Numerical Simulation of Wave Propagation

Sara Carcangiu; Augusto Montisci; Renato Forcinetti

doi:10.1007/978-3-319-10566-6_2

Ultrasonic Nondestructive Evaluation Systems pp 17-45 | Cite as

Numerical Simulation of Wave Propagation

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Sara CarcangiuEmail author
Augusto Montisci
Renato Forcinetti

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First Online: 05 December 2014

Abstract

Wave equations are hyperbolic partial differential equations (PDEs) which describe the propagation of various types of waves, such as acoustic, elastic, and electromagnetic waves. In order to solve PDEs, the finite element method (FEM) can be used. After a brief introduction to the mathematical method used by FEM to evaluate the solution in nodes, where the polynomial curve that interpolates the differential equation has to be solved, we will describe the methodology used to solve the wave propagation problem described by the Helmholtz’s equation. The wave propagation problem is analyzed by following specific steps: construction of the geometry to study, application of the boundary conditions, and meshing of the domain to be solved. The same procedure is used to simulate the behavior of piezoelectric transducers and the problem of wave propagation in medium with defects. Finally, the procedure followed for the simulation of acoustic problems using a specific software, i.e., COMSOL Multiphysics, is illustrated.

Keywords

Finite Element Method Piezoelectric Material Acoustic Pressure Flight Time Finite Element Method Model

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Appendix: Developing the Model with COMSOL

In this book, all the simulations are done using the FEM software package called “COMSOL Multiphysics” [7]. This software is used for various physics and engineering applications and it is focused especially in coupling different physics together (e.g., acoustics and solid mechanics). COMSOL Multiphysics offers an extensive interface to MATLAB and its toolboxes for a large variety of programming and preprocessing and postprocessing possibilities. In comparison to conventional physics-based user interfaces, COMSOL Multiphysics is highly flexible and it allows to program in your own PDEs if they are not already implemented.

In order to study a wave propagation problem using COMSOL Multiphysics, the analysis of the model can be divided in different fundamental steps that will be illustrated in the following focusing on acoustics as application [6]. Every step is managed with specific tools provided by the software. Of course, the steps are similar for other physics.

1.

Start COMSOL Multiphysics by clicking the icon.
2.
When COMSOL starts, the Model Wizard will open automatically. It will require you to select:
- the coordinate system for the model, i.e., choose how many dimensions to work in (Fig. 2.13);
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  Fig. 2.13
  Selection of space dimension
- the relevant physics to the problem taking into account that multiple physics can be added to a single model if you want coupling (Fig. 2.14);
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  Fig. 2.14
  The acoustics module physics interfaces
- the type of study you wish to perform: time domain, frequency domain, stationary, eigenvalue, … (Fig. 2.15).
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  Fig. 2.15
  Selection of study type
3.

Construction of the geometry. The geometry could be drawn directly inside the finite element software, using the internal CAD tool: this is the easiest and fastest way, if and only if the geometry is not very complicated.

Default units are mks units (International System (SI) units). It is possible to change units by selecting the root object in the model tree. For axisymmetric geometries, the axis of symmetry is drawn as a line in the graphics window (see Fig. 2.16).
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Fig. 2.16
Graphical user interface

The other way is to design the device using specialized CAD software and then import the geometry in COMSOL .
4.

Assignment of the materials to the subdomains of the geometry. The type of material can be chosen among those already present in the database, or one can create his/her own materials with the option “+Material,” specifying the proprieties that have to be used during the analysis (see Fig. 2.17 ).
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Fig. 2.17
Material properties
5.
The initial and boundary conditions. The default initial conditions are \(p=0 \mathrm{[Pa]}\), for the sound pressure, and \(\mathrm{d} p/\mathrm{d} t=0 \mathrm{[Pa/s]}\), for the pressure time derivative. As shown in Fig. 2.18 for acoustic pressure mode, it is possible to choose among many boundary conditions .
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Fig. 2.18
Boundary conditions
- Sound hard boundary (Neumann condition)
- Sound soft boundary (Dirichlet condition )
- Pressure (Dirichlet condition ) \(p=p_0\)
- Normal acceleration (Neumann condition )
- Impedance condition
- Radiation condition
- Sources, etc.
6.

Discretizations of the domain through the use of the finite elements. The software has an efficient automatic algorithm for mesh generation with triangular elements (Lagrange second order) or rectangular ones.

In Fig. 2.19, the mesh options are shown. Mesh sizes and distribution can be parameterized in order to modify the mesh varying the frequency of analysis.
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Fig. 2.19
Mesh settings
7.

Solving and postprocessing of results. Finally, in order to solve the problem, one needs to simply right click the “study” tab and select “compute” (see Fig. 2.20).
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Fig. 2.20
Solving options

There are lots of results that can be displayed through the proper postprocessing tools in order to find the information desired. Using the “results branch” (see Fig. 2.21), it is possible to define and use data sets, derived values, and tables.
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Fig. 2.21
Postprocess the results

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Authors and Affiliations

Sara Carcangiu
- 1
Email author
Augusto Montisci
- 1
Renato Forcinetti
- 1

1.Dipartimento di Ingegneria Elettrica ed (DIEE)University of CagliariCagliariItaly

Numerical Simulation of Wave Propagation

Abstract

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References

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