Problem Description

I am setting up a model where I want to include a region of infinite extent. How should such situations be modeled and meshed?

Solution

Overview

There are three options for modeling a domain that is meant to represent a region of infinite extent. They each have different areas of applicability:

  • The Infinite Element domain functionality is meant for governing equations that are diffusion-like in nature. The Heat Transfer in Solids physics interface is one such case. Infinite Elements represent a region that is stretched along certain coordinate axes with the effect of approximating an infinitely large domain.

  • The Perfectly Matched Layer (PML) domain functionality is meant for stationary governing equations that are wave-like in nature, wherein the fields describe a radiation of energy. The Electromagnetic Waves, Frequency Domain interface is one such case. The PML acts as a domain that is a nearly ideal absorber or radiation.

  • The Absorbing Layer functionality is the time-domain analogue of the PML. It is also meant for governing equations that are wave-like in nature but are solved via a time-explicit approach. The Electromagnetic Waves, Time Explicit interface is one such case.

Schematic of a situation where a region of interest (green) is within a region of infinite extent (blue).

The most typical usage of these features is to model the case of a region of interest that is fully encapsulated within an region of infinite extent, as described in the image above. To accurately capture the behavior in the region of interest one must solve the relevant governing equations in that region, as well as the region of infinite extent. However, solving for the fields in an infinitely large region is computationally impossible, so various strategies are used to truncate the model to a reasonable size. The Infinite Elements, PML's, and Absorbing Layers are one such truncation strategy that share similar setup, usage, and (with the exception of Absorbing Layers) similar meshing requirements. This article addresses the geometry and meshing requirements of these three features.

To determine if the physics you are using supports any of the above options, first add the physics to your model, then right-click on the Component > Definitions branch, or go to the Definitions toolbar. Depending upon which physics are present in your model, one, some, or none of the above options will be present.

Geometry Setup

Regardless of which of the three (Infinite Elements, PML's, Absorbing Layers) are being used, the geometry setup is the same. If modeling in 2D, then the geometry should be set up as one of the two cases shown below, describing a Cartesian or Cylindrical infinite domain.

Visualization of geometry of the Cartesian (left) and Cylindrical (right) infinite domains in 2D.

If modeling in 2D axisymmetry, the geometry should be set up as one of these two cases, describing a Spherical or Cylindrical infinite domain:

Visualization of geometry of the Spherical (left) and Cylindrical (right) infinite domains in 2D axisymmetry.

If modeling in 3D, the geometry should be set up as one of these three cases, representing a Spherical, Cartesian, or Cylindrical domain:

Visualization of geometry of the Spherical (left) Cartesian (middle) and Cylindrical (right) infinite domains in 3D. Some of the Infinite Domains, and the interior domain of interest, are omitted for visualization.

Note that in 2D the Rectangle, Circle, and in 3D the Sphere, Block, and Cylinder geometry features all include the option to introduce Layers which will simplify the setup of the above cases.

It is typical to make the thickness of these domains about one-tenth of the overall dimensions of the modeling space. The actual thickness of these domains should not affect their performance. It is important that there be separate corner domains for the Cartesian and Cylindrical cases.

The distance from the region of interest to the infinite domain (Infinite Element, Perfectly Matched Layer, or Absorbing Layer) is a parameter that does need to be studied. For wave-type problems, the PML should typically by at least one-quarter wavelength away. For Infinite Element domains, the distance from the objects to interest to the infinite domain can be small.

Special Considerations for Cylindrical and Spherical cases

When the geometry is either Cylindrical or Spherical, in the 3D case, the Infinite Element, Perfectly Matched Layer, or Absorbing Layer will all offer the option to define a Center Coordinate and (for the Cylindrical case) the Center Axis Direction. These should be adjusted based upon where and how the geometry is oriented. Although not necessary, it is often good practice to center the model about the origin and z-axis. Similarly, in 2D and 2D Axisymmetry, make sure that the geometry orientation matches the feature settings.

Meshing Considerations

For the case of the Infinite Element and Perfectly Matched Layers, it is important that the mesh matches the coordinate stretching direction, the direction of absorption. Meshes should look similar to the plots below. Use Mapped meshes in 2D, and Swept meshes in 3D, to produce these types of meshes. For numerical reasons it is good for the elements in these domains to not be too distorted or stretched. Start with at least five elements through these domains and always perform a Mesh Refinement Study.

Visualization of appropriate Infinite Element or Perfectly Matched Layer meshes for the 2D Cartesian (left) and Cylindrical (right) cases.

Visualization of appropriate Infinite Element or Perfectly Matched Layer meshes for the 2D Axisymmetric Spherical (left) and Cylindrical (right) cases.

Visualization of appropriate Infinite Element or Perfectly Matched Layer meshes for the 3D Spherical (left) Cartesian (middle) and Cylindrical (right) cases. Meshes on other domains are not shown.

Note that the Absorbing Layer, used in the Time Explicit approach, should be meshed with triangular (in 2D) or tetrahedral (in 3D) elements, and not with a swept mesh.

Computational Requirements

These Infinite Elements, Perfectly Matched Layers, and Absorbing Layers do introduce additional computational cost. Depending upon the physics, and situation being modeled, it is sometimes possible to avoid using these features, albeit at possibly lower accuracy. Also, the corner regions within the Cylindrical and Cartesian types do introduce a relatively large computation cost compared to the Spherical types. In general, unless dealing with very high-aspect ratio cases, the Spherical type will be the least expensive in terms of memory needed.

Further Resources

  1. The COMSOL Multiphysics Reference Manual chapter on Infinite Elements, Perfectly Matched Layers, and Absorbing Layers.
  2. Automated Meshing for Electromagnetic Waves, Frequency Domain Simulations
  3. Automated Meshing for Infinite Element Domains
  4. For an example where an Infinite Element can be replaced with a boundary condition in Magnetic Field modeling, see: How to Choose Between Boundary Conditions for Coil Modeling
  5. For a comparison of of the Perfectly Matched Layer to a Scattering Boundary Condition in wave electromagnetics, see: Using Perfectly Matched Layers and Scattering Boundary Conditions for Wave Electromagnetics Problems