An Iterative Approach for Improving the Performance of Existing Beam and Triangular Elements
Date Issued
2005
Date
2005
Author(s)
Chen, Chung-Wei
DOI
en-US
Abstract
When solving a structural problem by the finite element method, we first discretize the structure of concern into a number of elements. Each of the elements has to satisfy not only the equilibrium conditions but also interelement compatibilities, so that accurate solutions can be computed for the structure under applied loads. In some cases, an element is derived such that the equilibrium condition is satisfied only within element, whereas the compatibilities along the interelement boundaries are not strictly maintained due to geometric complexities. An element stiffness matrix so derived is approximate, which, when employed in structural analysis, will yield only approximate solutions. If the stiffness matrix is inherently defective, it is not always possible to improve the accuracy of the solution obtained by merely using more elements or finer meshes.
In order to solve this kind of linear problems, an iterative procedure is presented in this study, and the accuracy of the solutions obtained by the finite element method can be improved greatly, especially for the cases with stiffness matrices that are inherently defective. The iterative procedure can be classified into three major phases. The predictor phase refers to solution of the structural displacements from the structural stiffness equations. The corrector phase is concerned with the computation of element forces from the element displacements obtained in the predictor phase. In the equilibrium-checking phase, the element forces summed up for each node of the structure are compared with the applied loads. The differences between the applied loads and internal structural forces are regarded as the unbalanced forces.
It has been demonstrated that the predictor affects only the speed of convergence of iterations, but not the accuracy of solutions. The corrector, however, determines entirely the accuracy of the iterative solutions. In this study, the traditional step of solving the structural displacements from the structural equations will be referred to as the predictor. The stiffness matrix involved in this step is approximate or ill-behaved due to the difficulties encountered in the formulation. Because of this, the structural displacements are also approximate. In this study, an iterative procedure will be employed to improve the accuracy of solutions obtained by approximate or ill-behaved stiffness matrices for the analysis of linear problems.
The key point hinges on the use of an accurate corrector, namely, accurate force-displacement relations, for computing the element forces from the element displacements that are made available by the predictor. Once the element forces are computed using a qualified corrector, they can be summed up and compared with the applied loads for evaluation of the unbalanced forces. By treating the unbalanced forces as applied loads, the original structural equations will be solved for corrected displacements, and the element forces can be updated accordingly. Such a procedure of iteration should be repeated until the unbalanced forces can be neglected. It will be demonstrated that even for very rough stiffness matrices used in the predictor, very accurate solutions can be obtained by the iterative procedure, if the corrector used is accurate enough.
As an illustration, the iterative procedure will be incorporated in the linear analysis for dealing with the plane beam problems, and the membrane and plate problems. From the numerical examples, the effectiveness of the procedure in remedying the inherent defects of some finite elements for linear analysis is fully demonstrated.
Subjects
迭代法
預測子
校正子
梁元素
三角板元素
iterative procedure
predictor
corrector
beam element
triangular plate element
Type
thesis
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