Here since were dealing with a single element we

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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...
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