Keywords: 3D structural linear dynamic analysis, high-rise buildings, HPC techniques, direct time integration methods, numerical libraries, parallel direct and iterative solvers.

This paper describes a High Performance Computing (HPC) based application for the three-dimensional linear dynamic analysis of large-scale buildings, where all the nodes of the structure are taken into account and six degrees of freedom per each node are considered. Direct time integration algorithms [1], modal analysis and frequency domain analysis are different effective techniques, widely employed for the numerical solution of the high computational demand equations that governs the motion of structural dynamic problems. Although for certain applications modal based techniques continue being the chosen alternative by analysts, especially for linear systems, time integration methods have been widely used in many commercial packages because of their inherent advantages. Moreover, their applicability to modern HPC platforms is automatically implied.

In this work, the following eight well-known direct time integration methods have been efficiently parallelized: Newmark, Wilson-, Central Difference, single-step Houbolt, HHT-, WBZ-, Generalized- and SDIRK. A consistent mass matrix has been assumed and Rayleigh damping has been employed. Distributed-memory paradigm has been applied, where communication among the processors is carried out by explicit message passing commands implemented in the MPI library. Furthermore, the MPI-2 IO development of ROMIO has been employed to provide good performance in the access to secondary storage. Therefore, although the application has been run on a cost-effective Beowulf cluster, it can be easily migrated to a great variety of parallel architectures.

First of all, the stiffness, mass, damping and effective stiffness matrices are generated in parallel. These matrices will be distributed into the processors following a row-wise block-striped distribution. Then, for each time step, the following phases have been also parallelized: the calculation of the displacements, velocities and accelerations at the joints, by means of the chosen time integration method, the computation of member end forces, and the calculation of deformations and moments at any point of the structure. Different basic linear algebra operations have been efficiently parallelized, such as a constant times a vector or a matrix, vector or sparse matrix sums, and sparse matrix-vector product.

The WSMP [2], MUMPS [3] and PETSc [4] parallel public domain numerical libraries have been employed for solving the resulting systems of linear equations, where the coefficient matrix is sparse, symmetric and positive definite. These libraries have been selected by their availability, good performance and state-of-the art capabilities. Since stiffness, matrix and damping matrices are constant in a linear analysis, the resulting coefficient matrix can be just factorized once and a forward-backward substitution is carried out for each time step if a direct method is applied. PETSc routines have been exploited as well as for matrix assembly and matrix-vector operations.

One medium-sized building was selected for assessing the performance of the algorithms developed on a cluster of PCs composed of 20 Pentium XEON (2 GHz biprocessors with 1 GByte of RAM) interconnected by a SCI network. A triangular dynamic load was applied to the building within the first 1.5 seconds. The response of the structure was simulated during 4 seconds, with the time step equal to 0.01 seconds. The shortest simulation times were achieved with MUMPS library, together with the QAMD ordering. As an example, if the Newmark method is selected, the structural analysis required 16.83 minutes when just one processor was employed and 3.69 minutes for 16 processors. Good results were also obtained with the WSMP, but simulations with 1 processor overcame the RAM memory available. Worse results were achieved by means of the PETSc, employing as a linear solver a combination of Conjugate Gradient, as iterative method, with block Jacobi preconditioning, where incomplete Cholesky factorization was used as sub-block preconditioner.

As a conclusion, it can be said that the HPC-based application performs 3D realistic dynamic linear analysis of high-rise buildings, where all the nodes of the structure are considered, providing comprehensive results in very reasonable response times.

1

T.C. Fung, "Numerical Dissipation in Time-Step Integration Algorithms for Structural Dynamic Analysis", Progress in Structural Engineering and Materials, 5, 167-180, 2003. doi:10.1002/pse.149

2

A. Gupta, "WSMP: Watson Sparse Matrix Package Part I - Direct Solution of Symmetric Sparse System", Technical Report IBM Research Report RC 21886(98462), IBM (2000).

3

P. Amestoy, I. Duff, J.Y. L'Excellent, J. Koster, "MUltifrontal Massively Parallel Solver (MUMPS Version 4.3) Users Guide", 2003.

4

S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M. Knepley, L. Curfman-McInnes, B.F. Smith, H. Zhang, "PETSc Users Manual", Technical Report ANL-95/11 - Revision 2.2.0. Argonne National Laboratory, 2004.

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