By definition the vector of connection forces g is zero at the internal subsystemDoF. Between the substructures we can define the intermediate interface displacement field uγ, to govern the compatibility of substructural displacements at the interface. This condition then writes: u(s) b −L (s) b uγ =0 (2) HereL (s) b is a Booleanmatrix localizing the DoF fromthe global intermediate displacement field corresponding to the substructure boundary DoF. As a result, we end up with a three field formulation of the substructuring problem, having as independent unknowns the substructure DoF field u(s), the field of interface connection forces g (s) b and the intermediate interface displacement field uγ. Taking a variational approach we can now obtain the assembled equations. To this end, we can set up the Lagrangian of this problemas: L u(s),g (s) b ,uγ =∑ s 1 2 u(s) T Z(s)u(s) −f(s) T u(s) +g (s) T b L (s) b uγ−u (s) b (3) For the sake of simplicity we will now consider the assembly problemfor two substructures only (s =1,2), although the concepts presented can be easily generalized to the assembly of an arbitrary number of components. This situation is depicted in figure 1 (a). To find the assembled equations we take the variation with respect to the free variables to find the equations of motion of the assembly of two components: ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢ ⎣ Z (1) ii Z (1) ib 0 0 0 0 0 Z (1) bi Z (1) bb 0 0 −I 0 0 0 0 Z (2) ii Z (2) ib 0 0 0 0 0 Z (2) bi Z (2) bb 0 −I 0 0 −I 0 0 0 0 L(1) b 0 0 0 −I 0 0 L(2) b 0 0 0 0 L (1) T b L(2) T b 0 ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥ ⎦ ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎢⎣ u(1) i u(1) b u(2) i u(2) b g (1) b g (2) b uγ ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎦ = ⎡ ⎢⎢⎢ ⎢⎢⎢ ⎢⎢⎢ ⎣ f (1) i f (1) b f (2) i f (2) b 0 0 0 ⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎦ (4) In the above equations we can recognize the fifth and sixth equations as the compatibility conditions, governing the compatibility between the u(s) b and uγ. The last row is the equilibriumcondition on the interface, stating that the sumof the substructure connection forces must be zero. Note that when starting fromthis three-field formulation, we only required compatibility at the substructure interfaces; the equilibriumconditions followed naturally fromthe compatibility condition. The above assembled equations of motion still contain the full three fields, which is inefficient for most analyses froma computational point of view. Therefore, it is desired to simplify the equations. In essence two ways exist to do this, namely so calledprimal or dual assembly, as discussed next. 2.1.1 Dual Assembly In dual assembly, one eliminates the interface connection force fields by realizing that the interface forces should be equal and opposite to satisfy the interface equilibrium. To a priori satisfy this condition, we can introduce a unique field of interface forces λ, as follows: g (s) b =−B (s) T b λ Here B (s) b is a signed Boolean matrix acting on the substructure interface DoF and λcorresponds physically to the interface force intensities. Note that the minus sign is chosen to stress the fact that whereas g (s) b was seen as an external force for the substructure, λis considered an internal force. Due to the construction of the Booleanmatrices it holds that [14]: ∑ s B (s) b L(s) b =0 (5) Hence this choice for the interface connection forces satisfies the interface equilibriumfor any λ. This choice gives rise to the following transformation: S.N. Voormeeren, P.L.C. van der Valk and D.J. Rixen 332
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