以迭代法修正梁及三角形元素之缺陷

Abstract

在以有限元素法分析結構問題時，我們是將結構切割成許多元素來表示。每個元素不僅需滿足本身之平衡條件外，亦需符合元素間之諧和性。如此之元素方能準確的計算出結構受外力作用下之變形量。然而在某些情況下，元素只能滿足本身之平衡條件，而邊界之連續性卻因為幾何程度上的複雜而無法完全的維持。這樣的元素導致出近似的勁度矩陣，當使用在結構分析時，僅能得到近似解。而且如果勁度矩陣具有內在缺陷時，單靠細分元素是無法改善此一結果。 為了解決此問題，本文發展了一套迭代流程，可以大幅提升傳統有限元素線性分析的精度，特別適用於元素勁度矩陣具有內在的缺陷的情況。本文之迭代流程可區分成三個主要階段：預測階段主要是由結構之平衡方程式求解位移；校正階段則由預測階段所求出之位移來求算元素內力；平衡檢測階段則是檢核疊加於節點上之內力和外力之比較，而二者間之差稱為不平衡力。 經前人證明可知預測子只影響迭代的收斂速度，而不是解的精度。校正子卻可完全地決定計算結果的準確度。本文中，傳統求解位移步驟中所使用的勁度矩陣將視為預測子。此步驟中使用的勁度矩陣由於推導上的困難僅為近似。因此求得的位移亦為近似值。本文將使用迭代流程於線性分析上來改善由不良勁度矩陣所得的解之精度。 此迭代之重點在於使用準確之校正子，即準確之內力與位移之關係，用以計算由預測子解出的位移來求得元素內力。元素內力經由良好的校正子求出後，則可疊加與外力荷載比較以求出不平衡力。再將此不平衡力視為外力，代入結構平衡方程式可得出修正位移，元素內力亦可因此而更新。此迭代流程需不斷重覆進行待不平衡力可忽略為止。如此可證明出即使使用很差的勁度矩陣作為預測子，若校正子夠準確的話，則經由迭代出的結果亦是準確的。 本文將此迭代流程應用於平面梁、膜及板等問題之分析作為說明。由數值範例分析中，經由此有效的迭代流程於線性分析可彌補某些具有內在缺陷之元素。

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.