The development of a multicolored successive over relaxation (SOR) program for the finite element machine is discussed. The multicolored SOR method uses a generalization of the classical Red/Black grid point ordering for the SOR method. These multicolored orderings have the advantage of allowing the SOR method to be implemented as a Jacobi method, which is ideal for arrays of processors, but still enjoy the greater rate of convergence of the SOR method. The program solves a general second order self adjoint elliptic problem on a square region with Dirichlet boundary conditions, discretized by quadratic elements on triangular regions. For this general problem and discretization, six colors are necessary for the multicolored method to operate efficiently. The specific problem that was solved using the six color program was Poisson's equation; for Poisson's equation, three colors are necessary but six may be used. In general, the number of colors needed is a function of the differential equation, the region and boundary conditions, and the particular finite element used for the discretization

Ortega, J. M.

English

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An Annual Report
NUMERICAL ALGORITHMS FOR FINITE ELEMENT
COMPUTATIONS ON ARRAYS OF MICROPROCESSORS
Submitted to:
Division of Structures and Dynamics
NASA/Langley Research Center
Hampton, VA 23665
r	 Attention: Mr. David Loendorf
Submitted by:
J. M. Ortega
Professor and Chairman
EC c
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Report No. UVA/528190/AMCS81/101
March 1981
(NASA-Ca-164008) NUMERICAL ALGORITHMS FOA
FINITE ELEMENT C:OMPUTATIUNS UN ARRAYS OF
!MICROPROCESSORS semiannual Report, 1 Sep.
1980 - 28 Feb. 1981 (Virginia Univ.) 14 p
HC A02/MF AU1	 CSCL 09B G3/61
No1-197.94
UIICIdS
41d4u
SCHOOL OF ENGINEERING AND
APPLIED SCIENCE
DEPARTMENT OF APPLIED MATHEMATICS
AND COMPUTER SCIENCE
UNIVERSITY OF VIRGINIA
CHARLOTTESVILLE, VIRGINIA 22901
An Annual Report
NUMERICAL ALGORITHMS FOR FINITE ELEMENT
COMPUTATIONS ON ARRAYS OF MICROPROCESSORS
Submitted to:
Division of Structures and Dynamics
NASA/Langley Research Center
Hampton, VA 23665
Attention: Mr. David Loendorf
Submitted by:
J. M. Ortega
Professor and Chairman
Department of Applied Mathematics and Computer Science
RESEARCH LABORATORIES FOR THE ENGINEERING SCIENCES
SCHOOL OF ENGINEERING AND APPLIED SCIENCE
UNIVERSITY OF VIRGINIA
CHARLOTTESVILLE, VIRGINIA
I
3
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Report No. UVA/528190/AMC581/101 	 Copy No.
March 1981
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IThis report summarizes work performed under NASA Grant NAG1-46 during
the period September 1, 1980 to February 28, 1981 and constitutes the second
semi-annual report. This work has been done under the technical monitorship
of David Loendorf of Langley Research Center.
The primary accomplishment during this period has been the development
of a multi-colored SOR program for the Finite Element Machine. The multi-
colored SOR method, described in detail in the last semi-annual report, uses
a generalization of the classical Red/Black grid point ordering for the SOR
method. These "multi-colored" orderings have the advantage of allowing the
SOR method to be implemented as a Jacobi method, which is ideal for arrays
of processors, but still enjoy the greater rate of convergence of the SOR
method.
i
The multi-colored SOR program was written in PASCAL and has run success-
;
	
	 fully on the current four processor version of the Finite Element Machine.
The program has been written for a general nxn array of processors, however,
and should be easily convertible to the 36 processor array when that is
operational.
The current program was written to solve a general second order
E
	 self-adjoint elliptic problem on a square region with Dirichlet
i	 boundary conditions, discretized by quadratic elements on triangular
regions. This problem is described in more detail in the Appendix to
this report. We note that for this general problem and discretization,
six colors are necessary (and sufficient) for the multi-colored method
to operate efficiently. The specific problem that was solved using the
six-color program was Poisson's equation; for Poisson's equation, three
colors are necessary and sufficient but six may be used.
In general, the number of colors needed will be a function of the
differential equation, the region and boundary conditions, and the
particular finite elements used for the discretization. One of the
►:roblems currently being investigated is how to determine in a systematic
way the proper number of colors and the grouping of the unknowns into
- 2 -
the processors. Proper processor assignment is, of course, crucial for
the success of the method. In the Appendix a rough analysis is given of
some of the questions that arise in processor assignment. The six and
three color orderings may be found in Figures 1 and 2 respectively.
The program has thus far been run on two test problems:
Problem 1:	 Uxx+ Uyy= 4 in the unit square
U a
 x2 + y2 on the boundary
Problem 2:	 Uxx+ Uyy = 1	 in the unit square
U = 0	 on the boundary
Problem 1 is a standard mathematical test problem. Problem 2 is a model for
a fixed boundary membrane and was suggested by the project monitor, David
Loendorf. The results on both problems have been very encouraging.
In addition to the program development, a number of questions relating
to the efficiency of the multi-color method have been investigated.
1. Will the iterates produced by the multi-color SOR method converge?
2. What is the rate of convergence of the method, espec'ally as a function
of the overrelaxation factor?
3. How does one choose an optimum (or good) overrelaxation factor?
The answer to the first question is quite easy if the coefficient matrix
of the system is positive definite. In this case, the multi-color ordering
is just a permutation transformation of this matrix, and the permuted matrix
is also positive definite and SOR will converge.
The other two questions are more difficult. The coefficient matrix does
not have Property A, the classical condition of Young which allows a corre-
- 3 -
spondence between the eigenvalues of the SOR iteration matrix, and the Jacobi
iteration matrix. Nevertheless, the computer results so far indicate that the
rate of convergence of the multi-color SOR method applied to either Problem 1
or 2 above behaves, as a function of the relaxation factor, very much like the
SOR method applied to the classical 5-point finite difference discretization of
the Laplace's equation. These experimental results are very encouraging, since
SOR on the 5-point discretization is essentially a best case situation. Work
is continuing on attempting to understand why the experimental results are true.
It is anticipated that ir, the near future, additional more demanding probl
will be attempted on the Finite Element Machine; in particular, Ms. Adams is
planning to spend most of the summer of 1981 at Langley Research Center. We
also plan to continue our investigation of certain variants of the conjugate
gradient method, as an alternative to the multi-colored SOR method.
APPENDIX
The general problem considered is the second order self-adjoint
elliptic problem
Lu(x,y) _ -f(x,y)
	
in _11.. , the unit square
u(x,y) = g(x,y)
	
on a.L
where
Lu = au -(bux)x -(cuy)y
where a,b,c are functions of x and y.
MATHEMATICAL SOLUTION (Quadratic Elements on Triangular Regions)
S(au -(bux)x -(cuy)y) v = J -fv
-1L	 .rt..
Using the divergence theorem, the left side becomes:
S(auv + buxvx + cuyvy) - S (buxnl + cuyn 2 ) v
- .fL	 a4-
where n = n l el + n2e2 is the normal to D4.
Note that for Dirichlet boundary conditions, the natural boundary condition
buxn l + cuyn2 will always be zero on tJL. Therefore, the equation to be solved
is:
S(auv + buxvx + cuyvy )	 = J -fv
Now, to solve Poisson's Equation, U xx + Uyy = f , set a=0, b=c=1 to get,
(1)	 (uxvx + uyvy ) _	 -fv
Discrotize the unit square with M points in each row and column including the
1 2+-i
boundary points. Let u =I ^ u i j ^ i j + —75^' ubp V' by	 and v - ^kj
e_1j_1	 be
where the ^ 's are the basis functions for Galerkin's Finite Element Method
and the fl bp 's represent, those basis functions associated with boundary points.
t
i^	 1
With these substitutions and definitions, (1) now becomes the following
(2n-1)x(2n-1) system of linear equations for the solution of the interior
points (values of u at discrete points H/2 units apart in both the x and
y directions).
J 1'13%i ^'^x^ ... f rr^',r,»'r^ ••• J ^'t,^,+ i^,r.^ ux,n
f ^
nn♦• 	
`^
^
•
1^011
-I FO# Djo
bP
-1f0m ^rA; be
Ito
By using the six basis functions for quadratic elements and performing
Lit e integration over a standard triangle; the integrals of the coefficient
m.itrix can be evaluated. Likewise, by using a numerical technique, the
integrals on the right side can be evaluated.
CONN ECT I V i 'iY
1ho . 11.0tui a 4sf the basis functions ( 0's) will caust , the coeff icient
mtrix to be sparse. In particular, 0 ij can be nonzero only on triangles
the point i j is on. This implies that ^ i j is nonzero on either G or 2
triangles. This connectivity is illustrated for 
^A , ^b , ^` , and /L
below:
3t
r ^	 f a•
17 ^ 1f	
.r...
.^
ae
A: C'	 .
u
Note that 
r A 
is nonzero on triangles T 1 , T 2 , 1. 3 , TG, 
"'50 and Tb.
HvIlce, only J fOA • 0 1	 , (	 , ^/2 ,	 , an" f '^YA ^^^18
.!L I	 ^lJ 7'^	 A
need to be evaluated. 	 "Thus the iterative equation for the next value of
U  will only contain U 1 , U 2 , ... , and U fa . From the connectivity of
C O and tJ U a similar analysis can be made for the solutions of U u , UU , AND l,l).
.	
4
COLORING SCHEME
.In order to solve for any of the points, say U  for exany)le, the values
of all the points Wat point A is connected to must be known. These points
must either have been calculated on a previous iteration or must have been
calculated already on the current iteration.
If processors are to be working in parallel, with each processor calcul:,ting
values of a given number of these points, it is desired that the calculation he
ordered so that each processor will not have to wait for values from other
processors before computation can begin (no starvation in a sense). 	 This
can he accomplished by assigning to each point a color in such a way 'that thr
paint will be a different color from all other points it is connected to. To
illustrate this consider the simple connectivity of the 5-star:
Clrarly two colors, red and black, will produce tho desired result. This is
illustra -d hclow:
.K	 it	 .R	 .B
.K	 111	 .K	 .IS
. It	 . K	 .11	 . It
1i
	
5
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f
	
Note that each R point only needs values from B points, and each B point
only needs values from each R point. Hence, all R points are calculated first,
using B values that were available from the last iteration, and then all B
points are calculated by using the R points that were calculated this iteration.
The effect is that two Jacobi sweeps constitute one Gauss -Siedel sweep.
The same basic idea of coloring the points can be applied to the
connectivity described earlier by using six colors. This is illustrated
below and in Figure 1. Note from the figure that all the hypotenuse points
can be the same color, R, because they are not connected. Likewise, all the
vertical sides can be the same color, G, and all the horizontal size midponts
can be the same color, B. Three extra colors are needed to color the vertices.
6
ti -	 ft	 nil	 it	 Y
	
0R	 G	 R	 G	 R	 G
	
,k	 I 	 N.,R
	 JG \tt	 !G
Y-.,__U Nn	D	 till
	
R	 G `. R ^t\. R
	 G
11MCkSSt)k A SS IG NM1:N[
•	 0111. I lon•:i	 ion ill	 (lointS to processors is t.h(- anx ►unt of
work that will be T-quired for each processor. Ideally, cacti processor should
be k0j)t busy at all times; that is, each processor should do the same fraction
of thc work. In this vasu, the Ulove statements say that ideally each prorrssor
should havv the- sa m- number of points to calculate.
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Another consideration is the algorithm that will be used by each processor.
If the points can be assigned in such a way that the algorithm in each processor
will be the same (SIMD), the task of programming in parallel is greatly reduced,
as well as the guarghtee that each processor will be doing its fraction of th+,
work.
These considerations give motivation for assinninr; to vac :h processor
the same color structure. For the simple connectivity of the 5-star and two
colors and four processors, the assignment could be as follows:
	
^^.B	 .R	 .B	 .R
	
.R	 .B	 .R	 .B
	
.B	 .R	 .B	 .11
For the problem under consideration, pleasb note from Fiaur* 1 that
P
the color structure repeats every 611 by 6R square of points. Therefore, if
Prove ; ors are assigncNl to every GR by OR squaw, vach, processor will have
tluv i0vill.ical su-LIeture and an identical algorithm. This assignilwnt is
i l l urt rat rjl l ,rlow. Not i , that because of Lite LrianCul ar iiat ion or the square
	
rer,ion, an odd numtH-r
	 of points	 will	 in each row and rolimin.
V.
r
r.	 X
v	 Qj,	 ^	 Q	 '^	 bR	 ^	 ^'
7
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A few words should be said about the boundary processors. All boundary
processors have values that do not have to be calculated, values that need riot
be sent to other processors, and values that need not be received from other pr o-
cessors. In this sense, the boundary processors could have a different algorithm
than the other processors. However, from a programming standpoint, it is
easier to test to see if a processor is on the boundary and take appropriaLm
action than to write a separate algorithm for the boundary processvc. The
algorithm	 uses a PASCAL CASE statement to determine
the type of processor. As written, the algorithm is S1MV, but the MIMU
algorithm is easily extracted from the CASE st'ttementn,
ANALYSIS OF FRO(T.SSUR ASSIGNMENT
First of all, since each processor is doing its fraction of the work,
the bast Iveduj) over a uniprocessor one can hope to obtain is U(Nl'), where
Nl' = (K+1) 2 , the number of processors. In practice, however, this will not I)c
realized since speedup will be slowed down by the processor inner communications
(that is, data transmissions).	 An analytical ex{}t'essiun for T(K), this
number of local data transmissions needed when a K by K grid of G*R by OR
pt^+^ e^sc^rs is used, is derived below but
A
K -- # of rectangles
II -- width of each rectangle
M -- # of points in each row	 M - 2n + 1 - interior	 plus
boundary
It -- determines the tilt by 61( topolugy
U-A -- width of original syuate ( 1 for this exa14110
K -- # of tilt by 6A processors in a row (see picture)
M/bH = K +(blt+l )/tilt 	 or M = 61'K + bit + 1	 (I )
it-it M ­ 2(ls-AMI + 1	 (2)
aCombine ( 1) and ( 2) to get
R = (B-A)/H(3K + 3) = R/(3K + 3)
Since K is odd, 3K + 3 is even. This implies that R must be chosen to be
even and divisible by 3K + 3. At first this appears to be a resericti:)n, but
it is not really since, if a specified R is not divisible by 3K + 3, it may
be increased until it is, thereby improving the accuracy of the solution (Note
that R increasing means H is decreasing).
It is an easy task to add up the transmissions that occur between.
processors (two-way) to get for the K by K topology the following:
T(K) = 6K 2 - 6K + 12KR	 (3)
Also, it is easy to get an expression for the number of t.ran: ;mission links
(lines) needed:
L;K) = 4(2K 2 + K)
Uf these links, almost ', narely 4K 2 , will have either 1 or 2 transmissions only.
It is interesting to note that the number of transmissions for the bR Icy 61,.
topology is bounded above and below by 3 and 2 times the number of transmis:.twts
for a 12K by 1211 topology respectively:
2T(K) < T(2K+1)<3T'(K)
or	 ( ^)
" 1 1211 x t:'it	 166 x tilt < 31'1211 x 1211
1^1 ru,
too
	 (h)
41.	 5L
 x 12k
	 hOR x 6H ^ S1 12H x 12H
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Equation (S) is encouraging;, since reducing; the blocks from 12R 	 12R I.
6R by 6R gives a potential speedup of 4 in couq )utation time but. only require::
y
between 2 :u►d 3 tittw-s as many data transmissions. If the architecture of Ili..
pat-allel c:ompuLer is"such Lhal data transmission is rapid, a potential savinr,7,
should occur. Also, if Lite architecture permits transmissions (sands and ree e i vi -0
Lo occur simultaneously with computation, the coloring; of the taints sugge sl t I,.,1
many of t; ,cse data transmissions can occur while computation is being carri—I
In other words, the an ►ount of slowdown due to the transmissions is very
archilecture dependent and should be determined experimentally as well.
DATA STKI)(AllItG USGD FOR F.ACII PROCESSOR
It was discovered that writing the alg6rithm became fairly easy once
the "correct" data structute was discovered. AL first, it was believed that
o.tvh processor should only store its color structure. However , ' Llci s ict--a
W. r; '1 11i(AIy :ihan,ItInvII Will -I it was real iied Lhai each procv ::rot q w;I receive
v.rluer. frc,m other prore::::c,r::, and these values were u:;ed to petleaps calvu al,
uu,re than c,ue poiitL in Lite color structure. 	 It, was decided to use a 61t+3 by 40:1 1
art ay	 store Use 61! by 61t r.olur sttuclure, one column of receives from the e. ­ ::t
pros essur, two columns of receives from the east processor, ot ►e row of receive:;
11411:1 t he ::(lilt Ic prune :;::or, and two rows of receives irom the nc,t Lh processor.
Ilolow is in ilIuArat ion or tho daLa sltuc t-Urr u::rd.(k=1 in the illusUation).
LCC_r.R
STtiuc r 3 ^^
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Note that this data structure applies to the boundary processors as well. The
boundary values can be thought of as receives that are stored only once. Bence,
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!	 the same data structure can be used in all the processors.
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Numerical algorithms for finite element computations on arrays of microprocessors

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