φL = φL s φL f , φR = φR s φRf . (7) The same kind of relation can be obtained by considering the symmetrization concept outlined above as φsymT Msystemτφ sym¨η+φsymT Ksystemτφ symη= φsymT Fs Ff , (8) where τis the post-symmetrization matrix provided in equation (3). It is worth mentioning the fact that since equation (8) is the symmetrized form of equation (6), its associated left and right eigenvectors are identical. Obviously φsym = φR and that τφsym = φL. Further discussion on the relation between left and right eigenvectors can be found in Appendix A. A common practice in the reduction approaches is to use an approximation space, T, which is also called the Ritz basis. If we want to find an approximation of the eigensolutions of the system, one can project the symmetrized eigenvalue problem onto this reduction space as TT Ksystemτ −ω2Msystemτ Tη=0, (9) which can also be written as TT Ksystem−ω2Msystem τTη=0. (10) Equation (10) reveals an important result for the reduction and projection basis representations of the unsymmetric coupled problem, namely, one should always use different reduction and projection spaces on the right and left, respectively. This fact is also used in the explanation of the iterative correction algorithm in the following subsection. 2.3 Finding global enrichments: Iterative reduced correction algorithm, IRCA The iterative correction algorithm is outlined in this subsection. The details might be found in [9]. The starting point for the algorithm is the uncoupled system modes. These modes are usually cheap to compute compared to solving the full coupled eigenvalue problem. Starting with the uncoupled system modes, the algorithm tries to converge to the system eigenpairs. The name of the algorithm comes from the fact that the corrections on one physical domain are computed by taking into account the effects of the other physical subdomain. In this way, the uncoupled starting modes are corrected iteratively to approximate the system response for the coupled problem. The steps of the algorithm are going to be outlined below. Umut Tabak and Daniel J. Rixen 266
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