Keywords: domain decomposition method, structural analysis, coarse grid conjugate gradient method, enhanced CGCG method, conjugate orthogonal decomposition, parallel processing.

The paper shows a domain decomposition method applied to a structural analysis, the coarse grid conjugate gradient (CGCG) method [1] and its enhancement. Contrary to the ordinary iterative substructuring method in which the subdomains are considered as substructures, the CGCG method is a fully iterative method in the sense that the subdomains do not act as substructures in our method. Instead, the domain decomposition is used for the parallel processing and the global coarse grid motion of the decomposed subdomains. The CGCG method is implemented in a commercial code ADVENTURECluster. The largest analysis we have performed is a drop impact analysis of a cell phone with 305 million degrees of freedom on the IBM Blue Gene/L with 8 racks, 8192 nodes, and the same number of parallel processes [1].

In the CGCG method, a finite element mesh model is divided into the subdomains corresponding to the parallel processes. Then we decompose these subdomains into a number of the subdomains. We use these subdomains to bring the coarse motion. The coarse space is constructed with the rigid body motion of the subdomains. The basis of the rigid body motion is given by three-direction translations and three-axis rotations. Thus each subdomain has six degrees of freedom. The dimension of the coarse space is six times the number of the subdomains, which is as small as the direct method is applied. Then its conjugate orthogonal space is constructed, which is as large a model as the CG method is needed in general. Depending on the basis and the conjugate orthogonal projections, the stiffness matrix is given in a diagonal block form corresponding to the coarse space and the conjugate space. This leads to an extension to the decomposition of the layered coarse spaces and its conjugate orthogonal spaces. The direct method is applied to the coarse spaces independently, and CG method is applied to the deepest conjugate orthogonal space. We call this algorithm an enhanced CGCG method. Each layer is called a generation.

The effectiveness of the enhanced CGCG method is shown by a numerical example. The model is a simple single component model with about eight million degrees of freedom. The result shows a 30% improvement of the performance. Though the total performance is found to be almost given by the first generation decomposition, we can determine how much we depend on the direct method by taking the generation depth appropreately. Thus we can apply the enhanced CGCG method to problems that the iterative methods are hard to solve.

1

H. Akiba, T. Ohyama, Y. Shibata, K. Yuyama, Y. Katai, R. Takeuchi, T. Hoshino, S. Yoshimura, H. Noguchi, M. Gupta, J.A. Gunnels, V. Austel, Y. Sabharwal, R. Garg, S. Kato, T. Kawakami, S. Todokoro, J. Ikeda, "Large Scale Drop Impact Analysis of Mobile Phone Using ADVC on Blue Gene/L", Proceedings of SC06, IEEE, 2006. doi:10.1145/1188455.1188503

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