Unformatted text preview: ere are (p ; 5)+(p ; 4)+(p ; 3)+ =6 element modes for a polynomial
of order p.
Shephard et al. 6] also construct blending functions for pyramids, wedges, and prisms.
They display several shape functions and also present entity functions using the basis of
Carnevali et al. 4]. Problems 1. Construct the shape functions associated with a vertex, an edge, and a face node
for a cubic Lagrangian interpolant on the tetrahedron shown on the right of Figure
4.5.3. Express your answer in the tetrahedral coordinates (4.5.3). 30 Finite Element Approximation
η y
3 (x
11
00 3,y 3) 11
00
11
00
h2 α
3
h1 1
0
1
0 α
1 1 (x 1,y 1) α
2
h3 11
002 (x ,y )
11
00 2 2 x 11
003 (0,1)
11
00
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00
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11
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111111111
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111111111111
000000000000
ξ
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11
00
1 (0,0) 2 (1,0) Figure 4.6.1: Nomenclature for a nite element in the physical (x y)plane and for its
mapping to a canonical element in the computational ( )plane. 4.6 Interpolation Error Analysis
We conclude this chapter with a brief discussion of the errors in interpolating a function u
by a piecewise polynomial function U . This work extends our earlier study in Section 2.6
to multidimensional situations. Two and threedimensional interpolation is, naturally,
more complex. In one dimension, it was su cient to study limiting processes where mesh
spacings tend to zero. In two and three dimensions, we must also ensure that element
shapes cannot be too distorted. This usually means that elements cannot become too
thin as the mesh is re ned. We have been using coordinate mappings to construct
bases. Concentrating on twodimensional problems, the coordinate transformation from
a canonical element in, say, the ( )plane to an actual element in the (x y)plane must
be such that no distorted elements are produced.
Let's focus on triangular elements and consider a linear mapping of a canonical unit,
right, 45 triangle in the ( )plane to an element e in the (x y)plane (Figure 4.6.1).
More complex mappings will be discussed in Chapter 5. Using the transformation (4.2.8)
to triangular coordinates in combination with the de nitions (4.2.6) and (4.2.7) of the
canonical variables, we have
232
32 3 2
32
3
x
x1 x2 x3
x1 x2 x3
1; ;
1
4 y 5 = 4 y1 y2 y3 5 4 2 5 = 4 y1 y2 y3 5 4
5:
(4.6.1)
1
111
111
3
The Jacobian of this transformation is Je := x x :
yy (4.6.2a) 4.4. ThreeDimensional Shape Functions 31 Di erentiating (4.6.1), we nd the determinant of this Jacobian as
det(Je) = (x2 ; x1 )(y3 ; y1) ; (x3 ; x1)(y2 ; y1): Lemma 4.6.1. Let he be the longest edge and
then (4.6.2b) be the smallest angle of Element e, e h2 sin
e
2
(4.6.3)
e det(Je ) he sin e :
2
Proof. Label the vertices of Element e as 1, 2, and 3 their a...
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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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