Direct decoupling of substructures using primal and dual formulation Before deriving the equilibrium condition, it should be noted that the interface DoFs involved in the equilibrium condition are not necessarily the same used to enforce the compatibility condition, as long as controllability between equilibrium and compatibility DoFs is ensured. If the compatibility and the equilibrium DoFs are not the same, the approach is called non-collocated [8]. Note that a non-collocated approach requires an extended or mixed interface and therefore it is only possible in the decoupling problems (in coupling problems only standard interface can be defined). Obviously, the traditional approach, in which compatibility and equilibrium DoFs are the same, is called collocated. Let NE denote the number of interface DoFs on which equilibrium is enforced. The equilibrium condition for constraint forces implies that, when the connecting forces are added for a pair of matching DoFs, their sum must be zero, i.e. gAB r +g B s =0: this holds for any pair of matching DoFs. Furthermore, if DoF k on the coupled systemAB(or to the residual subsystemB) does not belong to the equilibrium interface, it must be gAB k =0 (g B k =0): this holds for any DoF not involved in the equilibrium condition. Overall, the above conditions can be expressed as: LAB E LB E T gAB gB ={0} (3) where the matrix [LE]= [L AB E ][L B E] is a Boolean localisation matrix. Note that the number of columns of [LE] is equal to the number NE of equilibrium interface DoFs plus the number NNE of DoFs not belonging to the equilibrium interface. Note that NNE =NAB+NB−2NE: in fact, the number of DoFs belonging to the equilibrium interface must be subtracted once fromNAB and once fromNB. Therefore, the size of [LE] is (NAB+NB)×(NAB+NB−NE). Eqs. (1-3) can be put together to obtain the so-called 3-field formulation: ⎧⎪ ⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ ZAB [0] [0] − ZB uAB uB = f AB f B + gAB gB (4) BAB C BB C uAB uB ={0} (5) LAB E LB E T gAB gB ={0} (6) 51
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