. A considerable decrease in computation time is
seen to exist for the vector computation over that of the scalar computa-
tion.
	
The last column of the table gives the ratio of scalar to vector com-
putation times. This column indicates that is the vector length increases; the
scalar to vector time ratio increases si gnificantly until the vector length
becomes too large and an asymptotic condition is approached, Tile vector
length used by Namrock and Jacobson (1982) in evaluating elastohydrodynamic
lubrication of rectangular contacts was 660. For this example, the table in-
dicates that over a 50-to-1 speedup in the film shape calculations can be ob-
tained on supercomputers that utilize vector computation. Figure 5 shows the
results presented in Table I in graphical form. here the leveling off of the
scalar to vector computation times as the vector length becomes very large is 	 i
,r	 quite apparent. In a typical elastohydrodynamic lubrication run the film
shape code shown in figure A is used 2000 times. 	 a'
Conclusions
A brief description of supercomputers" capabilities has been briefly presen-
ted. To demonstrate their usefulness in solving tribological problems, a sim-
ple kernel of the film shape calculations in an elastohydrodynamically lubri-
cated rectangular contact was presented. Both scalar and vector versions of
t;le film shape code were presented. The runtimes for the two types of code
indicate that over a 50-to-1 speedup of scalar to vector computation times is
obtained for vector lengths typically used in elastohydrodynamic lubrication
analysis. The significance of these results should prove useful in consider-
ing additional aspects in elastohydrodynamic lubrication such as:
(1) Thermal effects
2 Surface roughness effects
3 Non--Newtonian effects of the fluid
Q Side leakage effects
The use of supercomputers in this decade for the incorporation of these
coupled effects should reduce run times considerably and thereby make such
calculations more feasible.
References
Q
1. Namrock, B. J. and B. Jacobson (1982) Elastohydrodynamics Lubrication of
Rectangular Contacts, submitted to be presented at the 1982 Joint Lubri-
cation Conference and later published in JOLT of the American Society of
Mechanical Cngineers.
2. Le%,ine, R. 0., "Supercomputers," Sci. Am., 246, (1), 118-135 (1982).
3. Timoshenko, S. and Goodier, J. N., Theory of ElastiG^, 2nd ed., McGraw
Hill (1951).
5
Rx
ORIGINAL ^°^^ ^t--)
OF POOR QUALITY
TABLE 1. - COMPARISON OF SCALAR AND VECTOR COMPUTER RUN TIMES
FOR DIFFERENT VECTOR LENGTHS
Vector
length,
N
Scalar time,
nsec
Vector time,
nsec
 Scalar time
ect9r	 me
10 241 70 3.44
30 1 17a 155 7:61
50 2 726 243 11.22
100 9 252 483 19.28
300 73 357 1 933 37.95
500 198 282 4 188 47.35
700 383 997 7 242 53.02
1000 776 588 13 363 58.11
3000 6 890 354 99 988 68.91
5000 19 084 530 266 602 71.58
7000 37 358 446 513 218 72.79
10000 76 171 860 1 033 169 73.73
Figure 1, - Rollers and equivalent roller.
P
2b
r
P
X	 A	 +^	
X
X_X1	 X1
f	 Figure 2. - Surface deformation of semi-infinite body subjected to
uniform pressure over a rectangular area.
k,
i?
f'
DO8I-I,NX
SUM-,ODO
DO 1011 .
 1, NX
M - IABS (I-11) + I
" 10 SUM - SUM+ PR (II) *D(M)
Wt? - 2.ODOOSUM/Pi
H(I) - HOI+RX*(S(I)+W(I))
IF (H(I), GT. HMIN) GO TO 4000
HMIN - H(I)
NSAVE - I
400 CONTINUE
8 PHI(I) - Q. ODO-DEXP(-G*PR(I)))*(H(I)**I.50DO)IG
Figure 3. - Scalar film shape code,
GI . L OIG
C DO 500 I, NX
DD(I) -D(NX+I-I)
500 CONTINUEC
DO 510 1-1, NX
DD(I+NX-1) - D(I)
W(I) n PR(1)*DD(I+NX-1)
510 CONTINUE
C` DO 530 K - 2, NX
s° DO 520 I -1, NX
W(I) - W(I)+PR(K)*DD(I+NX-K)
520 CONTINUE
! 530 CONTINUE
C DO 5401 . 1, NX
$ WW - (2, 01PI)4(I)
H(I) - HOI+RXO(S(I)+W(I))
PHI(I) - (1.0-EXP(-G,^PR(I)))*(H(I)OSQRT(H(I)))*GI
C 540 CONTINUE
- NSAVE - Q8SMINI(H(1 NX))+1
HMIN - H(NSAVE)
t
C
' Figure 4. - Vector film shape code,
n,wngnn
4
101
	
2 3 4	 6 81
	 2	 4	 6 81	 2	 4	 6 81
	
x10 1	x102	 x103
VECTOR LENGTH
Figure 5, - Effect of scalar run time divided by vector run time on various vector
lengths,
80
70
60
tg
x
50
40
UN
30
20
. „...	
-^...,,..w.ica.ss ..^v. _ ,nx'rosrc--.._.G,..:....w .:wx.:dxaexi+T:+auamm-•.^-." 	 ^,....-..-«-..«,^.., ... ,a
	
...._...,....» ...... x+.
	
.s -;.. ^....^w..,,.........,._
f^


Film shape calculations on supercomputers

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