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Table 4.4.1: Dimension of the hierarchical basis of order p on square and triangular
elements.
1
03 (0,0,1)
1
0
60
1 1
0 5 1
0
1
0 7
11
00
11
00
11
00 ζ1
111
000
111
000
1111
0000
111
000
2 (0,1,0)
ξ
111
000
1 1111
0 0000
1111
0000
1111
0000
111
000
1 1111 1
0 0000 0
111
000
4
1111 1
0000 0
111
000
1111
0000
111
000
1111
0000 1
0
1
0 1 (1,0,0) ζ2 Figure 4.4.3: Node placement and coordinates for hierarchical approximations on a triangle.
Edge shape functions. For p 2 there are 3(p ; 1) edge shape functions which are
each nonzero on one edge (to which they are associated) and vanish on the other two.
Each shape function is selected to match the corresponding edge shape function on a
square element so that a continuous approximation may be obtained on meshes with
both triangular and quadrilateral elements. Let us construct of the shape functions N4k ,
k = 2 3 : : : p, associated with Edge 4. They are required to vanish on Edges 5 and 6
and must have the form N4k ( 1 2 3) = 1 2 k( ) k = 2 3 ::: p (4.4.6a) where k ( ) is a shape function to be determined and is a coordinate on Edge 4 that
has value 1 at Node 1, 0 at Node 4, and 1 at Node 2. Since Edge 4 is 3 = 0, we have N4k ( 12 0) = 1 2 k ( ) 1 + 2 = 1: 20 Finite Element Approximation The latter condition follows from (4.2.8) with 3 = 0. Along Edge 4, 1 ranges from 1 to
0 and 2 ranges from 0 to 1 as ranges from 1 to 1 thus, we may select
(4.4.6b)
1 = (1 ; )=2
2 = (1 + )=2
3 = 0:
While may be de ned in other ways, this linear mapping ensures that 1 + 2 = 1 on
Edge 4. Compatibility with the edge shape function (4.4.2) requires
N4k ( 1 2 0) = N k ( ) = (1 ; )4(1 + ) k ( )
where N k ( ) is the onedimensional hierarchical shape function (4.4.2e). Thus,
k
k ( ) = 4N ( ) :
(4.4.6c)
1; 2
The result can be written in terms of triangular coordinates by using (4.4.6b) to obtain
= 2 ; 1 hence,
N4k ( 1 2 3) = 1 2 k ( 2 ; 1)
k = 2 3 : : : p:
(4.4.7a)
Shape functions along other edges follow by permuting indices, i.e.,
N5k ( 1 2 3) = 2 3 k ( 3 ; 2)
(4.4.7b)
N6k ( 1 2 3) = 3 1 k ( 1 ; 3)
k = 2 3 : : : p:
(4.4.7c)
It might appear that the shape functions k ( ) has singularities at = 1 however, the
onedimensional hierarchical shape functions have (1 ; 2) as a factor. Thus, k ( ) is a
polynomial of degree k ; 2. Using (2.5.8), the rst four of them are
3 ( ) = ;p10
2 ( ) = ;p6
r
r
7 (5 2 ; 1)
4( ) = ;
5 ( ) = ; 9 (7 3 ; 3 ):
(4.4.8)
8
8
Interior shape functions. The (p ; 1)(p ; 2)=2 internal shape functions for p 3 are
products of the bubble function
N73 0 0 = 1 2 3
(4.4.9a)
and Legendre polynomials. The Legendre polynomials are functions of two of the three
triangular coordinates. Following Szabo and Babuska 7], we present them in terms of
2 ; 1 and 3 . Thus,
N74 1 0 = N73 0 0 P1( 2 ; 1)
(4.4.9b)
N74 0 1 = N73 0 0 P1(2 3 ; 1)
(4.4.9c)
N75 2 0 = N73 0 0 P2( 2 ; 1)
(4.4.9d)
N75 1 1 = N73 0 0 P1( 2 ; 1)P1(2 3 ; 1)
(4.4.9e)
::::
(4.4.9f)
N75 0 2 = N73 0 0 P2(2 3 ; 1) 4.4. ThreeDimensional Shape Functions 21 The shift i...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty
 The Land

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