有限元逆矩阵形函数构造方法及其编程

发布时间：2018-07-25 11:58

【摘要】：有限元的单元位移模式和插值函数是有限元计算非常重要的部分，，位移模式或插值形式的选择将直接影响单元的计算精度。现阶段使用的单元位移插值函数大多数采用的是多项式插值的方式，其好处是单元的完备性和协调性容易满足且容易检查求证。C0单元一般采用拉格朗日插值，C1或者更高阶的插值函数往往采用Hermit插值，因此形函数的归一性和自身性质容易满足。传统的有限元编程方法倾向与求解出形函数的显式表达式，其形函数的推导一般采用两种方法： ①从研究位移模式入手，选择适当的位移模式和适当的单元节点数目和分布，使单元的自由度和位移模式中的待定常数数目相匹配，建立方程组求解位移模式中的待定常数，从而推导出形函数。 ②从插值基函数入手，也就是形函数的自身性能，用配方法来求解形函数。本文的观点是形函数的显式表达式并不是非求不可，本文探讨直接从位移模式出发，对于形函数并不用显式表达而用逆矩阵构造形函数来代替，直接针对逆矩阵表达式来编程计算。这种逆矩阵表达形函数的好处是可以免去形函数的理论求解过程。对于多节点和高阶连续条件的单元其形函数求解往往是比较繁琐的过程，并且对于多节点单元的节点相对位置的少许改动，形函数显式表达式就需要相应改动。逆矩阵表达的形函数只需相应改动节点坐标就可以实现计算，这使构造变动节点相对坐标的单元也变得非常方便。本文根据逆矩阵形函数构造方法编写的单元都采用等参单元的编写形式，等参单元，主要有以下两点好处： ①通过限定相对规则的母单元可以控制矩阵运算求逆过程的误差。 ②由于限定了母单元，每一种类型的单元形函数逆矩阵求解过程在整个计算过程中只用计算一次，有效减小计算量。本文将采用逆矩阵形函数构造方法来构造单元，一维的情况将构造经典两节点梁单元和一个三节点的5次Herimt型的梁单元，二维的情况将构造等参三角形三节点平面单元和等参三角形六节点单元，对于六节点单元将改变其边中节点的位置来构造两种节点非均匀分布的六节点三角形平面单元并用来计算一个平面应力集中的算例。并将这些逆矩阵表达形函数的单元与显式表达形函数的单元计算结果进行了比较。
[Abstract]:The element displacement model and interpolation function of finite element are very important parts of finite element calculation. The choice of displacement mode or interpolation form will directly affect the accuracy of element calculation. Most of the element displacement interpolation functions used at the present stage are polynomial interpolation. The advantage of this method is that the completeness and coordination of the elements are easy to satisfy and the verification. C0 elements are generally using Lagrange interpolation C1 or Hermit interpolation for higher order interpolation functions, so the normalization and properties of shape functions are easy to satisfy. The traditional finite element programming method tends to solve the explicit expression of the shape function. The derivation of the shape function generally adopts two methods: (1) starting with the study of displacement mode, In order to match the degree of freedom of the element with the number of undetermined constants in the displacement mode, a set of equations is established to solve the undetermined constant in the displacement mode by selecting the appropriate displacement mode and the appropriate number and distribution of the nodes of the element. Therefore, the shape function is derived. 2 starting with the interpolation basis function, that is, the form function's own performance, the form function is solved by the matching method. The point of view in this paper is that the explicit expression of shape function is not necessary. In this paper, we discuss directly from the displacement mode, for shape function, we do not use explicit expression but use inverse matrix to construct shape function instead of form function. Directly for the inverse matrix expression to program calculation. The advantage of the inverse matrix representation is that the theoretical solution of the shape function can be avoided. For the element with multi-node and high-order continuity condition, the shape function solution is often a tedious process, and for the relative position of the multi-node unit, the explicit expression of the shape function needs to be changed accordingly. The shape function expressed by the inverse matrix can be calculated only by changing the coordinate of the node, which makes it convenient to construct the unit of the relative coordinate of the variable node. In this paper, the elements written according to the method of constructing inverse matrix shape functions are all written in the form of isoparametric elements, and isoparametric elements. There are two main advantages: (1) the error of matrix operation can be controlled by defining the relative rule of the parent unit. 2 because of the limitation of the master unit, Each type of inverse matrix of cell function is solved only once in the whole calculation process, which effectively reduces the computation cost. In this paper, the inverse matrix shape function is used to construct the element. In one dimensional case, the classical two-node beam element and a three-node Herimt beam element of order 5 are constructed. In the case of two dimensions, the isoparametric triangle three-node plane element and the isoparametric triangle six-node element will be constructed. For the six-node element, the position of the node in the edge is changed to construct the six-node triangular plane element with non-uniform distribution of two nodes and to calculate an example of the plane stress concentration. The results of the element of the inverse matrix expressing the shape function are compared with the results of the element calculation of the explicit form function.
【学位授予单位】：重庆大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TU311.41

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