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

The simulation of realistic structural engineering applications, such as non-linear dynamic analysis of structures, is often extremely time-consuming. Parallel and distributed processing provides an excellent opportunity for achieving the additional computational power needed to effectively simulate these systems. However, the efficient use of parallel and distributed computing architectures hinges on the quality of the computational algorithms being utilized in the solution of these problems. Therefore, it is imperative that new algorithms be developed to take advantage of this type of hardware.

The scope of the present research is on direct solution algorithms for the integration of the equations of motion arising in structural dynamics problems. Several concurrent time-stepping algorithms have been developed to date. In particular, two algorithms are the focus of the present research, namely the Group Implicit (GI) algorithm[1] and the Iterative Group Implicit (IGI) algorithm[2]. In both of these methods the solution is found on a domain-by-domain basis and compatibility at the interface between sub-domains is restored by means of an averaging scheme. Both of these algorithms are highly parallelisable and are unconditionally stable, as long as the underlying solution algorithm used to solve the sub-domains is unconditionally stable. However, the GI algorithm has been found to have severe limitations in accuracy [3,4]. These inaccuracies stem from the fact that the mass-averaging rule used to restore the compatibility at the interface Degrees Of Freedom (DOFs), disrupts the equilibrium at the interface and at the adjacent DOFs. Adjacent DOFs are those that share an element with interface DOFs but are not at the interface. To address this issue, the IGI algorithm considers the residual interface forces introduced by the enforcement of compatibility and eliminates it by means of an iterative procedure. Furthermore, both methods are highly modular and scalable, and therefore well suited for distributed and parallel computing.

In this work, the convergence characteristics of the IGI algorithm have been studied. It has been found through numerical studies that the IGI algorithm[2] does not always converge. This convergence problem limits the time step size to impractical values. Furthermore, the accuracy of the results is also affected by this problem.

The source of the convergence problem has been found to be the approximation, which has been made in the distribution of the corrective interface forces. The present study shows that approximating the distribution factors has an adverse effect on the IGI solution algorithm. The error due to the approximation can only be defeated under impractical conditions, i.e., very small time step sizes or very large mass values. As a solution to this problem, the approximation on the distribution factor has been removed in spite of a little increase in the communication time as well as a small extra computation time resulting from a matrix-vector multiplication rather than a vector-vector multiplication. The effects of this modification have been found to be successful for the linear transient analysis of structures. The numerical results show that the number of iterations required for convergence with a practical time-step size is dramatically reduced.

The mass averaging rule is required for the consistency of the GI algorithm. However, in the case of IGI algorithm, it is not only found to be redundant but also restrictive for some practical types of problems. In the MIGI this requirement has been eliminated and a straightforward averaging of the displacements is used. A thirty-storey steel frame structural application is developed to demonstrate the accuracy and performance of the MIGI algorithm. This demonstration has been performed on an IBM SP2 distributed computer. It has been found that the MIGI algorithm produces accurate and reliable results, as well as considerable speed-ups 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

Hajjar, J.F., Abel, J.F.,. "On the Accuracy of Some Domain-by-Domain Algorithms for Parallel Processing of Dynamics", International Journal for Numerical Methods in Engineering, 28, 1855-1874, 1989. doi:10.1002/nme.1620280811

4

Farhat, C., Sobh, N., "A Consistency Analysis of a Class of Concurrent Transient Implicit/ Explicit Algorithms", Computer Methods in Applied Mechanics and Engineering, 84(2), 147-162, 1990. doi:10.1016/0045-7825(90)90114-2

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