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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 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 11 00 11 00 111111111 000000000 111111111111 000000000000 ξ 11 00 11 00 111111111 000000000 11 00 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 multi-dimensional situations. Two- and three-dimensional 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 two-dimensional 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. Three-Dimensional 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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