Numerical Framework for the Automated Calibration of Physical Models for Musical Instruments

[Rodolphe-Vivant]

Numerical Framework for the Automated Calibration of Physical Models for Musical Instruments - M2 Project

Student : Vivant Rodolphe
Supervisors : Dr Emmanuel Franck, Dr Juliette Chabassier, Dr Laurent Navoret, Dr Michel Victor-Dansac
Hosting institution : Institut de Recherche Mathématique Avancée (I.R.M.A)
Time Period : October 2025 - January 2026

Context

In order to enable the automated calibration of physical models, it is essential to rely on a robust and efficient numerical solver for the physical equations describing sound production in reed instruments (such as clarinets, saxophones, bassoons, harmonicas, and bagpipes).

This project is carried out as a joint collaboration between IRMA and Modartt, a French company specialized in the development of artistic and technological software for musical acoustics.

The main objective of this work is to implement, within a JAX framework, a discontinuous Galerkin discretization of the mixed formulation of the wave equations for reed instruments with variable cross-sectional geometry. This numerical model is coupled with a radiation-impedance-type boundary condition at the instrument bell.

Physical Model

$$ \begin{aligned} \frac{S(x)}{c S^\ast}\partial_t p(x,t) + \partial_x v(x,t) &= 0, \\ \frac{S^\ast}{c S(x)}\partial_t v(x,t) + \partial_x p(x,t) &= 0, \\ Z^\ast_T\partial_t \phi + \sqrt{\alpha},p(L,t) &= 0, \\ -\frac{\beta}{Z^\ast_T}p(L,t) + v(L,t) + \sqrt{\alpha},\phi &= 0. \end{aligned} $$

The fist two equations are mixed form the wave equation in a variable section tube (S(x) : cross section area, c : sound's speed, $S^\ast$ : reference cross section)
The other two equation desribed the influence of exterior environment throug a radiation impedance condition ($\phi(t)$ : boundary variable, $Z^\ast_T$ : reference impedance, $\alpha$, $\beta$ : parameters of the acoustic load.

DG Method for 1D Wave Equation with Variable Cross-Section

Quasi-Conservative Form

We consider the 1D wave equation in mixed form:

with

Initial conditions:

Mesh and DG Space

Domain partition:

DG approximation:

Local DG Formulation

For test functions , the weak DG formulation reads:

with physical flux:

Numerical Flux (Rusanov / Explicit)

At interface :

DG Formulation for Each Element (K_j)

For test functions :

All dependence on (S(x)) is in the mass matrix (C(x)), ensuring a uniform CFL condition.

Boundary Condition (Radiation Impedance)

Right boundary ODE:

With characteristic variables:

The incoming wave is:

This $(w^-)$ is used in the DG numerical flux to impose the boundary condition consistently with the time integration scheme.

Results

Initial Condition

The initial pressure profile is defined as a smooth compactly supported bump function centered at (x = L/2). Introducing the rescaled variable:

the initial pressure reads:

p0(x) = (phi0/4) * exp(1 - 1/(1 - xi(x)^2)) for -1 < xi(x) < 1,
0 elsewhere

• p0(x) → initial pressure $(p_0(x)$
• phi0 → amplitude $(\phi_0)$
• xi(x) → rescaled variable $(\xi(x))$
• if xi(x)<1 and xi(x)>-1 else 0 → replaces cases and |xi(x)|<1

The initial acoustic volume velocity is set to zero:

Uniform cross-section

To assess the accuracy of the Discontinuous Galerkin (DG) implementation, we consider a uniform duct cross-section $S(x)=S^\ast=1$, for which an analytical solution is available.

Solution Plot

Figure: DG numerical solutions constant cross-section.

The numerical solution obtained using the Discontinuous Galerkin Method accurately captures the wave propagation over time described in. The method effectively resolves the wave fronts and maintains stability throughout the simulation.

Convergence Study (Inside the domain)

• Spatial convergence: order $(d+1)$ with polynomial degree $d$ (in our case P1 base so the expected spatial order of convergence is 2)
• Temporal convergence:
  • Explicit Euler: first order
  • Runge–Kutta 2: second order

Figure: Convergence of the DG method with Explicit Euler scheme.

Figure: Convergence of the DG method with RK2 scheme.

Convergence Study (At the $x=L$ boundary)

Figure: Convergence of the DG method with Explicit Euler scheme.

Figure: Convergence of the DG method with RK2 scheme.

The model both inside the domain and at $x=L$ boundary respect the expected total convergence order.

Variable Cross Section

Tested profile

Figure: Exponential Profile.

Figure: Conical Profile.

Solution Plot (Exponential Profile)

Figure: DG numerical solutions exponential cross-section.

Solution Plot (Conical Profile)

Figure: DG numerical solutions conical cross-section.

The analysis of the numerical results for a non-uniform cross-section is more qualitative due to the absence of an analytical solution for direct comparison. We can oberse that the wave propagation is influenced by the varying cross-sectional area, as it has stop to be symmetric. For example, the pressure is amplified in regions where the cross-sectional area decreases.
Note: Since we assume the wave speed to be the maximum eigenvalue of $C^{−1}(x)A$, the Rusanov flux does not directly account for the variable cross-section. As the system is not strictly conservative, some numerical energy dissipation is observed (e.g., the pressure becomes negative , and the acoustic velocity remains positive where it should vanish).

Performance Study

Figure: Explicit Euler.

Figure: RK2.

For larger N, the execution time scales $\sim N^2$, as expected for this king of explicit methods.

Conclusion

Future work will focus on extending the solver to handle a fully conservative system of
equations, as well as implementing the complete right-hand-side boundary conditions describing the effect of the instrument player which are essential for accurately modeling the reed–instrument interaction.
Additionally, further investigations will be required to develop an effective automated calibration strategy, including data preprocessing, parameter selection, and the choice of appropriate loss functions for optimization.

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