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Research and Implementation of Direct Methods for Large Sparse Matrix

Author: FengGuangXiang
Tutor: WangChengEn
School: Northeastern University
Course: Systems Engineering
Keywords: sparse matrix direct method LU factorization frontal method matrix storage
CLC: TP301.6
Type: Master's thesis
Year: 2010
Downloads: 355
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Abstract


Currently, large-scale scientific computation and data processing problems are frequently presented in many basic research and applications of key areas, such as science and technology, national economy, national defense reconstruction, etc. Whereas these problems are highly dependent on the computer environment, and whether the problems are solved directly determines the scientific research level. Lots of important practical problems can be directly or indirectly attributed to matrix solution, and linear equation group resolving is the most basic issues in these solving processes. With the development of computing technology, matrix solution method is also constantly updated. In the work of calculating large matrix used computer, there is storage problem first of all, and secondly the calculation time is too long, which often exceeds the time of machine stability resulting in failure. In order to resolve these contradictions, we must use the feature of matrix sparse and find efficient algorithm which can use less computer resources, such storage and running time.In this paper, on the background of the laboratory project "System of Aero-engine Thermo-solid Coupling Analysis", aiming at the problems in finite element model of aero-engine components thermosetting coupling, the sparse feature of stiffness matrix and the method for large sparse linear equations generating sparse matrix are researched. This project is run on the platform of serial computer. Taking into account the accuracy, numerical stability and the existing solver, the direct method is adopted in this solver, which can solve both symmetry and non-symmetric matrix. For five hundred thousand-order matrix or less, firstly the requirement of memory usage is decreased by optimizing sparse matrix storage scheme, then the non-zero element filling is reduced and the sparse characteristic is fully utilized by symbol decomposition on solving matrix, finally the numerical decomposition as well as element elimination is conducted and the solution process is achieved after back substitution. The solver designed tests and compares various types of sparse matrix, and the result solved can meet the requirements of the project.Based on some basic theories and algorithms for solving sparse matrix, the design scheme of solver is proposed. At the end of this paper, the actual operation situation of the solver in the project is analyzed and compared.

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CLC: > Industrial Technology > Automation technology,computer technology > Computing technology,computer technology > General issues > Theories, methods > Algorithm Theory
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