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Unformatted text preview: n 3 ensures that the range of the Legendre polynomials is ;1 1].
Like the edge shape functions for a square (4.4.2), the edge shape functions for a
triangle (4.4.7) are products of a function on the edge ( k ( i ; j )) and a function ( i j i 6=
j ) that blends the edge function onto the element. However, the edge functions for the
triangle are not the same as those for the square. The two are related by (4.4.6c). Having
the same edge functions for all element shapes simpli es construction of the element
sti ness matrices 6]. We can, of course, make the edge functions the same by rede ning
the blending functions. Thus, using (4.4.6a,c), the edge function for Edge 4 can be N k ( )
if the blending function is
41 2:
1; 2
In a similar manner, using (4.4.2a) and (4.4.6c), the edge function for the shape function
N3k 1 can be k ( ) if the blending function is N1 ( )(1 ; 2) :
4
Shephard et al. 6] show that representations in terms of k involve fewer algebraic
operations and, hence, are preferred.
The rst three edge and interior shape functions are shown in Figure 4.4.4. A degree
p hierarchical approximation on a triangle has 3+3(p ; 1)+ +(p ; 1)+(p ; 2)+ =2 unknowns
and shape functions. This function is listed in Table 4.4.1. We see that for p > 1, there are
two fewer shape functions with triangular elements than with squares. The triangular
element is optimal in the sense of using the minimal number of shape functions for a
complete polynomial of a given degree. This, however, does not mean that the complexity
of solving a given problem is less with triangular elements than with quadrilaterals. This
issue depends on the partial di erential equations, the geometry, the mesh structure, and
other factors.
Carnevali et al. 4] introduced shape functions that produce better conditioned element sti ness matrices at higher values of p than the bases presented here 7]. Adjerid
et al. 1] construct an alternate basis that appears to further reduce ill conditioning at
high p. 4.5 ThreeDimensional Shape Functions
Threedimensional nite element shape functions are constructed in the same manner as
in two dimensions. Common element shapes are tetrahedra and hexahedra and we will
examine some Lagrange and hierarchical approximations on these elements. 22 Finite Element Approximation 0 0.4 −0.1 0.3
0.2 −0.2 0.1
−0.3
0
−0.4
−0.1
−0.5 −0.2 −0.6 −0.3 −0.7
0 1
0.2 −0.4
0 1 0.8
0.4 0.2 0.6
0.6 0.8
0.4 0.4
0.8 0.6
0.6 0.2
1 0.4
0.8 0 0.2
1 0.25 0.04 0.2 0 0.035 0.15 0.03 0.1 0.025 0.05 0.02
0 0.015 −0.05 0.01 −0.1 0.005 −0.15
−0.2
0 1
0.2 0
0 1
0.2 0.8
0.4
0.6 0.8
0.4 0.6 0.6 0.4
0.8 0.6 1 0.4
0.8 0.2 0.2
1 0 0.015 0 0.01 0.01 0.005 0.005 0 0 −0.005 −0.005 −0.01 −0.01 −0.015 −0.015
0 1
0.2 0.8
0.4 0.6
0.6 0.4
0.8 0.2
1 0 −0.02
0 1
0.2 0.8
0.4 0.6
0.6 0.4
0.8 0.2
1 0 Figure 4.4.4: Hierarchical edge and interior shape functions N42 (top left), N43 (top right),
N44 (middle left), N73 0 0 (middle right), N74 1 0 (bottom left), N74 0 1 (bottom right...
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 Spring '14
 JosephE.Flaherty
 The Land

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