A stabilized finite element method for the mixed wave equation in an ALE framework with application to diphthong production

The archived file is not the final published version of the article. © (2016) S. Hirzel Verlag/European Acoustics Association The definitive publisher-authenticated version is available online at http://www.ingentaconnect.com/contentone/dav/aaua/2016/00000102/00000001/art00012 Readers must contact the publisher for reprint or permission to use the material in any form.Working with the wave equation in mixed rather than irreducible form allows one to directly account for both, the acoustic pressure field and the acoustic particle velocity field. Indeed, this becomes the natural option in many problems, such as those involving waves propagating in moving domains, because the equations can easily be set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference. Yet, when attempting a standard Galerkin finite element solution (FEM) for them, it turns out that an inf-sup compatibility constraint has to be satisfied, which prevents from using equal interpolations for the approximated acoustic pressure and velocity fields. In this work it is proposed to resort to a subgrid scale stabilization strategy to circumvent this condition and thus facilitate code implementation. As a possible application, we address the generation of diphthongs in voice production.Peer ReviewedPostprint (author's final draft

Residual-based stabilization of the finite element approximation to the acoustic perturbation equations for low Mach number aeroacoustics

This is the peer reviewed version of the following article: [Guasch, O., Sánchez-Martín, P., Pont, A., Baiges, J., and Codina, R. (2016) Residual-based stabilization of the finite element approximation to the acoustic perturbation equations for low Mach number aeroacoustics. Int. J. Numer. Meth. Fluids, 82: 839–857. doi: 10.1002/fld.4243], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/fld.4243/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.The acoustic perturbation equations (APE) are suitable to predict aerodynamic noise in the presence of a non-uniform mean flow. As for any hybrid computational aeroacoustics approach, a first computational fluid dynamics simulation is carried out from which the mean flow characteristics and acoustic sources are obtained. In a second step, the APE are solved to get the acoustic pressure and particle velocity fields. However, resorting to the finite element method (FEM) for that purpose is not straightforward. Whereas mixed finite elements satisfying an appropriate inf–sup compatibility condition can be built in the case of no mean flow, that is, for the standard wave equation in mixed form, these are difficult to implement and their good performance is yet to be checked for more complex wave operators. As a consequence, strong simplifying assumptions are usually considered when solving the APE with FEM. It is possible to avoid them by resorting to stabilized formulations. In this work, a residual-based stabilized FEM is presented for the APE at low Mach numbers, which allows one to deal with the APE convective and reaction terms in its full extent. The key of the approach resides in the design of the matrix of stabilization parameters. The performance of the formulation and the contributions of the different terms in the equations are tested for an acoustic pulse propagating in sheared-solenoidal mean flow, and for the aeolian tone generated by flow past a two-dimensional cylinder.Peer ReviewedPostprint (author's final draft

Unified solver for fluid dynamics and aeroacoustics in isentropic gas flows

The high computational cost of solving numerically the fully compressible Navier–Stokes equations, together with the poor performance of most numerical formulations for compressible flow in the low Mach number regime, has led to the necessity for more affordable numerical models for Computational Aeroacoustics. For low Mach number subsonic flows with neither shocks nor thermal coupling, both flow dynamics and wave propagation can be considered isentropic. Therefore, a joint isentropic formulation for flow and aeroacoustics can be devised which avoids the need for segregating flow and acoustic scales. Under these assumptions density and pressure fluctuations are directly proportional, and a two field velocity-pressure compressible formulation can be derived as an extension of an incompressible solver. Moreover, the linear system of equations which arises from the proposed isentropic formulation is better conditioned than the homologous incompressible one due to the presence of a pressure time derivative. Similarly to other compressible formulations the prescription of boundary conditions will have to deal with the backscattering of acoustic waves. In this sense, a separated imposition of boundary conditions for flow and acoustic scales which allows the evacuation of waves through Dirichlet boundaries without using any tailored damping model will be presented.Peer ReviewedPostprint (author's final draft

The solution of vibroacoustic linear systems as a finite sum of transmission paths

© 2021 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Linear systems are frequently encountered in low, mid and high vibroacoustics modelling of mechanical built-up structures. It has recently been proved that the solution to those systems can be always factorized as an infinite (weighted) Neumann series summation, which accounts for signal transmission through paths connecting system elements. The key to path expansion relies on the concept of direct transmissibility. In this work, we explore some additional theoretical aspects of transmissibility-based transmission path analysis (TPA), which is known to constitute a valuable tool to remedy noise and vibration problems. In particular, we show that it is also possible to expand the solution of a matrix linear system as a finite summation of transmission paths. Furthermore, our goal is to provide mathematical and physical insight into such path factorization. As regards the former, we exploit the relationship between graph theory and matrix algebra to interpret the terms appearing in the series expansion as combinations of open and closed paths in a graph. In what concerns the second, two benchmark examples are addressed that benefit from the graph theory outcomes. The first one consists of a mass-damping-stiffness system representative of vibroacoustic modelling at low frequencies. A relation is established between the relative weights of the paths, the global system resonances and the resonances of complementary systems, which contain elements not belonging to the paths. The second example involves a statistical energy analysis (SEA) model made of connected plates. The meaning of the relative weights of open paths in the finite expansion for energy transmission between SEA subsystems is analyzed and compared to the results of infinite SEA path factorization.Peer ReviewedPostprint (author's final draft

Finite element computation of diphthong sounds using tuned two-dimensional vocal tracts

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.Postprint (published version

Stabilized finite element formulation for the mixed convected wave equation in domains with driven flexible boundaries

A stabilized finite element (FEM) formulation for the wave equation in mixed form with convection is presented, which permits using the same interpolation fields for the acoustic pressure and the acoustic particle velocity. The formulation is based on a variational multiscale approach, in which the problem unknowns are split into a large scale component that can be captured by the computational mesh, and a small, subgrid scale component, whose influence into the large scales has to be modelled. A suitable option is that of taking the subgrid scales, or subscales, as being related to the finite element residual by means of a matrix of stabilization parameters. The design of the later turns to be the key for the good performance of the method. In addition, the mixed convected wave equation has been set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference to account for domains with moving boundaries. The movement of the boundaries in the present work consists of two components, an external prescribed motion and a motion related to the boundary elastic back reaction to the acoustic pressure, in the normal direction. A mass-damper-stiffness auxiliary equation is solved for each boundary node to include this effect. As a first benchmark example, we have considered the case of 2D simple duct acoustics with mean flow. More complex 3D examples are also presented consisting of vowel and diphthong generation, following a numerical approach to voice production. The numerical simulation of voice not only allows one to see how waves propagate inside the vocal tract, but also to collect the acoustic pressure at a node close to the mouth exit, convert it to an audio file and listen to it.Postprint (published version

Concurrent finite element simulation of quadrupolar and dipolar flow noise in low Mach number aeroacoustics

The computation of flow-induced noise at low Mach numbers usually relies on a two-step hybrid methodolgy. In the first step, an incompressible fluid dynamics simulation (CFD) is performed and an acoustic source term is derived from it. The latter becomes the inhomogeneous term for an acoustic wave equation, which is solved in the second step, often resorting to boundary integral formulations. In the presence of rigid bodies, Curie's acoustic analogy is probably the most extended approach. It has been shown that Curie's boundary dipolar noise contribution does in fact correspond to the diffraction of the quadrupolar aerodynamic noise generated by the flow past the rigid body. In this work, advantage is taken from this fact to propose an alternative computational methodology to get the individual quadrupolar and dipolar contributions to the total acoustic pressure. For any linear acoustic wave operator, the unknown acoustic pressure can be split into its incident and diffracted components and be computed simultaneously to the incompressible flow field, in a single finite element computational run. This circumvents the problem found in Curie's analogy of needing the total pressure at the body's boundary, which includes the acoustic pressure fluctuations. The latter cannot be obtained from an incompressible CFD simulation. The proposed unified strategy could be beneficial for a large variety problems such as those involving noise generated from duct terminations, or those related with the simulation of fricatives in numerical voice production, among many others.Peer Reviewe

Finite element computation of diphthong sounds using tuned two-dimensional vocal tracts

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

A stabilized finite element method for the mixed wave equation in an ALE framework with application to diphthong production

The archived file is not the final published version of the article. © (2016) S. Hirzel Verlag/European Acoustics Association The definitive publisher-authenticated version is available online at http://www.ingentaconnect.com/contentone/dav/aaua/2016/00000102/00000001/art00012 Readers must contact the publisher for reprint or permission to use the material in any form.Working with the wave equation in mixed rather than irreducible form allows one to directly account for both, the acoustic pressure field and the acoustic particle velocity field. Indeed, this becomes the natural option in many problems, such as those involving waves propagating in moving domains, because the equations can easily be set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference. Yet, when attempting a standard Galerkin finite element solution (FEM) for them, it turns out that an inf-sup compatibility constraint has to be satisfied, which prevents from using equal interpolations for the approximated acoustic pressure and velocity fields. In this work it is proposed to resort to a subgrid scale stabilization strategy to circumvent this condition and thus facilitate code implementation. As a possible application, we address the generation of diphthongs in voice production.Peer Reviewe

Stabilized finite element formulation for the mixed convected wave equation in domains with driven flexible boundaries

A stabilized finite element (FEM) formulation for the wave equation in mixed form with convection is presented, which permits using the same interpolation fields for the acoustic pressure and the acoustic particle velocity. The formulation is based on a variational multiscale approach, in which the problem unknowns are split into a large scale component that can be captured by the computational mesh, and a small, subgrid scale component, whose influence into the large scales has to be modelled. A suitable option is that of taking the subgrid scales, or subscales, as being related to the finite element residual by means of a matrix of stabilization parameters. The design of the later turns to be the key for the good performance of the method. In addition, the mixed convected wave equation has been set in an arbitrary Lagrangian-Eulerian (ALE) frame of reference to account for domains with moving boundaries. The movement of the boundaries in the present work consists of two components, an external prescribed motion and a motion related to the boundary elastic back reaction to the acoustic pressure, in the normal direction. A mass-damper-stiffness auxiliary equation is solved for each boundary node to include this effect. As a first benchmark example, we have considered the case of 2D simple duct acoustics with mean flow. More complex 3D examples are also presented consisting of vowel and diphthong generation, following a numerical approach to voice production. The numerical simulation of voice not only allows one to see how waves propagate inside the vocal tract, but also to collect the acoustic pressure at a node close to the mouth exit, convert it to an audio file and listen to it