Papers by Hector Espinoza
Computer Methods in Applied Mechanics and Engineering, 2014
In this paper we develop numerical approximations of the wave equation in mixed form supplemented... more In this paper we develop numerical approximations of the wave equation in mixed form supplemented with non-reflecting boundary conditions (NRBCs) of Sommerfeld-type on artificial boundaries for truncated domains. We consider three different variational forms for this problem, depending on the functional space for the solution, in particular, in what refers to the regularity required on artificial boundaries. Then, stabilized finite element methods that can mimic these three functional settings are described. Stability and convergence analyses of these stabilized formulations including the NRBC are presented. Additionally, numerical convergence test are evaluated for various polynomial interpolations, stabilization methods and variational forms. Finally, several benchmark problems are solved to determine the accuracy of these methods in 2D and 3D.
ABSTRACT A stabilized finite element (FEM) formulation for the wave equation in mixed form with c... more ABSTRACT 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.
The paper investigates the multi-objective design optimisation of stamping process to control bot... more The paper investigates the multi-objective design optimisation of stamping process to control both the shape and quality of final Advanced High Strength Steels (AHSSs) in terms of springback and safety using Distributed Multi-Objective Evolutionary Algorithm (DMOGA) coupled with Finite Element Analysis (FEA) based stamping analyser. The design problem of stamping process is formulated to minimise the difference between the desired shape and the final geometry obtained by a numerical simulation accounting elastic springback. In addition, the final product quality is maximised by improving safety factor without winkling, thinning, or failure. Numerical results show that a proposed methodology improves the final product quality while reducing its springback.
Acta Acustica united with Acustica, 2016
Working with the wave equation in mixed rather than irreducible form allows one to directly accou... more 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.
ABSTRACT Though most engineering problems in acoustics directly deal with the wave equation in it... more ABSTRACT Though most engineering problems in acoustics directly deal with the wave equation in its irreducible form, in many situations it becomes interesting to consider it in mixed form, so as to directly account for both, the acoustic pressure and the acoustic velocity fields. A particular case is that of waves propagating in domains with moving boundaries. When attempting a finite element solution to such problems, the mixed formulation naturally allows to set the equations in an arbitrary Lagrangian-Eulerian (ALE) framework. This results in the appearance of some extra terms involving the scalar product of the mesh velocity and the gradient of the pressure and velocity fields. As known, the finite element solution to the mixed wave equation needs to be stabilized so as to use equal interpolation for the pressure and velocity fields. Following the lines in, where algebraic and orthogonal subgrid stabilization were used, in this work a stabilized finite element method is proposed for the ALE wave equation in mixed form. As an application, we face the problem of the numerical generation of diphthongs. Much numerical work has recently been done with regard to static vocal tract acoustics i.e., generation of vowels and related phenomena, but little has been reported on dynamic vocal tract acoustics, most efforts being placed to date in the simulation of phonation. As a first step towards the generation of diphthongs, some 2D simulations will be presented based on simplified vocal tract geometries, which can be tuned to exhibit a 3D behavior.
Finite element methods (FEM) are increasingly being used to simulate the acoustics of the vocal t... more 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.
mixed wave equation, stabilized finite element methods, numerical analysis, Fourier analysis, von... more mixed wave equation, stabilized finite element methods, numerical analysis, Fourier analysis, von Neuman analysis, dispersion, dissipation, stability, convergence, voice simulation
Computer Methods in Applied Mechanics and Engineering, 2015
In this paper we analyze time marching schemes for the wave equation in mixed form. The problem i... more In this paper we analyze time marching schemes for the wave equation in mixed form. The problem is discretized in space using stabilized finite elements. On the one hand, stability and convergence analyses of the fully discrete numerical schemes are presented. On the other hand, we use Fourier techniques (also known as von Neumann analysis) in order to analyze stability, dispersion and dissipation. Additionally, numerical convergence tests are presented for various time integration schemes, polynomial interpolations (for the spatial discretization), stabilization methods, and variational forms. Finally, a 1D example is solved to analyze the behavior of the different schemes considered.
Wave equations usually arise from the manipulation of conservation laws, such as linearization or... more Wave equations usually arise from the manipulation of conservation laws, such as linearization or time derivation of these. In particular, the classical irreducible hyperbolic wave equation of second order in space and time is often obtained from a combination of two equations in two independent unknowns, of first order both in space and time. These equations constitute the so called mixed form of the wave problem. The mixed wave equation we consider has one scalar variable and a vector one, while the irreducible form can be written in terms of the scalar variable only. Using the former instead of the latter may be due to the wish of obtaining a better approximation for the vector field. However, there is one case in which the irreducible form cannot be obtained, and the mixed one is mandatory; this happens when the domain in which the problem is posed is time dependent. For example, if an Arbitrary Lagrangian-Eulerian (ALE) formulation is used, the two equations of the mixed form o...
A stabilized finite element (FEM) formulation for the wave equation in mixed form with convection... more 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...
In this work a novel edge-based finite element implementation applied to specific equations is pr... more In this work a novel edge-based finite element implementation applied to specific equations is presented. It contains a full description on how we obtained it for the diffusion equation, stabilized convection-diffusion equation and stabilized Navier-Stokes equations. Additionally, classical benchmark problems are solved to show the capabilities of the new implementation.
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Papers by Hector Espinoza