Microcomputers in Civil Engineering 12 (1997) 119–128 On Estimating Machine Dependency of Fine- and Coarse-Grained Parallel Structural Computations W. J. Watkins, D. Kennedy * & F. W. Williams Cardiff School of Engineering, University of Wales Cardiff, P.O. Box 917, The Parade, Cardiff CF2 1XH, United Kingdom Abstract: Fine- and coarse-grained parallel methods for the calculation of structural eigenvalues are evaluated using two distributed-memory parallel computers having similar architectures but a significant difference in the ratio between calculation speed and communication speed. A simple com- putational model is used to devise a simulation procedure that enables predictions to be made on one parallel computer of the solution times that would be obtained on parallel com- puters having different values of this ratio. Thus the speed-up and efficiency of parallel methods can be presented in a gen- eral form rather than being specific to the computer used to obtain them. 1 INTRODUCTION Critical buckling and undamped vibration problems of struc- tural analysis may be solved by the application of exact analytical solutions of the member stiffness equations. The Wittrick-Williams algorithm 21,22 provides a solution to the resulting eigenvalue problems that guarantees convergence on all required critical buckling loads or natural frequencies. Such analysis of three-dimensional frames 4 and prismatic plate assemblies 12 is often much more computationally effi- cient than analysis by the conventional finite-element method. The exact method locates the eigenvalues using an iterative scheme in which the real symmetric (or complex Hermitian) dynamic stiffness matrix K of the structure, whose elements are transcendental functions of the eigenparameter (i.e., the load factor or frequency), is assembled and reduced to upper- triangular form at successive trial values of the eigenparame- ter. The solution time is dominated by the triangulation of K at each iteration but may be reduced either by performing the triangulation more efficiently 19 or by using procedures 11,20 * To whom correspondence should be addressed. that reduce the number of iterations required to converge on the eigenvalues. We have recently developed efficient forms of the Wittrick- Williams method to run on parallel MIMD (multiple-instruc- tion, multiple-data) computers. These parallel forms include a “coarse-grained” parallelization 16 of the iterative process used to converge on eigenvalues, a “fine-grained” paralleliza- tion 17 of the matrix triangulation procedure, and a hybrid method 10 employing both kinds of parallelism simultane- ously. Parallel methods often have been assessed 6,7,15 by mea- suring the speed-up and efficiency obtained when sample problems are solved on a particular parallel computer using different numbers of processors. It will be shown in this paper that such an assessment may provide an unreliable indication of a method’s performance when it is run on a different par- allel computer, even one of a similar hardware configuration. A computational model is developed for distributed-memory systems with message passing and is used to show that the per- formance of a method on such a machine depends crucially on the ratio of its calculation speed to its communication speed. By introducing redundant calculations or communications, it is possible to simulate the performance of the method on ma- chines with differing values of this ratio. Numerical results have been obtained on two different parallel computers, indi- cating the validity of the computational model and its ability to predict solution times on a range of hardware. 2 OVERVIEW OF THE EIGENVALUE EXTRACTION ALGORITHM The Wittrick-Williams algorithm 22 states that J , the number of eigenvalues exceeded at any trial value F of load factor or frequency, is given by J = J 0 + s {K} (1) © 1997 Microcomputers in Civil Engineering. Published by Blackwell Publishers, 350 Main Street, Malden, MA 02148, USA, and 108 Cowley Road, Oxford OX41JF, UK.