augmentation modes helps rejecting those spurious eigensolutions to the higher frequencies [13]. 6. Here we have chosen to use Gas attachment modes. One could also have chosen to use K+ instead. In that case the attachment modes would not have been M-orthogonal to the rigid-body modes R, which would have introduced an additional coupling term between rigid body modes and attachment modes in the reduced mass matrix ¯M(see (26)). 7. Finally let us note that in this new formulation (25), the statically condensed part of the reduced matrices, namely the parts pertaining to the attachment and rigid-body modes, do not change when the approximation space is enriched with vibration modes Θ. Thus was not the case in the original formulation. This will simplify the application of interface reduction as explained in the next section. 4 Interface modes to reduce the interface problem One of the major bottlenecks of substructuring techniques is that as the number of substructure increases, the size of the interface problem, namely the number of interface dofs λ in the case of the Dual Craig-Bampton, also increases so that for a large number of substructure the reduced problem has not significantly less dofs than the original problem. This issue is not proper to the dual approach but is known to be an issue for any substructuring method. For the classical (primal) Craig-Bampton method an interface reduction method was proposed in [4], and was further analyzed in [1, 2]. The technique proposed in [4] consists in representing the interface dofs (the displacements for the classical Criag-Bampton) in terms of a reduced number of modes. The modes to represent the interface behavior is usually obtained by solving the statically condensed problem, i.e. a Guyan reduction of the system equivalent to a Craig-Bampton with no internal modes. This method was applied and modified in [9]. An a posteriori error analysis of the Craig-Bampton method with interface reduction and an adaptive enrichment of the reduction basis was proposed in [10]. In order to reduce the interface problem for the Dual Craig Bampton, let us consider the case where no vibration modes are included in the reduction basis, namely we consider only the static modes so that the approximation writes u λ = R −GBT 0 I α λ =Tdual,stat α λ (30) so that the reduction of the dual problem leads to I 0 0 MI ¨α ¨λ + 0 RTBT BR −FI α λ =TT dual,statf (31) Clearly in this case it does not matter if we consider the first or second variant of the Dual Craig-Bampton (25) or (25) since when no vibration modes are included Gres =G. Similar to the procedure proposed in [4] for the primal Craig-Bampton, Interface Reduction in the Dual Craig-Bampton method 319
RkJQdWJsaXNoZXIy MTMzNzEzMQ==