A Truly Hybrid Approach to Substructuring Problems Using Mixed Assembly and Implicit Solving Strategies one simplycombines the Lagrangians found earlier for the stiffness and flexibility assembly cases in eqs. (3) and (11) and takes their variation. Let us consider the case where we want to assemble substructure 1, expressed in dynamic stiffness form, with component 2, which has a receptance representation. This is schematically shown in figure 1 (c). The following three field assembled equations are then found froma variational approach: ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎣ Z (1) ii Z (1) ib 0 0 0 Z (1) bi Z (1) bb −I 0 0 0 −I 0 0 L(1) b 0 0 0 Y (2) bb L (2) b 0 0 L(1) T b L (2) T b 0 ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎦ ⎡ ⎢⎢⎢ ⎢⎢ ⎣ u(1) i u(1) b g (1) b g (2) b uγ ⎤ ⎥⎥⎥ ⎥⎥ ⎦ = ⎡ ⎢⎢⎢ ⎢⎢ ⎣ f (1) i f (1) b 0 p (2) b 0 ⎤ ⎥⎥⎥ ⎥⎥ ⎦ (14) In this equation, the third row is the compatibility condition for component 1 while the fourth row can be recognized as the (possibly weakened) compatibility condition for component 2. The fifth row constitutes the equilibriumcondition. As in the previous sections, the three field assembled equations of motion can be simplified to find a more compact expression. However, since we are considering mixed assembly, we cannot simply apply the primal or dual assembly methods of the previous sections. Instead, we should alsomix the transformations. First it should be realized that in the case of mixedassembly we need both a unique interface displacement uγ field and unique interface force field λto facilitate the interaction between the force and displacement DoF of both substructures. One can then devise a transformation in the form: ⎡ ⎢⎢⎢ ⎢⎢ ⎣ u (1) i u (1) b g (1) b g (2) b uγ ⎤ ⎥⎥⎥ ⎥⎥ ⎦ = ⎡ ⎢⎢⎢ ⎢⎢ ⎣ I 0 0 0 L (1) b 0 0 0 −B (1) T b 0 0 −B (2) T b 0 I 0 ⎤ ⎥⎥⎥ ⎥⎥ ⎦ ⎡ ⎣ u(1) i uγ λ ⎤ ⎦ This transformation corresponds to primal assembly for the first structure as in the previous section, whereas the second component is subject to Dirichlet-Neumann assembly as outlined in section 2.1.3. Substitution of this transformation in eq. (14) introduces both the unique interface force field and eliminates u(2) b . Pre-multiplication is needed only for the sake of symmetry and gives the mixed assembled system: ⎡ ⎢⎣ Z (1) ii Z (1) ib L (1) b 0 L(1) T b Z (1) bi L(1) T b Z (1) bb L(1) b L(1) T b B (1) T b 0 −B (2) b L (2) b −B (2) b Y (2) bb B (2) T b ⎤ ⎥⎦ ⎡ ⎣ u (1) i uγ λ ⎤ ⎦ =⎡ ⎢⎣ f (1) i L (1) T b f (1) b −B (2) b p (2) b ⎤ ⎥⎦ (15) Note that, as expected, the above mixed assembled systemis symmetric, since we know that: ∑ s B (s) b L (s) b =∑ s L (s) T b B (s) T b T =0 7ZAR $ Z $Bt 1 ` 2 1 / wdD K ` 2 /w1D K .wdD `wdD K Vw1D .wdD Vw1D Fig. 6: Simplification of the three-field formulation of the mixed assembly case. 337
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