99560 - FINITE DIFFERENCE METHODS FOR DIFFERENTIAL EQUATIONS M

Learning outcomes

The students learn and are able to implement the finite difference method for the simulation of linear and nonlinear acoustic systems. These include lumped systems such as the harmonic oscillator, the Duffing oscillator, the van der Pol oscillator, oscillators with nonlinear damping, as well as distributed vibrating systems such as cables, strings, bars, membranes and plates, in both linear and nonlinear regimes. Furthermore, the students learn the mathematical methods used in the analysis of such systems, such as Fourier and Laplace methods for linear systems, and pertubation methods for nonlinear systems.

Course contents

The course introduces analytical and numerical techniques aimed at studying and simulating acoustics and vibroacoustics problems. It is therefore a course in computational acoustics. 

 The course is aimed at understanding the physical processes responsible for the propagation of elastic waves, their mathematical modeling, and their simulation. Through practical examples and numerous computer labs, the students will have the opportunity to appreciate the interactions between these three domains in relatively simple application cases, using the tools learned in class to build the numerical schemes from scratch. On the other hand, mainstream numerical simulation software will not be used which, while allowing the solution of complex problems, does not allow learning the numerical scheme design techniques.

Course topics are listed below.

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

2. The Laplacian and biharmonic operators. Definitions, examples of use in typical problems in acoustics and vibroacoustics. Examples of boundary-value problems. Eigenvalues and eigenfunctions. Modal representation.

3. Discretization of the Laplacian and biharmonic operators under various boundary conditions. Examples in one and two dimensions. Cartesian grids; use of polar grids in two dimensions for problems with circular geometry.

4. Lagrange interpolation. Use of Cartesian grids for problems defined on non-rectangular domains.

5. Numerical computation of eigenvalues and eigenfunctions of the Laplacian and biharmonic for problems with and without analytic solution.

6. Wave propagation. Models in one and two dimensions. Equation for cables, rods, membranes, plates. 2D acoustics, visualization of the acoustic field.

7. Time differences operators. Harmonic Oscillator. Explicit and implicit schemes. Truncation error and frequency warping. Exact integrator. Application to wave propagation problems. Stability.

8. Examples of nonlinear problems in acoustics. Duffing oscillator, nonlinear wave equation.

The syllabus may change in due course as required.

A video presentation of the course contents is available via the link provided the at the bottom of the page.

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 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