The Nonlinear Pressure Acoustics, Time Explicit Interface
The Nonlinear Pressure Acoustics, Time Explicit (nate) interface (), found under the Acoustics>Ultrasound subbranch () when adding a physics interface, is used to model the propagation of nonlinear finite amplitude acoustic waves in computational domains that contain many wavelengths. It is suited for time-dependent simulations with arbitrary time-dependent sources and fields. Absorbing layers are used to set up effective nonreflecting-like boundary conditions. Application areas include biomedical applications, for instance, ultrasonic imaging and high-intensity focused ultrasound (HIFU). The applications are not restricted to ultrasound. The interface exists in 2D, 2D axisymmetric, and 3D.
The interface is based on the discontinuous Galerkin (dG or dG-FEM) method and uses a time explicit solver. The method is very memory efficient and can solve problems with many million degrees of freedom (DOFs). The method is also well suited for distributed computing on clusters.
In the linear case it can be advantageous to use The Pressure Acoustics, Time Explicit Interface instead, for example, when modeling scattering phenomena using a scattered field formulation and a background acoustic field. This split is not possible for nonlinear problems. In the presence of a stationary background flow and linear propagation the The Convected Wave Equation, Time Explicit Interface should be used, for example, when modeling ultrasonic flow meters.
For modeling acoustic-structure interaction (ASI) or vibroacoustic problems the interface is fully multiphysics enabled and can be coupled to the The Elastic Waves, Time Explicit Interface, using either the Acoustic–Structure Boundary, Time Explicit or the Pair Acoustic–Structure Boundary, Time Explicit multiphysics couplings.
The interface solves the second-order nonlinear governing equations for the acoustic pressure p and the acoustic velocity perturbations u. The interface is suited for modeling progressive wave propagation phenomena when the cumulative nonlinear effects surpass the local nonlinear effects. Thus the model is consistent with the second-order Westervelt equation for the acoustic pressure (see also the Nonlinear Acoustics (Westervelt) Contributions feature available with The Pressure Acoustics, Transient Interface). General bulk dissipation (volumetric damping) can be added to model real fluids.
Several features are available to help solve the nonlinear and highly nonlinear problems including the use of a numerical Limiter (to capture shocks) but also the use of Adaptive Mesh Refinement.
Settings
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 nate.
Filter Parameters for Absorbing Layers
To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box. In the Filter Parameters for Absorbing Layers section you can change and control the values set for the filter used in the Absorbing Layers. The values of the filter parameters defined here are used in all absorbing layers added to the model and they override the value of filter parameters enabled in the material model (Nonlinear Pressure Acoustics, Time Explicit Model). The default values of the filter parameters α, ηc, and s are set to 0.1, 0.01, and 2, respectively. Inside the absorbing layer it is important to use a filter that is not too aggressive since this will result in spurious reflections.
For general information about the filter see the Filter Parameters section under Wave Form PDE in the COMSOL Multiphysics Reference Manual.
Numerical Flux
To display this section, click the Show More Options button () and select Stabilization in the Show More Options dialog box. Only one option exists for the Numerical flux formulation used in the numerical scheme when solving the dG problem. For Lax–Friedrichs (the only option) enter the Lax–Friedrichs flux parameter τLF (default is 0.2). The value of the parameter τLF should be between 0 and 0.5. A parameter value of 0 represents a central flux, which is the least dissipative but also the least stable numerical flux. A parameter value of 0.5 gives a maximally dissipative global Lax–Friedrichs flux (see The Lax–Friedrichs Flux).
Limiter
To display this section, click the Show More Options button () and select Stabilization in the Show More Options dialog box. The limiter is used to control and stabilize highly nonlinear problems with shock formation. The limiter can only be used with linear discretization. For details see the Solving Highly Nonlinear Problems section in Modeling with the Nonlinear Pressure Acoustics, Time Explicit Interface.
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For an example of a highly nonlinear problem that uses the WENO limiter see the Nonlinear Propagation of a Cylindrical Wave — Verification Model tutorial model. Application Library path Acoustics_Module/Nonlinear_Acoustics/ nonlinear_cylindrical_wave
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For an example of a nonlinear problem with pulse propagation that uses mesh adaptation see the High-Intensity Focused Ultrasound (HIFU) Propagation Through a Tissue Phantom tutorial model. Application Library path Acoustics_Module/ Nonlinear_Acoustics/hifu_tissue_sample
Transient Mesh Settings
Enter a value for the Maximum frequency to resolve fmax (the default is 1000[Hz]). This value is used to set up the physics-controlled mesh for the Nonlinear Pressure Acoustics, Time Explicit interface. Remember that this value should be set in order to resolve harmonics generated due to nonlinear effects.
Discretization
In this section you can select the Element order for the acoustic pressure and acoustic velocity (the same is used for both). Per default the order is set to Quartic (4th order). Using quartic elements together with a mesh size equal to approximately one and a half of the wavelength, leads to the best performance when using the dG method. For highly nonlinear problems it can be necessary to use a Limiter in which case it is necessary to change the Element order to Linear.
When solving Nonlinear Pressure Acoustics, Time Explicit, it is important to have consistent settings for the Geometry Shape Function and the Discretization of the physics. The Automatic setting for the Geometry shape function (in the Curved Mesh Elements section on the Components node’s settings) may results in a linear geometry representation, if other physics are present in the model. This can lead to numerical errors when solving as the default is to use fourth-order spatial discretization of the dependent variables. To remedy this change the Geometry shape function to Quadratic Lagrange.
If the settings in a model are inconsistent, a warning is given in the solver with the text:
The geometry shape function order is not consistent with the discretization of the dependent variables.
Consider increasing the geometry shape function order to Quadratic Lagrange in the Curved Mesh Element section in the Component node's settings.
Dependent Variables
The dependent variables are the Acoustic pressure, and the Acoustic velocity. The names can be changed, but the names of fields and dependent variables must be unique within a model. The name for the Acoustic velocity, components can also be selected individually.
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For information specific to modeling nonlinear problems using the time explicit interface, see Modeling with the Nonlinear Pressure Acoustics, Time Explicit Interface.
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For information about modeling strategies, meshing, the absorbing layer and more, see the relevant information in the Modeling with the Convected Wave Equation Interface.