COMSOL Multiphysics offers some unique operators that can couple different parts of different dimensions, such as the Extrusion and Projection Operators. While they can be somewhat difficult to get into, once you get the hang of it, they are extremely useful, and they can enable you to do operations that other software packages cannot (unless you do some manual coding, if even possible).

One way to get into to using them and their underlying mahtematics, is to see them being used for some cases, so I will briefly show a few examples, where I have used these operations with great success. You can see more details in the __documentation__.

__Topology Optimization with Periodicity__

To save time you should always try and exploit any symmetry/periodicity in your simulation setup. In this case I was dealing with an acoustic cloaking structure that repeated several times around the circumference, and so I could apply topology optimization to one sector only, and extrude the result to the other regions, without explicitly doing the optimization here. Without extrusion operations, this would not be possible.

__Integration and Mapping to Lower or Same Dimension __

With projection operations there is an inherent integration taking place, but you can choose the coordinates involved in the integration freely. This example is treated in a __recent post__, but I will show here again how you can do a circumferential integration to split a total displacement into a radial part and a circumferential part via projection and extrusion operators. Below, I have done it in one go, but in the previous post I show how to do it in steps. The total displacement is shown alongside the radially and circumferentially decomposed displacements.

__3D Integration on 2Daxi Models__

Sometimes the standard operations available in a physics with a certain dimension is not enough for what you want to do, although the needed information in inherently available. This could be the case for 2Daxi case, where you want to do integration taking into account the full 3D nature of the case, for example in you have a 'dest' operator in the integrand due to a Green's function or similar. You can save time exploiting the axisymmetry with multiple physics involved, while at the same time getting the needed 3D-related results.

__Partial Symmetry__

For some cases parts of the whole assembly have an inherent symmetry that can be considered even if the symmetry is not found throughout. This could be the case when an axisymmetric driver is placed in a car or a loudspeaker enclosure in general. Mappings can be done via projections and extrusions that can even incorporate the proper coupling, while lessening the computational burden.

Once you get the hang of it, these operators become extremely useful. If you think that you have might have a need for these operators in your simulation setup, feel free to reach out for consultancy from Acculution ApS.