Keywords: 3D modal analysis, eigenvalue problem, eigensolvers, parallel numerical libraries, HPC techniques, large dimension buildings.

The two basic methods for computing the response of linear structural models to dynamic loads are time integration and modal analysis [1]. Although time integration methods have been widely used in many commercial packages, because of their inherent advantages, modal based techniques continue being the chosen alternative by many analysts, especially for linear systems.

Actually, the modal analysis method is an appropriate superposition of the modal deformational shapes of the structural model. Its starting point is the analysis of the free vibration problem, in which there is no damping or external forcing. It leads to the solution of a generalized eigenvalue problem [2], where modal frequencies (eigenvalues) and associated modal shapes (eigenvectors) should be computed to define the natural vibration characteristics and avoid the resonance.

In this paper, an MPI-based parallel application for the 3D modal analysis of large scale buildings has been implemented, where six degrees of freedom for each node have been assumed and where all the nodes of the structure without movement constraints have been considered. Since the communications are based on the MPI library, the application can be easily migrated to a wide variety of parallel platforms. Moreover, it has been developed on top of the PETSc environment, which offers parallel implementations of matrix assembly routines, different basic matrix and vector linear algebra operations, and iterative methods for solving linear systems of equations.

First of all, the stiffness and mass matrices are generated in parallel following a block-wise block-striped distributions. A consistent mass matrix has been considered. Then, the lowest natural frequencies and mode shapes of vibration are computed by solving the generalized eigenvalue problem. In this paper, the SLEPc library has been employed to obtain them [3]. This numerical library has been chosen thanks to its good parallel performance and its easiness of usage. In addition to its own implemented eigensolvers (Lanczos, Arnoldi, etc.), SLEPc offers wrappers to easily use the solvers implemented in other external eigenvalue packages, such as ARPACK [4]. Moreover, SLEPc is based on PETSc environment [5]. Therefore, PETSc can be used for solving the linear systems, appearing in each step of the eigensolver process, by means of its different parallel iterative methods and preconditioners, or as interface to other external packages composed of direct methods, such as MUMPS [6].

Once the eigensolver problem has been solved, and for each time step, a set of uncoupled equations must be solved in parallel, each of them composed of one degree of freedom. For that, the Duhamel integral, and eight different time integration methods have been implemented: Newmark, Wilson-, Central Difference, Single-step Houbolt, HHT-, WBZ-, Generalized- and SDIRK. Next, the displacement, velocity and acceleration vectors at the joints of the structure are evaluated in parallel. Finally, the member end forces, and the stresses and deformation at any point of the structure are also updated.

Free vibration problem of three buildings have been performed, showing that AR-PACK and the implementations of the Arnoldi and Lanczos methods in the SLEPc library present similar execution times and parallel performance. The combination of these eigensolvers with shift-and-invert spectral transformation provided the lowest execution times. Besides, the MUMPS and PETSc libraries were employed to solve the linear systems of equations, where MUMPS was demonstrated to be much more appropriate in terms of accuracy and response times. Therefore, it can be said that the ARPACK/SLEPc + shift-and-inverse + MUMPS compose a very powerful tool for solving the large scale generalized eigenvalue problem in structural analysis.

Finally, the behaviour of two buildings, under a seismic load acting during 7 seconds, was simulated in very reasonable times. It demonstrates that a realistic 3D dynamic analysis of large-scale structures can be carried out, even in a cost-effective cluster of PCs, without the necessity of using inadequate simplifications.

1

R.W. Clough, J. Penzien, "Dynamics of Structures", Second Edition, McGraw-Hill, Inc, 1993.

2

Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, H. van der Vorst, "Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide", SIAM, Philadelphia, 2000.

3

V. Hernández, J.E. Román, A. Tomás, V. Vidal, "SLEPc Users Manual. Scalable Library for Eigenvalue Problem Computations", Technical Report DSIC-II/24/02, Universidad Politécnica de Valencia, 2006.

4

R.B. Lehoucq, D.C. Sorensen, C. Yang, "ARPACK Users' Guide, Solution of Large-Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods", SIAM, Philadelphia, PA, 1998.

5

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 2.3.1 Argonne National Laboratory, 2006.

6

P. Amestoy, I. Duff, J.Y. L'Excellent, J. Koster, "MUltifrontal Massively Parallel Solver (MUMPS version 4.6.2) Users Guide", 2006.

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