A parallel Preconditioned Conjugate Gradient (PCG) iterative solver has been developed and implemented on the iPSC-860 scalable hypercube. This new implementation makes use of the Parallel Automated Runtime Toolkit at ICASE (PARTI) primitives to efficiently program irregular communications patterns that exist in general sparse matrices and in particular in the finite element sparse stiffness matrices. The iterative PCG has been used to solve the finite element equations that result from discretizing large scale aerospace structures. In particular, the static response of the High Speed Civil Transport (HSCT) finite element model is solved on the iPSC-860

Sobh, Nahil Atef

English

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DEPARTMENT OF MECHANICAL ENGINEERING & MECHANICS
COLLEGE OF ENGINEERING & TECHNOLOGY
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA 23529
ITERATIVE METHODS FOR LARGE SCALE STATIC ANALYSIS OF
STRUCTURES ON A SCALABLE MULTIPROCESSOR SUPERCOMPUTER
By
Nahil A. Sobh, Principal Investigator
Final Report
For the period ended May 31, 1992
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Prepared for
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665
Under
Research Grant NAG-I-1291
Ronnie E. Gilliaa, Technical Monitor
SMD-Computational Mechanics Branch
__,_ ._ _"_
Submitted by the
Old Dominion University Research Foundation
P.O. Box 6369
Norfolk, Virginia 23508-0369
June 1992
(NASA-CR-190369) ITERATIVE METHODS FOR
LARGE SCALE STATIC ANALYSIS OF STRUCTURES ON
A 5CALAqLE MULTIPRDCESSOR SUPERCOMPUTER
Fin_l Report, period ended 31 May 1992 (Old
Dominion Univ.) 15 p G3/&l
N92-26018
Uncles
0092877
https://ntrs.nasa.gov/search.jsp?R=19920016775 2020-03-17T10:53:45+00:00Z
Final Report
ITERATIVE METHODS FOR LARGE SCALE STATIC ANALYSIS OF
STRUCTURES ON A SCALABLE MULTIPROCESSOR SUPERCOMPUTER
NAHIL ATEF SOBH 1
A parallel Preconditioned Conjugate Gradient (PCG) iterative solver has been developed
and implemented on the iPSC-860 scalable hypercube. This new implementation makes
use of the PARTI (Parallel Automated Runtime Toolkit at ICASE) primitives to efficiently
program irregular communications patterns that exist in general sparse matrices and in
particular in the finite element sparse stiffness matrices.
The iterative PCG has been used to solve the finite element equations that result
from discretizing large scale aerospace structures. In particular, the static response of
the High Speed Civil Transport (HSCT) finite element model cited in reference [3] is
solved on the iPSC-860.
The preconditioned conjugate gradient algorithm implemented on the iPSC is
outline below.
We have parallelizied the following three crucial steps:
1. The matrix vector multiplication (Kp).
2. The inner products (zTr and pT(Kp) ).
3. The solution of the preconditioning step (Mz = r).
l Principal Investigator, Assistant Professor Department of Mechanical Engineering and Mechanics
1. 120
2. ro = f - Ku
3. for
4.
5.
6.
7.
8.
9.
10.
k = 1,2,.. do
if llrk_all < tolerance,
Solve Mz__l = rk-1
_T_ITk--I
Pk - zk-1 +/3kPk-x
gkT_l rk--I
uk = ttk-I + otkpk
r k = rk_ 1 -- otkKpk
11. end of for loop
exit
(Initialize displacements)
(Initialize residual 'out of balance forces')
(Iteration loop on k)
(If equilibrium is satisfied 'exit')
(Solve for the iteration vector)
(Evaluate the scaling parameter)
(Update search direction)
(Evaluate new step length)
(Update displacements)
(Update residuals 'out of balance forces')
(go to 3)
Good preconditioners are those which approximate or mimic the operational role
of the original problem. Thus the preconditioning step given in the PCG algorithm
involves solving a system, Mz = r, which is very close to the original system, Ku = f.
The communications overhead constraint imposed by MIMD machines limit the use of
highly coupled direct solvers. Therefore we have used an approximation to K that has
a loosely coupled structure. In particular, we have used a diagonal, i.e., point Jacobi
preconditioner. Jacobi type preconditioners parallelize very well on MIMD and SIMD
supercomputer architectures. Although they may look simple it has been shown that their
overall performance can be very competitive when compared to other preconditioners. We
have also implemented a block SSOR family of preconditioners. The SSOR family of
preconditioners accelerate the convergence rate but at the expense of extra computations and
less communications. The current work is focusing on the trade off between communications
and computations when using the SSOR precondifioners.
Computer listing of the C program is given in the appendix.
2
This researchhasaccomplishedthe following tasks :
1. Implementeda conjugategradientiterative solverfor symmetricandsparsematrices
on the Intel iPSC-860hypercubeat ICASE.
2. ImplementedSSORandpoint -Jacobipreconditionersto accelerateconvergenceof the
conjugategradientiterative solver and reducecommunications.
3. Generatedlarge-scalefinite elementmodelsof structuresusingthe COMET software
at NASA Langley ResearchCenter. Performedlinear staticanalysison the generated
structuralmodelson the Intel iPSC-860at ICASE_ASA-Langley.
4. A generalpurposesoftwarehasbeenimplementedto generatefull stiffnessmatrices
given the symmetricpart from COMET.
No. of CPU time (seconds)
Processors diagonal preconditioning
High Speed Civil Transport (HSCT) (16,146 equations)
CPU time (seconds)
SSOR
8 1342
16 786 764
32 565 438
Blade Siffened Panel (BSP) (1,824 equations, )
1 2.5 3.0
2 2.1 2.1
4 1.8 1.4
8 1.9 1.3
16 1.7 1.2
The table shown abovesummarizesthe performanceof the preconditionedconjugate
gradientiterative solver. The first columnis the numberof processorsusedto solve the
abovementionedproblem, the secondcolumn correspondsto the amountof cpu time in
secondsrequired to solve a given problemusingthe diagonalof the stiffnessmatrix as
a preconditioner.The third columnis the cpu time in secondsrequiredto solvea given
problemusing a symmetricsuccessiveover relaxationpreconditioner.
The speedupperformanceon the large problem (HSCT) is modest. The speedup
performance is expected to be better as the number of equations is increased. Although one
may think that a direct solver will be much more efficient on these problems, one should
not ignore the fact that direct solvers require extensive amount of storage which limit the
amount of equations one can solve. Thus iterative solvers wiU find most of their efficient
use on super large problems which exceed in their size the core memory available on a
given supercomputer. This limitation should drive one to use out-of-core solvers which
are slow compared to iterative solvers and are not easy to parallelize. Therefore efforts
should concentrate on developing efficient parallel out-of-core solvers and iterative solvers
with emphasis on parallelizable preconditioners to efficiently use the future technology
of massively parallel supercomputers.
References
,
,
*
Berryman, H. and Saltz, L A Manual for PARTI Runtime Primitives, ICASE Interm
Report 13, September 1990.
Golub, G. H. and Van Loan, C. F., Matrix Computations, The John Hopkins University
Press, 1983.
Poole, E. L., Knight, N. F. and Davis, D. D., High-Performance Equation Solvers
and Their Impact on Finite Element Analysis, Finite Element Methods in the Design
Process, ed. by Robinson, J., England, 1990
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Iterative methods for large scale static analysis of structures on a scalable multiprocessor supercomputer

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