This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ment, we number the 4.2. Lagrange Shape Functions on Triangles 5 3 1
03
1
0 1
0
1
0
6 1
0
1
0 1
0
1
0
1 1
07
1
0
1
0 8
11
00
11
00 5 1
0
1
0 9 1
0
1
0 1
0
1
0 4 1
0
1
0 1 1
0
1
0 10 1
0
1
0 11
00
11
00
4 2 11
00
11
006
11
00
11
00
11
00
1
0
11
00
1
0
5
2 Figure 4.2.3: Arrangement of nodes for quadratic (left) and cubic (right) Lagrange nite
element approximations.
nodes from 1 to 6 as shown in Figure 4.2.3. The shape functions have the form (4.2.5)
with n2 = 6
Nj = a1 + a2x + a3y + a4x2 + a5 xy + a6y2
and the six coe cients aj , j = 1 2 : : : 6, are determined by requiring Nj (xk yk ) =
The basis jk j k = 1 2 : : : 6: = N=1 Nj e(x y)
e
is continuous by virtue of the placement of the nodes. The shape function Nj e is a
quadratic function of a local coordinate on each edge of the triangle. This quadratic
function of a single variable is uniquely determined by the values of the shape functions
at the three nodes on the given edge. Shape functions on shared edges of neighboring
triangles are determined by the same nodal values hence, ensuring that the basis is
globally of class C 0.
The construction of cubic approximations would proceed in the same manner. A
complete cubic in two dimensions has 10 parameters. These parameters can be determined by selecting 10 nodes on each element. Following the reasoning described above,
we should place four nodes on each edge since a cubic function of one variable is uniquely
determined by prescribing four quantities. This accounts for nine of the ten nodes. The
last node can be placed at the centroid as shown in Figure 4.2.3.
The construction of Lagrangian approximations is straight forward but algebraically
complicated. Complexity can be signi cantly reduced by using one of the following two
coordinate transformations.
j 6 Finite Element Approximation
11
00
11
00 3 (x 3,y 3) y η
N 1=
3 N 1=
2 1 0 N 1= 1
2 11
00
11
00
1 (x 1,y 1) 11
002 (x 2,y 2)
11
00 N 1= 0
3
x 11
003 (0,1)
111111111
000000000
11
00
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111
000000000
111111111111
000000000000
ξ
11
00
11
00
111111111
000000000
11
00
11
00
1 (0,0) 2 (1,0) Figure 4.2.4: Mapping an arbitrary triangular element in the (x y)plane (left) to a
canonical 45 right triangle in the ( )plane (right).
1. Transformation to a canonical element. The idea is to transform an arbitrary
element in the physical (x y)plane to one having a simpler geometry in a computational
( )plane. For purposes of illustration, consider an arbitrary triangle having vertex
nodes numbered 1, 2, and 3 which is mapped by a linear transformation to a unit 45
right triangle, as shown in Figure 4.2.4.
Consider N21 and N31 as de ned by (4.2.2). (A superscript 1 has been added to
emphasize that the shape functions are lin...
View
Full
Document
This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty
 The Land

Click to edit the document details