A simultaneous iterative procedure for the Kron's component modal synthesis approach

Abstract

A simultaneous iterative procedure for the Kron's component modal synthesis method is developed in this paper for fast calculating the modal parameters of a large-scale and/or complicated structure. To simultaneously improve the precision of all the concerned modal parameters, the modal transformation matrix is chosen as the iterative term. For a faster convergence rate, the reduced system matrices are consistent with the modal transformation matrix. With a mathematically proved consistency and reasonably justified convergence, the proposed method can provide highly accurate results after a few iterations. The implementation details are presented for reference together with some computational considerations. Compared with other methods for obtaining modal parameters, the proposed method has such merits as high computational efficiency and still in a substructuring scheme. Two numerical examples are provided to illustrate and validate the proposed method. SIMULTANEOUS ITERATIVE COMPONENT MODAL SYNTHESIS METHOD 991 the dual [13,18] or field-consistent [19] assembling; and so on. Among these approaches, the Kron's CMS method [13] has some distinct advantages in dealing with large-scale structures. Firstly, it is a free-interface method, and thus, the size of the reduced model is not related to the number of interface degrees of freedom (DOFs). Besides, it can be combined with the test data [5,6,24,25]. Secondly, it assembles the components in dual form [5,18] and thus can deal with cases when two or more components are assembled under complicated interface conditions. The Kron's CMS method was firstly developed by Kron in his book Diakoptics [26] for electrical problems and then modified for structural problems via the receptance matrix by Simpson et al. [27]. Some early works improved the method from the perspectives of theoretical [28,29] and numerical [30][31][32] considerations. Recently, Weng et al. improved the Kron's method for practical engineering applications. Instead of directly following the definition, their methods calculate the receptance matrix with either the first-order approximation [33] or an iterative procedure [34]. This improvement reduces the computational cost and in the meantime provides the feasibility for further eigensensitivity analyses [35]. Moreover, compared with the approximation-based methods, the iterative methods can be more effective when accurate results are required [34]. Nevertheless, in the existing iterative procedure, the precision of modes has to be improved one by one as a result of the eigenvalue-dependent reduced stiffness matrix, which may include extra computational cost [36]. Thus, some further improvement in the iterative procedure may still be needed for enhancing the computational efficiency.This paper suggests a simultaneously iterative procedure for the Kron's CMS method. With this method, the modal transformation matrix, instead of eigenvalues, is chosen to be improved iteratively. Consequently, the precision of all the interested modes can be improved simult...

An eigenvector‐based iterative procedure for the free‐interface component modal synthesis method

“…To compare the computational cost, the CPU time for the SIC method, EIC method, and global Lanczos method is counted and listed in Table 3. The SIC method is programmed according to the descriptions in the work of Cui et al 28 In Table 3, the CPU time in the columns "Comp-Eig," "Constant-Matrices," "Updating," and "Orthogonalization" mainly corresponds to the operations in Lines (3), (4a-4c), (6a-6d), and (7)-(9) of Algorithm 1, respectively. For the SIC method, (1,4,5) of the implementation in the work of Cui et al, 28 respectively.…”

“…The SIC method is programmed according to the descriptions in the work of Cui et al 28 In Table 3, the CPU time in the columns "Comp-Eig," "Constant-Matrices," "Updating," and "Orthogonalization" mainly corresponds to the operations in Lines (3), (4a-4c), (6a-6d), and (7)-(9) of Algorithm 1, respectively. For the SIC method, (1,4,5) of the implementation in the work of Cui et al, 28 respectively. Table 3 suggests that the EIC method is more efficient than the conventional SIC method since their convergence rates are similar, but the computational cost per iteration of the former is lower.…”

“…This can be explained as follows: (i) for a given truncation frequency band, there exists an upper boundary for the difference between the cost of global and component eigenanalysis as n increases; (ii) the additional cost for high-precision initial guess can be much higher than the cost of iterative synthesizing, as pointed out in the previous work. 27,28 Moreover, the proposed method is slightly more efficient than (still of the same order as) the global method when the cube is partitioned into 16 components, mainly due to the distribution of the computations for the factorization of stiffness matrices (b ( ) K decreases with n, as can be seen in Table 4). Extra computational cost for enabling substructuring is also much lower than conventional free-interface CMS methods.…”

“…In general, the proposed EIC method is a Kron's subspace iteration started with free-interface component modes. Notice that high-order free-interface approximation-based series expansion, for instance, the one employed in the SIC method, 28 has a close relationship with Kron's inverse iteration equation (3), whose derivations are given in Appendix B. Therefore, replacing the iterative updating of modal transformation matrices in conventional free-interface CMS methods with Kron's inverse iteration does not reduce the convergence rate; this will be validated via a numerical example in Section 5.…”

“…Nevertheless, its efficiency may still be improved since only one global mode can be obtained from a single round of iterations. As such, the authors developed a simultaneous iterative Kron's CMS (SIC) method 28 to improve the efficiency of Weng's iterative method. In this method, the iterative term is an eigenvalue-independent matrix that shares the same projections with the exact modal transformation matrix in the reduced modal space.…”