Hutton - Fundamentals of Finite Element Analysis (523155), страница 33

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This feature ofpolynomials is utilized to a significant extent in following the development ofvarious element interpolation functions.As in the two-dimensional case, to satisfy the geometric isotropy requirements, the polynomial expression of the field variable in three dimensions mustbe complete or incomplete but symmetric. Completeness and symmetry can alsobe depicted graphically by the “Pascal pyramid” shown in Figure 6.6. While thethree-dimensional case is a bit more difficult to visualize, the basic premise1xzyx2xzz2xyyzy 2 xz 2x 2zx3z3xyzx 2yyz 2x y2y 2zy3Figure 6.6 Pascal “pyramid” forpolynomials in three dimensions.175Hutton: Fundamentals ofFinite Element Analysis1766.

Interpolation Functionsfor General ElementFormulationCHAPTER 6© The McGraw−HillCompanies, 2004TextInterpolation Functions for General Element Formulationremains that each independent variable must be of equal “strength” in the polynomial. For example, the 3-D quadratic polynomialP (x , y, z) = a0 + a1 x + a2 y + a3 z + a4 x 2 + a5 y 2 + a6 z 2+ a7 x y + a8 x z + a9 yz(6.29)is complete and could be applied to an element having 10 nodes. Similarly, anincomplete, symmetric form such asP (x , y, z) = a0 + a1 x + a2 y + a3 z + a4 x 2 + a5 y 2 + a6 z 2(6.30)P (x , y, z) = a0 + a1 x + a2 y + a3 z + a4 x y + a5 x z + a6 yz(6.31)orcould be used for elements having seven nodal degrees of freedom (an unlikelycase, however).Geometric isotropy is not an absolute requirement for field variablerepesentation [1], hence, interpolation functions.

As demonstrated by manyresearchers, incomplete representations are quite often used and solution convergence attained. However, in terms of h-refinement, use of geometrically isotropicrepresentations guarantees satisfaction of the compatibility and completenessrequirements. For the p-refinement method, the reader is reminded that the interpolation functions in any finite element analysis solution are approximations tothe power series expansion of the problem solution. As we increase the numberof element nodes, the order of the interpolation functions increases and, in thelimit, as the number of nodes approaches infinity, the polynomial expression ofthe field variable approaches the power series expansion of the solution.6.5 TRIANGULAR ELEMENTSThe interpolation functions for triangular elements are inherently formulated intwo dimensions and a family of such elements exists. Figure 6.7 depicts the firstthree elements (linear, quadratic, and cubic) of the family.

Note that, in the caseof the cubic element, an internal node exists. The internal node is required to(a)(b)(c)Figure 6.7 Triangular elements:(a) 3-node linear, (b) 6-node quadratic,(c) 10-node cubic.Hutton: Fundamentals ofFinite Element Analysis6. Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004Text6.5 Triangular Elements3 (x3, y3)y2 (x2, y2)x1 (x1, y1)Figure 6.8 A general three-node triangular elementreferred to global coordinates.obtain geometric isotropy, as is subsequently discussed. The triangular elementsare not limited to two-dimensional problems. In fact, the triangular elements canbe used in axisymmetric 3-D cases (discussed later in this chapter) as well as instructural analyses involving out-of-plane bending, as in plate and shell structures.

In the latter cases, the nodal degrees of freedom include first derivatives ofthe field variable as well as the field variable itself. While plate and shell problems are beyond the scope of this book, we allude to those problems again brieflyin Chapter 9.Figure 6.8 depicts a general, three-node triangular element to which weattach an element coordinate system that is, for now, assumed to be the same asthe global system. Here, it is assumed that only 1 degree of freedom is associatedwith each node.

We express the field variable in the polynomial form␾(x , y) = a0 + a1 x + a2 y(6.32)Applying the nodal conditions␾(x 1 , y1 ) = ␾1␾(x 2 , y2 ) = ␾2(6.33)␾(x 3 , y3 ) = ␾3and following the general procedure previously outlined, we obtain   1 x1 y1  a0   ␾1  1 x2 y2  a1 = ␾2   a2␾31 x2 y2(6.34)To solve for the polynomial coefficients, the matrix of coefficients in Equation 6.34 must be inverted. Inversion of the matrix is algebraically tedious butstraightforward, and we finda0 =1[␾1 (x 2 y3 − x 3 y2 ) + ␾2 (x 3 y1 − x 1 y3 ) + ␾3 (x 1 y2 − x 2 y1 )]2A177Hutton: Fundamentals ofFinite Element Analysis1786. Interpolation Functionsfor General ElementFormulationCHAPTER 6a1 =© The McGraw−HillCompanies, 2004TextInterpolation Functions for General Element Formulation1[␾1 ( y2 − y3 ) + ␾2 ( y3 − y1 ) + ␾3 ( y1 − y2 )]2A(6.35)1a2 =[␾1 (x 3 − x 2 ) + ␾2 (x 1 − x 3 ) + ␾3 (x 2 − x 1 )]2ASubstituting the values into Equation 6.32 and collecting coefficients of the nodalvariables, we obtain[(x2 y3 − x3 y2 ) + ( y2 − y3 )x + (x3 − x2 ) y]␾11+ [(x3 y1 − x1 y3 ) + ( y3 − y1 )x + (x1 − x3 ) y]␾2␾(x, y) =2A + [(x1 y2 − x2 y1 ) + ( y1 − y2 )x + (x2 − x1 ) y]␾3(6.36)Given the form of Equation 6.36, the interpolation functions are observed to beN 1 (x , y) =1[(x 2 y3 − x 3 y2 ) + ( y2 − y3 )x + (x 3 − x 2 ) y]2AN 2 (x , y) =1[(x 3 y1 − x 1 y3 ) + ( y3 − y1 )x + (x 1 − x 3 ) y]2AN 3 (x , y) =1[(x 1 y2 − x 2 y1 ) + ( y1 − y2 )x + (x 2 − x 1 ) y]2A(6.37)where A is the area of the triangular element.

Given the coordinates of the threevertices of a triangle, it can be shown that the area is given by1 1 x 1 y1 A = 1 x 2 y2 (6.38)2 1 xy33Note that the algebraically complex form of the interpolation functionsarises primarily from the choice of the element coordinate system of Figure 6.8.As the linear representation of the field variable exhibits geometric isotropy,location and orientation of the element coordinate axes can be chosen arbitrarilywithout affecting the interpolation results.

If, for example, the element coordinate system shown in Figure 6.9 is utilized, considerable algebraic simplificationresults. In the coordinate system shown, we have x 1 = y1 = y2 = 0, 2 A = x 2 y3 ,3 (x3, y3)y2(x2, 0)1 (0, 0)xFigure 6.9 Three-nodetriangle having an elementcoordinate system attachedto the element.Hutton: Fundamentals ofFinite Element Analysis6. Interpolation Functionsfor General ElementFormulation© The McGraw−HillCompanies, 2004Text6.5 Triangular Elementsand the interpolation functions becomeN 1 (x , y) =1[x 2 y3 − y3 x + (x 3 − x 2 ) y]x 2 y3N 2 (x , y) =1[y3 x − x 3 y]x 2 y3N 3 (x , y) =yy3(6.39)which are clearly of a simpler form than Equation 6.37.

The simplification isnot without cost, however. If the element coordinate system is directly associated with element orientation, as in Figure 6.9, the element characteristic matrices must be transformed to a common global coordinate system during modelassembly. (Recall the transformation of stiffness matrices demonstrated for barand beam elements earlier.) As finite element models usually employ a largenumber of elements, the additional computations required for element transformation can be quite time consuming. Consequently, computational efficiencyis improved if each element coordinate system is oriented such that the axesare parallel to the global axes.

The transformation step is then unnecessarywhen model assembly takes place. In practice, most commercial finite elementsoftware packages provide for use of either type element coordinate as a useroption [6].Returning to Equation 6.32, observe that it is possible for the field variableto take on a constant value, as per the completeness requirement, and that the firstpartial derivatives with respect to the independent variables x and y are constants.The latter shows that the gradients of the field variable are constant in both coordinate directions.

For a planar structural element, this results in constant straincomponents. In fact, in structural applications, the three-node triangular elementis commonly known as a constant strain triangle (CST, for short). In the case ofheat transfer, the element produces constant temperature gradients, therefore,constant heat flow within an element.6.5.1 Area CoordinatesWhen expressed in Cartesian coordinates, the interpolation functions for thetriangular element are algebraically complex. Further, the integrations requiredto obtain element characteristic matrices are cumbersome. Considerable simplification of the interpolation functions as well as the subsequently requiredintegration is obtained via the use of area coordinates.

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