It is worth mentioning that when reducing the substructures on the component level, the same ideas outlined in Section 3.1 are used for the left and right projection spaces. The difference between the strategy proposed in this subsection and in the previous is only coming from the fact that the generalized dofs are also assembled globally with this scheme. 3.3 Enrichment of the reduction space with pseudo-vectors The third improvement/enrichment approach is to include the pseudo-vectors into the reduction basis together with the fixed interface modes. The right reduction basis expansion is given as qb qi = I 0 φP b,R −K−1 ii Kib φi,R φPi,R ⎡ ⎣ ˜qb ηi ηPi ⎤ ⎦ . (33) Following the similar reasoning of an intermediate transformation matrix from section 3.1, one can write qb qi = I 0 0 −K−1 ii Kib φi,R φPi,R+K−1 ii Kib φPi,R ⎡ ⎣ qb ηi ηPi ⎤ ⎦ . (34) For the projection basis, the modes that should be integrated into the basis are the left eigenvectors of the according problems. A mixture of fixed interface and global modes should be used. Taking these factors into account the left basis can be represented as TL cb T =⎡ ⎢⎣ I −KbiK−1 ii 0 φT i,L φP b,L T φPi,L T ⎤ ⎥⎦, (35) TL cb T =⎡ ⎢⎣ I −KbiK−1 ii 0 φT i,L 0 φPi,L T +φP b,L TKbiK−1 ii ⎤ ⎥⎦. (36) In order not to include linearly dependent vectors in the reduction basis, an orthogonalization through an interaction problem is accomplished for the vectors that are integrated into the left and right basis. Representing the mode vectors in the right basis as YR = φi,R φPi,R+K−1 ii Kib φPi,R , (37) and the mode set vectors in the left basis as Globally enriched substructuring techniques for vibro-acoustic simulation 273
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