This allows us to use a mesh with an element size that is about half of the wavelength for the highest-frequency component that needs to be resolved. In its default formulation, the Convected Wave Equation, Time Explicit interface uses quartic (fourth-order) shape functions, a sweet spot for speed and efficiency in wave problems solved with the DG method. Still, FEM methods are more flexible and can easily be coupled in multiphysics applications. The latter is due to the fact that the smallest wavelength in the system must be resolved with a certain number of mesh elements. The challenge is that the RAM consumption for implicit methods grows rapidly when increasing the model size or frequency. The FEM-based physics interfaces, such as the Pressure Acoustics, Transient or Linearized Navier-Stokes, Transient interfaces, require the use of a time-implicit method. On the other hand, due to the discontinuous elements, the DG method is more accurate for the same polynomial order and mesh. When using an implicit method, it’s tempting to use a time step that is too large, which introduces errors from the time method. When doing a comparison, it’s important to look at the error level. Implicit methods are sometimes thought to be faster on small to medium problems that fit in the RAM. As for the DG method, computing the local flux vector and divergence of the flux is a time-consuming process that can be run efficiently with BLAS level 3 operations. In DG-FEM, only a few mass matrices are inverted for a reference mesh element (making them small in size) before evolving in time. In contrast, time-implicit methods require inverting this matrix, which consumes a lot of memory when solving large problems. With this method, it isn’t necessary to invert a full system matrix when stepping forward in time. The Convected Wave Equation, Time Explicit interface is, as mentioned, based on the DG method, a time-explicit formulation that is memory efficient. ![]() Increasing Efficiency with a Time-Explicit Method Note that there are no physical loss mechanisms in the interface and the above equations are set on a conservative form. When the background velocity is set to zero, the equations are effectively reduced to modeling the classical wave equation. The background flow can be a stationary flow with a velocity gradient that ranges from small to moderate. The speed of sound is c 0 and the steady-state mean background flow variables are defined with a subscript 0 through the density ρ 0, pressure p 0, and velocity field u 0. The acoustic pressure p, as well as the acoustic velocity perturbation u, are the dependent variables. The mass and momentum conservation equations read: These equations assume an adiabatic equation of state (see Ref. The governing equations solved by the Convected Wave Equation, Time Explicit interface are the linearized Euler equations. It is also useful for nonultrasound applications like room acoustics and auto cabins that experience the transient propagation of audio pulses. This interface is useful for linear ultrasound applications, such as ultrasound flow meters and ultrasound sensors in which the time-of-flight parameter is considered. ![]() ![]() This makes the Convected Wave Equation, Time Explicit interface suited for modeling the propagation of linear acoustic signals that span large distances in relation to the wavelength. With the absorbing layer technology to truncate the computational domain and the memory-efficient formulation of DG-FEM, you can set up and solve very large problems in the time domain - measured in terms of the number of wavelengths that can be resolved. A turbulent flow model is also present to calculate the background flow through the flow meter. It also includes absorbing layers (sponge layers) that can act as effective nonreflecting boundary conditions.Ī model of an ultrasound flow meter that uses the Convected Wave Equation, Time Explicit interface. Using the Convected Wave Equation, Time Explicit interface enables you to efficiently solve large transient linear acoustics problems that contain many wavelengths in a stationary background flow. ![]() The technology behind this interface comes from the discontinuous Galerkin (DG) method, also called DG-FEM, which relies on a solver that is time explicit and very memory lean. The new Convected Wave Equation, Time Explicit interface builds on the functionality of the Acoustics Module. Today, we will explore how to use this interface with the example of an ultrasound flow meter.Ībout the Convected Wave Equation, Time Explicit Physics Interface It can include a stationary background flow and is suited for modeling linear ultrasound applications. To make solving these types of problems easier, we’ve added a new physics interface based on this method to the Acoustics Module: the Convected Wave Equation, Time Explicit interface. Modeling acoustically large problems requires a memory-efficient approach like the discontinuous Galerkin method.
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