the internal nodes of substructure B. Note that the denotation 1, 2 and 3 indicates the function of the nodes/DoFs rather than the total amount. Let us define the sets of DoFs for the two substructures A and B in concatenated form. The displacements and applied forces are represented by the sets and . For the purpose of substructuring, a set of interface forces is introduced which only contains non-zero entries on the interface DoFs: The relation between dynamic displacements and applied forces of the uncoupled problem is governed by a particular dynamic equation, such as presented in the table above. The uncoupled equations of motion are augmented by extra terms/equations for compatibility and equilibrium, as discussed next. Compatibility The compatibility condition requires that the interface DoFs have the same sign and value at both sides of the interface: . This condition can be expressed using a so-called signed Boolean matrix, denoted by . For the given example this can be expressed as: In some cases the interface nodes of the substructures are non-conforming, e.g. when two substructures are meshed separately. In such cases a non-Boolean matrix has to be used in order to enforce a weak interface compatibility. A second form in which the compatibility condition can be expressed is by means of coordinate substitution by a set of generalised coordinates . The set contains the unique coordinates that remain after assembly of the substructures Every matching pair of interface DoFs is described by a single generalised coordinate, which means that the compatibility condition is automatically enforced. Expressing using gives: Matrix is referred to as the Boolean localisation matrix. A useful relation between matrix and can be exposed by noting that compatibility should hold for any set of physical coordinates expressed by . Indeed, substituting in the equation : Hence represents the nullspace of : This means in practice that one only needs to define or ; the other Boolean matrix is calculated using the nullspace property. Equilibrium The second condition that has to be satisfied for substructure assembly is the force equilibrium for matching interface forces . For the current example, this condition can be written as . Similar to the compatibility equation, the force equilibrium condition can be expressed using a Boolean matrix. Use is made of the transpose of the Boolean localisation matrix that was introduced to write compatibility: The equations for and state that the