Acoustic Wave Propagation Problems Research Articles

Solution of time-domain acoustic wave propagation problems using a RBF interpolation model with “a priori” estimation of the free parameter

In the last decade, Meshless Methods have found widespread application in different fields of engineering and science. Beyond novelty, their mathematical simplicity and numerical accuracy have been the key of their rapid dissemination. Among meshless techniques, RBF (Radial Basis Functions) based methods can be simple and general to solve the problems related to multiple areas of applied physics and engineering. In the specific field of acoustics, there are usually two possible approaches for solving a problem: time- and frequency-domain. In this paper, the authors propose a local time-domain approach to establish an efficient methodology for the solution of large-scale acoustic wave propagation problems. For this purpose, a local interpolation scheme, based on the reproduction of the local wave field using RBFs (MultiQuadric and Gaussian), is implemented and its accuracy is verified against known closed-form solutions. An explicit time-domain marching procedure is adopted, and the quality of the numerical results is also compared with that obtained using standard space-time Finite–Difference schemes. Additionally, the RBF interpolation model is used to simulate the propagation of a Ricker pulse in two simple test cases, and applied to simulate a more complex configuration, corresponding to an underwater sound propagation problem. In this frame, the results are also compared with those computed using a fourth-order in space and second-order in time Finite–Difference scheme.

Error Analysis for a Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation

Finite element methods for acoustic wave propagation problems at higher frequency result in very large matrices due to the need to resolve the wave. This problem is made worse by discontinuous Galerkin methods that typically have more degrees of freedom than similar conforming methods. However hybridizable discontinuous Galerkin methods offer an attractive alternative because degrees of freedom in each triangle can be cheaply removed from the global computation and the method reduces to solving only for degrees of freedom on the skeleton of the mesh. In this paper we derive new error estimates for a hybridizable discontinuous Galerkin scheme applied to the Helmholtz equation. We also provide extensive numerical results that probe the optimality of these results. An interesting observation is that, after eliminating the internal element degrees of freedom, the condition number of the condensed hybridized system is seen to be almost independent of the wave number.

Augmented Galerkin schemes for the numerical solution of scattering by small obstacles

We are interested in the problem of a bidimensional acoustic wave propagation in a medium including a small obstacle with homogeneous Dirichlet boundary condition. We present and analyse a numerical scheme suitable for finite elements that does not suffer from numerical locking, and takes the presence of the small obstacle into account. It is based on the fictitious domain method combined with matched asymptotic expansions.

A meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks

The numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered. Since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods, based on global interpolation by analytic functions, show low accuracy. In order to circumvent this issue, a meshfree modification of the method of fundamental solutions is developed, where the approximation basis is enriched by an extra span of corner adapted non-smooth shape functions. The high accuracy of the new method is illustrated by solving several boundary value problems for the Helmholtz equation, modelling physical phenomena from the fields of room acoustics and acoustic resonance.

Coupling the BEM/TBEM and the MFS for the numerical simulation of acoustic wave propagation

The coupling of the boundary element method (BEM)/the traction boundary element method (TBEM) and the method of fundamental solutions (MFS) is proposed for the transient analysis of acoustic wave propagation problems in the presence of multi-inclusions to overcome the limitations posed by each method. The full domain is divided into sub-domains which are modeled using the BEM/TBEM and the MFS, and the sub-domains are coupled with the imposition of the required boundary conditions. The accuracy of the proposed algorithms, using different combinations of BEM/TBEM and MFS, is verified by comparing the solutions against reference solutions. The applicability of the proposed method is shown by simulating the acoustic behavior of a rigid acoustic screen in the vicinity of a dome and by computing the acoustic attenuation provided by a fluid-filled thin inclusion separating two railway tracks in an underground train station.

Untwisting the Campbell diagrams of weakly anisotropic rotor systems

A brake can be modeled as an axi-symmetric rotor perturbed by dissipative, conservative, and non-conservative positional forces originated at the frictional contact with the anisotropic stator. The Campbell diagram of the unperturbed system is a mesh-like structure in the frequency-speed plane with double eigenfrequencies at the nodes. The diagram is convenient for the analysis of the traveling waves in the rotating elastic continuum. Computing sensitivities of the doublets we find that at every particular node the untwisting of the mesh into the branches of complex eigenvalues is generically determined by only four 2×2 sub-blocks of the perturbing matrix. Selection of the unstable modes that cause self-excited vibrations in the subcritical speed range, is governed by the exceptional points at the corners of the singular eigenvalue surface–`double coffee-filter'–which is typical also in the problems of electromagnetic and acoustic wave propagation in non-rotating anisotropic chiral media. As a mechanical example a model of a rotating shaft is studied in detail.

Estimation of number of modes in periodic graphs

The problem of acoustic wave propagation in a one-periodic system of thin pipes is studied. A dispersion equation linking the time frequency ω with the propagation constant (quasimomentum) γ is derived. A particular question is studied, namely how many different values of γ can obey the dispersion equation for a fixed ω. A simple estimation of this amount is obtained in terms of the properties of the graph representing a single period of the system.

A fictitious domain method for acoustic wave propagation problems

Based on the essence of Lagrange multiplier, a fictitious domain method, especially for acoustic wave propagation problems in exterior domain, is proposed, where all computational tasks are carried out in an auxiliary simply shaped domain, and the original domain is embedded into it. The simplicity of the auxiliary domain enables us to use uniform meshes to construct the finite element space and form a special structured stiffness matrix. Finally, the stability, convergence and error estimates are obtained for the numerical schemes.

The problem of surface wave propagation in a reduced Cosserat medium

This article continues a series of publications devoted to the study of waves in the framework of the asymmetric theory of elasticity, where the deformed state of the medium is characterized by independent vectors of translation and rotation. The problem of acoustic Rayleigh wave propagation in half space is considered within a model of the reduced Cosserat medium. A general analytic solution of this problem is obtained. The analysis of this solution is compared with the corresponding solution for a classical elastic medium and full linear Cosserat medium. It is shown that the Rayleigh wave is characterized by a range of forbidden frequencies, where this wave cannot propagate. The dispersion curve consists of two branches. One of them has a cut-off frequency and cut-off wavenumber.

Nonreflecting boundary conditions based on nonlinear multidimensional characteristics

AbstractNonlinear characteristic boundary conditions based on nonlinear multidimensional characteristics are proposed for 2‐ and 3‐D compressible Navier–Stokes equations with/without scalar transport equations. This approach is consistent with the flow physics and transport properties. Based on the theory of characteristics, which is a rigorous mathematical technique, multidimensional flows can be decomposed into acoustic, entropy, and vorticity waves. Nonreflecting boundary conditions are derived by setting corresponding characteristic variables of incoming waves to zero and by partially damping the source terms of the incoming acoustic waves. In order to obtain the resulting optimal damping coefficient, analysis is performed for problems of pure acoustic plane wave propagation and arbitrary flows. The proposed boundary conditions are tested on two benchmark problems: cylindrical acoustic wave propagation and the wake flow behind a cylinder with strong periodic vortex convected out of the computational domain. This new approach substantially minimizes the spurious wave reflections of pressure, density, temperature, and velocity as well as vorticity from the artificial boundaries, where strong multidimensional flow effects exist. The numerical simulations yield accurate results, confirm the optimal damping coefficient obtained from analysis, and verify that the method substantially improves the 1‐D characteristics‐based nonreflecting boundary conditions for complex multidimensional flows. Copyright © 2009 John Wiley & Sons, Ltd.

A new combined stable and dispersion relation preserving compact scheme for non-periodic problems

A new compact scheme is presented for computing wave propagation problems and Navier–Stokes equation. A combined compact difference scheme is developed for non-periodic problems (called NCCD henceforth) that simultaneously evaluates first and second derivatives, improving an existing combined compact difference (CCD) scheme. Following the methodologies in Sengupta et al. [T.K. Sengupta, S.K. Sircar, A. Dipankar, High accuracy schemes for DNS and acoustics, J. Sci. Comput. 26 (2) (2006) 151–193], stability and dispersion relation preservation (DRP) property analysis is performed here for general CCD schemes for the first time, emphasizing their utility in uni- and bi-directional wave propagation problems – that is relevant to acoustic wave propagation problems. We highlight: (a) specific points in parameter space those give rise to least phase and dispersion errors for non-periodic wave problems; (b) the solution error of CCD/NCCD schemes in solving Stommel Ocean model (an elliptic p.d.e.) and (c) the effectiveness of the NCCD scheme in solving Navier–Stokes equation for the benchmark lid-driven cavity problem at high Reynolds numbers, showing that the present method is capable of providing very accurate solution using far fewer points as compared to existing solutions in the literature.

The mean-field method in the problem of acoustic wave propagation in a turbulent atmosphere

We analyze the problem of acoustic-wave propagation in a turbulent atmosphere using the mean-field method. The equation for the sound pressure is written with accuracy up to terms that are linear with respect to the Mach number of the turbulent air flow. An expression for the attenuation constant of the mean field is obtained. For the turbulence model described by the von Karman correlation function of fluctuations, the attenuation coefficient of the mean field is numerically studied in detail. It is shown that under typical conditions of the near-ground atmospheric layer, the predominant contribution to scattering of acoustic waves is given by the turbulent motion of the air masses.

Numerical approximation of acoustic waves by spectral element methods

In this paper we consider the approximation of acoustic wave propagation problems. Precisely, we investigate the stability and convergence of the classical Newmark schema in time and spectral element discretization in space for the wave problems. A special attention is payed to the non-homogeneous boundary data. Some detailed error estimates are obtained. From these results, the spectral accuracy and influences of the non-homogeneous boundary data are made evident. Several numerical examples are provided to confirm our theoretical analysis. The advantage of the present method is demonstrated by a numerical comparison with the finite element method.

Fast acoustic integral-equation solver for complex inhomogeneous media

We describe elements and representative applications of an integral-equation solver for large-scale computations in acoustic wave propagation problems. In the solver construction we used elements of our previously developed fast integral-equation solver for Maxwell’s equations. In comparison with the conventional integral equation approach (method of moments), our solver achieves significant reduction of execution time and memory through the FFT-based matrix compression. One particular aspect of the solver we discuss, pertinent to its high efficiency and accuracy, is an efficient treatment of problems associated with subwavelength discretization. We illustrate the approach and its application on the example of a numerical simulation of acoustic wave propagation through the human head. [Work was supported by a grant from AFOSR.]

A posteriori error estimation for acoustic wave propagation problems

The main purpose of this paper is to review a posteriori error estimators for the simulation of acoustic wave propagation problems by computational methods. Residual-type (explicit and implicit) and recovery-type estimators are presented in detail in the case of the Helmholtz problem. Recent work on goal-oriented error estimation techniques with respect to so-called quantities of interest or output functionals are also accounted for. Fundamental results from a priori error estimation are presented and issues dealing with pollution error at large wave numbers are extensively discussed.

Performance of parallel incomplete LDLt factorizations for solving acoustic wave propagation problems from industry

AbstractParallel incomplete LDLt (ParILDLt) factorizations are used to solve highly indefinite complex‐symmetric linear systems that arise from finite element discretization of acoustic wave propagation problems. The parallelization strategy is a generalized domain decomposition type approach in which adjacent subdomains have to exchange data during the construction of the incomplete factorization preconditioning matrix, as well as during each local forward and backward substitution. Comparison with the SYSNOISE (LMS International NV) direct solver, and the finite element tearing and interconnecting method for the Helmholtz equation (FETI‐H), is done in terms of execution time and memory usage. Challenging industrial problems are tested, showing that high performance is achieved with ParILDLt. Copyright © 2004 John Wiley & Sons, Ltd.

DISCRETE HUYGENS' MODELLING APPROACH TO WAVE PROPAGATIONS IN A HOMOGENEOUS ELASTIC FIELD

The application of the discrete Huygens' modelling has been discussed for acoustic wave propagation problems, in which the scalar wave field problems have been focused. The present paper extends the application of the modelling to the elastic wave propagation in a homogeneous elastic medium in which two types of waves, the longitudinal wave and the shear wave, are independent except at the boundary. Each wave can be treated like a scalar wave until the two waves reach the boundary where they couple so as to satisfy the displacement or stress boundary condition. We propose the approach confining ourselves to the two-dimensional field. Some examples are demonstrated, whose solutions are compared with the vectorial wave modelling and finite difference modelling solutions whenever they are available.

Applications of High-Order Optimized Upwind Schemes for Computational Aeroacoustics

High-order optimized upwind dispersion-relation-preserving finite difference schemes are used for solving multidimensional acoustic wave propagation problems. The accuracy and the robustness of the schemes are tested for acoustic problems with various geometry and boundary conditions. Excellent agreements between the numerical and the analytical solutions indicate that the high-order optimized upwind schemes minimize not only the dispersion error but also the dissipation error while retaining the numerical stability. The simulation results have also demonstrated that the optimized upwind schemes damp out the spurious short waves automatically without the need of filter or explicit dissipation terms. The overall effectiveness of the high-order optimized upwind schemes is verified. The good numerical result of the problem of acoustic scattering by a sphere suggests that the optimized upwind schemes may be used to solve acoustic problems involving a relatively complex geometry.

Dispersion-corrected explicit integration of the wave equation

A simple dispersion correction is introduced into the centered difference explicit time integration scheme for wave propagation problems. The extra term introduces a change of the wave speed, depending on local wave characteristics. A rigorous theoretical analysis is provided for the scalar wave equation on a regular mesh, and a notable improvement in the ability to conserve wave fronts is demonstrated by two numerical examples. The correction is reinterpreted to apply to a typical finite element formulation in terms of mass and stiffness matrices, resulting in a general dispersion correction of the explicit centered difference time integration algorithm for acoustic and elastic wave propagation problems.

Rigorous solutions of acoustic wave diffraction by penetrable wedges

A well-known problem of acoustic wave propagation in two fluid wedges with different densities and wave speeds is addressed by a recently developed approach to more general problems of wave propagation in bi-material solid wedges. The obtained solutions are valid for arbitrary frequencies and the approach is also extendable to wedge structures in which one or both wedges support propagation of shear waves.