⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ u(1) i u(1) b u(2) i u(2) b g (1) b g (2) b uγ ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦ = ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢ ⎣ I 0 0 0 0 0 0 0 0 L (1) b 0 I 0 0 0 0 0 0 0 L (2) b 0 0 I 0 0 0 0 0 I 0 0 0 0 0 I ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥ ⎦ ⎡ ⎢⎢⎢ ⎢⎢ ⎣ u (1) i u(2) i g (1) b g (2) b uγ ⎤ ⎥⎥⎥ ⎥⎥ ⎦ Again, this transformation is substituted in (4) thereby eliminating the substructure boundary DoF sets u(s) b and satisfying the interface compatibility condition. Pre-multiplication is then needed to obtain a symmetric systemof equations. Using again the relation between the Boolean matrices Lb and Bb in eq. (5), we see that by doing so the equilibriumcondition is also satisfied and drops out of the equation. The procedure is illustrated in figure 3 and results in the following expression for the assembled system: ⎡ ⎢⎣ Z (1) ii 0 Z (1) ib L (1) b 0 Z (2) ii Z (2) ib L (2) b L (1) T b Z (1) bi L (2) T b Z (2) bi L (1) T b Z (1) bb L (1) b +L (2) T b Z (2) bb L (2) b ⎤ ⎥⎦ ⎡ ⎢⎣ u (1) i u (2) i uγ ⎤ ⎥⎦= ⎡ ⎢⎣ f (1) i f (2) i L (1)T b f (1) b +L (2)T b f (2) b ⎤ ⎥⎦ (8) The above equations formthe so calledprimal assembled system; the most compact formof the assembled equations of motion using a minimumnumber of DoF. Note that this type of assembly is the way individual elements are classically assembled in a finiteelement method. However, as we noted previously regarding the dual formof theassembled equations, the above equation (8) usually needs to be inverted to a receptance formto be useful in an experimental FBS analysis. The resulting expression for the assembled receptance matrix is in the literature commonly termed the “impedance coupling” method [11, ?, 14]. 2 1 ` 2 1 ` 2 1 7ZAR $ Z $Bt _T8Zj $}j${s $Bt ` .wdD .w1D .wdD .w1D .wdD .w1D Fig. 3: Simplification of the three-field formulation for stiffness assembly according to primal assembly. 2.1.3 Amix of both: Dirichlet-Neumannassembly In addition to the primal and dual assembly methods treated above, the assembled equations in (4) can also be simplified by combining both methods, that is, by choosing both a unique set of interface DoF and interface forces. Thereby, both the equilibriumand compatibility condition on the interface are satisfieda priori. This gives rise to the following transformation: ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ u (1) i u (1) b u (2) i u (2) b g (1) b g (2) b uγ ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦ = ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ I 0 0 0 0 0 0 L (1) b 0 I 0 0 0 0 0 L (2) b 0 0 −B (1) T b 0 0 0 −B (2) T b 0 0 0 0 I ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦ ⎡ ⎢⎢ ⎣ u(1) i u(2) i λ uγ ⎤ ⎥⎥ ⎦ Substitution of this transformation in the three field assembled equations of motion in (4) simultaneously introduces the interface force field λand eliminates the substructure boundary DoFu(s) b , as illustrated in figure 4. This gives: S.N. Voormeeren, P.L.C. van der Valk and D.J. Rixen 334
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