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

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft

ParMooN - a modernized program package based on mapped finite elements

{\sc ParMooN} is a program package for the numerical solution of elliptic and parabolic partial differential equations. It inherits the distinct features of its predecessor {\sc MooNMD} \cite{JM04}: strict decoupling of geometry and finite element spaces, implementation of mapped finite elements as their definition can be found in textbooks, and a geometric multigrid preconditioner with the option to use different finite element spaces on different levels of the multigrid hierarchy. After having presented some thoughts about in-house research codes, this paper focuses on aspects of the parallelization for a distributed memory environment, which is the main novelty of {\sc ParMooN}. Numerical studies, performed on compute servers, assess the efficiency of the parallelized geometric multigrid preconditioner in comparison with some parallel solvers that are available in the library {\sc PETSc}. The results of these studies give a first indication whether the cumbersome implementation of the parallelized geometric multigrid method was worthwhile or not.Comment: partly supported by European Union (EU), Horizon 2020, Marie Sk{\l}odowska-Curie Innovative Training Networks (ITN-EID), MIMESIS, grant number 67571

Numerical simulation of aeroacoustics using the variational multiscale method : application to the problem of human phonation

The solution of the human phonation problem applying computational mechanics is covered by several research branches, such as Computational Fluid Dynamics (CFD), biomechanics or acoustics, among others. In the present thesis, the problem is approached from the Computational Aeroacoustics (CAA) point of view and the first main objective consists in developing numerical methods of general application that can take part in the solution of any scenario related to human phonation with a reasonable cost. In this sense, only the compressible Navier-Stokes equations can describe all flow and acoustic scales without any modeling, which is known as Direct Numerical Simulation (DNS), but its computational cost is usually unaffordable. Even in the case of a Large Eddy Simulation (LES), where the small scales are modeled, the cost can still be a handicap due to the complexity of the problem. This drawback gets worse in the low Mach regime due to the large disparity between flow velocity and sound speed, which leads to an ill-conditioning of the system of equations, specially for conservative schemes. At this point, it makes sense to move towards the incompressible flow approximation, bearing in mind the low velocities expected in human phonation problems. Incompressible flows do not yield any acoustics, for which a second problem containing the propagation of the sound sources needs to be modeled and solved. These are the so called hybrid methods, which allow a better conditioning of the problem by segregating flow and acoustic scales. Lighthill's analogy has been taken as starting point for the present work, but its restriction to free-field scenarios has motivated the extension of the method to arbitrary geometries and non-uniform flows. The first development in this direction consists in a splitting of Lighthill's analogy into a quadrupolar and dipolar component, which does not change the original problem but allows assessing the contribution of solid boundaries to the generation of sound. The second step consists in the development of a stabilized Finite Element (FEM) formulation for the Acoustic Perturbation Equations (APE) which account for non-uniform flows and perform a complete filtering of the acoustic scales. The final step assumes the compressible approach but omitting the energy equation and thus considering both flow and acoustic propagation as isentropic. In this case the solver is unified and hence a method for applying compatible boundary conditions for flow and acoustics has been developed. Moreover, the whole numerical framework has been extended to dynamic phonation cases, which require using an Arbitrary Lagrangian Eulerian (ALE) reference. Also, a novel remeshing strategy with conservative interpolation between meshes is presented. In the last chapter a challenging case in human phonation has been chosen for testing the developed computational framework: the fricative phoneme /s/. Unlike vowels, which are voiced sounds defined by a few characteristic frequencies, fricatives cannot be simulated as the propagation of a known analytic solution (glottal pulse) because the sound sources correspond to a wide range of turbulent scales. Therefore, a CFD calculation is mandatory in order to capture all relevant eddies behind the generation of sound. This problem is solved with an LES together with the Variational Multiscale (VMS) stabilization method as turbulence model, which is supplemented with several acoustic formulations when using incompressible flow. The analysis of the results focuses on the numerical representation of turbulence and the acoustic signal at the far-field, which has been compared to experimental recordings. Finally, the role of the upper incisors in the generation of the fricative sound has been evaluated. All simulations have been run with the parallel multiphysics FEM code FEMUSS, based on FORTRAN Object-Oriented-Programming land the OpenMPI parallel library.La solució del problema de la veu humana des de la mecànica computacional és objecte d'estudi per part de diverses disciplines, com per exemple la Dinàmica de Fluids Computacional (CFD), la biomecànica o l'acústica. En la present tesi s'encara el problema des de l'Aeroacústica Computacional (CAA) i el primer objectiu consisteix en desenvolupar mètodes numèrics d'aplicació general que puguin ser part de la solució, amb un cost computacional raonable, de qualsevol escenari relacionat amb la fonació humana. En aquest sentit, només les equacions de flux compressible de Navier-Stokes aconsegueixen descriure totes les escales alhora, tant les dinàmiques com les acústiques, sense recórrer a cap modelització, conegut com a Simulació Numèrica Directa (DNS), però el seu cost computacional és normalment inassumible. Fins i tot en el cas d'una Large Eddy Simulation (LES), on les escales petites són modelades, el cost pot resultar excessiu a causa de la complexitat del problema. Aquest fet encara és més accentuat en el règim de baix nombre de Mach donada la gran disparitat entre la velocitat del fluid i la del so i el conseqüent mal condicionament del sistema d'equacions, sobretot en esquemes conservatius. Per tant, tenint en compte les baixes velocitats de l'aire al tracte vocal, té sentit recórrer a l'aproximació de flux incompressible. Els fluids incompressibles no inclouen la part acústica, de manera que cal calcular un segon problema que descrigui la propagació de les fonts de so. Aquests són els anomenats mètodes híbrids, que permeten un millor condicionament del problema gràcies a la segregació de les escales acústiques de les dinàmiques. S'ha pres l'analogia de Lighthill com a punt de partida, però la seva restricció a casos en camp obert ha motivat l'extensió del mètode cap a geometries arbitràries i fluxos no uniformes. El primer desenvolupament en aquesta direcció consisteix en la divisió de l'analogia de Lighthill en una component quadrupolar i una altra de dipolar, fet que no altera el problema original però que permet analitzar la contribució de cossos sòlids en la generació de so. El segon pas consisteix en el desenvolupament d'una formulació estabilitzada en elements finits (FEM) de les Acoustic Perturbation Equations (APE), que incorporen la propagació en fluxos no uniformes i que realitzen un filtrat complet de les escales acústiques. El pas final assumeix la compressibilitat del fluid però omet l'equació d'energia, i per tant considera la dinàmica i l'acústica fenòmens isentròpics. En aquest cas el solver és unificat i per tant s'ha desenvolupat un mètode per imposar condicions de contorn compatibles entre ambdues escales del fluid. Finalment, les formulacions numèriques han estat adaptades a casos de fonació dinàmica mitjançant una referència Arbitrària Lagrangiana Euleriana (ALE). A més, es presenta una estratègia de remallat amb interpolació conservativa entre malles. En l'últim capítol es presenta un cas de fonació humana que suposa un repte per la seva complexitat i que ha servit per validar les formulacions numèriques presentades: la fricativa sorda /s/. A diferència de les vocals, que són sons sonors definits per unes poques freqüències característiques, les fricatives no poden ser simulades com la propagació d'una funció analítica coneguda (pols glotal) perquè les fonts de so corresponen a un rang ampli d'escales turbulents. Per tant és necessària una simulació CFD per tal de capturar-les. El problema se soluciona amb un model de turbulència LES amb el mètode d'estabilització Variational Multiscale. L'anàlisi se centra en la representació numèrica de la turbulència i en el senyal acústic al camp llunyà, tot comparant-lo amb dades experimentals. Finalment, s'avalua la contribució dels incisius superiors en la generació del so fricatiu sord /s/. Totes les simulacions han estat realitzades amb el codi FEM multi-físic en paral·lel FEMUSS, basat en programació orientada a objectes en FORTRAN i en OpenMPI

The LifeV library: engineering mathematics beyond the proof of concept

LifeV is a library for the finite element (FE) solution of partial differential equations in one, two, and three dimensions. It is written in C++ and designed to run on diverse parallel architectures, including cloud and high performance computing facilities. In spite of its academic research nature, meaning a library for the development and testing of new methods, one distinguishing feature of LifeV is its use on real world problems and it is intended to provide a tool for many engineering applications. It has been actually used in computational hemodynamics, including cardiac mechanics and fluid-structure interaction problems, in porous media, ice sheets dynamics for both forward and inverse problems. In this paper we give a short overview of the features of LifeV and its coding paradigms on simple problems. The main focus is on the parallel environment which is mainly driven by domain decomposition methods and based on external libraries such as MPI, the Trilinos project, HDF5 and ParMetis. Dedicated to the memory of Fausto Saleri.Comment: Review of the LifeV Finite Element librar

On parallel scalability aspects of strongly coupled partitioned fluid-structure-acoustics interaction

Multi-physics simulations, such as fluid-structure-acoustics interaction (FSA), require a high performance computing environment in order to perform the simulation in a reasonable amount of computation time. Currently used coupling methods use a staggered execution of the fluid and solid solver [6], which leads to inherent load imbalances. In [12] a new coupling scheme based on a quasi-Newton method is proposed for fluidstructure interaction which coupled the fluid and solid solver in parallel. The quasi- Newton method requires approximately the same number of coupling iterations per time step compared to a staggered coupling approach, resulting in a better load balance when running in a parallel environment. This contribution investigates the scalability limit and load-balancing for a strongly coupled fluid-structure interaction problem, and also for a fluid-structure-acoustics interaction problem. The acoustic far field of the fluid-structure-acoustics interaction problem is loosely coupled with the flow field

Shell-crossing in quasi-one-dimensional flow

Blow-up of solutions for the cosmological fluid equations, often dubbed shell-crossing or orbit crossing, denotes the breakdown of the single-stream regime of the cold-dark-matter fluid. At this instant, the velocity becomes multi-valued and the density singular. Shell-crossing is well understood in one dimension (1D), but not in higher dimensions. This paper is about quasi-one-dimensional (Q1D) flow that depends on all three coordinates but differs only slightly from a strictly 1D flow, thereby allowing a perturbative treatment of shell-crossing using the Euler--Poisson equations written in Lagrangian coordinates. The signature of shell-crossing is then just the vanishing of the Jacobian of the Lagrangian map, a regular perturbation problem. In essence the problem of the first shell-crossing, which is highly singular in Eulerian coordinates, has been desingularized by switching to Lagrangian coordinates, and can then be handled by perturbation theory. Here, all-order recursion relations are obtained for the time-Taylor coefficients of the displacement field, and it is shown that the Taylor series has an infinite radius of convergence. This allows the determination of the time and location of the first shell-crossing, which is generically shown to be taking place earlier than for the unperturbed 1D flow. The time variable used for these statements is not the cosmic time $t$ but the linear growth time $\tau \sim t^{2/3}$. For simplicity, calculations are restricted to an Einstein--de Sitter universe in the Newtonian approximation, and tailored initial data are used. However it is straightforward to relax these limitations, if needed.Comment: 9 pages; received 2017 May 24, and accepted 2017 June 21 at MNRA