For convenience lets also place a cartesian

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Unformatted text preview: ce, let's also place a Cartesian coordinate system at Vertex 2. We proceed as usual, constructing shape functions on each element. Although not really needed for our present development, those bilinear shape functions that are nonzero on Edge 1-2 follow. 4 Adaptive Finite Element Techniques 1 1 0y 1 0 1 0 1 0 11 00 1 1 0 11 00 11 00 2 11 00 11 00 3 11 00 3 11 00 11 00 11 00 2 x 1111 0000 Figure 8.2.2: Irregular node at the intersection of a re ned element. On Element 1: y N11 = ( h + x )( h ) h N21 = ( h + x )( h ; y ): h h On Element 2: 2 N12 = ( h=h=; x )( y ; h=2 ) 2 h=2 2 ; N32 = ( h=h=; x )( hh=2y ): 2 On Element 3: 2 2 N23 = ( h=h=; x )( h=h=; y ) 2 2 2 y N33 = ( h=h=; x )( h=2 ): 2 As in Chapter 2, the second subscript on Nje denotes the element index. The restriction of U on Element 1 to Edge 1-2 is U (x y) = c1N11 (x y) + c2N21 (x y): Evaluating this at Node 3 yields U (x3 y3) = c1 + c2 2 x < 0: The restriction of U on Elements 2 and 3 to Edge 1-2 is h=2 U (x y) = c1N12 (x y) + c3N32 (x y) iif y < h=2 : c2N23 (x y) + c3N33 (x y) f y In either case, we have U (x3 y3) = c3 x > 0: Equating the two expressions for U (x3 y3) yields the constraint condition c = c1 + c2 : 3 2 (8.2.1) 8.2. h-Re nement 5 Figure 8.2.3: The one-irregular rule: the intended re nement of an element to create two irregular nodes on an edge (left) necessitates re nement of a neighboring element to have no more than one irregular node per element edge (right). Thus, instead of determining c3 by Galerkin's method, we constrain it to be determined as the average of the solutions at the two vertices at the ends of the edge. With the piecewise-bilinear basis used for this illustration, the solution along an edge containing an irregular node is a linear function rather than a piecewise-linear function. Software based on this form of adaptive re nement has been implemented for elliptic 27] and parabolic 1] systems. One could guess that di culties arise when there are too many irregular nodes on an edge. To overcome this, software developers typically use Bank's 9, 10] \one-irregular" and \three-neighbor" rules. The one-irregular rule limits the number of irregular nodes on an element edge to one. The impending introduction of a second irregular node on an edge requires re nement of a neighboring element as shown in Figure 8.2.3. The three-neighbor rule states that any element having irregular nodes on three of its four edges must be re ned. A modi ed quadtree (Section 5.2) can be used to store the mesh and solution data. Thus, let the root of a tree structure denote the original domain . With a structured grid, we'll assume that is square, although it could be obtained by a mapping of a distorted region to a square (Section 5.2). The elements of the original mesh are regarded as o spring of the root (Figure 8.2.4). Elements introduced by adaptive re nement are obtained by bisection and are regarded as o spring of the elements of the original mesh. This structure is de...
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