interface forces on internal nodes are zero, hence not present. The equation for correctly establishes the force equilibrium between a matching pair of interface DoFs according to Newton's third law. A second notation in which the equilibrium condition can be expressed is by introducing a set of Lagrange multipliers . The substitution of these Lagrange multipliers is possible as and differ only in sign, not in value. Using again the signed Boolean matrix : The set defines the intensity of the interface forces . Each Lagrange multiplier represents the magnitude of two matching interface forces in the assembly. By defining the interface forces using Lagrange multipliers , force equilibrium is automatically satisfied. This can be seen by substituting into the first equilibrium equation: Again, the nullspace property of the Boolean matrices is used here, namely: . The two conditions as presented above can be applied to establish coupling / decoupling in a myriad of domains and are thus independent of variables such as time, frequency, mode, etc. Some implementations of the interface conditions for the most common domains of substructuring are presented below. Substructuring in the physical domain The physical domain is the domain that has the most straightforward physical interpretation. For each discrete linearised dynamic system one is able to write an equilibrium between the externally applied forces and the internal forces originating from intrinsic inertia, viscous damping and elasticity. This relation is governed by one of the most elementary formulas in structural vibrations: represent the mass, damping and stiffness matrix of the system. These matrices are often obtained from finite element modelling (FEM), and are referred to as the numerical model of the structure. Furthermore, represents the DoFs and the force vector which are dependent on time . This dependency is omitted in the following equations in order to improve readability. Coupling in the physical domain Coupling of substructures in the physical domain first requires writing the uncoupled equations of motion of the substructures in block diagonal form: Next, two assembly approaches can be distinguished: primal and dual assembly. Primal assembly For primal assembly, a unique set of degrees of freedom is defined in order to satisfy compatibility, . Furthermore, a second equation is added to enforce interface force equilibrium. This results in the following coupled dynamic equilibrium equations: Pre-multiplying the first equation by and noting that , the primal assembly reduces to: The primally assembled system matrices