Large-scale Structural Analysis Using General Sparse Matrix Technique

Transcription

1 Large-scale Structural Analysis Using General Sparse Matrix Technique Yuan-Sen Yang 1), Shang-Hsien Hsieh 1), Kuang-Wu Chou 1), and I-Chau Tsai 1) 1) Department of Civil Engineering, National Taiwan University, Taipei, Taiwan, R.O.C. ABSTRACT This paper investigates the general sparse matrix technique for solving simultaneous equilibrium equations encountered in large-scale structural analyses. The General Sparse Matrix (GSM) technique is introduced first. Numerical experiments are then conducted on structural analyses of several finite element models to compare the GSM approach with the more traditional SKyline Matrix (SKM) approach. The numerical studies show that the GSM approach requires less time and memory storage than the SKM approach. Finally, discussions are given on factors that make the GSM approach more advantageous than the SKM approach. INTRODUCTION The strategy of solving the equilibrium equations in matrix form (e.g., K d = f) plays an important role in finite element analysis. In structural analyses, the direct method (i.e., factorizing the matrix K into the form of L L T or L D L T ) is usually preferred because the iterative method may have the slow-convergence problem caused by an ill-conditioned system. For solving a linear system using the direct method, it is often computationally expensive (in terms of time and storage requirements) to factorize the effective stiffness matrix K in the equilibrium equations, especially when a large-scale structural problem is analyzed. In order to reduce the time and storage requirements for factorizing a large system matrix K, the SKyline Matrix (SKM) technique has been employed by most of the finite element packages. The SKM approach stores and computes the matrix items within the skyline of the matrix. Nevertheless, there is still considerable number of zero items stored within the skyline. To use the computer resources more efficiently, the General Sparse Matrix (GSM) technique was proposed (George and Liu, 1981), which only stores items that are required in matrix factorization. However, it has not been commonly used in the field of structural engineering. In this work, the GSM technique is studied and employed for solving the equilibrium equations in large-scale structural engineering problems. The storage

2 requirements and computational efficiency of using the GSM approach to store and factorize the matrix K are also compared with those of using the SKM approach in a series of numerical experiments. GENERAL SPARSE MATRIX TECHNIQUE The GSM technique stores only the nonzero items that may be arbitrarily distributed in a matrix. However, the distribution pattern of the nonzero pattern in the matrix K is different from that in the factorized matrix L. Some zero items in the matrix K would become nonzero (which is also called as fill-ins ) during factorization. Therefore, the nonzero items in the matrix L are always more than those in the matrix K. In order to factorize the matrix K correctly, enough memory should be allocated based on the storage requirement of the matrix L, not just the matrix K. The symbolic factorization is used required in the GSM approach for predicting the distribution of nonzero items in the matrix L after the matrix K has been factorized. However, it is not needed in the SKM approach because both the matrix L and the matrix K have the same skyline. Like for the SKM approach, matrix renumbering is usually needed for the GSM approach (George and Liu, 1981). The matrix renumbering must be performed before the symbolic factorization because the distribution of the nonzero items in the matrix L depends on the numbering of the matrix. The purpose of the GSM renumbering is to reduce as many as possible the nonzero items in the matrix L. The fewer the nonzero items are, the less time and storage is required for factorizing the matrix K. Several algorithms, such as Minimum-Degree (MD) algorithm (George and Liu, 1981), Nested-Dissection (ND) algorithm (George, 1973), and their modified algorithms (e.g., Liu, 1985; Bui and Jones, 1993) are usually used for matrix renumbering in the GSM approach. Here we use an example to demonstrate the differences in storage pattern between the GSM and SKM approaches. A finite element mesh of an L-shaped building, named as L-BUILDIN, is shown if Fig. 1. Figures 2 and 3 show the storage patterns of the nonzero items in the matrices using the SKM and GSM approaches, respectively. The skyline matrix (see Fig. 2) is renumbered by the Reverse Cuthill-McKee (RCM) algorithm (Liu, 1976) and contains 81,504 items. The general sparse matrix (see Fig. 3) is renumbered by the Modified Minimum-Degree (MMD) algorithm (Liu, 1985), and contains 66,204 nonzero items. It can be seen that the renumbering in the SKM approach tends to gather the nonzero items closer to the diagonal, while the one in the GSM approach tends to scatter them over the matrix. 182 nodes 496 beam-column elements 1,008 degree of freedom Fig.1 The LBUILDIN model (Hsieh, 1995)

3 Fig.2 Storage pattern of the SKM approach Fig.3 Storage pattern of the GSM approach NUMERICAL COMPARATIVE STUDIES Ashcraft et al. (1987) and Poole et al. (1992) compared the GSM approach with the SKM approach numerically by using several structural analysis examples. However, these two groups of researches had different findings. Ashcraft et al. showed that the GSM approach is much superior to the SKM approach in terms of computational counts and storage requirement, while Poole et al. showed that the GSM approach does not have much advantages over the SKM approach. Because both of them did not clearly describe the characteristics of the finite element models used, it is therefor difficult for us to judge their correctness. In this work, the GSM approach is investigated for solving the equilibrium equations of several structural analyses. The characteristics of the finite element models are emphasized. The time and storage required for factorizing the matrix are measured and compared between the GSM and SKM approaches. In the numerical studies, the factorization time measured does not include time for matrix renumbering and symbolic factorization. This is because they usually cost little time when compared with matrix factorization and become even less significant in nonlinear finite element analyses where they are usually performed only once at the very beginning of the analysis (even though the stiffness matrix K may be updated and factorized several times). The measured storage requirement indicates the size of memory used to store the entire matrix, including the storage for the nonzero items and information associated with their distribution pattern. A sparse matrix package called SPARSPAK (George and Liu, 1981) has been used for performing the matrix factorizations in this work. The SPARSPAK package provides both the skyline matrix and general sparse matrix computations for solving a linear system. The RCM and MMD renumbering algorithm are used for the SKM and GSM approaches, respectively. An object-oriented finite element analysis package called FE++ (Lu, 1994) has been modified to implement both the SKM and GSM approaches for the comparative studies. All of the numerical tests have been performed on a personal computer running the Windows NT operating system. The computer has a Pentium II-233 CPU and 128 Megabytes SDRAM. The computer programs are compiled using the Microsoft Visual C++ compiler. In addition, it should be noted that the virtual memory capability is not used in all of the tests. The results of the numerical comparative studies are summarized in Table 1. R T and R M denote the ratios of the time and storage requirements, respectively, needed by the GSM approach over the SKM approach. The results show that, in all examples studied in this work, the GSM approach is superior to the SKM approach in terms of both time and storage

4 requirements for matrix factorization in the finite element analyses. More detailed descriptions about the comparative tests and discussions on the differences of performance between the GSM and SKM approaches are provided below. Table 1 Comparisons between the GSM and SKM approaches for time and storage requirements Factorization Time (sec.) Finite Element GSM SKM RT (=GSM/SKM) Models 10-STORY STORY C-BUILDIN E- BUILDIN T7070 T Memory Requirements (megabytes) GSM SKM RM (=GSM/SKM) Two finite element models are used for testing the influence of the scale of the finite element models on the GSM and SKM approaches. Figures 4 and 5 show the two finite element models used (named 10-STORY and 20-STORY, respectively) which have almost the same shape but different sizes. As already shown in Table 1, the GSM approach spends 21% less time and 33% less memory than the SKM approach on the 10-STORY model, while spends 41% less time and 52% less memory on the 20-STORY model. Therefore, the result seems to indicate that the GSM approach would become more advantageous on finite element models with larger sizes. 396 nodes 960 beam-column elements 2,160 DOF 2,541 nodes 6,820 beam-column elements 14,520 DOF Fig.4 The 10-STORY model Fig.5 The 20-STORY model The C-BUILDIN (see Fig. 6) and the E-BUILDIN (see Fig. 7) models are used here to investigate the influence of the branches in the finite element models on the GSM and SKM approaches. The major difference between these two finite element models, C-BUILDIN and E-BUILDIN, is that the E-BUILDIN model has a branch at the middle of the structure while the C-BUILDIN model does not. As already shown in Table 1, the GSM approach spends 45% less time and 49% less memory than the SKM approach on the C-BUILDIN model,

5 while spends 60% less time and 57% less memory on the E-BUILDIN model. Therefore, the result seems to indicate that the GSM approach would become more advantageous on finite element models with irregular shapes. 2,856 nodes 7,400 beam-column elements 16,320 degrees of freedom Fig.6 The C-BUILDIN model 3,276 nodes 8,500 beam-column elements 18,720 degrees of freedom Fig.7 The E-BUILDIN model The finite element models with different aspect ratio are used to compare the GSM and SKM approaches. The T40020 model (see Fig. 8) is a slender truss structure with 20 horizontal spans and 400 vertical spans. The T7070 model (see Fig. 9) is a squared truss structure with 70 horizontal and vertical spans. As already shown in Table 1, the GSM approach spends 37% less time and 50% less memory than the SKM approach on the T7070 model, while spends 16% less time and 31% less memory on the T40020 model. Therefore, the result seems to indicate that the advantage of the GSM approach over the SKM approach would become less obvious on finite element models with slender shapes. 16,421 nodes 64,000 truss elements 46,743 degrees of freedom Fig.8 The T40020 model 9,941 nodes 39,200 truss elements 28,983 degrees of freedom Fig.9 The T7070 model CONCLUSIONS Based on the numerical comparative studies in this work, it has been found that the GSM technique can really help to reduce the time and memory storage in the finite element structural analyses, especially when the finite element model is large-scale and with irregular shapes (e.g., with branches). In all the examples studied, the GSM approach performs better than the SKM approach and has up to 60% and 57% savings on time and memory,

6 respectively, over the SKM approach. As the size and complexity of engineering problems increase with time, it is believed that the GSM approach will play a more important role in the analysis of the increasingly large finite element models with complicated shapes. ACKNOWLEDGEMENTS We thank Prof. T. W. Lin of National Taiwan University for providing his experiences on finite element analysis using both the SKM and GSM approaches, and Dr. K. N. Chiang of National Center for High-performance Computing for providing the experiences on the GSM technique. We also acknowledge the support from the National Science Council of Republic of China under Grant No E REFERENCES Ashcraft, C. C., Grimes, R. G., Peyton, B. W., and Simon, H. D. (1987). "Progress in Sparse Matrix Methods for Large Linear Systems on Vector Supercomputers," The International Journal of Supercomputer Applications, 1(4), Bui, T. N. and Jones, C. (1993). A Heuristic for Reducing Fill-in in Sparse Matrix Factorization, Proceedings of the 6 th SIAM Conference on Parallel Processing for Scientific Computing, , Virginia, USA, March George, A. and Liu, J. W. H. (1981). Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, USA. George, A., (1973). "Nested Dissection of a Regular Finite Element Mesh," SIAM Journal on Numerical Analysis, 10, Hsieh, S. H. (1995). Parallel Nonlinear Dynamic Analysis of Framed Structures with Flexible Floor Using Networked Workstations, Proceedings of the 5 th KU-KAIST-NTU Trilateral Seminar/Workshop on Civil Engineering, , Taipei, Taiwan, ROC, November Liu, J. W. H. (1976). "Comparative Analysis of The Cuthill-McKee and The Reverse Cuthill-McKee Ordering Algorithms for Sparse Matrices," SIAM Journal on Numerical Analysis, 13(2), Liu, J. W. H. (1985). "Modification of The Munimum-Degree Algorithm by Multiple Elimination," ACM Transactions on Mathematical Software, 11(2), Lu, J. (1994). "FE++: An Object-Oriented Application Framework for Finite Element Programming," Proceedings of the 2 nd Annual Object-Oriented Numerics Conference, , Sunriver, OR. Poole, E. L., Knight, N. F., Jr, and Davis, D. D., Jr. (1992). "High-performance Equation Solvers and Their Impact on Finite Element Analysis," International Journal for Numerical Methods in Engineering, 33,