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camber and the artificial density converge, the use of an iterative procedure using the conjugate gradient method should be more efficient and it is chosen over direct solution approach. 3.8 The Convective Equation Solver The governing equation for the artificial density, Eq.(2.19), and the governing equa- tion for the blade camber, Eq.(2.28), are two first order equations of hyperbolic type. It can be solved by using the method of characteristics, or alternatively, it can be solved by using direct numerical differentiation and integration using the finite element method. The RHS of these equations are obtained from the previous iteration or from initial guess. The discrete approximations to operator W, a + W,.- can be separated into two parts. The parts that contain the given initial nodal points will be transposed to the RHS, and the other parts that contain the unknown nodal points are retained on the LHS. As an illustration, Eq.(2.28) can be written as Of Of W,- +W, q (3.11) where qequals[ - w+Vb*Val- Now if we use an element with four sides with fi and f4 given, then the unknowns f2 and fs are solved for by satisfying the above equation at nodes 1 and 4 (or alterna- tively, at two Gaussian points, which are preferable, since they give a better weighted accuracy). Using Eq.(3.7) we get [W,(I11N,fi+IIN.,f) +Wz,(I 1Nf +I22Ni,f)] 1 = Jq14 [Wri(11Nifi + I12Ni,nfi) +W,(I 21N,ef, + 22N,nfA)]= 1 , J q=J (3.12) ur azPDF Image | Radial Inflow Splitter Blades in Three-Dimensional
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