The accuracy of the MSCS reduction relies on how unlikely it is for the phase index components from two different partitioned sets to couple with each other. In the previous section, it is shown that the mixing scheme of phase index components can not guarantee that the resulting MSCS reduction can captured the coupling between the phase index components. To improve the accuracy of eigensolutions obtained from the MSCS reduction method, an iterative method based on the subspace iteration [112] is developed for multi-stage bladed disks.
5.3.1 General Subspace Iteration Method
The subspace iteration method developed by Bathe [112] is can solve general eigenvalue problems that take the form of
Kφ= λMφ, (5.22)
where K and M are the stiffness and mass matrices of a finite element system. The eigenvalue problems of the systems with gyroscopic effect can also be transformed into this general form when it is represented in the state space. The subspace iteration method can solve the smallest p eigenvalues λn (n = 1, 2, ..., p) and their corresponding eigenvectors φn (n = 1, 2, ..., p), where the eigenvalues are ordered as
0 < λ1 6 λ2 6 · · · 6 λp. (5.23)
The number of eigensolutions to be solved (the number p in Eq. (5.23)) is arbitrary as long as it does not exceeds the size of the matrix K in Eq. (5.22), but in most applications this number is much smaller than the size of K. If the smallest eigenvalue is zero (such case is easy to identify because it indicates rigid body modes exist), a shift can be used to reach the condition in Eq. (5.23) [115].
To solve the p eigensolutions that are ordered as Eq. (5.23), one should start with q starting iteration vectors (q = max{p + 8, 2p} according to Ref. [112]) to establish X0 and iterate in the following process:
Y(m−1) = MX(m−1), (5.24a)
K ˜X(m) = Y(m−1), (5.24b)
K˜(m) = ˜XH(m)K ˜X(m) = ˜XH(m)Y(m−1), (5.24c) M˜(m) = ˜XH(m)M ˜X(m), (5.24d) K˜(m)Q(m) = ˜K(m)Q(m)Λ(m), (5.24e)
X(m) = ˜X(m)Q(m), (5.24f)
where m = 1, 2, 3, ... labels the iteration times. For each iteration, the columns of Xm−1 for m > 1 in Eq. (5.24a) are the estimated eigenvectors obtained from the last iteration. After the matrix Y(m−1) is calculated from Eq. (5.24a), the matrix ˜X(m) is obtained by solving Eq. (5.24b). The columns of the matrix ˜X(m) are used as the basis to span a subspace of the full eigenspace, and the full system matrices M and K are projected to the subspace through Eqs. (5.24c-5.24d). The dimension of the reduced matrix operators ˜M and ˜K is q × q. By solving the eigenvalue problem in Eq. (5.24e), the diagonal matrix Λ(m) contains
the updated estimation of the q eigenvalues (with subscript (m)),
Λ(m) =
λ(m),1
λ(m),2 . ..
λ(m),q
, (5.25)
and Q(m) is a square matrix containing the reduced eigenvectors in the mth iteration.
Through Eq. (5.24f), the updated full system eigenvectors are obtained as the columns of X(m). Convergence of eigenvalues is checked after each iteration. The iterative process will stop if all the ratios
λ(m),n− λ(m−1),n /
λ(m),n
for n = 1, 2, . . . , p are within the desired tolerance. Otherwise, the iteration will continue.
According to the iteration process in Eq. (5.24), the quality of starting iteration vectors (X0 in Eq. (5.24)) is important for the rate of convergence. If the subspace spanned by the starting iteration vectors contains the exact vectors, a single iteration can provide the exact eigenvalues and eigenvectors. A source of starting iteration vectors with good quality is the modes of a system that is similar to the system to be solved, and such cases occur in structure optimization problems where the natural frequencies are calculated as the structure changes [115]. In this work, the full system modes obtained from the MSCS reduction method are used as the starting iteration vectors (Xi,(0) in Eq. (5.24)). These vectors are considered as ideal starting iteration vectors because solving the coupled systems assembled by sector models in Eq. (5.16) are equivalent to solving the full system with constraints at the inter-stage interface. These constraints are resulted from the facts that not all the phase index components are considered simultaneously within a coupled system assembled by sector models, and the full original systems is equivalent to the coupled system that assembled by all sector models with all phase index components. These constraints differentiate the
coupled systems in Eq. (5.16) from the original full system model, but the modes from the MSCS reduction method are starting iteration vectors with reliable quality. And this will be validated by the later numerical example.
With starting iteration vectors from the MSCS reduction method, better estimation of eigensolutions can be obtained through the iteration in Eq. (5.24), and the estimation will finally converge to the eigensolutions in Eq. (5.22)). When the iteration process in Eq.
(5.24) applies, the matrix operators in the iterative process (M and K in Eq. (5.24)) are those for the full multi-stage model. This path to improving accuracy works, however, it has a significant drawback. The full system matrix operators in Eq. (5.24) can be very large in industrial applications, so it is not convenient to conduct iterations with respect to these huge matrices. In practice, building and storing such huge matrices can be inconvenient.
Therefore, it is necessary to improve the subspace iteration method so that the iterations with respect to these huge full system matrices can be avoided.
5.3.2 Combination of Subspace Iteration and Multi-Stage Cyclic Symmetry Reduction
The combination of the subspace iteration with the MSCS reduction is more than sim-ply using the estimated full system modes obtained from the MSCS reduction method as the starting iteration vectors in the general subspace iteration in Eq. (5.24). In this section, an improved subspace iteration that only requires sector-level-sized calculations is formulated.
Parallel computation can also be applied to the proposed iterative process to further im-prove the computational speed. The matrix operators for the full multi-stage system or its component stages are not necessary in the final formulation, although they are used in the following derivation leading to the final formulation.