21 mapping of a simply connected region left onto a

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Unformatted text preview: connected region (left) onto a rectangular computational domain (right). In Figure 5.2.1, we show a region with four segments f ( 0), f ( 1), f (0 ), and f (1 ) that are related to the computational lines = 0, = 1, = 0, and = 1, respectively. (The four curved segments may involve di erent functions, but we have written them all as f for simplicity.) Also consider the projection operators x = P (f ) = N1 ( )f (0 ) + N2( )f (1 ) (5.2.2a) x = P (f ) = N1 ( )f ( 0) + N2( )f ( 1) (5.2.2b) N1 ( ) = 1 ; (5.2.2c) N2 ( ) = (5.2.2d) where and 4 Mesh Generation and Assembly are the familiar hat functions scaled to the interval 0 1. As shown in Figure 5.2.2, the mapping x = P (f ) transforms the left and right edges of the domain correctly, but ignores the top and bottom while the mapping x = P (f ) transforms the top and bottom boundaries correctly but not the sides. Coordinate lines of constant and are mapped as either curves or straight lines on the physical domain. 2,2 1,2 2,2 1,2 y y 1,1 2,1 2,1 1,1 x Figure 5.2.2: The transformations x = P (f ) (left) and x = P (f ) (right) as applied to the simply-connected domain shown in Figure 5.2.1. 2,2 2,2 1,2 1,2 y y 1,1 1,1 x 2,1 x Figure 5.2.3: Illustrations of the transformations x = P P (f ) (left) and x = P (right) as applied to the simply-connected domain shown in Figure 5.2.1. 2,1 P (f ) With a goal of constructing an e ective mapping, let us introduce the tensor product and Boolean sums of the projections (5.2.2) as x = P P (f ) = 2 2 XX i=1 j =1 Ni( )Nj ( )f (i ; 1 j ; 1) (5.2.3a) 5.2. Mesh Generation 5 x=P P (f ) = P (f ) + P (f ) ; P P (f ): (5.2.3b) An application of these transformations to a simply-connected domain is shown in Figure 5.2.3. The transformation (5.2.3a) is a bilinear function of and while (5.2.3b) is clearly the one needed to map the simply connected domain onto the computational plane. Lines of constant and become curves in the physical domain (Figure 5.2.3). Although these transformations are simple, they have been used...
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