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IJSTR >> Volume 9 - Issue 10, October 2020 Edition

International Journal of Scientific & Technology Research  
International Journal of Scientific & Technology Research

Website: http://www.ijstr.org

ISSN 2277-8616

A New Modified Version Of Gauss-Seidel Iterative Method Using Grouping Relaxation Approach

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Baher A. Haleem, Ihab M. El Aghoury, Bahaa S. Tork, Hisham A. El-Arabaty



Systems of linear equations, Finite element method, Direct techniques, Indirect techniques, Stationary iterative methods, Jacobi, Gauss-Seidel.



Systems of linear equations appear in many areas either directly as in modeling physical situations or indirectly as in the numerical solutions of other mathematical models. The solution of the linear equations’ system is probably the most important issue in numerical methods like the finite element method (FEM). Using the finite element method in modeling various structures, with either simple or complicated configuration of elements, in structural engineering became prevalent many years ago. There are two main types of solvers depending on whether the used method is direct or iterative (indirect) method. In contrast to the iterative techniques, the direct techniques provide almost exact solutions, however they are not convenient for some situations, including but not limited to huge systems of equations. In such situations, the iterative solvers are favored as they have privileges concerning solving speed and storage requirements. In addition, indirect solvers are simpler to program. This research focuses on using the Classical (Stationary) iterative techniques for solving linear systems of equations. The main objective of this research is to develop a new modified version of the well-known Gauss-Seidel (GS) iterative technique which is adapted to solving problems in structural engineering. The proposed technique remarkably outperforms GS technique regarding the required number of iterations and the convergence speed. In this paper, the differences between the direct and iterative approaches have been discussed, along with a quick overview of some of the methods underlying these two classes. Then, the idea and algorithm of the new proposed “Modified Gauss-Seidel” (MGS) technique have been elucidated. Afterward, the algorithm has been programmed and used to solve some 2D Practical Examples, besides using the conventional Jacobi and GS techniques. Finally, the obtained results have been compared to assess the proposed MGS; it outperformed both Jacobi and GS.



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