It is important to note that, from this point on, Mf ρ , Kf ρ and Ff ρ will be renamed Mf , Kf andFf respectively. 2.2 Symmetrization concept and its relation to reduction Symmetrization approaches of the coupled vibro-acoustic problem defined in equation (2) exist and these are outlined in [7] and [8]. Therein, matrix scaling and eigenvector augmentation methods have been used to obtain symmetric formulations of the original unsymmetric coupled eigenvalue problem. In our work, we consider the post-multiplication approaches from [7] where the following post-transformation matrices are used1 : τ= K−1 s Ms −K−1 s Ksf 0 I . (3) Applying then a change of variable u p = τ ˜u p , (4) leads to the symmetric system MsK−1 s Ms −MsK−1 s Ksf −KT sf K−1 s Ms (Mf +KT sf K−1 s Ksf ) ¨˜u ¨p + Ms 0 0 Kf ˜u p = Fs Ff . (5) The new system matrices are thus obtained by post-multiplication with τ. The new variable ˜ucan be interpreted as an acceleration. The use of the post-multiplier matrix yields a symmetric system and will be useful to obtain real eigenvalues in the interaction problem of the reduced system (see the iterative correction algorithm in the next subsection). However, explicit construction of the symmetric system matrices given in (5) is not necessary in the implementation of the algorithm. A common way to reduce the dynamic equations consists in representing the solution as a truncated series of modes of the full system. Using truncated left and right modal spaces one obtains the reduced problem φLT Msystemφ R ¨η+φL T Ksystemφ Rη= φL T Fs Ff , (6) which leads to a diagonal problem due to the orthogonality properties of the left and right eigenvectors. In the equations, superscript L represents the left eigenvectors, and the superscript Rrepresents the right eigenvectors of the problem. The left and right modes are composed of a fluid and a structural part, namely 1 This representation has an advantage for the forthcoming computations where the factorization of Ks is often available from the structural eigenvalue solver. Globally enriched substructuring techniques for vibro-acoustic simulation 265
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