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** FDTD** stands for

The FDTD method is one of the simulation techniques for the investigation of the wave propagation in a given field, which can be 1-D, 2-D, or 3-D. For the purpose of an acoustic wave simulation, some types of FDTD method have been proposed: ** Acoustic FDTD**,

In the

The followings are some examples and demonstrations of the FDTD method.

[Download .mp4 file (3,328 kB)]

This movie is an example of the 3-D Elastic FDTD simulation in a cancellous bone model. The model was prepared using X-ray CT images of an actual bone sample. The connecting network is solid (trabecular bone) and the remaining part around the bone is liquid (water). A pulse ultrasound at 1 MHz is transmitted from the bottom part of the field.

(The original data come from: Yoshiki NAGATANI *et al*., Ultrasonics 48 (2008) pp.607-612.)

The codes below are truly physical simulation programs of sound wave propagation
in a two-dimensional field. These programs solve the equations of *3-D and 2-D Acoustic FDTD* simulation.

- JavaScript version (3-D Acoustic FDTD) -
*A Helium Balloon in the Air* **A realtime 3-D rendering**of the 3-D Acoustic FDTD simulation written in JavaScript. The 3-D rendering is based on the WebGL library*three.js*. This script can be run on your HTML5 and WebGL-compatible web browsers (Blink, WebKit, Gecko, IE, and Spartan including recent smartphones).

[published on: Jan. 1st, 2015 / updated on: Apr. 5th, 2015]

[**⇒ Go to the page**]- JavaScript version (2-D Acoustic FDTD) -
*Sound Propagation in Lena* -
Pure JavaScript simulation of the 2-D Acoustic FDTD method. This can be run on your HTML5-compatible web browsers (Blink, WebKit, Gecko, IE, and Spartan including smartphones).

[published on: Nov. 28th, 2014 / updated on: Apr. 5th, 2015]

[**⇒ Go to the page**] - Processing version (2-D Acoustic FDTD)

A sketch (program) of 2-D Acoustic FDTD for Processing 2 & 3. This sketch has been tested with Processing 2.2.1 and 3.1.2 (64bit) both on Windows and OSX.**You can easily change the model by preparing your own image files.**You can also change the acoustic properties (wave speed etc.) by changing the parameters in the source code directly. Please see the "Stability Conditions" section below.

Prior to running this sketch, please setup Processing 2 or 3. Then, open and run "acoustic_2d_fdtd_processing2.pde" or "acoustic_2d_fdtd_processing3.pde" file. No additional library is required.

- First vertion for Processing 2.x published on: Nov. 28th, 2014

- Version for Processing 3.0: Aug. 2nd, 2016

- CC-BY-SA 3.0 License is specified: Aug. 2nd, 2016

[⇒**Download Sketch from GitHub**(github.com/nagataniyoshiki/FDTD_processing)]

- Other Languages
- - MATLAB: 2-D Acoustic FDTD simulation with homogeneous media surrounded by total reflecting walls.

(This is one of the simplest but the slowest program of the FDTD method.)

[published on: Jan. 12th, 2015]

- Stability Conditions of the FDTD method
- For the stable and precise calculation of FDTD, the
*Δx*[m] (spatial resolution) and*Δt*[s] (temporal resolution = time step) should satisfy the relationships below:

, (1)

. (2)

The condition (1) is empirically recommended for good precision of the calculation. In contrast, the condition (2), called*Courant–Friedrichs–Lewy (CFL) condition*, is strictly required for the stable simulation.

The wavelength*λ*[m] is calculated by

, (3)

and the wave speed (longitudinal wave)*c*[m/s] in fluid media and solid media are calculated by the following equations, respectively:

. (4)

**If you change the acoustic parameters and/or the frequency of the initial wave in the demo programs**, please choose the values of*Δx*(dx) and*Δt*(dt) carefully to satisfy the conditions (1) and (2). - Acoustic FDTD
- The acoustic FDTD is a method for the calculation of
**the sound in fluid media**(i.e. air or liquid).

In the acoustic FDTD method, particle velocity and sound pressure (scalar value) in the simulation field are aligned alternately, which is called*staggered grids*. Then, pressure and particle velocity are updated alternately, which is called*leapfrog algorithm*. Thanks to this elegant design, the temporal evolution of the sound field in the model can be calculated step by step.

cf. The governing equations of the acoustic FDTD are:

. - Elastic FDTD
- The elastic FDTD is a method for the calculation of
**the sound in solid media, where shear waves can propagate as well as the longitudinal waves**.

In the elastic FDTD method, particle velocity, normal stress, and shear stress (vector values) in the simulation field are aligned alternately. Then, stresses and particle velocity are updated alternately.

cf. The governing equations of the elastic FDTD are:

.

Yoshiki NAGATANI (Ph.D. (Engineering))

- Associate Professor at Kobe City College of Technology, Japan (Department of Electronics)

- Visiting Researcher, Université Paris-Est Créteil, France (UPEC)

@nagataniyoshiki

www.researchgate.net/profile/Yoshiki_Nagatani

www.linkedin.com/in/nagataniyoshiki

[ Please see my website also: http://ultrasonics.jp/nagatani/e/ ]

I deeply appreciate Dr. Ryosuke TACHIBANA (@ro_tachi) (The University of Tokyo) providing an insightful idea of creating visualized simulation projects and testing the programs.