## Section: New Results

### Modeling and design of wind musical instruments

#### Full Waveform inversion for bore reconstruction of woodwind instruments

Participants : Juliette Chabassier, Augustin Ernoult.

Several techniques can be used to reconstruct the internal geometry of a wind instrument from acoustics measurements. One possibility is to simulate the passive linear acoustic response of an instrument and to use an optimization process to fit the simulation to the measurements. This technique can be seen as a first step toward the design of wind instruments, where the targeted acoustics properties come no more longer from measurements but are imposed by the designer. We applied the FWI methodology, along with 1D spectral finite element discretization in space [19], to the woodwind instruments (with tone holes, losses and radiation). The algorithm have been implemented in Pyhton3 and is know operational to reconstruct the bore of real instrument. This functionality will be available in an upcoming version of the toolbox OpenWind.

#### Computation of the entry impedance of a wind instrument with toneholes and radiation

Participants : Guillaume Castera, Juliette Chabassier, Augustin Ernoult, Alexis Thibault, Robin Tournemenne.

Modeling the entry impedance of wind instruments pipes is essential for sound synthesis or instrument qualification. Based on a one-dimensional model of acoustic propagation (“telegraphist's equations”) we find approximate solutions using a high-order Finite Element Method (FEM1D). Contrary to the more standard semi-analytic Transfer Matrix Method (TMM), the FEM1D can take into account arbitrarily complex and variable coefficients [19]. It is therefore well-suited for the realistic cases involving boundary losses, smooth waveguide geometry, and possibly even a temperature gradient along the instrument's bore. We model toneholes as junctions of one-dimensional waveguides, and an acoustic radiation impedance models the radiation of all open tube ends. A global matrix is assembled to connect all these elements together, and solved for each frequency to compute the impedance curve. Source code is available in our Python3 toolbox OpenWind.

#### Time-domain simulation of a reed instrument with toneholes

Participants : Juliette Chabassier, Augustin Ernoult, Alexis Thibault, Robin Tournemenne.

As part of the project aiming at providing practical tools for instrument design, we have been developing a sound synthesis module for reed instruments. We model a reed music instrument, such as the oboe or the bassoon, as a coupled system composed of a nonlinear source mechanism (the reed), and a linear resonating part (the air within the instrument's bore). Acoustic wave propagation inside of the instrument is reduced to a one-dimensional model, on which a variational approximation is performed, yielding high-order finite elements in space. Tone holes on the side of the instrument are taken into account using junctions of one-dimensional waveguides. The acoustic radiation impedance is written as a positive Padé approximation, so that it leads to a stable time domain model even when opening and closing holes during the simulation. For the reed, a one-degree-of-freedom lumped model is used, in which the reed opening follows a second order ODE and acts as a valve, modulating the flow that enters the pipe based on the pressure difference between the player's mouth and the inside of the instrument. Energy-consistent time discretization schemes have been derived for each component, so that it is possible to simulate instruments with an arbitrary geometry with good numerical accuracy and stability. The simulations have been implemented in Python3 and will be made available in an upcoming version of the toolbox OpenWind.

#### Time-domain simulation of 1D acoustic wave propagation with boundary layer losses

Participants : Juliette Chabassier, Augustin Ernoult, Alexis Thibault.

Energy dissipation effects are of critical importance in musical acoustics. Boundary layer losses occurring in acoustic waveguides are usually modeled in the frequency domain, leading to slowly-decreasing kernels in the time domain similar to fractional derivatives. We have developed an energy-consistent time-domain model based on positive fraction approximation of the dissipative operators, leading to the use of $2N+1$ additional variables per degree of freedom. Coefficients for these new variables depend only on $N$ so that they can be tabulated without any prior knowledge of the waveguide geometry. They are found with an optimization procedure. This model can be discretized using 1D finite elements [19], and an energy-consistent time-stepping scheme can be found. The resulting numerical scheme has been implemented numerically, and source code will be made available in an upcoming version of the toolbox OpenWind. An article is being written and will be submitted soon to JASA.

#### Numerical libraries for hybrid meshes in a discontinuous Galerkin context

Participants : Hélène Barucq, Aurélien Citrain, Julien Diaz.

Elasticus team code has been designed for triangles and tetrahedra mesh cell types. The first part of this work was dedicated to add quadrangle libraries and then to extend them to hybrid triangles-quadrangles (so in 2D). This implied to work on polynomials to form functions basis for the (discontinuous) finite element method, to finally be able to construct reference matrices (mass, stiffness, ...).

A complementary work has been done on mesh generation. The goal was to encircle an unstructured triangle mesh, obtained by third-party softwares, with a quadrangle mesh layer. At first, we built scripts to generate structured triangle meshes, quadrangle meshes and hybrid meshes (triangles surrounded by quadrangles). In 2018, we have implemented the coupling between Discontinuous Galerkin methods (using the triangles/tetrahedra) and Spectral Element methods (using quadrangles/hexahedra). We have also implemented the PML in the SEM part, and we are now working on the local time-stepping feature.