A Truly Hybrid Approach to Substructuring Problems Using Mixed Assembly and Implicit Solving Strategies ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ 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 I 0 0 0 0 0 0 I 0 0 0 0 0 0 I 0 0 0 0 0 0 −B (1) T b 0 0 0 0 0 −B (2) T b 0 0 0 0 0 0 I ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎦ ⎡ ⎢ ⎢⎢ ⎢ ⎢⎢ ⎢ ⎣ u (1) i u (1) b u (2) i u (2) b λ uγ ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦ (6) Substituting this transformation in the three-field assembled equations of motion (4) replaces the local connection forces by the unique global field λ, satisfying the interface equilibriumcondition. To end up with a symmetric systemone can use the above transformation to subsequently pre-multiply the equations. This eliminates the intermediate displacement field uγ, due to the relation between the Boolean matrices in (5). The procedure is illustrated in figure 2 and the simplified assembled equations become: ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎣ Z (1) ii Z (1) ib 0 0 0 Z (1) bi Z (1) bb 0 0 B (1) T b 0 0 Z (2) ii Z (2) ib 0 0 0 Z (2) bi Z (2) bb B (2) T b 0 B (1) b 0 B (2) b 0 ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎦ ⎡ ⎢⎢⎢ ⎢⎢ ⎣ u(1) i u(1) b u(2) i u(2) b λ ⎤ ⎥⎥⎥ ⎥⎥ ⎦ = ⎡ ⎢⎢⎢ ⎢⎢ ⎣ f (1) i f (1) b f (2) i f (2) b 0 ⎤ ⎥⎥⎥ ⎥⎥ ⎦ (7) The above systemis called the dual assembled system, since the unknowns defining the interface problemare forces which are mathematically dual to the original displacement unknowns. As a result, the compatibility condition is present explicitly in the assembled equations of motion, i.e. the last row in eq. (2.2). In a practical setting however, where measured receptance matrices are used to describe the component dynamics, this formof the assembled equations is not very relevant. One can therefore transformthe above equation (2.2) to a receptance formby eliminating the Lagrange multipliers. The resulting expression is the well known LM FBSmethod for assembly of receptance FRFmatrices [13, 14]. 2 1 `wdD K ` w1D K ` 2 1 ` wdD K ` w1D K ` 2 1 4 wdD K ` wdD K 4 w1D K ` w1D K 7ZAR $ Z $Bt _T8Zj $}j${s $Bt .wdD .w1D .wdD .w1D .wdD .w1D Fig. 2: Simplification of the three-field formulation for stiffness assembly according to dual assembly. 2.1.2 Primal Assembly Another way to obtain a simplified expression for the assembled system(4) can be obtained by realizing that the compatibility condition can be a priori satisfied by choosing a unique set of substructure interface DoF as: u (s) b =L (s) b uγ This choice gives the following transformation: 333
RkJQdWJsaXNoZXIy MTMzNzEzMQ==