Extracting Reduced Order State-Space Matrices

COMSOL Multiphysics models often have a large number of degrees of freedom. This leads to large state-space model when exported using mphstate. COMSOL Multiphysics provides model reduction functionality, which can reduce the number of states using an eigenvalue (or eigenfrequency) study leading to a low number of states that can be used for simulation and analysis.

The function mphreduction returns state-space matrices or a MATLAB state-space model using the ss function of a given reduced-order model set in the COMSOL model.

See the Modal Reduced-Order Model and The Modal Solver Algorithm in the COMSOL Multiphysics Reference Manual for more information.

In this section you will find an example that illustrate how to use the mphreduction function to extract the reduced order model state-space matrices.

The problem studied here is a heat transfer model set with a heat source. In the expected state space system the heat source is set as input. The temperature at specified location is used as output. The tutorial shows how to extract the state space matrices and solve the system using the MATLAB functionalities.

This is the same problem solved as in Extracting State-Space Matrices so you can compare the solution, and computational performance when solving the problem with full state space system matrices.

The following example sets up a model using commands from MATLAB. You need the Heat Transfer Module or MEMS Module to run the model, as reduced order modeling requires eigenvalue of the physical problem. If you don’t have access to one of these modules, you can find the commands that sets up and solve the same problem using the General Form PDE interface in the section Code for use with MATLAB® - General Form PDE.

First, load the model model_tutorial_llmatlab from the Application Library:

This is model is used as base model for documentation tutorial. The geometry, mesh, physics is set but for this specific problem you need to edit the model to use parameter for the initial and external temperature as they will be used later in the reduced order model:

This model is defined with a temperature constraint, you need to replace it with by a heat flux condition as constraints are not supported as input for reduced order models:

You need now to add a time dependent study:

Add a domain point probe plot, which will defines the output of the reduced order model:

Run the study and create a plot group to display the probe:

A reduced order model requires eigenvalue solution of the problem. To create an eigenfrequency study and specify the number of eigenvalue to 30 enter:

Before adding a reduced-order study to your model you need to set the input, here in this example the power variable power, the bottom temperature Temp, and the exterior temperature Text:

Now you can add a model reduction study.

Here the model reduction study generates automatically a Time Dependent, Modal Reduced-Order Model node you can identify with the tag rom1. To get the newly generated reduced order model you can either type the command:

or inspect the model using mphnavigator window by typing:

	To visualize the reduced order node in the mphnavigator window, you need to go the Settings menu and select Advanced.

A call to mphreduction creates the state-space matrices needed to simulate the reduced-order system.

Now the reduced-order system can be simulated using the function ode23s.

Compare the solution computed with the system and the one computed with COMSOL Multiphysics (see Figure 4-3):

Evaluate the steady-state temperature value:

As an alternative to using ode23s, you can use functions in the Control System Toolbox, which is an add-on to MATLAB. The matrices stored in MR can be used to manually construct a state-space system using the function ss, or you can call mphreduction using the return option to specify that mphreduction should do the conversion.