Finite element computation of diphthong sounds using tuned two-dimensional vocal tracts Marc Arnela and Oriol Guasch GTM Grup de recerca en Tecnologies M` edia, La Salle, Universitat Ramon Llull, Barcelona, Catalonia, Spain Ramon Codina and Hector Espinoza Universitat Polit` ecnica de Catalunya, Barcelona, Catalonia, Spain Summary Finite element methods (FEM) are increasingly being used to simulate the acoustics of the vocal tract. For vowel production, the irreducible wave equation for the acoustic pressure is typically solved. However, diphthong sounds require moving vocal tract geometries so that the wave equation has to be expressed in an Arbitrary Lagrangian-Eulerian (ALE) framework. It then becomes more convenient to directly work with the wave equation in its mixed form, which not only involves the acoustic pressure but also the acoustic velocity. In turn, this entails some numerical difficulties that require resorting to stabilized FEM approaches. In this work, FEM simulations for the wave equation in mixed form are carried out to produce some diphthongs. Tuned two-dimensional vocal tracts are used which mimic the behavior of three-dimensional vocal tracts with circular cross-section. PACS no. 43.70.Bk 1. Introduction Finite Element approaches are becoming popular to simulate the acoustics of the vocal tract. They allow to overcome some of the classical limitations of one- dimensional (1D) models, such as the plane wave as- sumption, and to consider complex shapes for the vo- cal tract. With regard to vowel production, the acous- tic pressure is the magnitude of interest, so the stan- dard (irreducible) wave equation, or its Fourier trans- form, the Helmholtz equation, are the usual equations being solved (e.g., [1, 2, 3, 4]). However, approaches dealing with the wave equation in mixed form that involves both, the acoustic pressure and the acoustic velocity, can also be considered (see e.g. [5, 6]). Actu- ally, it will be shown herein that to generate diphthong sounds it precisely becomes more convenient to deal with the later. In the case of diphthongs, one has to consider moving domains (dynamic vocal tracts) and consequently it becomes necessary to express the in- volved equations in an Arbitrary Lagrangian-Eulerian (ALE) framework ([7, 8]). This can be done more nat- urally for the wave equation in mixed form. Consequently, in this work we will begin by set- ting the wave equation in mixed form in an ALE frame of reference, and then solve it to synthesize (c) European Acoustics Association diphthong sounds using a stabilized Finite Element Method ([9, 10, 11]). This will allow us to use equal interpolations for the acoustic pressure and velocity fields. Moving vocal tract geometries will be gener- ated using tuned two-dimensional (2D) vocal tract models, which can recover to some extent the acoustic behavior of three-dimensional (3D) vocal tracts with circular cross-sections [12, 13]. The work is organized as follows. In Section 2 the proposed methodology for diphthong production will be presented, whereas some numerical exemples will be shown in Section 3. Conclusions will close the paper in Section 4. 2. Methodology 2.1. Mixed wave equation in an ALE frame- work The linearized continuity and momentum equations for sound propagation in a stationary inviscid fluid read, in a spatial (Eulerian) frame of reference, 1 ρ 0 c 2 0 ∂ t p + ∇· u =0, (1a) ρ 0 ∂ t u + ∇p =0. (1b) p(x,t) and u(x,t) respectively denote the acoustic pressure and the acoustic velocity, while c 0 and ρ 0 stand for the speed of sound and for the air density.