Alberto Torin

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Percussion Instrument Modelling In 3D: Sound Synthesis Through Time Domain Numerical Simulation

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Ap. B - Finite precision effects in FDTD schemes

This appendix offers novel insights on some round off effects that appear in finite difference schemes. First, it is clarified what is meant by conservation of energy "to machine precision", and how energy variations should be calculated in order to highlight the correct quantisation. This is an original contribution of this work. Then, the implementation of a numerical scheme for the simple harmonic oscillator is analysed, both in the lossless and lossy case. A second order implementation is compared with the corresponding first order system. It is found that the first order system behaves better than the second order implementation from an energy conservation point of view, and that some numerical errors can be avoided with a different implementation. The case of a free particle is analysed after. The second order implementation and the first order system are compared numerically, and an explanation for the quantised jumps that can be seen in the second order energy plot is proposed. Finally, round off errors are analysed in the case of the 3D wave equation. It is demonstrated how a naive implementation can lead to a linear drift in the energy calculation, and how it is possible to cure this artefact.


Table of contents

Abstract

Chapter 1 - Introduction

Chapter 2 - Linear plate vibration

Chapter 3 - Nonlinear plate vibration

Chapter 4 - Collisions

Chapter 5 - Coupling with the acoustic field in 3D

Chapter 6 - Virtual instruments

Chapter 7 - Summary and conclusions

Appendix A - Experimental investigation of nonlinearities in drum membranes

Appendix B - Finite precision effects in FDTD schemes

Appendix C - Iterative methods for linear systems