u λ = ⎡ ⎢⎢⎢ ⎢⎣ R(1) Θ(1) 0 −G(1) resB(1)T . . . . . . .. . 0 R(Ns) Θ(Ns) −G(Ns) res B(Ns) T 0 ··· 0 I ⎤ ⎥⎥⎥ ⎥⎦ ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ α(1) η(1) .. . α(Ns) η(Ns) λ ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦ = R Θ −GresBT 0 0 I ⎡⎣ α η λ ⎤ ⎦ =Tdual ⎡ ⎣ α η λ ⎤ ⎦ (16) Finally using this approximation to reduce the dual dynamic equations (4), one finds the reduced system ˜M⎡ ⎣ ¨α ¨η ¨λ ⎤ ⎦ +˜K⎡ ⎣ α η λ ⎤ ⎦ =TT dual f 0 (17) where the reduced matrices are ˜M=TT dual M0 0 0 Tdual =⎡ ⎣ I 0 0 0 I 0 0 0Mres ⎤ ⎦ (18) ˜K=TT dual KBT B 0 Tdual =⎡ ⎣ 0 0 RTBT 0 Ω2 ΘTBT BR BΘ −Fres ⎤ ⎦ (19) where Fres =BGresB T = Ns ∑s=1 B(s)G(s) resB(s)T (20) Mres =BGresMGresB T = Ns ∑s=1 B(s)G(s) resM(s)G(s) resB(s)T (21) The reduced form (17) was first introduced in [12] and was named the Dual Craig-Bampton method since the form highly resembles the classical Craig-Bampton method, but now using free interface modes and using a dual assembly procedure. Note that the reduction basis (16) is identical to what is used in the MacNeal and Rubin methods [11, 14], but here we obtain a nice block diagonal form of the reduced matrices since, contrary to the latter methods, we consider the dually assembled problem. This implies that not only the equilibrium equations are weakened by the reduction, but also the interface compatibility. Let us note that the reduced form (17) can be computed by first reducing the operators per substructures, then assemble the different components in the global ˜Kand ˜M. Finally let us note that, in order to improve the reduction efficiency, one can use so-called truncation augmentation of the local basis as explained in [13]. Interface Reduction in the Dual Craig-Bampton method 315
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