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Unformatted text preview: located on a mesh entity. where (v f ) = A(v u) = ZZ ZZ vfdxdy (5.5.1b) p(vxux + vy uy ) + qvu]dxdy (5.5.1c) As usual, is a two-dimensional domain with boundary @ = @ smooth solutions of (5.5.1) satisfy ;(pux )x ; (puy )y + qu = f (x y) 2 E @ N . Recall that (5.5.2a) 5.5. Element Matrices and Their Assembly 25 u= (x y) 2 @ E (5.5.2b) un = 0 (x y) 2 @ N (5.5.2c) where n is the unit outward normal vector to @ . Trivial natural boundary conditions are considered for simplicity. More complicated situations will be examined later in this section. Following the one-dimensional examples of Chapters 1 and 2, we select nite-dimensional N N 1 subspaces SE and S0 of HE and H01 and write (5.5.1b,c) as the sum of contributions over elements (V f ) = A(V U ) = N X e=1 N X e=1 (V f )e (5.5.3a) Ae(V U ): (5.5.3b) Here, N is the number of elements in the mesh, (V f )e = ZZ V fdxdy (5.5.3c) e is the local L2 inner product, Ae(V U ) = ZZ p(VxUx + Vy Uy ) + qV U ]dxdy (5.5.3d) e is the local strain energy, and e is the portion of occupied by element e. The evaluation of (5.5.3c,d) can be simple or complex depending on the functions p, q, and f and the mesh used to discretize . If p and q were constant, for example, the local strain energy (5.5.3d) could be integrated exactly as illustrated in Chapters 1 and 2 for one-dimensional problems. Let's pursue a more general approach and discuss procedures based on transforming integrals (5.5.3c,d) on element e to a canonical element 0 and evaluating them numerically. Thus, let U0 ( ) = U (x( ) y( )) and V0 ( ) = V (x( ) y( )) and transform the integrals (5.5.3c,d)) to element 0 to get (V f )e = ZZ 0 Ae(V U ) = V0( ZZ 0 )f (x( p(V0 x ) y( + V0 x)(U0 )) det(Je)d d : x + U0 x)+ (5.5.4a) 26 Mesh Generation and Assembly p(V0 y + V0 y )(U0 y + U0 y ) + qV0U0] det(Je)d d where Je is the Jacobian of the transformation (cf. (5.4.1)). Expanding the terms in the strain energy Ae (V U ) = ZZ g1eV0 U0 + g2e(V0 U0 + V0 U0 ) + g3eV0 U0 + qV0 U0] det(Je)d d 0 (5.5...
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