# #032: Periodicity and Exterior Field Calculation

March 28, 2020

I have talked to several people in my network recently who would like a new feature in COMSOL Multiphysics(R), so that the 'Exterior Field Calculation' node will work for structures with cyclic symmetry. That is understandable, since you can save a lot of computational time and memory by exploiting this symmetry, while at the same time having access to the full three-dimensional exterior sound field. In the following post I will briefly demonstrate that there indeed is a very elegant way to this, so that plots, animations, and any other evaluations can be done, just as for a full 3D model, without solving for more than a single geometry sector.

Imagine a geometry with cyclic symmetry consisting of three equal sector as shown below.

A blue disc area is to be given an acceleration boundary condition, while the cones are sound hard and will cause scattering of the sound field. As of current, to get the exterior sound field, you will apply the Exterior Field Calculation to a closed surface, and the underlying Kirchhoff-Helmholtz integral will be solved accordingly.

In this example we can take advantage of the fact that a baffle is included by checking one of the 'Symmetric/Infinite sound hard boundary' check boxes. However, we cannot exploit the circular symmetry by applying periodicity conditions while at the same time using the Exterior Field calculation.

I have seen other attempt at solutions to this problem, but none were very elegant, and I was asked if I could perhaps come up with an alternative. And indeed I did:

Only one sector is needed for the Finite Element analysis, which cuts the number of degrees of freedom to 1/3, or 1/N for N sectors. This will dramatically reduce the computational effort, and cases that could not even run before, can now be swept across many frequencies within a reasonable time frame.

Below is first the radiation pattern when no measures have been taken, so periodicity is exploited and the exterior field is calculated as out of the box:

Now that is better. The same result will be found using the full three-dimensional geometry, but it takes much longer to solver. Again, the more sectors, the more this will benefit the user.

Armed with the proper knowledge of the underlying mathematics, combined with the openness of the simulation software COMSOL Multiphysics, it is possible to further expand on the already immensely comprehensive and brilliant functionalities of this software.

The approach will be available as a product, please see www.acculution.com/products.