Improvement on the iterated IRS method for structural eigensolutions
Introduction
In the structural analysis with the finite element (FE) method, a very large number of degrees of freedom (d.o.f.s) (several hundreds or thousands) are usually required to describe the structure accurately. In this situation, it is often necessary to reduce d.o.f.s for a variety of engineering and mechanical problems [1]. For example, although many eigensolution algorithms exist, model reduction method (or condensation, economization) is still an efficient technique to give fast computation of some lowest natural frequencies and corresponding mode shapes of large structures [2], [3], [4], [5], [6], [7]. In recent years, it has also been used in the experimental modal analysis and related fields [8], [9] since the number of measured points in experiments is much less than that of d.o.f.s in the FE analysis and thus it is necessary to reduce the complete system matrices to the size of the experimental model or expand the measured mode shapes to the full size of the FE model.
The strategy of model reduction in solving eigenproblem is to remove some d.o.f.s (called slaves) of the original FE model and retain a much smaller set of d.o.f.s (called masters), then to solve the eigenfunction of the reduced model and approximate the eigensolutions of the original model. The pivot task of various reduction methods is to estimate the transformation matrix between the mode shapes corresponding to the masters and those to the slaves. Guyan [2] and Irons [3] firstly proposed the static condensation technique nearly 40 years ago, which neglects the inertia terms of the slavery d.o.f.s. Later some dynamic approaches were proposed to increase the accuracy of the condensation method. For example, Paz [4] studied Guyan's method with a shifted eigenvalue; O’Callahan [5] proposed the improved reduced system (IRS) by adding an extra term in the transformation matrix of Guyan's method.
Recently some iterative dynamic schemes have been developed which update a transformation matrix repeatedly until the eigenpairs meet the required precision. In particular, Friswell et al. [6] proposed an iterated IRS (IIRS) technique and the convergence was proved later [10]. Unfortunately, the convergence speed of this method cannot be comparable to that of the subspace iteration method (SIM), a commonly used algorithm in the structural community. Even some improvements have been made to increase the convergence [7], the convergence property has not been proved so far. Dependence on the master selection also prevents this kind of method from becoming a more popular eigensolver in engineering.
In this paper, an improvement on the IIRS is presented by modifying the iterative formula of the transformation matrix. We will demonstrate that this modification is equivalent to the standard SIM. The convergence proof is also given, in a similar way to that of Friswell et al. [10]. The effectiveness of this improvement is demonstrated by applications to two numerical examples.
Section snippets
Iterated improved reduced system (IIRS) method
The generalized eigenvalue problem of a system with N d.o.f.s is described as follows, in the block form, according to the chosen master d.o.f.s (retained) and slavery d.o.f.s (removed),where K and M are the N×N symmetric stiffness and mass matrices; Φ consists of the mass-normalized eigenvectors and Λ is a diagonal matrix containing corresponding eigenvalues, , on the diagonal. Only the first m modes are included in the above equation.
The present improvement of IIRS
In this part, we present a modification of the current IIRS and demonstrate its connection with the standard SIM, one of the most powerful and popular technique in the structural community.
We rewrite Eq. (5) asandSubstituting Eq. (14) into KR=TTKT, one can get the reduced stiffness matrixNoting
Convergence of the present method
In this section, we will roughly prove that the present iterative formula for T(k) in the form of , converges to the actual T, in a similar way to that of Friswell et al. [10] in proving the convergence of IIRS that takes form of , .
First we rewrite , , aswhere S isDefine an error matrixSimilarly let , is expanded as the first order Taylor series in terms of ,
Numerical examples
Two structures are applied to illustrate the effectiveness and accuracy of the proposed algorithm. It was found that the selection of master d.o.f.s certainly affects the convergence speed or accuracy of the reduction methods. Some strategies have been studied in the master d.o.f.s selection for condensation [13], [14], sensor placement [15], [16] or damage identification [17]. However, this is not the focus of the present paper. In this paper, the master d.o.f.s are selected as those with
Conclusions and discussions
A new effective model reduction method has been developed for structural eigensolutions. This technique modifies the current iterated IRS technique and is found equivalent to the widely used SIM. The convergence of the method is mathematically verified.
The present algorithm has been applied to two practical examples. Numerical results have showed that the proposed technique can accurately predict the frequencies and the mode shapes of interest. As compared with the other commonly used
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