Perfectly Matched Layers (PMLs)
The perfectly matched layer (PML) is a domain or layer (sometimes called sponge layer) that is added to an acoustic model to mimic an open and nonreflecting infinite domain. It sets up a perfectly absorbing domain as an alternative to nonreflecting boundary conditions. The PML works with all types of waves, not only plane waves. It is also efficient at very oblique angles of incidence. In the frequency domain the PML imposes a complex-valued coordinate transformation to the selected domain that effectively makes it absorbing at a maintained wave impedance, and thus eliminating reflections at the interface. In the time domain additional equitations are solved in the PML for the inverse Laplace transformed equations.
A Perfectly Matched Layers node is added to the model from the Model>Definitions node. In the frequency domain the PMLs can be used for the Pressure Acoustics, Acoustic-Structure Interaction, Aeroacoustics, and Thermoviscous Acoustics interfaces. In the time domain the PMLs only exist for the Pressure Acoustics, Transient interface.
In this section:
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Meshing the PMLs
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Coordinate Stretching, Scaling, and Curvature
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Geometry Type (User Defined)
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Limitations of the Perfectly Matched Layers
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Time Domain Perfectly Matched Layers
Meshing the PMLs
Optimal behavior of the PML is achieved when the mesh inside the PML domain is structured. Use a mapped mesh in 2D models and a swept mesh in 3D models. Use at least 8 layers when using the default polynomial stretching option. As a good starting point for the rational stretching use 5 or 6 mesh layers inside the PML. When creating the geometry for your model, it is also advantageous to use the layer feature to create the PML domains. This ensures that the geometry is suited for a structured mesh.
The following is a good tutorial model on the use of perfectly matched layers (PMLs): Acoustic Scattering off an Ellipsoid: Application Library path Acoustics_Module/Tutorials/acoustic_scattering
Coordinate Stretching, Scaling, and Curvature
The choice of the Coordinate stretching type and the PML scaling factor and the PML curvature parameter depends on the problem at hand. A detailed description is given in the PML Implementation section of the COMSOL Multiphysics Reference Manual. In general, the Rational stretching option is used for open radiation problems for propagating waves (it is efficient for many angles of incidence). The Polynomial stretching option is good for systems with a mix of different wave types (propagating and evanescent), for example, for terminating a waveguide or in multiphysics problems. Note that when solving a model using an iterative solver the Polynomial scaling should be used to ensure convergence.
There is also a User Defined coordinate stretching type which allow users to define advanced stretching functions to handle special cases. The stretching can in this way be optimized to a special problem.
To ensure that the PML is working optimally, it is good practice to make a mesh convergence test by refining (or most probably adding more layers to) the mesh in the PML domain. This is especially important at low frequencies, where evanescent waves may interact with the PML and give erroneous solutions.
The behavior of the PMLs at low frequencies is discussed in the following model. Plotting the total radiated power can be a good indicator of possible issues. Lumped Loudspeaker Driver: Application Library path Acoustics_Module/Electroacoustic_Transducers/lumped_loudspeaker_driver
Infinite Elements, Perfectly Matched Layers, and Absorbing Layers in the COMSOL Multiphysics Reference Manual.
The PMLs damp a certain wavelength existing in the system. The wavelength is deducted from the frequency and a reference wave speed cref. The wave speed is defined in the Typical Wave Speed section. Set cref equal to the speed of sound of the material in the PML.
When multiple physics and materials are involved, be cautious. Depending on the configuration, use several PMLs and PML domains or a single PML. For details, see Configuration of Perfectly Matched Layers (PMLs) for Acoustic-Structure Interaction Models.
Geometry Type (User Defined)
When setting up a PML, you also select the geometry type of the layer. Typically, the predefined options Cartesian, Cylindrical, or Spherical can apply in most situations. Using these COMSOL will automatically detect the layer thickness and define the local coordinates inside the PML. In some cases the automatic detection can fail (this can, for example, happen for certain imported CAD geometries). The automatic detection also fails if the domain is not the outer most entity in the geometry. A workaround is then to use the User defined geometry type. This advanced option makes is possible to define the local Distance functions and layer Thickness manually. For example, for a spherical PML geometry the typical distance function is sqrt(x^2+y^2+z^2)-r0, where r0 is the radius of the inner domain. The user-defined option can also be used for spacial layer shapes.
Infinite Elements, Perfectly Matched Layers, and Absorbing Layers in the COMSOL Multiphysics Reference Manual.
Limitations of the Perfectly Matched Layers
When a model contains a Background Pressure Field and PMLs, certain configurations will create incompatibilities that lead to erroneous behavior. The problem arises if a domain with a background pressure field is next to a domain without the feature (for example when setting up absorption problems) and the two domains have a common PML attached to them. Meaning that the PML next to the background pressure field touches the PML next to the domain without the background pressure field. In this case, there is an incompatibility at the common edge of the PMLs. In one PML domain the pressure DOF is interpreted as a scattered field, while it is the total field in the other. Note that you can set up models that contain this feature configuration as long as the PMLs do not touch.
Time Domain Perfectly Matched Layers
In the time domain the PML does not include a real stretching component. This means that the geometric thickness needs to be set adequately. Typically the layer should have a thickness which is between λ/8 and λ/4, where λ is the lowest wavelength in the signal. The PML also needs to be places roughly λ/4 away from the sources.
When meshing the PMLs for time domain simulations use a structured mesh in the same way as in the frequency domain. Use at least 8 mesh layers for the rational scaling and 6 for the polynomial scaling.
See Theory for the Perfectly Matched Layers in the Time Domain for details about the implementation.