Cray-2 Supercomputer Research Articles

Static analysis of cable networks and their supporting structures

A nonlinear static analysis method for cable structures, particularly emphasizing cable networks, is presented. Cable strains are measured from the current geometry and compressed cables are analytically disassembled with the construction of the equilibrium equation and the stiffness matrix. Finite rotations of cable intersections, referred to as nodes, and cable elements passing through more than two nodes are included in the formulation. An integrated analysis with a linear finite element method is also presented to account for the elastic deformations of supporting structures for the cable networks. The formulation is programmed for a CRAY2 supercomputer using parallel processing to solve the linear equations. Applications of the method used for mesh antenna development are also presented.

Iterative methods for nonsymmetric systems in DAEs and stiff ODEs codes

Past research demonstrated that the Newton method coupled with Krylov subspace iterative methods is more efficient than the Newton method coupled with Gaussian elimination when used to solve the corrector equation in stiff systems of Ordinary Differential Equations (ODEs)-with large and sparse Jacobian. Some Krylov subspace methods have been incorporated in the mathematical software codes (such as LSODE), which solve numerical stiff systems of ODEs. In this article we review the past results on this topic and three iterative methods for solving nonsymmetric linear systems of equations. We then incorporate these iterative methods in DASSL, which is a program for numerical integration of systems of stiff ODEs and Differential-Algebraic Equations (DAEs). We present numerical tests (with systems of stiff ODEs) and timing comparisons on the CRAY-2 supercomputer.

An efficient nonsymmetric Lanczos method on parallel vector computers

In this paper, we introduce the s-step biorthogonal Lanczos method for finding a few eigenvalues of a large sparse nonsymmetric matrix, and we prove that the s-step method generates reduction matrices which are similar to reduction matrices generated by the standard method. We prove that the breakdown conditions of the s-step method are less stringent than the standard one. One iteration of the s-step biorthogonal Lanczos algorithm corresponds to s iterations of the standard biorthogonal Lanczos algorithm, and the s-step method has improved data locality and minimized global communication and superior parallel properties to the standard one on parallel machines (Chronopoulos and Gear (1989) and Kim and Chronopoulos (1991)). We implement the s-step biorthogonal Lanczos method on the CRAY-2 super computer and discuss the breakdown conditions and demonstrate the superior performance of the s-step method to the standard one

The Stefan Problem Solved Via Conjugate Gradient—Like Iterative Methods On a Parallel Vector Machine

The aim of this paper is to illustrate the validity and efficiency of iterative methods for solving large linear systems arising from the finite element discretizations of the equation governing conduction-controlled solidi fication processes. Starting from the basic enthalpy equation, two alternative formulations are obtained and fixed-grid finite element discretizations are devel oped. These discretizations yield a set of nonlinear equations that are linearized using the Newton-Raph son scheme. The linearized equations are used as a basis for evaluating different iterative methods of the conjugate gradient type. Symmetric scaling and in complete factorization preconditioning of the linear equations are used to improve the convergence prop erties of the iterative methods. Vectorization and paral lelization are also employed to make full use of the CRAY-2 supercomputer. The results indicate that the implementation of currently available iterative solvers leads to efficient solution methodologies for phase change problems.

Using local memory to boost the performance of FFT algorithms on the CRAY-2 supercomputer

One of the many interesting architectural features of the CRAY-2 supercomputer is that each processor has access to 16K 64-bit words of local memory. This is in addition to the extremely large, 268-million-word common memory that is accessible by all four processors. By using local memory judiciously, it is possible to achieve increased performance on the CRAY-2. This is partly because accesses to local memory can be done simultaneously with accesses to common memory and other operations. Also, it is slightly faster to start up a vector access to local memory, and a processor does not have to compete with other processors when accessing its local memory. In this paper, we present an algorithm for computing the fast Fourier transform that takes advantage of the CRAY-2's local memory. It operates by solving subproblems, which are themselves Fourier transforms, entirely within local memory. By doing so it achieves a performance increase of between 25 and 40 percent over an equivalent algorithm that uses only common memory, and for some input sizes is able to outperform the CRAY-2 library FFT.

The solution of large-scale least-squares problems on supercomputers

We discuss methods for solving medium to large-scale sparse least-squares problems on supercomputers, illustrating our remarks by experiments on the CRAY-2 supercomputer at Harwell. The method we are primarily concerned with is an augmented system approach which has the merit of both robustness and accuracy, in addition to a kernel operation that is just the solution of a symmetric indefinite system. We consider extensions to handle weighted and constrained problems, and include experiments on systems similar to those arising in the Karmarkar algorithm for linear programming. We indicate how recent improvements to the kernel software could greatly improve the performance of the least-squares code.

Restricted Maximum Likelihood Estimation of Variance Components in Animal Model Using Sparse Matrix Inversion and a Supercomputer

The estimation of variance components by EM-type REML algorithms requires repeated inversion of the coefficient matrix of the mixed model equations. In the animal model, the rank of the coefficient matrix is usually larger than the number of records. If inversion is done by dense matrix algorithms, equations are limited to fewer than 2000. This study investigated the inversion of large coefficient matrices by direct sparse matrix solvers in analyzing conformation final scores of selected Holstein herds. The largest model contained 36,771 animal effects, 30,739 permanent environmental effects, 6102 herd × sire interactions, 532 herd-time classes, and 55 unknown parent groups, resulting in a coefficient matrix of order 74,199. Inversion was by sparse matrix package SMPAK, partly vectorized for the Cray-2 supercomputer. The convergence rate was accelerated by modified Aitken's extrapolation. For most rounds, the traces were based on a sample of inverse elements, resulting in up to 25 times lower cost per round. The largest model required 8 MWords (64 Mbytes) memory and used 5h central processing unit time for one inverse or 10h CPU time for complete run. Estimates were .67, 6.08, 4.62, and 3.47 for variance of herd by sire, permanent environment, animal, and residual, respectively.

COMPUTATION OF CONJUGATE THREE-DIMENSIONAL NATURAL CONVECTION HEAT TRANSFER FROM A TRANSVERSELY FINNED HORIZONTAL CYLINDER

A vectorized finite-difference code has been developed for the Cray-2 supercomputer which has the capability of simulating a wide class of three-dimensional coupled conduction-convection flows. This program numerically solves the transient form of the Navier-Stokes equations of motion for laminar flow using the vorticity-vector potential method. One problem that has been considered is three-dimensional natural convection from a heated horizontal cylinder with an attached circular conducting fin array. Numerical solutions have been obtained for a Rayleigh number of 106 andaPrandtl number of 5. A parametric study has been performed by varying the fin length, fin spacing, fin thickness, and the fin conductivity parameter. Plots showing the local surface heal flux distribution, temperature distributions, and flow fields were developed using color graphics packages available on the Cray-2 that revealed the existence of very complex fin-cylinder interactions. The local heat flux in the upper section of the fins...

Navier-Stokes calculations of hovering rotor flowfields

Unsteady, thin-layer Navier-Stokes equations written in rotor coordinates are solved using a partially flux-split, implicit numerical algorithm to calculate the flowfields of a hovering rotor blade at subsonic and transonic conditions. Numerical results are in good agreement with experimental data for both nonlifting and lifting rectangular blades. For the lifting case, the rotor wake effects are modeled by applying a correction to the geometric angle of attack of the blades. Alternate methods of calculating the hovering rotor flowfields in blade-fixed mode that have the same circulation distribution as a hovering blade are explored. All of the results presented in this paper were computed on a Cray-2 supercomputer.

Modelling the propagation and scattering of elastic waves in inhomogeneous anisotropic media

A finite-difference numerical model for studying the propagation and scattering of elastic waves in three-dimensional inhomogeneous and anisotropic media of arbitrary symmetry is presented. The model takes account of short pulses and finite-width beams and incorporates mode conversion at boundaries. The model, called SWAM3D (scattering of elastic waves in inhomogeneous anisotropic media in 3D), can handle a region of size equivalent to 250*250*250 nodes, that is a cube of side 25 shear wavelengths. It runs on a CRAY-2 supercomputer and represents a significant advance in the numerical treatment of wave propagation and scattering in inhomogeneous and anisotropic materials. As an indication of the power of the model, it is applied to the scattering of elastic waves of different polarisations from a cylindrical hole in a homogeneous transversely isotropic material. The propagation and scattering of elastic wave in inhomogeneous materials is an important problem in ultrasonic inspection of welds in liquid-metal-cooled fast reactors. A-scans and differential scattering cross sections are presented. They are used to explain some experimental results in the ultrasonic inspection of austenitic welds. The present results confirm the validity and applicability of the model.

Multitasking case study on the cray-2: The Q2R cellular automaton

The Q2R cellular automaton which may be used for microcanonical simulation of the Ising model is implemented in parallel on the four processors of a Cray-2 supercomputer. The simulation speed is 4.3 GHz as opposed to 670 MHz for the previously fastest reported implementation using one processor of a Cray-XMP. We also simulated the largest Ising system ever studied, 15, 130, 968, 192 spins. A number of subtleties in implementing and optimizing the parallel algorithm are discussed, as well as problems in defining performance measurements. Performance and results from the Q2R algorithm are compared to those from the standard Metropolis algorithm, and a number of dynamic exponents are measured for the first time.

Large-scale, explicit wave simulations on the cray-2

Most time-domain, wave modeling problems in geophysics are intractable by classical analysis methods, due principally to nonseparability and to a lesser extent material nonlinearity. Therefore discrete numerical solutions are often necessary for the simulation of realistic models. Applications in 2-D and 3-D geophysical modeling are the subject of this paper, particularly as solved on a CRAY-2 supercomputer. Implementation and performance differences between earlier CRAYs and the CRAY-2 are described, including the discrepancy between scalar fetch and vector processing speeds. Explicit finite element solvers are applied to applications involving 2-D simulation of a seismic refraction experiment across the state of Maine, 3-D simulation of near-source scattering experiments, and both linear and nonlinear axisymmetric source simulation. Results show that the CRAY-2 allows cost-effective 2-D simulations of truly large-scale models, but only begins to be effective in 3-D for models of interest in geophysics. The large memory (256 megawords) is adequate but a speed increase of at least an order of magnitude is necessary for cost-effective 3-D. True multiprocessor capability (rather than ‘multicomputer’) provides an alternative to raw speed but involves another set of difficulties.

Numerical results on the transcendence of constants involving 𝜋,𝑒, and Euler’s constant

Let x = ( x 1 , x 2 , … , x n ) x = ({x_1},{x_2}, \ldots ,{x_n}) be a vector of real numbers. x is said to possess an integer relation if there exist integers a i {a_i} such that a 1 x 1 + a 2 x 2 + ⋯ + a n x n = 0 {a_1}{x_1} + {a_2}{x_2} + \cdots + {a_n}{x_n} = 0 . Recently, Ferguson and Forcade discovered practical algorithms [7], [8], [9] which, for any n, either find a relation if one exists or else establish bounds within which no relation can exist. One obvious application of these algorithms is to determine whether or not a given computed real number satisfies any algebraic equation with integer coefficients (where the sizes of the coefficients are within some bound). The recursive form of the Ferguson-Forcade algorithm has been implemented with multiprecision arithmetic on the Cray-2 supercomputer at NASA Ames Research Center. The resulting computer program has been used to probe the question of whether or not certain constants involving π \pi , e, and γ \gamma satisfy any simple polynomial equations. These computations established that the following constants cannot satisfy any algebraic equation of degree eight or less with integer coefficients whose Euclidean norm is 10 9 {10^9} or less: e / π e/\pi , e + π e + \pi , log e π {\log _e}\pi , γ \gamma , e γ {e^\gamma } , γ / e \gamma /e , γ / π \gamma /\pi , and log e γ {\log _e}\gamma . Stronger results were obtained in several cases. These computations thus lend credence to the conjecture that each of the above mathematical constants is transcendental.

Distributed Interactive Graphics Applications in Computational Fluid Dynamics

Implementation of two distributed graphics pro grams used in computational fluid dynamics is discussed. Both programs are interactive in na ture. They run on a CRAY-2 supercomputer and use a Silicon Graphics Iris workstation as the front-end machine. The hardware and supporting software are from the Numerical Aerodynamic Simulation project The supercomputer does all numerically intensive work and the workstation, as the front-end machine, allows the user to perform real-time interactive transformations on the displayed data. The first program was written as a distributed program that computes particle traces for fluid flow solutions existing on the su percomputer. The second is an older post-pro cessing and plotting program modified to run in a distributed mode. Both programs have realized a large increase in speed over that obtained using a single machine. By using these pro grams, one can learn quickly about complex features of a three-dimensional flow field. Some color results are presented.

NASA launches wind-tunnel tests by Cray-2 supercomputer

NASA launches wind-tunnel tests by Cray-2 supercomputer

Molecule–corrugated surface scattering calculations using the close coupling wave packet method

Numerically exact quantum calculations for scattering of N2 off a model corrugated rigid lattice are reported. The computations were done using the recently developed close coupling-wave packet method for treating quantum scattering. Results for all energies in the range 0.010–0.040 eV are obtained from a single wave packet propagation. The calculations included even N2 rotor j states j=0–12, all mj’s, and sufficient grid points to describe diffraction states −8≤m≤8, −4≤n≤4, for a total of 13 923 channels. The computations, which required a total of 230 minutes of computer time, were carried out on the CRAY2 supercomputer at the University of Minnesota Supercomputer Center under National Science Foundation support.

Simulation of large Eden clusters.

Simulations of Eden clusters with more than ${10}^{9}$ sites by the use of a Cray Research Cray-2 supercomputer show that the thickness of the surface layer of round two-dimensional clusters, averaged over all directions, increases as the radius for sizes above 50?0?0 sites. In simulations of flat surfaces, on the other hand, the thickness increases as the square root of the height for strip widths above 500. The initial growth of such strips is not described by one single power law and asymptotically has a ``dynamic'' exponent close to (3/2), i.e., it increases as predicted theoretically with (height${)}^{1/3}$. In the range covered, our three-dimensional flat-surface thickness is not consistent with a logarithmic law, and also in four dimensions the thickness does not approach a constant. Round clusters with more than ${10}^{9}$ sites on the square lattice confirm the expected behavior due to anisotropy. Higher dimensions up to d=12 were also studied.