Unformatted text preview: he approximation. Unknowns associated
with interior shape functions are only coupled to unknowns on the element and can
easily be eliminated by a variety of techniques. Considering the biquadratic polynomial in the form (4.3.3a), we might determine c3 3 so that the coe cient of the 4.4. Hierarchical Shape Functions 15 quartic term x2 y2 vanishes. Show how this may be done for a 2 2 square canonical element. Polynomials of this type have been called serendipity by Zienkiewicz
8]. In the next section, we shall see that they are also a part of the hierarchical
family of approximations. The parameter c3 3 is said to be \constrained" since it is
prescribed in advance and not determined as part of the Galerkin procedure. Plot
or sketch shape functions associated with a vertex and a midside. 4.4 Hierarchical Shape Functions
We have discussed the advantages of hierarchical bases relative to Lagrangian bases for
onedimensional problems in Section 2.5. Similar advantages apply in two and three dimensions. We'll again use the basis of Szabo and Babuska 7], but follow the construction
procedure of Shephard et al. 6] and Dey et al. 5]. Hierarchical bases of degree p may
be constructed for triangles and squares. Squares are the simpler of the two, so let us
handle them rst. 4.4.1 Hierarchical Shape Functions on Squares
We'll construct the basis on the canonical element f( )j ; 1
1g, indexing
the vertices, edges, and interiors as described for the biquadratic approximation shown
in Figure 4.3.1. The hierarchical polynomial of order p has a basis consisting of the
following shape functions.
Vertex shape functions. The four vertex shape functions are the bilinear functions
(4.3.1ce) Ni1j = Ni( )Nj ( ) i j=1 2 (4.4.1a) where N1 ( ) = 1 ;
2 N2 ( ) = 1 + :
2 (4.4.1b) The shape function N11 1 is shown in the upper left portion of Figure 4.4.1.
Edge shape functions. For p 2, there are 4(p ; 1) shape functions associated with
the midside nodes (3 1), (2 3), (3 2), and (1 3): N3k 1(
N3k 2(
N1k 3(
N2k 3( )
)
)
) =
=
=
= N1(
N2(
N1(
N2( )N k (
)N k (
)N k (
)N k ( )
)
)
) k = 2 3 ::: p (4.4.2a)
(4.4.2b)
(4.4.2c)
(4.4.2d) 16 Finite Element Approximation where N k ( ), k = 2 3 : : : p, are the onedimensional hierarchical shape functions given
by (2.5.8a) as
r
Z
k ( ) = 2k ; 1
P ( )d :
(4.4.2e)
N
2 ;1 k;1
Edge shape functions N3k 1 are shown for k = 2 3 4, in Figure 4.4.1. The edge shape
functions are the product of a linear function of the variable normal to the edge to which
they are associated and a hierarchical polynomial of degree k in a variable on this edge.
The linear function (Nj ( ), Nj ( ), j = 1 2) \blends" the edge function (N k ( ), N k ( ))
onto the element so as to ensure continuity of the basis.
Interior shape functions. For p 4, there are (p ; 2)(p ; 3)=2 internal shape functions
associated with the centroid, Node (3 3). The rst internal shape function is the \bubble
function" N34 30 0 = (1 ; 2)(1 ; 2): (4.4.3a) The remaining shape functions are products of N34 30 0 and the Legendre polynomials as N35 31 0
N35 30 1
N36 32 0
N36 31...
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 Spring '14
 JosephE.Flaherty
 Polynomials, The Land, Quadratic equation, NJ, Degree of a polynomial, shape functions

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