Unformatted text preview: 1 2 3 ∆x ∆x Figure 7 One-Dimensional Mesh
φι Ni N i+1 φ (x) i Element i i+1 φ ι+1 i Element i i+1 Figure 8 Linear Shape Functions on Element i and Corresponding Variation of 3.2.2 Finite Element Method. We consider again the one-dimensional diffusion equation, Equation 8. There are different kinds of ﬁnite element methods. Let us look at a popular variant, the Galerkin ﬁnite element method. More detailed information about this class of numerical techniques may be found in (Reddy, 1993; Zienkiewicz, 2000; Heubner & Thornton, 1982). We start again with the non-conservation form of the governing equation, Eq. 9. The computational domain is divided into N-1 elements corresponding to N nodes, as shown in Fig. 7. Let be an approximation to . Since is only an approximation, it does not satisfy Equation 9 exactly, so that there is a residual R: (17) We wish to ﬁnd a such that (18) is a weight function, and Equation 18 requires that the residual become zero in a weighted sense. In order to generate a set of discrete equations we use a family of weight functions , , where N is the number of nodes. Thus, we require (19) The weight functions are typically local in that they are non-zero in the vicinity of node , but are zero everywhere else in the domain. Further, we assume a shape function for , i.e., assume how varies between nodes. Thus, (20) U V¦ F VWa`XY b Y SY U V¦ T4 F 6 S U V¦ S4 T50 RFCQECCD9BP9§7I 6 23 ¦§¢ H0 )('%#! @ 4 &$ " GFECECDBA§8 6 50 [email protected] 4 2¦ 3 §¢ 1)('%#! 0 &$ " ¤ §¢ ¦ © ¨ ¥£¡ ¤¢ 0 The Galerkin ﬁnite element method requires that the weight and shape functions be the same, i.e., . Typically the shape function variation is also local, as shown for the case of a linear shape function in Fig. 8. Here, Furthermore, the source term is also interpolated on the domain as: (21) Thus, under the Galerkin ﬁnite element formulation, The next step is to integrate the ﬁrst term in Eq. 22 by parts. This procedure yields: (23) Using Eq. 20 we may write: The ﬁrst term in Eq. 23 is given by are the heat ﬂuxes in the positive x direction at the boundaries.Here by deﬁnition. If the shape function is local, it is non-zero only in the vicinity of the node . Thus, the ﬁrst term in Eq. 23 is (26) (27) (28) Here Thus the overall equation may be written as: The discrete equation for a node j may thus be written as: ¦ §¢ Ax W Ax U U ¦U U ¦ §¢ F!Q9CEQCC9BA9§7i 6 ¦ S i ¦ S F ¥¢ V¦ b V¦ S @ b 4 F b 4 Y © S Y F VWa`YX 4 F x ¨ Y Y F VWa`YX 4 F ¢ ¡ x W h 7 i PQ9ECECDBgfQd 2 W h [email protected] e U i ¦ U 7e
fQd ¦ S 4 F b 4 F S e yEdHb 6 W U V¦ 7 U ¦ S W ¦ S F F U S 4 F b W ¦ b U i ¦ U S 4 F b T4 F S ¦ §¢ ¦ §¢ ¢ VWa`XY ¢ b YF Y ¦ Ax ¦¦ ¦ §¢ §¢ §¢ Ax i ¥¢ U FPE9CQECCD9BP§I 6 §¢ © V¦ S @97 2¦U Ax V¦ S 4 F x ¨ ¢ 4 F ¢ ¡ x x ¢ 4 F ¡ ¤¦ Ax Ax ¦ ¥¢ U F [email protected] PE9ECQCDA§
6 3 ¥¢ © V¦ 4 F ¨ §¢ ¤ ¢ V¦ 4 F ¡ y 2 ¦ U S x S x and W h 3 7 i hPE9CQECCD9B3yEd 2 @e e fQd 7
e fQdHb U ¦i7 #f spU r f ¦ S 'f su¨ ¦ S U ¦i U#f sS pU 7 f ¦ ¦i S qpU r¨ geS ¦ ¦ i h¨ f d 7 U U vw© V¦ S F bVWa`YX V¦ S © Y Y e© U V¦ S F b VatY U V¦ S F Y 4 m¨ !4 BVWa`XY l k jb YY Y F c Y0 (22) (24) (25) (29) (30) (31) (32) ∆x W w P E e δ xw δ xe Figure 9 Arrangement of Control Volumes In the above equations, when and are given, the equations at nodes and may be used to evaluate the ﬂuxes and . On the other hand, when and are speciﬁed, the same equations are used to ﬁnd and . By choosing speciﬁc shape functions a coupled algebraic equation set may be derived for the nodal values . Since is local, the matrix is sparse. We should note here that because the Galerkin ﬁnite element method only requires the residual to be zero in some weighted sense, it does not enforce the conservation principle in its original form. We now turn to a method which employs conservation as a tool for developing discrete equations. 3.2.3 Finite Volume Method. The ﬁnite volume method (sometimes called the control volume method) divides the domain in to a ﬁnite number of non-overlapping cells or control volumes over which conservation of is enforced in a discrete sense. We start with the conservative form of the scalar transport equation, Equation 8. Consider a one-dimensional mesh, with cells as shown in Figure 9. Let us store discrete values of at cell centroids, denoted by , and . The cell faces are denoted by and . Let us assume the face areas to be unity. We focus on the cell associated with . We start by integrating Equation 8 over the cell P. This yields (35) t w §¢ u s w §¢ u ¦ t i¦ 2¦s 3 §¢ © ¨ ¢ ¡ ¢ ¡ so that (36) b 2 6 p o ¦U §¢ © V¦ F W V¦ F U b Va`XY S Y S Y ¦ ¦ 4 ¦ §¢ ¥¢ ¢ §¢ ¢ 4F YF 0 4Y j W b o Here w¦ u ¦ t ¦ x§¢ v§¢ t 2¦s 3 §¢ © y ¨ §¢ ¢ ¡ ¢ s Ax x Ax x UF Y V¦ S F Y 4j Y 4l r Y q W F n 6 (33) (34) ...
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