Elements of numerical analysis with application to sound synthesis Antoine Chaigne ENSTA UME, France

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Elements of numerical analysis with application
to sound synthesisAntoine ChaigneENSTA UME,
Francechaigne_at_ensta.fr
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Summary

• Introduction
• Examples of continuous models
• Strategies for discrete formulations
• Finite difference methods
• Stability of a numerical scheme
• Numerical dispersion or frequency warping
• Discussion and comparison with other methods

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Introduction

• Physical modeling
• Boundary-value problems
• Time and space discretization
• Predicting the properties of a discrete model

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1st example a 1D ideal string clamped at both
ends

• Wave equation
• Boundary conditions
• Initial conditions

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2nd example a 1D vibrating bar free at both ends

• Bar equation (flexion)
• Boundary conditions
• Initial conditions

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Strategies for discrete formulations

• Direct discretization of the spatial and time
  derivatives
• Finite Difference Methods (FDM)
• Integral ( weak ) formulation of the problem
  variational approach
• Finite Element Methods (FEM)
• Boundary Element Methods (BEM)
• Modal truncation

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Finite Difference Methods (FDM)

• Approximations based on Taylor series
• Examples of 2nd order FDM schemes

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Higher-order approximations
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String equation with the 2nd order FDM scheme
Equivalent to
Notations
We obtain here an EXPLICIT SCHEME
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Stability of the explicit 2nd order FDM scheme
Fourier method
Wave equation
Stability condition
CFL condition
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Discussion on the stability condition
Stability condition for the 2nd order explicit
FDM scheme
Ideal vibrating string
Time sampling frequency
Equivalence with Shannons theorem
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A practical example simulation of a guitar string
H1 the time sampling frequency is given
We want to simulate the note A2
We must select a number of spatial steps
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Homework

• The stability condition for the ideal bar
  equation (Euler-Bernoulli) discretized with a 2nd
  order explicit FDM scheme yields the stability
  condition
• Try do demonstrate it using the Fourier method!
• What do you think of this result compared to the
  stability condition for the ideal string ?

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Short summary of FDM properties

• The recurrence equation and its stability
  properties depend on the order and type of
  approximations used for the derivatives
• The FDM method is also applicable to more complex
  problems
• Ex.1 heterogeneous string
• Ex.2 nonlinearity due to tension
• It is a particularly convenient method for simple
  geometrical shapes (lines, rectangles, circles,
  cubes)
• There are other methods for deriving a stability
  condition
• Ex energetic methods
• However, with increasing complexitiy of the
  model, it is not always possible to find a
  stability condition analytically
• Other strategies tests of convergence, use of
  implicit schemes
• The FDM is not optimal for curved boundaries
  (ex. guitar body) and for irregular mesh

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Numerical dispersion of the 2nd order FDM scheme
Harmonic wave
Continuous wave equation
Recurrence equation
Numerical wave velocity
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Dispersion curve for the 2nd order FDM scheme
(wave equation)
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Discussion on dispersion properties

• For this FDM scheme, the numerical wave velocity,
  and thus the eigenfrequencies of the string, are
  underestimated, except for r1.
• For more complex models, and for stability
  reasons, it could be necessary to have rlt1. In
  this case, time oversampling will limit the
  dispersion.

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Conclusions

• Numerical analysis techniques are widely used in
  various domains geophysics, mechanics,
  electromagnetic waves and others.
• Complex equations from the physics, including
  models of musical instruments, are still
  challenging problems for applied mathematicians.
• For sound synthesis, it can be used as a good
  starting point to test complex models and hear
  how it sounds.
• Once a model has been validated, there are many
  exciting challenges for people who want to
  develop hybrid techniques taking advantages of
  the properties of other fields (ex waveguides).