The Electromagnetic Waves, Boundary Elements (ebem) interface () is used to solve for time-harmonic electromagnetic field distributions. This interface is found under the Wave Optics branch () when adding a physics interface,. The formulation is based on the boundary element method (BEM) and is available in 2D and 3D. The physics interface solves the vector Helmholtz equation for piecewise-constant material properties and uses the electric field as dependent variable.

The advantage of the boundary element method is that only boundaries need to be meshed and the degrees of freedom (DOFs) solved for are restricted to the boundaries. This introduces some clear ease-of-use for handling complex geometries. However, the BEM technique results in fully populated or dense matrices that need dedicated numerical methods. The BEM method is so to speak more expensive per DOF than the FEM method, but has fewer DOFs. Assembling and solving these can be very demanding. This means that when solving models of small and medium size, The Electromagnetic Waves, Frequency Domain Interface will often be faster, than solving the same problem with the BEM interface. The challenge for the FEM interface is to set up open boundaries, for example, using Perfectly Matched Layers (PMLs) in an efficient way. When the geometries are complex or two structures are far apart, large air domains need to be meshed. This costs a lot on the computational side as the frequency is increased.

For large models (problems that contain many wavelengths, at high frequency or for large domains) the stabilized formulation option (see Stabilization) ensures efficient convergence at the cost of some additional degrees of freedom. For low to medium frequencies (small to medium models), running without stabilization is more efficient. The stabilized formulation only gives a benefit in computing time for the large models.

For this physics interface, the maximum mesh element size should well resolve the complex electric field on the boundaries. Thus, if the wave propagates tangentially to the boundary, the maximum mesh size should be a fraction of the wavelength. However, if the wave propagates essentially in the normal direction to the boundary, the maximum mesh element size can be larger. The physics interface supports the Frequency Domain and Wavelength Domain study types. The Frequency Domain and Wavelength Domain study types are used for source driven simulations for a single frequency/wavelength or a sequence of frequencies/wavelengths.

When this physics interface is added, these default nodes are also added to the Model Builder — Wave Equation, Electric, Perfect Electric Conductor, and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click Electromagnetic Waves, Boundary Elements to select physics features from the context menu.

The physics-controlled mesh only defines mesh settings for the boundaries. It is controlled from the Settings window for the Mesh node (if the Sequence type is Physics-controlled mesh). In the table in the Physics-Controlled Mesh section, find the physics interface in the Contributor column and select or clear the check box in the Use column on the same row for enabling (the default) or disabling contributions from the physics interface to the physics-controlled mesh.

When the Use check box for the physics interface is selected, this invokes a parameter for the maximum mesh element size in free space. The physics-controlled mesh automatically scales the maximum mesh element size as the wavelength changes in different dielectric and magnetic regions.

When the Use check box is selected for the physics interface, in the section for the physics interface below the table, choose one of the four options for the Maximum mesh element size control parameter — From study (the default), User defined, Frequency, or Wavelength. When From study is selected, 1/5 of the vacuum wavelength from the highest frequency defined in the study step is used for the maximum mesh element size. For the option User defined, enter a suitable Maximum element size in free space. For example, 1/5 of the vacuum wavelength or smaller. When Frequency is selected, enter the highest frequency intended to be used during the simulation. The maximum mesh element size in free space is 1/5 of the vacuum wavelength for the entered frequency. For the Wavelength option, enter the smallest vacuum wavelength intended to be used during the simulation. The maximum mesh element size in free space is 1/5 of the entered wavelength.

The Label is the default physics interface name.

The Name is used primarily as a scope prefix for variables defined by the physics interface. Refer to such physics interface variables in expressions using the pattern <name>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the name string must be unique. Only letters, numbers, and underscores (_) are permitted in the Name field. The first character must be a letter.

The default Name (for the first physics interface in the model) is ebem.

From the Selection list, select any of the options — Manual, All domains, All voids, or All domains and voids (the default). The geometric entity list displays the selected domain entity numbers. Edit the list of selected domain entity numbers using the selection toolbar buttons to the right of the list or by selecting the geometric entities in the Graphics window. Entity numbers for voids can be entered by clicking the Paste () button in the selection toolbar and supplying the entity numbers in the dialog box. The entity number for the infinite void is 0, and finite voids have negative entity numbers.

Selections can also be entered using the Selection List window, available from the Windows menu on the Home toolbar.

Select the Electric field components solved for — Three-component vector, Out-of-plane vector, or In-plane vector. Select:

From the Formulation list, select whether to solve for the Full field (the default) or the Scattered field.

For Scattered field select a Background wave type according to the following table:

Enter the component expressions for the Background electric field Eb (SI unit: V/m). The entered expressions must be differentiable.

For Gaussian beam select the Gaussian beam type — Paraxial approximation (the default) or Plane wave expansion.

When selecting Paraxial approximation, the Gaussian beam background field is a solution to the paraxial wave equation, which is an approximation to the Helmholtz equation solved for by the Electromagnetic Waves, Boundary Elements (ebem) interface. The approximation is valid for Gaussian beams that have a beam radius that is much larger than the wavelength. Since the paraxial Gaussian beam background field is an approximation to the Helmholtz equation, for tightly focused beams, you can get a nonzero scattered field solution, even if you do not have any scatterers. The option Plane wave expansion means that the electric field for the Gaussian beam is approximated by an expansion of the electric field into a number of plane waves. Since each plane wave is a solution to the Helmholtz equation, the plane wave expansion of the electric field is also a solution to the Helmholtz equation. Thus, this option can be used also for tightly focused Gaussian beams.

For Plane wave expansion select Wave vector distribution type — Automatic (the default) or User defined. For Automatic also check Allow evanescent waves, to include evanescent waves in the plane wave expansion. For User defined also enter values for the Wave vector count Nk (the default value is 13) and Maximum transverse wave number kt,max (SI unit: rad/m, default value is (2*(sqrt(2*log(10))))/ebem.w0). Use an odd number for the Wave vector count Nk to make sure that a wave vector pointing in the main propagation direction is included in the plane-wave expansion. The Wave vector count Nk specifies the number of wave vectors that will be included per transverse dimension. So for 3D the total number of wave vectors will be Nk·Nk.

A plane wave expansion with a finite number of plane waves included will make the field periodic in the plane orthogonal to the main propagation direction. If the separation between the transverse wave vector components, given by 2kt,max/(Nk − 1), is too small, replicas of the Gaussian beam background field can appear. To avoid that, increase the value for the Wave vector count Nk.

For more information about the Gaussian beam theory, see Gaussian Beams as Background Fields and Input Fields.

The initial background wave is predefined as E0 = exp(−jkxx)z. This field is transformed by three successive rotations along the roll, pitch, and yaw angles, in that order. For a graphic representation of the initial background field and the definition of the three rotations compare with Figure 3-1 below.

To display this section, click the Show More Options button () and select Stabilization in the Show More Options dialog box.

For large models (problems that contain many wavelengths, at high frequency or for large domains) enable the Use stabilization option (enable by default) to ensure efficient convergence at the cost of some additional degrees of freedom.

When Use stabilization is selected, a text field for the Stabilization parameter is enabled with the default value sqrt(abs(ebem.k[m])). This is a parameter that should scale inversely with the wavelength. The default gives good performance in most cases.

To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box.

When Use far-field approximation for matrix assembly is selected, a text field for the Minimum near field range in vacuum for preconditioning is enabled with the default value ((2*pi)/ebem.k0)/10 (one tenth of a wavelength). For problems having a wide distribution of mesh element sizes, including mesh elements that are much smaller than the wavelength, a smaller value for this parameter may make the iterative solver convergence faster.

To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box.

From the Electric field/Flux field list, choose from predefined options for the boundary element discretization order for the electric field variable and the flux field (magnetic field) variable, respectively. The predefined options represent the suitable combinations of element orders such as Quadratic/Linear (the default). For more information about the Discretization section, see Settings for the Discretization Sections in the COMSOL Multiphysics Reference Manual.

The dependent variables (field variables) are for the Electric field E and its components (in the Electric field components fields). The name can be changed but the names of fields and dependent variables must be unique within a model.