有限元逆矩陣形函數(shù)構造方法及其編程
發(fā)布時間：2018-07-25 11:58
【摘要】：有限元的單元位移模式和插值函數(shù)是有限元計算非常重要的部分，，位移模式或插值形式的選擇將直接影響單元的計算精度�，F(xiàn)階段使用的單元位移插值函數(shù)大多數(shù)采用的是多項式插值的方式，其好處是單元的完備性和協(xié)調(diào)性容易滿足且容易檢查求證。C0單元一般采用拉格朗日插值，C1或者更高階的插值函數(shù)往往采用Hermit插值，因此形函數(shù)的歸一性和自身性質容易滿足。傳統(tǒng)的有限元編程方法傾向與求解出形函數(shù)的顯式表達式，其形函數(shù)的推導一般采用兩種方法： ①從研究位移模式入手，選擇適當?shù)奈灰颇Ｊ胶瓦m當?shù)膯卧?jié)點數(shù)目和分布，使單元的自由度和位移模式中的待定常數(shù)數(shù)目相匹配，建立方程組求解位移模式中的待定常數(shù)，從而推導出形函數(shù)。 ②從插值基函數(shù)入手，也就是形函數(shù)的自身性能，用配方法來求解形函數(shù)。 本文的觀點是形函數(shù)的顯式表達式并不是非求不可，本文探討直接從位移模式出發(fā)，對于形函數(shù)并不用顯式表達而用逆矩陣構造形函數(shù)來代替，直接針對逆矩陣表達式來編程計算。這種逆矩陣表達形函數(shù)的好處是可以免去形函數(shù)的理論求解過程。對于多節(jié)點和高階連續(xù)條件的單元其形函數(shù)求解往往是比較繁瑣的過程，并且對于多節(jié)點單元的節(jié)點相對位置的少許改動，形函數(shù)顯式表達式就需要相應改動。逆矩陣表達的形函數(shù)只需相應改動節(jié)點坐標就可以實現(xiàn)計算，這使構造變動節(jié)點相對坐標的單元也變得非常方便。 本文根據(jù)逆矩陣形函數(shù)構造方法編寫的單元都采用等參單元的編寫形式，等參單元，主要有以下兩點好處： ①通過限定相對規(guī)則的母單元可以控制矩陣運算求逆過程的誤差。 ②由于限定了母單元，每一種類型的單元形函數(shù)逆矩陣求解過程在整個計算過程中只用計算一次，有效減小計算量。 本文將采用逆矩陣形函數(shù)構造方法來構造單元，一維的情況將構造經(jīng)典兩節(jié)點梁單元和一個三節(jié)點的5次Herimt型的梁單元，二維的情況將構造等參三角形三節(jié)點平面單元和等參三角形六節(jié)點單元，對于六節(jié)點單元將改變其邊中節(jié)點的位置來構造兩種節(jié)點非均勻分布的六節(jié)點三角形平面單元并用來計算一個平面應力集中的算例。并將這些逆矩陣表達形函數(shù)的單元與顯式表達形函數(shù)的單元計算結果進行了比較。
[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.
【學位授予單位】：重慶大學
【學位級別】：碩士
【學位授予年份】：2014
【分類號】：TU311.41
本文編號：2143752
[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.
【學位授予單位】：重慶大學
【學位級別】：碩士
【學位授予年份】：2014
【分類號】：TU311.41
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