behave dynamically. This approach is expected to improve the computational results especially when strong coupling exist between components. As will be detailed in the next subsections, there are several possible variants in using global modes. The first approach consists of using the vectors generated by IRCA in a standard Craig-Bampton basis to replace the fixed interface modes. The second approach is the same as the first approach however in this approach separate generalized dofs are not retained per component basis but the assembly of the generalized dofs is accomplished over one set of global generalized coordinates. In the third approach, one might select to use a combination of fixed interface modes and pseudo-vectors. These schemes are outlined in Sections 3.1, 3.2 and 3.3. 3.1 Replacing fixed interface modes by pseudo-vectors The replacement of the fixed interface modes with pseudo-vectors is outlined in this subsection. The representation is provided starting from a standard Craig-Bampton basis. Use of the left and right projection ideas are provided in [11] and written as TT cb,L = I −KbiK−1 ii 0 φT i,L , (26) Tcb,R = I 0 −K−1 ii Kib φi,R . (27) In equations (26) and (27), the subscript Rand L represents the right and left fixed interface modes, respectively. Similarly, subscript i represents the internal modes. For the right projection basis, one uses the pseudo-vectors, given by superscript P, in the second column instead of the fixed interface mode vectors, namely, qb qi = I φP b,R −K−1 ii Kib φPi,R ˜qb ηi . (28) It is important to note that this transformation does not provide interface connectors as in standard substructuring strategies since the boundary degrees of freedom have been transform in ˜qb. One thus needs an intermediate transformation to end up with the connector boundary dofs : using the first line of (28) to write ˜qb =qb −φ P b,Rηi, and substituting in the second line of (28) one finds the equivalent transformation qb qi = I 0 −K−1 ii Kib φPi,R+K−1 ii Kib φP b,R qb ηPi . (29) Globally enriched substructuring techniques for vibro-acoustic simulation 271
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