 # 2 Basics of Substructuring Assembly

This page introduces the basics for combining two substructures. Therefore, consider two substructures A and B as depicted in the following figure:

The two substructures have a total of six nodes; the displacements of the nodes are described by a set of Degrees of Freedom (DoFs)  which have different subscript indices depending on their function:

• Index 1: DoFs of the internal nodes of substructure A.
• Index 2: DoFs of the coupling nodes of substructures A and B, i.e. interface DoFs.
• Index 3: DoFs of the internal nodes of substructure B.

For the purpose of substructuring, the set of external forces $$\mathbf{f}$$ and motion $$\mathbf{u}$$ is extended by a set of interface forces $$\mathbf{g}$$ that keep the structures together:

$$\mathbf{u} \overset{\Delta}{=} \begin{bmatrix} \mathbf{u}_{1}^\mathrm{{A}}\\ \mathbf{u}_{2}^\mathrm{{A}}\\ \mathbf{u}_{2}^\mathrm{{B}}\\ \mathbf{u}_{3}^\mathrm{{B}} \end{bmatrix}, \; \; \; \; \mathbf{f} \overset{\Delta}{=} \begin{bmatrix} \mathbf{f}_{1}^\mathrm{{A}}\\ \mathbf{f}_{2}^\mathrm{{A}}\\ \mathbf{f}_{2}^\mathrm{{B}}\\ \mathbf{f}_{3}^\mathrm{{B}} \end{bmatrix}, \; \; \; \; \mathbf{g} \overset{\Delta}{=} \begin{bmatrix} \mathbf{g}_{1}^\mathrm{{A}}\\ \mathbf{g}_{2}^\mathrm{{A}}\\ \mathbf{g}_{2}^\mathrm{{B}}\\ \mathbf{g}_{3}^\mathrm{{B}} \end{bmatrix}$$

The relation between dynamic displacements $$\mathbf{u}$$ and external forces $$\mathbf{f}$$ of the uncoupled problem is governed by the corresponding dynamic equation, such as presented in the modeling domains chapter.

### Force & displacement equilibria

When two or more substructures are to be coupled, two conditions must always be satisfied, regardless of the coupling method used:

• Force equilibrium on the substructures’ interface Degrees of Freedom $$(\mathbf{g}_{2}^{\mathrm{A}} = – \mathbf{g}_{2}^{\mathrm{B}})$$. This can also be expressed using the Boolean localisation matrix $$\mathbf{L}$$, localizing the interface forces $$\mathbf{g}_{2}$$ in the global set of DoFs:

$$\mathbf{L} = \begin{bmatrix} \mathbf{I} & \mathbf{0} & \mathbf{0}\\ \mathbf{0} & \mathbf{I} & \mathbf{0}\\ \mathbf{0} & \mathbf{I} & \mathbf{0}\\ \mathbf{0} & \mathbf{0} & \mathbf{I} \end{bmatrix}, \;\;\; \mathbf{L}^\mathrm{T}\mathbf{g} = \mathbf{0} \rightarrow \left\{\begin{matrix} \mathbf{g}_1^\mathrm{A} = \mathbf{0}\\ \mathbf{g}_2^\mathrm{A} = – \mathbf{g}_2^\mathrm{B}\\ \mathbf{g}_3^\mathrm{B} = \mathbf{0} \end{matrix}\right.$$

• Compatibility of the substructures’ motion at the interface DoFs $$(\mathbf{u}_{2}^{\mathrm{A}} = \mathbf{u}_{2}^{\mathrm{B}})$$. Note that this can be expressed based on the signed Boolean matrix $$\mathbf{B}$$:

$$\mathbf{B} = \left [ \mathbf{0 -I \; \; I \;\; 0} \right ], \;\;\; \mathbf{Bu} = \mathbf{0} \rightarrow \left\{\begin{matrix} \mathbf{u}_{2}^\mathrm{{B}} – \mathbf{u}_{2}^\mathrm{{A}} = \mathbf{0} \end{matrix}\right.$$

### Primal and dual assembly

Depending on whether one chooses the motion $$\mathbf{u}$$ or the forces $$\mathbf{g}$$ as unknowns at the interface, a primal or dual assembled system of equations is obtained.

• In a primal assembly formulation, a unique set of interface DoFs is defined to satisfy compatibility directly. The interface forces are eliminated as unknowns during the assembly.
• In a dual assembly formulation, the full set of global DoFs is retained. The dually assembled system introduces Lagrange multipliers to satisfy the force equilibrium directly.