Direct decoupling of substructures using primal and dual formulation The dynamic stiffness matrices of the residual subsystemB and of the coupled systemABcan be partitioned as follows: ZB = ZB MM Z B MS ZB SM Z B SS =⎡ ⎢⎣ ZB cc Z B ci Z B cu ZB ic Z B ii Z B iu ZB uc Z B ui Z B uu ⎤ ⎥⎦ (42) ZAB = ZAB MM Z AB MS ZAB SM Z AB SS = ⎡ ⎢⎢⎢ ⎢⎣ ZA cc + Z B cc Z B ci Z B cu Z A ca ZB ic Z B ii Z B iu [0]ia ZB uc Z B ui Z B uu [0]ua ZA ac [0]ai [0]au Z A aa ⎤ ⎥⎥⎥ ⎥⎦ (43) where the zero blocks arise from the fact that the unknown subsystemA is joined to the residual subsystemB only through the coupling DoFs c. Note that, when standard interface is considered, i is an empty set, and u≡b. Using dynamic condensation, the condensed dynamic stiffness matrix of the residual subsystemBis: ˆZB = ZB cc Z B ci ZB ic Z B ii − ZB cu ZB iu ZB − 1 uu ZB uc Z B ui = =⎡ ⎣ ZB cc Z B ci ZB ic Z B ii ⎤ ⎦− ⎡ ⎣ ZB cu Z B − 1 uu Z B uc Z B cu Z B − 1 uu Z B ui ZB iu Z B − 1 uu Z B uc Z B iu Z B − 1 uu Z B ui ⎤ ⎦ (44) and the condensed dynamic stiffness matrix of the coupled subsystemABis: ˆZAB = ZA cc + Z B cc Z B ci ZB ic Z B ii − − ZB cu Z A ca ZB iu [0]ia ZB uu [0]ua [0]au Z A aa −1 ZB uc Z B ui ZA ac [0]ai = = ZA cc + Z B cc Z B ci ZB ic Z B ii − −⎡ ⎣ ZB cu Z B − 1 uu Z B uc + Z A ca Z A − 1 aa Z A ac Z B cu Z B − 1 uu Z B ui ZB iu Z B − 1 uu Z B uc Z B iu Z B − 1 uu Z B ui ⎤ ⎦(45) 75
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