Keywords: nonlinear transient analysis, parallel processing, distributed computing, parallel algorithms, group implicit algorithm, domain decomposition.

The transient dynamics analysis of structures is useful in areas such as earthquake and wind engineering. In particular, it is performed to help understand the behaviour of structures under dynamic loading considering second-order effects and material nonlinearities. It is also used to determine design rules for structural components and to derive the response spectra of a structure. The latter requires repetitive time- history analyses under various ground motion records. Nonlinear analyses usually require frequent update of the structural stiffness matrix. Furthermore, in these cases if an implicit scheme is used to integrate the equations of motion, the solution of simultaneous dynamic equilibrium equations are required at each analysis iteration. Thus, analysis of nonlinear structural dynamics problems using an unconditionally stable implicit method requires very long computer processing time.

Parallel group implicit methods for the solution of transient dynamics problems on a domain-by-domain basis have been found very attractive for the solution of linear structural dynamic systems. A great deal of research has been carried out on these algorithms. The Group Implicit (GI) algorithm [1] obtains the solution of each subdomain in parallel and enforces displacement compatibility at the subdomain interfaces. However, the algorithm disregards the loss of force equilibrium in the interface degrees-of-freedom that result from the compatibility enforcement. When the GI algorithm is used for dynamic analysis of framed structures, this force equilibrium inaccuracy becomes very significant; therefore, the time step size required for an accurate solution becomes impractically small. The Iterative Group Implicit (IGI) [2] solves this problem by considering the residual interface forces resulting from the enforcement of the compatibility at the interface degrees-of- freedom (DOF). However, it was found that the IGI algorithm does not always converge due to the approximation made in the distribution of the corrective interface forces [3]. The Modified Iterative Group Implicit Algorithm (MIGI) was develop to correct the approximation made in the calculation of the interface distribution factor [3]. Furthermore, in the MIGI algorithm a direct averaging rule for averaging of interface DOF displacements is adopted. The MIGI algorithm has been shown to significantly speed-up the computations for linear structural dynamics problems along with a great accuracy.

In this work, the MIGI algorithm is extended for nonlinear transient structural dynamics problems. Because of the geometric and/or material nonlinearities, the solution may not be obtained without dynamic equilibrium iterations (nonlinear iterations). Along with the nonlinear iterations, force equilibrium and displacement compatibility of the interface DOF need to be satisfied through the MIGI iterative procedure. At each nonlinear iteration, all the structural elements need to update their state using the computed displacements; therefore, the solution for the interior DOF must be recovered from the interface DOF. Two strategies have been developed to integrate the MIGI interface iterations and the nonlinear iteration, which are referred to as the Separated Iteration Procedure and the Combined Iteration Procedure. Both procedures have been implemented and tested in this work. It is found that both procedures can successfully solve the nonlinear problem. The efficiency and convergence characteristics of the developed algorithms have been studied through numerical studies. It is found that the isolation of the iterations produces a more efficient scheme than the one resulting from combination of the two loops.

The performance of the developed algorithm has been tested for a 20-storey steel model frame, which was designed for the city of Los Angeles, California in the USA, subjected to an existing earthquake motion record applied at the foundation level. This is a fully nonlinear application, i.e., both material and geometric nonlinearities are present. The application is tested on both a network of SUN workstations and on the IBM SP distributed computer. The MIGI algorithm converged to accurate (compatible and in equilibrium) interface solution in a few number of iterations and the convergence of the interior solution was almost unaffected by that. The responses obtained with different number of subdomains are found to be very accurate for the problem analysed. Significant speedup's have been achieved on both IBM SP and network of Sun workstations, when compared to the sequential version.

1

Ortiz, M., Sotelino, E.D., and Nour-Omid, B., "Efficiency of Group Implicit Concurrent Algorithms", International Journal for Numerical Methods in Engineering, 28, 2761-2776, 1989. doi:10.1002/nme.1620281204

2

Modak, S., Sotelino, E.D., "The Iterative Group Implicit Algorithm for Nonlinear Structural Analysis", International Journal for Numerical Methods in Engineering 47(4), 869-885, 2000. doi:10.1002/(SICI)1097-0207(20000210)47:4<869::AID-NME803>3.3.CO;2-7

3

Dere, Y., Sotelino, E.D., "Convergence of the Iterative Group-Implicit Algorithm for Parallel Transient Finite Element Analysis", Proceedings of the Eighth International Conference on Civil and Structural Engineering Computing, paper no 92, Vienna, AUSTRIA, September 19-21, 2001, Civil-Comp Press, Stirling, UK. doi:10.4203/ccp.73.92

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