Unformatted text preview: shes on the two faces not containing the edge to maintain continuity. Thus, if i = 1
and j = 2, the blending function for Edge (1 2) (which is marked with a 5 on the left of
Figure 4.5.3) vanishes on the faces 1 = 0 (Face A234 ) and 2 = 0 (Face A134 ).
The blending function for a face is
ijk ( 1 2 3 4 ) = i j k (4.5.10b) when the vertices on the face are i, j , and k. Again, the blending function ensures that
the shape function vanishes on all faces but Aijk . Again referring to Figures 4.5.2 or
4.5.3, the blending function 123 vanishes when 1 = 0 (Face A234 ), 2 = 0 (Face A134 ),
and 3 = 0 (Face A124 ).
The cubic element blending function for an edge is more di cult to write with our
notation. Instead of writing the general result, let's consider an edge parallel to the
axis. Then
2
(
) = 1 ; Nj ( )Nk ( ):
(4.5.11a)
1;2 j k
4
The factor (1 ; 2)=4 adjusts the edge function to (4.5.9) as described in the paragraph
following (4.4.9). The onedimensional shape functions Nj ( ) and Nk ( ) ensure that the
shape function vanishes on all faces not containing the edge. Blending functions for other
edges are obtained by cyclic permutation of , , and and the index. Thus, referring
to Figure 4.5.4, the edge function for the edge connecting vertices 2 1 1 and 2 2 1 is
2
(
) = 1 ; N2 ( )N1( ):
2 1;2 1
4
Since N2(;1) = 0 (cf. (4.5.7b)), the shape function vanishes on the rear face of the cube
shown in Figure 4.5.4. Since N1(1) = 0, the shape function vanishes on the top face of 4.4. ThreeDimensional Shape Functions 29 the cube of Figure 4.5.4. Finally, the shape function vanishes at = 1 and, hence, on
the left and right faces of the cube of Figure 4.5.4. Thus, the blending function (4.5.11a)
has ensured that the shape function vanishes on all but the bottom and front faces of
the cube of Figure 4.5.4.
The cubic face blending function for a face perpendicular to the axis is
i j k( ) = Ni ( )(1 ; 2)(1 ; 2): (4.5.11b) Referring to Figure 4.5.4, the quadratic terms in and ensure that the shape function vanishes on the right, left ( = 1), top, and bottom ( = 1) faces. The onedimensional shape function Ni( ) vanishes on the rear ( = ;1) face when i = 1 and on
the front ( = 1) face when i = 2 thus, the shape function vanishes on all faces but the
one to which it is associated.
Finally, there are elemental shape functions. For tetrahedra, there are (p ; 1)(p ;
2)(p ; 3)=6 elemental functions for p 4 that are given by N0k ( 1 2 3 4) = 1 2 3 4P ( 2 ; 1)P (2 3 ; 1)P (2 4 ; 1)
8 + + =k;4
k = 4 5 : : : p: (4.5.12a) The subscript 0 is used to identify the element's centroid. The shape functions vanish
on all element faces as indicated by the presence of the multiplier 1 2 3 4. We could
also split this function into the product of an elemental function involving the Legendre
polynomials and the blend involving the product of the tetrahedral coordinates. However,
this is not necessary.
For p 6 there are the following elemental shape functions for a cube N0k ( ) = (1 ; 2)(1 ; 2)(1 ; 2)P ( )P ( )P ( ) 8 + + = k ; 6:
(4.5.12b) Again, the shape function vanishes on all faces of the element to maintain continuity.
Adding, we see that th...
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 Spring '14
 JosephE.Flaherty
 Polynomials, The Land, Quadratic equation, NJ, Degree of a polynomial, shape functions

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