The Pressure Acoustics, Boundary Elements (pabe) interface (), found under the Pressure Acoustics branch () when adding a physics interface, is used to compute the pressure variation for the propagation of acoustic waves in gases and liquids. It is suited for all frequency-domain simulations with harmonic variations of the pressure field. The formulation is based on the boundary element method (BEM) and the interface is available in 2D and 3D. The physics interface solves Helmholtz equation for constant-valued material properties and uses the pressure as the dependent variable.

The interface is fully multiphysics enabled and can be coupled seamlessly with the physics interfaces that are based on the finite element method (FEM). This includes coupling to vibrating structures with the Acoustic–Structure Boundary multiphysics coupling, to FEM acoustic domains, with the Acoustic FEM–BEM Boundary multiphysics coupling, to thermoviscous acoustic domains with the Acoustic–Thermoviscous Acoustic Boundary multiphysics coupling, and to poroelastic waves domains with the Acoustic–Porous Boundary multiphysics coupling. This approach allows modeling in a FEM-BEM framework, using the strength of each formulation adequately. The BEM-based interface is especially well suited for radiation and scattering problems.

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 acoustic models of small and medium size, The Pressure Acoustics, 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 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 acoustically 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 acoustically large models.

where pt is the total acoustic pressure, keq is the wave number, ρc is the density, and cc is the speed of sound. The subscript “c” denotes that these can be complex-valued quantities in models with damping. The governing equations and boundary conditions are formulated using the total pressure pt with a so-called scattered field formulation. In the presence of a Background Pressure Field defining a background pressure wave pb (this could, for example, be a plane wave), the total acoustic pressure pt is the sum of the pressure solved for p (which is then equal to the scattered pressure ps) and the background pressure wave: pt = p+pb. The equations then contain the information about both the scattered field and the background pressure field.

When this physics interface is added, these default nodes are also added to the Model Builder — Pressure Acoustics, Sound Hard Boundary (Wall), and Initial Values.

Then, from the Physics toolbar, add other nodes that implement boundary conditions. You can also right-click Pressure Acoustics, Boundary Elements to select physics features from the context menu. Infinite conditions like a symmetry plane or an infinite sound hard boundary are defined in the Symmetry/Infinite Boundary Condition section.

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 pabe.

Expand the Equation section to see the equations solved for with the Equation form specified. The default selection is Equation form is set to Study controlled. The available studies are selected under Show equations assuming. If Frequency domain is selected as the Equation form the Frequency of the study can be either taken From solver or User defined.

Select the Enable physics symbols check box to display the infinite boundary condition lines or planes in the geometry.

The zero level on the dB scale varies with the type of fluid. That value is a reference pressure that corresponds to 0 dB. This variable occurs in calculations of the sound pressure level Lp based on the root mean square (rms) pressure prms, such that

where pref is the reference pressure and the star (*) represents the complex conjugate. This is an expression valid for the case of harmonically time-varying acoustic pressure p.

Select a Reference pressure for the sound pressure level based on the fluid type:

In this section you can specify infinite boundary conditions for the boundary element problem. These are conditions that are not added on a boundary but apply to a line in 2D or a plane in 3D. The planes where the conditions are applied are visualized if the Enable physics symbols is turned on (the default) in the Physics Symbols section.

For pressure acoustics a symmetry condition is mathematically equivalent to a sound hard boundary, and an antisymmetry condition is equivalent to a sound soft boundary. In this way you can include a Symmetry/Infinite sound hard boundary or an Antisymmetry/Infinite sound soft boundary along “infinite lines” (in 2D) or “infinite planes” in (3D). Only one condition per Cartesian direction can be applied.

For the infinite conditions to be mathematically valid it is important that all sources and BEM boundaries are located on the same side of (or on) the infinite conditions. The condition can, for example, not be used as an infinite baffle with sound radiating through a hole. In this case, the propagation domain is on a different side than the source. The Exclude Boundary feature can be used to exclude boundaries from the BEM problem for advanced symmetry configurations.

Choose an option from the Condition for the x = x0 plane, Condition for the y = y0 plane, and Condition for the z = z0 plane lists (when applicable). In 2D, these are out-of-plane surfaces. Choose one of the following options:

Then enter the value for the plane location x0, y0, or z0 (the default is 0 m). This allows an offset of the infinite condition planes along the main coordinate axes.

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

The Use far-field approximation check box is selected by default in order to accelerate the solution process. If the check box is cleared, the solution will be slightly more accurate but the computational time and memory consumption may become prohibitively high.

The Approximation type can be either ACA+ or ACA. These alternatives correspond to two different versions of the adaptive-cross-approximation (ACA) method, which is a fast matrix multiplication method based on far-field approximations.

In this section, you specify the condition to apply at infinity for an unbounded problem. For the Helmholtz equation, solved here, choose to specify an Outgoing wave (the default) or an Incoming wave. In almost all cases the Outgoing wave option should be selected.

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

Select Thin gap handling if narrow regions exist in the model. The Thin gap handling check box is used to increase the accuracy of the quadrature between almost-touching mesh element pairs, that is, pairs that are extremely close with respect to element size. The Threshold distance (relative to mesh size) controls the distance at which elements are considered almost-touching. A larger value should provide more accurate results, but at an increased computational cost. Under the Integration order settings, the field Integration order, very close elements (thin gap) becomes active (when Manual is selected), allowing a higher-order rule for such elements. In 3D, an additional variable transformation is applied to improve the accuracy even further.

The Integration orders are by default set to Automatic. This means that the quadrature integration order values will follow the element order selection in the Discretization section. Higher element orders automatically generate higher values for the quadrature integration orders. To see the used values or to set user-defined values, switch to Manual. Note that the values are reset to default when switching back and forth between the Automatic and Manual settings. The Integration order, very close elements (thin gap) is only editable when Thin gap handling is selected.

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

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

When Stabilized formulation is selected, a text field for the Stabilization parameter is enabled with the default value sqrt(abs(pabe.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.

Enter an expression for the Interpolation distance from boundary. The entered value defines the distance beyond which the kernel evaluation is used. The default is 0.5*h which gives reasonable results. This default corresponds to half the mesh size h. If the value is set to 0 no interpolation is done.

From the Dependent variable/Normal boundary flux list, choose from predefined options for the boundary element discretization order for the dependent variable and the normal boundary flux. The predefined options represent the suitable combinations of element orders such as Quadratic/Linear (the default).

The settings under Value types when using splitting of complex variables are important for sensitivity and optimization computations. See the description of the built-in operators fsens and fsensimag.

This physics interface defines one dependent variable (field), the Pressure p. If required, edit the name, which changes both the field name and the dependent variable name. The new field name must not coincide with the name of a field of another type, or with a component name belonging to some other field.