3 Are Rotational DoFs Essential in Substructure Decoupling? 29 • standard interface, including only the coupling DoFs (c) between subsystems U andR; • extended interface, including the coupling DoFs and a subset of internal DoFs (i r) subsystemR; • mixed interface, including subsets of coupling DoFs (d c) and/or internal DoFs (i r) of subsystemR. The use of a mixed interface may allow to substitute rotational coupling DoFs with translational internal DoFs. The compatibility condition at the (standard, extended, mixed) interface DoFs implies that any pair of matching DoFs uRU l and uR m, i.e. DoF l on the coupled systemRU andDoF mon subsystemRmust have the same displacement, that is uRU l uR m D0. Let the number of interface DoFs on which compatibility is enforced be denoted as NC. The compatibility condition can be generally expressed as: hBRU C BR Ci ( uRU uR ) D 0 (3.2) where each row of BC D BRU C BR C corresponds to a pair of matching DoFs. Note that BC has size NC .NRU CNR/ and is, in most cases, a signed Boolean matrix. It should be noted that the interface DoFs involved in the equilibrium condition need not to be the same as DoFs used to enforce compatibility. In this case, the approach is called non-collocated [7], whereas the traditional approach, in which compatibility and equilibrium DoFs are the same, is called collocated. Let NE denote the number of interface DoFs on which equilibrium is enforced. The equilibrium of constraint forces implies that their sum must be zero for any pair of matching DoFs, i.e. gRU r Cg R s D0. Furthermore, for any DoF k not involved in the equilibrium condition, it must be gRU k D0andgR k D0. Overall, the above conditions can be expressed as: "LRU E LR E # T (gRU gR ) D 0 (3.3) where the matrix LE D LRU E LR E is a Boolean localisation matrix. Note that the number of columns of LE is equal to the number NE of equilibrium interface DoFs plus the number NNE of DoFs not belonging to the equilibrium interface. Note that NNE DNRU CNR 2NE: in fact, the number of DoFs belonging to the equilibrium interface must be subtracted once fromNRU and once fromNR. Therefore, the size of LE is .NRU CNR/ .NRU CNR NE/. Equations (3.1)–(3.3) can be gathered to obtain the so-called three-field formulation. Starting from the three-field formulation, several assembly techniques can be devised: • dual assembly [1, 3] where equilibrium is satisfied exactly by defining a unique set of connecting force intensities; • primal assembly [1, 8] where compatibility is satisfied exactly by defining a unique set of interface DoFs; • hybrid assembly [9] where both compatibility and equilibrium are satisfied exactly. In the sequel, only the dual assembly is recalled. It can be shown [9] that whenever NC D NE, i.e. the number of compatibility DoFs is the same as the number of equilibrium DoFs, all assembly techniques provide the same result. 3.2.1 Dual Assembly In the dual assembly, the equilibrium conditiongRU r Cg R s D0at a pair of equilibrium interface DoFs is ensured by choosing gRU r D andg R s D . If a Boolean matrixBE related to interface equilibrium DoFs is defined similarly toBC, the overall interface equilibrium can be ensured by writing the connecting forces in the form: (gRU gR ) D 2 4 BRU E T BR E T 3 5 (3.4) where are Lagrange multipliers corresponding to connecting force intensities andBE is a NE .NRU CNR/ matrix. Since there is a unique set of connecting force intensities , the interface equilibrium condition is satisfied automatically for any , i.e.
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