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.