Component Mode Synthesis

Component Mode Synthesis (CMS) is a method used for Model Order Reduction. It is used for complex and large numerical models, which require a substantial computational effort and large amounts of memory. Model order reduction methods reduce the size of the numerical models. The reduced models approximate the full system's response characteristics with less computational effort and memory requirements. Good reduced models are able to replace the original model.

For CMS, the transformation is a combination of physical and generalized coordinate transformation, some degrees of freedom are retained in physical domain, while the remaining are transformed into generalized coordinates. Reduction using physical coordinates means that a suitable subset of physical coordinates is removed from the model, whereas for the reduction using generalized coordinates, the models are transformed from physical space to e.g. modal space and possibly the target space is truncated.

The general concept is:

Divide the system into substructures (components)
Define significant modes for the components
Couple the components to form the reduced-order model

Some advantages of the component mode synthesis are:

A large model is subdivided and the smaller parts can be treated individually
The different substructures can be handled and maintained by different engineers
Identical substructures have to be solved only once
A change in one substructure does not necessarily affect the other substructures

An often used component mode synthesis method is the Craig-Bampton method.

Craig-Bampton method

Theory

Consider the equation of motion of a substructure $\begin{array}{l}A\end{array}$

$\begin{array}{l}\displaystyle \mathbf{M}^A\mathbf{\ddot{x}}\left(t\right) + \mathbf{C}^A\mathbf{\dot{x}}\left(t\right) + \mathbf{K}^A\mathbf{x}\left(t\right)=\mathbf{f}^A\left(t\right)\end{array}$

partitioned into boundary $\begin{array}{l}b\end{array}$ and internal $\begin{array}{l}i\end{array}$ degrees of freedom:

$\begin{array}{l}\begin{bmatrix} \mathbf{M}_{bb}^A &\mathbf{M}_{bi}^A \\ \mathbf{M}_{ib}^A &\mathbf{M}_{ii}^A \end{bmatrix} \begin{bmatrix} \mathbf{\ddot{x}}_b^A\\ \mathbf{\ddot{x}}_i^A \end{bmatrix} + \begin{bmatrix} \mathbf{C}_{bb}^A &\mathbf{C}_{bi}^A \\ \mathbf{C}_{ib}^A &\mathbf{C}_{ii}^A \end{bmatrix} \begin{bmatrix} \mathbf{\dot{x}}_b^A\\\mathbf{\dot{x}}_i^A \end{bmatrix} + \begin{bmatrix} \mathbf{K}_{bb}^A &\mathbf{K}_{bi}^A \\\mathbf{K}_{ib}^A &\mathbf{K}_{ii}^A \end{bmatrix} \begin{bmatrix} \mathbf{x}_b^A\\\mathbf{x}_i^A \end{bmatrix} = \begin{bmatrix} \mathbf{f}_b^A \\\mathbf{f}_i^A \end{bmatrix}.\end{array}$

At the boundary dofs, forces or boundary conditions can be applied and other substructures can be coupled here. Component $\begin{array}{l}A\end{array}$ (the superscript $\begin{array}{l}A\end{array}$ is omitted in the following) is transformed into a mixed space consisting out of physical coordinates at the boundaries ( $\begin{array}{l}\mathbf{x}_b\end{array}$ ) and generalized coordinates ( $\begin{array}{l}\mathbf{q}\end{array}$ ) using the transformation matrix

$\begin{array}{l}\mathbf{V} = \begin{bmatrix} \mathbf{I}&0\\ \pmb{\Phi}_c & \pmb{\Phi}_n \end{bmatrix},\end{array}$

where $\begin{array}{l}\pmb{\Phi}_c\end{array}$ are constraint modes and $\begin{array}{l}\pmb{\Phi}_n\end{array}$ normal modes. The physical displacement at the interior dofs can be obtained again with

$\begin{array}{l}\mathbf{x}_i = \pmb{\Phi}_c \mathbf{x}_b + \pmb{\Phi}_n \mathbf{q}_n\end{array}$ .

Example for component modes - elastically supported beam

The deformation patterns of an elastically supported beam can be described through normal modes $\begin{array}{l}\pmb{\phi}_n\end{array}$ and constraint modes $\begin{array}{l}\pmb{\phi}_c\end{array}$ as follows:

The system response results from superposition of the different modes

$\begin{array}{l}\begin{bmatrix} \pmb{\phi}_{n,1}& \pmb{\phi}_{n,2} & \cdots & \pmb{\phi}_{n,p} \end{bmatrix} \mathbf{q}_n + \begin{bmatrix} \pmb{\phi}_{c,1}& \pmb{\phi}_{c,2} \end{bmatrix} \mathbf{x}_{b} = \mathbf{x}\end{array}$

Obtaining normal modes

The normal modes represent the free harmonic response of the interior dofs of the component. The normal modes $\begin{array}{l}\pmb{\Phi}_n = \left[\pmb{\phi}_{n,1},\dots,\pmb{\phi}_{n,p}\right]\end{array}$ are represented by the eigenvectors of the undamped component with fixed interfaces

$\begin{array}{l}\displaystyle \left(-\omega_j^2 \mathbf{M}_{ii} + \mathbf{K}_{ii}\right)\pmb{\phi}_{n,j} = 0,\ \text{with }j=1,\dots,p\end{array}$

Obtaining component modes

The constraint modes provide the deformation pattern, when the boundary is displaced. They can be obtained by imposing unit loads at the boundaries. We obtain the static equilibrium

$\begin{array}{l}\begin{bmatrix} \mathbf{K}_{bb} &\mathbf{K}_{bi} \\ \mathbf{K}_{ib} &\mathbf{K}_{ii} \end{bmatrix} \begin{bmatrix} \mathbf{I} \\\pmb{\Phi}_c \end{bmatrix} = \begin{bmatrix} \mathbf{R} \\0 \end{bmatrix}.\end{array}$

From this the constraint modes can be obtained as

$\begin{array}{l}\displaystyle \pmb{\Phi}_c = \left[ -\mathbf{K}_{ii}^{-1}\mathbf{K}_{ib} \right],\end{array}$

which is identical to the transformation matrix of the static condensation (see (Condensation:2)).

Remarks

The quality of the reduced model depends on the number of chosen normal modes.
The number of constraint modes is fixed in the classic Craig-Bampton method.
When coupling different substructures, attachment modes between the substructures and rigid body modes of the individual substructures have also to be considered.
The compatibility between some components $\begin{array}{l}A\end{array}$ and $\begin{array}{l}B\end{array}$ can be ensured using Lagrange multipliers. See (Bampton and Craig, 1968) for further details.

Example

An example for the Craig-Bampton method can be found here:

Example - Craig-Bampton method

References

Craig Jr, R. R., & Bampton, M. C. (1968). Coupling of substructures for dynamic analyses. AIAA journal, 6(7), 1313-1319

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