Pierre CARRÉ defended his thesis on the 20th of March, 2022, at 10.30AM at IRCAM. His thesis has been made in the S3AM team (STMS Laboratory: IRCAM/CNRS/SORBONNE UNIVERSITE/MINISTERE DE LA CULTURE) and the title is: Geometrically inspired numerical methods for physical model sound synthesis; application to a geometrically accurate string model.
You can follow this thesis through this link: https://medias.ircam.fr/xffe7b6_pierre-carre
Aziz HAMDOUNI, professor, Université de la Rochelle, UMR 7356
Jacky CRESSSON, professor, Université de Pau et des Pays de l'Adour, UMR 5142
Juliette CHABASSIER, researcher, INRIA Bordeaux S-O
Thomas HÉLIE, researcher, CNRS - STMS UMR 9912
Brigitte D'ANDREA-NOVEL, professor, Sorbonne Université - STMS UMR 9912
Joël BENSOAM, researcher, Universidada Nova de Lisboa - NET-MD
Physical model sound synthesis consists in numerically solving the differential equations describing the systems that produce vibrations. This approach allows the simulation of rich and complex effects, capturing the nature of the phenomena involved in a realistic way. However, the numerical treatment inevitably gives rise to stability problems, which may render the integration methods inoperative. The approach chosen in this thesis is to use the tools offered by the so-called geometric numerical methods, in order to compute the solutions of the modeled systems in a way that guarantees the verification of various fundamental physical principles.
Discrete variational mechanics, which allows the treatment of systems formulated with a Lagrangian, ensures by construction guarantees on energy, symplecticity, and conservation of moments resulting from the existence of symmetries. Besides, the choice of a particular coordinate system sometimes leads to non-linearities in the equations, which can cause difficulties for numerical methods; an intrinsic formulation of these equations using Lie groups allows to solve this problem. We have implemented a general framework for all these numerical approaches in an open source C++ library. The covariant formulation of mechanics extends the Lagrangian approach to the treatment of partial differential equations. We then propose to take into account the forces and boundary conditions under a unified variational principle extending the Lagrange-d'Alembert principle. It then becomes possible to express in the numerical domain the laws of evolution of the moments as a function of the forces and the boundary conditions.
The different tools studied in this thesis are finally combined and applied to the numerical resolution of a string model. Our three-dimensional model is geometrically exact, non-linear, takes into account the six degrees of freedom (translation and rotation), and remains valid for large amplitude deformations. A special attention is given to the establishment of the boundary conditions following a covariant treatment. The eigenfrequency spectra of the system are obtained under the assumption of small deformations, and compared to several well-known string models. The numerical resolution of the string dynamics is performed by applying a covariant numerical method with Lie group. A triple validation of the results is proposed: conservation of the physical invariants for an isolated system, comparative spectral analysis for small amplitude vibrations, and simulation of the string motion for a large magnitude excitation force.