Inverse Mtrix Shape Function Of Finite Element And Programming

Finite element displacement mode and interpolation function is a very importantpart in the finite element calculation.Displacement mode or form of interpolation willdirectly affect the calculation accuracy of the element. At this stage most displacementinterpolation functions using polynomial interpolation, completeness and coherence areeasily pleased and easily check for confirmation. Some elements generally useLagrangeinterpolation,otherhigher-order interpolation function use Hermit interpolation.In this way, the normalization of the shape function can be easily met. Traditional finiteelement programming used to solve the explicit expression of shape functions andderivation of the shape functions generally use two methods:①Starting from the study on displacement mode, select the appropriate mode ofdisplacement and the appropriate range of nodes number and distribution,trying to makethe number of undetermined constants in the displacement mode match with thefreedom of element.Set up the equations solve undetermined constants in thedisplacement mode, so as to derive the shape function.②Starting from the interpolation function, which is based on the function’s ownperformance,solve the shape function by method of undetermined coefficients.This view of this article is that the explicit expression of the shape functions is notnecessary.This article try to express the shape function by inverse matrix and completeprogramming work with the inverse matrix expression.We can avoid the process ofderiving the explicit expression.As a multi-node, high-level continuous conditions ofthe element-function solver is often a cumbersome process,and for a relatively minorchanges to the location of the node, explicit expression of the function would requirecorresponding changes.On the other way,inverse matrix expression simply change nodecoordinates can be calculated, making tectonic node relative coordinates of the elementbecomes very easy.All of the element in this article use the isoparametric element and there are twomain benefits:①Through a limited relative rules of the parent element can control matrix andprocess of error.②A s defined the standard element,the process to calculate the inverse matrix canbe just one time of each type of element. This document will use inverse-function construction methods to constructelements.On one-dimensional situation will construct classic two-node beam elementand a three-node beam element.On two-dimensional situation,we will construct atriangle three-nodes element and a triangle six-nodes element.We also conduct sometriangle six-nodes element with nodes non-enenly distributed. And it will comparethecalculation results of shape-funtion expressed by explicit expression and inverse matrixexpression.