A new method for the numerical solution of the meridional through-flow equations in an axial flow machine is presented based on the finite-element method. A rigorous derivation of the pitch-averaged flow equations is presented and the assumption of axisymmetric flow leads, with the introduction of a stream function, to the equation to be solved. A description is given of the finite-element technique which is applied in this problem. The method of solution allows the calculation of transonic stages. Numerical results are compared with experimental data and show very satisfactory agreement. This method appears, therefore, to compare very favorably with the other methods used up to now. Although the present results pertain to axial flow machines, the method is easily applicable to radial flow machines as well and the way of solution for this case is indicated.

A quasi-3D calculation program based on finite elements is presented in the spirit of Wu’s approach. In this work, however, the flow along the S2 surface is replaced by the calculation of the exact mass averaged-pitch averaged flow in a meridional plane. Extra terms appear in this equation which result from the deviations from axisymmetry and which can be calculated from the knowledge of the blade-to-blade flows. Due to the mass-averaging, these terms represent the only interaction from blade-to-blade S1 surfaces to the meridional flow. The complete program is integrated in a single package requiring only ten percent more computer storage than each of the composing S1 or S2 codes taken alone. The various parts of the program are described as well as the interaction process and specific approximations. Example of calculations compared with experimental data are given, showing good agreement with experimental data.

Incomplete factorizations are popular preconditioning techniques for solving large and sparse linear systems. In the case of highly indeÿnite complex-symmetric linear systems, the convergence of Krylov subspace methods sometimes degrades with increasing level of ÿll-in. The reasons for this disappointing behaviour are twofold. On the one hand, the eigenvalues of the preconditioned system tend to 1, but the 'convergence' is not monotonous. On the other hand, the eigenvalues with negative real part, on their move towards 1 have to cross the origin, whence the risk of clustering eigenvalues around 0 while 'improving' the preconditioner. This makes it risky to predict any gain when passing from a level to a higher one. We examine a remedy which consists in slightly moving the spectrum of the original system matrix along the imaginary axis. Theoretical analysis that motivates our approach and experimental results are presented, which displays the e ciency of the new preconditioning techniques.

SUMMARYThe preconditioned conjugate gradient algorithm is a well-known and powerful method used to solve large sparse symmetric positive definite linear systems. Such systems are generated by the finite element discretization in structural analysis but users of finite elements in this context generally still rely on direct methods. It is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods.

No abstract