The Pressure Acoustics, Boundary Elements Interface

The interface (

), found under the 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 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 good modeling strategies, meshing, solvers, postprocessing information, as well as tips and tricks, see the Modeling with the Pressure Acoustics Branch (BEM-Based Interface) section.

	Head and Torso HRTF Computation: Application Library path Acoustics_Module/Tutorials,_Pressure_Acoustics/head_torso_hrtf

The governing Helmholtz equation defined by the Pressure Acoustics, Boundary Element interface is given by:

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.

	The boundary element method (BEM) used in the Pressure Acoustics, Boundary Element interface is based on the direct method with Costabels symmetric coupling. The adaptive cross approximation (ACA) fast summation method is used for solving the resulting linear system. This method uses partial assembly of the matrices where the effect of the matrix vector multiplication is computed. The default iterative solver used is GMRES. With the built-in multiphysics couplings, it is easy and seamless to set up problems that combine FEM- and BEM-based physics. When solving these coupled models, the default approach is to use hybridization with the ACA for BEM and an appropriate preconditioner for the FEM part of the problem (direct or multigrid). More details about the BEM formulation can be found in The PDE, Boundary Elements Interface documentation in the COMSOL Multiphysics Reference Manual.

	For an tutorial that uses a BEM-FEM coupling to model an acoustic problem see the Bessel Panel. Application Library path Acoustics_Module/Tutorials,_Pressure_Acoustics/bessel_panel

When this physics interface is added, these default nodes are also added to the — , , and .

Then, from the toolbar, add other nodes that implement boundary conditions. You can also right-click 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.

	Physics Nodes — Equation Section in the COMSOL Multiphysics Reference Manual

The is the default physics interface name.

The 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 field. The first character must be a letter.

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

Expand the section to see the equations solved for with the specified. The default selection is is set to . The available studies are selected under . If is selected as the the of the study can be either taken or .

Select the 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 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 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 or an 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.

An important aspect is how the results are visualized, specifically when setting the in the grid data set. Say the condition represents a symmetry plane then visualize the solution in the entire domain. On the other hand if the condition represents an infinite sound hard boundary, like the ground or an infinite baffle, then only visualize the solution above that plane by setting the in the or data set in .

Choose an option from the 0, 0, and 0 (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 button (

) and select .

These settings are used for matrix assembly and postprocessing. They allow characterization of interactions occurring in the boundary element method as near-field or far-field interactions. While the near-field interactions are represented explicitly, the far-field interactions can be represented in an approximate way. This approach results in considerable memory and performance improvements when used in combination with iterative solvers using matrix-free format or during postprocessing. The near-field part of the stiffness matrix is used as input by the Direct and Sparse Approximate Inverse preconditioners.

The 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 can be either or . 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.

	More details are found under Far-Field Approximation Settings in The PDE, Boundary Elements Interface documentation in the COMSOL Multiphysics Reference Manual.

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 (the default) or an . In almost all cases the option should be selected.

The quadrature settings are by default set to . This means that the quadrature integration order values will follow the element order selection in the section. Higher element orders automatically generate higher values for the quadrature integration orders.

	More details are found under Quadrature in The PDE, Boundary Elements Interface documentation in the COMSOL Multiphysics Reference Manual.

To display this section, click the button (

) and select in the dialog box.

This setting controls the interpolation that is done close to boundaries (typically in the last mesh element) between variables on boundaries (these are the variables solved for in BEM based interfaces) and their value evaluated with the kernel in the domains. Numerically the kernel evaluation is prone to errors near the boundaries while the DOFs solved for are exact on the boundaries. Using an interpolation in the inner mesh element removes singularities in the results when postprocessing values in the domain near a boundary.

Enter an expression for the . 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.

	See also Postprocessing BEM Results in the Modeling with the Pressure Acoustics Branch (BEM-Based Interface) section.

From the 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 (the default).

The settings under 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 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.