26 the three linear shape functions nj1 j 1 2 3 can be

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Unformatted text preview: known as \triangular" or \barycentric" coordinates. To be more speci c, consider an arbitrary triangle with vertices numbered 1, 2, and 3 as shown in Figure 4.2.7. Let 1 1 = N1 1 2 = N2 1 3 = N3 and de ne the transformation from triangular to physical coordinates as 232 32 3 x x1 x2 x3 1 4 y 5 = 4 y1 y2 y3 5 4 2 5 : 1 111 3 (4.2.7) (4.2.8) Observe that ( 1 2 3) has value (1,0,0) at vertex 1, (0,1,0) at vertex 2 and (0,0,1) at vertex 3. An alternate, and more common, de nition of the triangular coordinate system involves ratios of areas of subtriangles to the whole triangle. Thus, let P be an arbitrary point in the interior of the triangle, then the triangular coordinates of P are AP 23 AP 31 AP 12 (4.2.9) 1= A 2= A 3= A 123 123 123 where A123 is the area of the triangle, AP 23 is the area of the subtriangle having vertices P , 2, 3, etc. 4.2. Lagrange Shape Functions on Triangles 9 1.2 1.2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2 0 −0.4 0 1 0.2 1 0.8 0.4 0.2 0.8 0.6 0.6 0.4 0.6 0.4 0.8 0.6 0.4 0.2 1 0.8 0.2 0 1 0 1 0.8 0.6 0.4 0.2 0 0 1 0.2 0.8 0.4 0.6 0.6 0.4 0.8 0.2 1 0 Figure 4.2.6: Cubic Lagrange shape functions associated with a vertex (left), an edge(right), and the centroid (bottom) of a right 45 triangular element. The triangular coordinate system is redundant since two quantities su ce to locate a point in a plane. This redundancy is expressed by the third of equations (4.2.8), which states that 1 + 2 + 3 = 1: This relation also follows by adding equations (4.2.9). Although seemingly distinct, triangular coordinates and the canonical coordinates are closely related. The triangular coordinate 2 is equivalent to the canonical coordinate and 3 is equivalent to , as seen from (4.2.6) and (4.2.7). Problems 1. With reference to the nodal placement and numbering shown on the left of Figure 4.2.3, construct the shape functions for Nodes 1 and 4 of the quadratic Lagrange polynomial. Derive your answer using triangular coordinates. Having done this, also express your answer in terms of the canonical ( ) coordinates. Plot or sketch 10 Finite Element Approximation 3 (0,0,1) ζ1 = 1 ζ2 = 0 P( ζ1 ,ζ2 ,ζ3 ) 1 (1,0,0) ζ1 = 0 ζ3 = 0 2 (0,1,0) Figure 4.2.7: Triangular coordinate system. the two shape functions on the canonical element. 2. A Lagrangian approximation of degree p on a triangle has three nodes at the vertices and p ; 1 nodes along each edge that are not at vertices. As we've discussed, the latter placement ensures continuity on a mesh of triangular elements. If no additional nodes are placed on edges, how many nodes are interior to the element if the approximation is to be complete? 4.3 Lagrange Shape Functions on Rectangles The triangle in two dimensions and the tetrahedron in three dimensions are the polyhedral shapes having the minimum number of edges and faces. They are optimal for de ning complete C 0 Lagrangian polynomials. Even so, Lagrangian interpolants are simple to construct on rectangles and hexahedra by taking products of one-dimensional Lagran...
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