idea is to make use of some global information and integrate this information in the reduced representation. These enrichment vectors are computed from an iterative algorithm which at convergence represents the global mode approximations. The ultimate aim is not to compute global modes but some enrichment vectors which might provide important dynamic information on how the other components behave and how this might be reflected on the component level. From the results presented here we can conclude that the proposed methods yield an acceptable accuracy level. However, there are important computational points to mention. Namely, the most important computational burden results from the computation of the correction vectors that are meant to complement the uncoupled system modes since expensive factorizations should be performed per physics due to the update of the operator matrices. Investigation of iterative solution methods in this respect might give a cure when the system sizes become exceedingly large. Integration of the global mode vectors in the basis brings important information and has shown to improve the results on a large portion of the spectrum. Results also reveal that finding a trade-off when using a mix of the fixed interface modes with global pseudo modes is also significantly difficult. This is due to the fact that there is no mechanism to reveal some information on the coupling beforehand. This information is also dependent on the problem type, fluid which is in contact with the structure and geometry of the overall problem. Acknowledgements This work is supported by the Technologiestichting STWof the Dutch Government and the authors gratefully acknowledge the means of support. Appendix A: Relation of left and right eigenvector and the symmetrization concept Let us consider the symmetrized vibro-acoustic problem and define its eigenvectors by (Ksystem−ω2Msystem) τφ sym φR =0 (45) The left eigenvector of this problem is, by symmetry, identical to φsym, namely φsym = φL. Consider now the non-symmetrized problem, the system exhibits left and right eigenvectors, which can be deduced from (45): φR = τφsym = K−1 s Ms φL s −K−1 s Ksf φL f φL f (46) The left eigenvalue problem can be written as ! Ks Ksf 0 Kf T −ω2 Ms 0 Mfs Mf T" φL s φL f =0 (47) Umut Tabak and Daniel J. Rix 278
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