Complex and large numerical models 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.

The main idea of Model Order Reduction is to describe the system in a lower dimensional subspace containing the required solution. The choice of a proper subspace is vital for the performance of the reduction procedure.

Theory

Dynamic systems are described using a differential equation of motion

\mathbf{M}\ddot{\mathbf{x}}\left(t\right) + \mathbf{C}\dot{\mathbf{x}}\left(t\right)+\mathbf{K}\mathbf{x}\left(t\right) = \mathbf{f}\left(t\right)

with system mass, damping, and stiffness described by the time-invariant matrices \mathbf{M},\mathbf{C},\mathbf{K}\in\mathbb{R}^{n\times n}. The force acting on the system is denoted by \mathbf{f}\left(t\right)\in\mathbb{R}^{n} and the degrees of freedom \ddot{\mathbf{x}}, \dot{\mathbf{x}}, \mathbf{x}\in\mathbb{R}^{n} are acceleration, velocity, and displacement. Applying a Fourier transformation leads to the equation of motion in frequency domain

s^2\mathbf{Mx}(s) + s\mathbf{Cx}(s) + \mathbf{Kx}(s) = \mathbf{f}(s)

with complex frequency s\in\mathbb{C}.

Coordinate Transformation

Model order reduction is often performed by transforming the system matrices to another space using some transformation matrix \mathbf{V}\in\mathbb{R}^{n\times r} where r is the dimension of the reduced order model. Transforming the degrees of freedom to \mathbf{\ddot{x}}_r, \mathbf{\dot{x}}_r, \mathbf{x}_r\in\mathbb{R}^{r}

\begin{equation} \mathbf{x}\left(t\right) = \mathbf{Vx}_r\left(t\right), \mathbf{\dot{x}}\left(t\right) = \mathbf{V\dot{x}}_r\left(t\right), \mathbf{\ddot{x}}\left(t\right) = \mathbf{V\ddot{x}}_r\left(t\right) \end{equation}

leads to the frequency domain equation of motion in the reduced space

\begin{align} s^2\mathbf{MV}\mathbf{x}_r(s) + s\mathbf{CV}\mathbf{x}_r(s) + \mathbf{KV}\mathbf{x}_r(s) &= \mathbf{f}(s)\\ s^2\mathbf{M}_r\mathbf{x}_r(s) + s\mathbf{C}_r\mathbf{x}_r(s) + \mathbf{K}_r\mathbf{x}_r(s) &= \mathbf{f}_r(s) \end{align}

with the transformed system matrices and force vector

(1) \begin{equation} \mathbf{M}_r = \mathbf{V}^H\mathbf{MV},\ \mathbf{C}_r=\mathbf{V}^H\mathbf{CV},\ \mathbf{K}_r =\mathbf{V}^H\mathbf{KV},\ \mathbf{f}_r=\mathbf{V}^H\mathbf{f}. \end{equation}

Typically, r is much smaller than n and so, analyses where the repeated evaluation of the system is necessary (e.g. a frequency sweep analysis) can be performed much more efficient.

Reduction Methods

Reduction using physical coordinates

Remove a suitable subset of physical coordinates from the model.

Examples:

Reduction using generalized coordinates

Transform the models from physical space to e.g. modal space and possibly truncate the target space.

Examples:

Subspace Methods

Project the full system onto a low-dimensional subspace containing the desired solution.

Examples:

  • Moment matching methods
  • proper orthogonal decomposition (POD)
  • balanced truncation (BT)