Imposing conditions 421 produces 232 32 3 1 1 xj yj a

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Unformatted text preview: tions (4.2.1) produces 232 32 3 1 1 xj yj a 4 0 5 = 4 1 xk yk 5 4 b 5 k 6= l 6= j j k l = 1 2 3: 0 1 xl yl c Solving this system by Crammer's rule yields k 6= l 6= j j k l=1 2 3 (4.2.2a) Nj (x y) = DkCl (x y) jkl where 2 3 1xy Dk l = det 4 1 xk yk 5 (4.2.2b) 1 xl yl where e 4.2. Lagrange Shape Functions on Triangles 3 1 02 1 0 1 (x 2 ,y 2) 1 0 1 0 1 03 1 0 (x 1 ,y 1) (x 3 ,y 3) Figure 4.2.1: Triangular element with vertices 1 2 3 having coordinates (x1 y1), (x2 y2), and (x3 y3). φ1 N1 3 3 1 1 2 2 Figure 4.2.2: Shape function N1 for Node 1 of element e (left) and basis function 1 for a cluster of four nite elements at Node 1. 2 3 1 xj yj Cj k l = det 4 1 xk yk 5 : (4.2.2c) 1 xl yl Basis functions are constructed by combining shape functions on neighboring elements as described in Section 2.4. A sample basis function for a four-element cluster is shown in Figure 4.2.2. The implicit construction of the basis in terms of shape function eliminates the need to know detailed geometric information such as the number of elements sharing 4 Finite Element Approximation a node. Placing the three nodes at element vertices guarantees a continuous basis. While interpolation at three non-colinear points is (necessary and) su cient to determine a unique linear polynomial, it will not determine a continuous approximation. With vertex placement, the shape function (e.g., Nj ) along any element edge is a linear function of a variable along that edge. This linear function is determined by the nodal values at the two vertex nodes on that edge (e.g., j and k). As shown in Figure 4.2.2, the shape function on a neighboring edge is determined by the same two nodal values thus, the basis (e.g., j ) is continuous. The restriction of U (x y) to element e has the form U (x y) = c1 N1(x y) + c2N2 (x y) + c3N3 (x y) (x y) 2 e: (4.2.3) Using (4.2.1), we have cj = U (xj yj ), j = 1 2 3. The construction of higher-order Lagrangian shape functions proceeds in the same manner. In order to construct a p th-degree polynomial approximation on element e, we introduce Nj (x y), j = 1 2 : : : np, shape functions at np nodes, where np = (p + 1)(p + 2) 2 (4.2.4) is the number of monomial terms in a complete polynomial of degree p in two dimensions. We may write a shape function in the form np X aiqi (x y) = aT q(x y) (4.2.5a) qT (x y) = 1 x y x2 xy y2 : : : yp]: (4.2.5b) Nj (x y) = i=1 where Thus, for example, a second degree (p = 2) polynomial would have n2 = 6 coe cients and qT (x y) = 1 x y x2 xy y2]: Including all np monomial terms in the polynomial approximation ensures isotropy in the sense that the degree of the trial function is conserved under coordinate translation and rotation. With six parameters, we consider constructing a quadratic Lagrange polynomial by placing nodes at the vertices and midsides of a triangular element. The introduction of nodes is unnecessary, but it is a convenience. Indexing of nodes and other entities will be discussed in Chapter 5. Here, since we're dealing with a single ele...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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