B4936 - Computational Acoustics M

Learning outcomes

Upon completing the course, the student possesses the analytical and numerical techniques for simulating acoustics problems. In particular, the student is familiar with and capable of applying: - The finite difference technique for spatial discretisation of the Laplacian and biharmonic operators in one and two dimensions. - The finite difference technique for temporal derivative discretisation. - Analytical tools for constructing numerical schemes for simulating wave equations in cables, bars, plates, and membranes.

Course contents

The course offers students both analytical and numerical techniques aimed at the study and simulation of problems in acoustics and vibroacoustics. It is, therefore, a course in computational acoustics.

The course focuses on understanding the physical processes responsible for the propagation of elastic waves, their mathematical modeling, and their simulation. Through practical examples and numerous computer labs, students will have the opportunity to explore the interactions among these three domains in relatively simple applications, using the tools learned in class to build numerical schemes from scratch. Commercial simulation software ("black box" tools) will not be used, as they may allow solving complex problems but do not help in learning how to design numerical schemes.

Industrial applications covered include:

1. Analysis and simulation of acoustic black holes (attenuation of incident waves via thickness variation of vibrating panels)

2. Analysis and simulation of acoustic tubes (Kundt’s tubes) for measuring the acoustic impedance of materials

3. Analysis and simulation of a plasma-based active acoustic absorber

Course topics include:

1. Introduction to finite differences. Construction of finite difference operators via Taylor series. Truncation error and order of accuracy.

2. Laplacian and biharmonic operators. Definitions, examples of typical acoustic and vibroacoustic problems. Boundary value problems. Eigenvalues and eigenfunctions. Modal representation.

3. Discretization of the Laplacian and biharmonic operators with various boundary conditions. Examples in one and two dimensions. Cartesian grids, with a brief introduction to polar grids in 2D for circular geometries.

4. Lagrange interpolation. Use of Cartesian grids for problems with non-rectangular geometry.

5. Numerical computation of eigenvalues and eigenfunctions of the Laplacian and biharmonic operators for problems with and without analytical solutions.

6. Wave propagation. One- and two-dimensional models. Equations for strings, rods, membranes, and plates. 2D acoustics, visualization of the acoustic field.

7. Finite differences in time. Harmonic oscillator. Explicit and implicit schemes. Truncation error and frequency distortion. Exact schemes. Application to wave propagation problems. Stability.

8. Introduction to nonlinear acoustic problems. Duffing oscillator, nonlinear wave equation.

The program may be adjusted during the course to better suit the needs of the class.

A video presentation of the course is available at the bottom of the page, under "Links to additional information."

Readings/Bibliography

Class hand-outs by the lecturer. Other useful texts:

- On finite differences

• R.J. LeVeque, Finite Difference Methods for Ordinary and Partia lDifferential Equations. Steady State and Time Dependent Problems. SIAM, Philadelphia, USA, 2007.
• J. Strikwerda, Finite Difference Schemes and Partial Differential Equations. SIAM, Philadelphia, USA, 2004.
• S. Bilbao, Numerical Sound Synthesis. Wiley, Chichester, UK, 2009.

- On theoretical acoustics

• A. D. Pierce, Acoustics: An Introduction to Its Physical Principles and Applications (Third Edition). Springer Nature, Cham, Switzerland 2019
• P.M. Morse and K.U. Ingard, Theoretical Acoustics. Princeton University Press, Princeton, USA, 1968
• L.E. Kinsler, A.R. Frey, A.B. Coppers, J.V. Sanders, Fundamentals of Acoustics (Fourth Edition). Wiley, Hoboken, USA, 2000.

- On dynamical systems

• L. Meirovitch, Fundamentals of Vibrations. Waveland, Long Grove, USA, 2001.
• A. H. Nayfeh, Professor D. T. Mook, Nonlinear Oscillations. Wiley, Weinheim, Germany, 2004.

Teaching methods

Class (3hrs/week)

Matlab tutorials (2hrs/week). During the tutorials, the students will implement the numerical methods seen during class.

Assessment methods

The exam is project-based. Each student will work independently and submit a Matlab project, to be discussed with the lecturer during an oral exam. To finalise the assessment of the course topics, the lecturer may ask further questions during the exam.

Teaching tools

Class hand-outs. Matlab demos. Powerpoint presentations. Accelerometric measurement demo.

Since the course involves computer lab sessions, the students must take modules 1 and 2 on health and safety in the workplace, available at [https://elearning-sicurezza.unibo.it/]

Links to further information

https://youtu.be/r-vtQfrefZ8

Office hours

See the website of Michele Ducceschi