Direct decoupling of substructures using primal and dual formulation ˆHAB − ˆHB = ˆZAB − 1 ˆZAB ˆHAB − ˆHB ˆZB ˆZB − 1 = = ˆZAB − 1 ˆZB − ˆZAB ˆZB − 1 = ˆHAB ˆZB − ˆZAB ˆHB (27) In the following, the influence of the choice of compatibility and equilibrium DoFs on singularity and ill-conditioning of ([ ˆHAB] −[ ˆHB]) will be analysed. 2.3 Singularity of the interface flexibility matrix (dual formulation) As shown in [3], and recalled in the Appendix, ˆZB and ˆZAB differ only in the upper left ccblock, i.e. that relative to the coupling DoFs, and they can be conveniently written in block matrix form as: ˆZAB =⎡ ⎣ ˆZAB cc ˆZB ci ˆZB ic ˆZB ii ⎤ ⎦ ˆZB =⎡ ⎣ ˆZB cc ˆZB ci ˆZB ic ˆZB ii ⎤ ⎦ (28) where subscripts c and i represent coupling DoFs and internal DoFs, respectively. Therefore: ˆZB − ˆZAB = ˆZB cc − ˆZAB cc [0]ci [0]ic [0]ii (29) Therefore, the interface flexibility matrix in Eq. (27) can be expanded as: ⎡ ⎣ ˆHAB cc ˆHAB ci ˆHAB ic ˆHAB ii ⎤ ⎦− ⎡ ⎣ ˆHB cc ˆHB ci ˆHB ic ˆHB ii ⎤ ⎦ = =⎡ ⎣ ˆHAB cc ˆHAB ci ˆHAB ic ˆHAB ii ⎤ ⎦ ⎡ ⎣ ˆZB cc − ˆZAB cc [0]ci [0]ic [0]ii ⎤ ⎦ ⎡ ⎣ ˆHB cc ˆHB ci ˆHB ic ˆHB ii ⎤ ⎦ (30) Note that the interface flexibility matrix is expressed as a product of three matrices, and it is singular if just one of these matrices is singular. In fact, the determinant of a matrix product equals the product of the determinants. 2.3.1 Singularity using collocated extended interface When using collocated approach with extended interface, the interface flexibility matrix can be singular for several reasons. 1. Because ([ ˆZB] −[ ˆZAB]) is singular at all frequencies: this is true if i is not an empty set as assumed for the extended interface. Note that the matrix is truly 57
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