Acoustic Transmission Problem Research Articles

Skeleton Integral Equations for Acoustic Transmission Problems with Varying Coefficients

Skeleton Integral Equations for Acoustic Transmission Problems with Varying Coefficients

Time-domain multiple traces boundary integral formulation for acoustic wave scattering in 2D

We present a novel computational scheme to solve acoustic wave transmission problems over two-dimensional composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. The continuous problem is cast as a local multiple traces time-domain boundary integral formulation. For discretization in time and space, we resort to convolution quadrature schemes coupled to a non-conforming spatial spectral discretization based on second kind Chebyshev polynomials displaying fast convergence. Computational experiments confirm convergence of multistep and multistage convolution quadrature for a variety of complex domains.

A new periodic FM-BEM for solving the acoustic transmission problems in periodic media

This paper presents a new fast multipole boundary element method (FM-BEM) for solving the acoustic transmission problems in 2-D periodic media. We divide the periodic media into fundamental blocks, and then construct the boundary integral equations in the fundamental block. The fast multipole algorithm is proposed for the square periodic and hexagonal periodic structures, the convergence and complexity of the algorithm are analyzed. Our approach is applied to solve the acoustic transmission problems for liquid phononic crystals, the acoustic band gaps of the phononic crystals are derived.

The skeleton equation method for acoustic transmission problems with varying coefficients

AbstractIn this paper we describe the skeleton equation method, to transform a three‐dimensional acoustic transmission problem with variable coefficients and mixed boundary conditions to a non‐local equation on a two‐dimensional skeleton, without relying on an explicit knowledge of Green's function. This is achieved introducing and analyzing abstract layer potentials as solutions of auxiliary coercive full space transmission problems; they satisfy jump conditions across domain interfaces. The novelty in this paper is that we reduce acoustic scattering problems with possible variable coefficients to boundary integral equation on the domain skeleton, and we prove frequency exlicit continuity and coercivity estimates. The resulting equations are stated in a form such that standard boundary element methods can be applied for the discretization.

Frequency-robust preconditioning of boundary integral equations for acoustic transmission

The scattering and transmission of harmonic acoustic waves at a penetrable material are commonly modelled by a set of Helmholtz equations. This system of partial differential equations can be rewritten into boundary integral equations defined at the surface of the objects and solved with the boundary element method (BEM). High frequencies or geometrical details require a fine surface mesh, which increases the number of degrees of freedom in the weak formulation. Then, matrix compression techniques need to be combined with iterative linear solvers to limit the computational footprint. Moreover, the convergence of the iterative linear solvers often depends on the frequency of the wave field and the objects' characteristic size. Here, the robust PMCHWT formulation is used to solve the acoustic transmission problem. An operator preconditioner based on on-surface radiation conditions (OSRC) is designed that yields frequency-robust convergence characteristics. Computational benchmarks compare the performance of this novel preconditioned formulation with other preconditioners and boundary integral formulations. The OSRC preconditioned PMCHWT formulation effectively simulates large-scale problems of engineering interest, such as focused ultrasound treatment of osteoid osteoma.

A Stable Boundary Integral Formulation of an Acoustic Wave Transmission Problem with Mixed Boundary Conditions

In this paper, we consider an acoustic wave transmission problem with mixed boundary conditions of Dirichlet, Neumann, and impedance type. We will derive a formulation as a direct, space-time retar...

Wavefield reconstruction inversion: an example

Nonlinear least squares data-fitting driven by physical process simulation is a classic and widely successful technique for the solution of inverse problems in science and engineering. Known as ‘full waveform inversion (FWI)’ in application to seismology, it can extract detailed maps of earth structure from near-surface seismic observations, but also suffers from a defect not always encountered in other applications: the least squares error function at the heart of this method tends to develop a high degree of nonconvexity, so that local optimization methods (the only numerical methods feasible for field-scale problems) may fail to produce geophysically useful final estimates of earth structure, unless provided with initial estimates of a quality not always available. A number of alternative optimization principles have been advanced that promise some degree of release from the multimodality of FWI, amongst them wavefield reconstruction inversion (WRI), the focus of this paper. Applied to a simple 1D acoustic transmission problem, both full waveform and WRI methods reduce to minimization of explicitly computable functions, in an asymptotic sense. The analysis presented here shows explicitly how multiple local minima arise in FWI, and that WRI can be vulnerable to the same ‘cycle-skipping’ failure mode.

Evaluation of Nonlinear System Identification to Model Piezoacoustic Transmission

Piezoeletric materials are used on high-precision and high-dynamics applications, such as for acoustic transmission. This paper covers the challenges of creating a black-box model for a simultaneous acoustic transmission problem, with data acquired in a laboratory setup. The system performance is analyzed for three different models: AutoRegressive Moving Average with eXogenous inputs (ARMAX) model, Nonlinear AutoRegressive with eXogenous inputs (NARX) model with artificial neural network structure, and Nonlinear AutoRegressive Moving Average with eXogenous inputs (NARMAX) models. The best results of each models are compared with respect to precision in free-run simulation. The prediction results show that the most complex NARMAX model had the best results, what encourages further research in creating nonlinear mathematical data-driven abstractions for the piezoacoustic transmission application.

Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions

We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) [Formula: see text] norm of the solution is bounded by the [Formula: see text] norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacles with strictly positive curvature, while here we assume only Lipschitz regularity and star-shapedness with respect to a point. Furthermore, our bounds are obtained using identities first introduced by Morawetz (essentially integration by parts), whereas the existing bounds use the much-more sophisticated technology of microlocal analysis and propagation of singularities. We also adapt existing results to show that if the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible.

Incorporating the Havriliak–Negami model in wave propagation through polymeric viscoelastic core in a laminated sandwich cylinder

This paper presents the fundamental problem of three-dimensional acoustic wave transmission through a sandwich structure with viscoelastic core excited by an obliquely plane wave. The structure consists of a functionally graded shell as an outer layer, an isotropic layer as an inner layer and also a viscoelastic core. This work considers Havriliak-Negami model for description of complex modulus of viscoelastic core. The Havriliak-Negami model is a new mathematical method for the simulation of polymeric relaxation behavior in the frequency domain. In this model, both of complex Young’s and shear moduli are frequency dependent. Moreover, considering the effective roles of shear deformation and rotary inertia, dynamic governing equations of the functionally graded cylinder, viscoelastic core and isotropic shell are derived within the frameworks of the three-dimensional theory of elasticity. Furthermore, analytical model of acoustic wave transmission based on transfer matrix method is composed of a set of fluid and structural equations which taking into account the frequency dependency of mechanical characterization for the viscoelastic core. A good agreement can be observed, comparing the present results with those of other authors. Besides, the results show that a double-walled cylinder with viscoelastic core has a better acoustic insulation in comparison with isotropic shell with the same mass. Contrary to elastic material, by thickening the viscoelastic layer, sound transmission loss decreases in mass-controlled region due to shear deformation and rotary inertia. It has been also proved that polymer with a larger loss factor has good performance in energy dissipation.

Coupling of boundary integral equation and finite element methods for transmission problems in acoustics

In this paper, we propose a coupling of finite element method (FEM) and boundary integral equation (BIE) method for solving acoustic transmission problems in two dimensions. The original transmission problem is firstly reduced to a nonlocal boundary value problem by introducing an artificial boundary and defining a transparent boundary condition from the relation between Dirichlet data and Neumann data on the artificial boundary. In this work, such relationship is described in terms of boundary integral operators. Then, essential mathematical analysis for the weak formulation corresponding to the nonlocal boundary value problem is discussed. Three different algorithms are utilized for the solution of boundary integral equations to be involved in the computational formulations, and numerical results are presented to demonstrate the efficiency and accuracy of the schemes.

The Helmholtz equation in heterogeneous media: A priori bounds, well-posedness, and resonances

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation ∇⋅(A∇u)+k2nu=−f where both A and n are functions of position. We prove new a priori bounds on the solution under conditions on A, n, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in k, A, n, and geometric parameters of the domain. We then show that these a priori bounds hold when A and n are L∞ and satisfy certain monotonicity conditions, and thereby obtain new results both about the well-posedness of such problems and about the resonances of acoustic transmission problems (i.e. A and n discontinuous) where the transmission interfaces are only assumed to be C0 and star-shaped; the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for C∞ convex interfaces with strictly positive curvature.

A better-performing Q-learning game-theoretic distributed routing for underwater wireless sensor networks

Underwater sensor networks have recently emerged as a promising networking technique for various underwater applications. However, the acoustic routing of underwater sensor networks in the aquatic environment presents challenges in terms of dynamic structure, high rates of energy consumption, long propagation delay, and narrow bandwidth. Therefore, it is difficult to adapt traditional routing protocols, which are known to be reliable in terrestrial wireless networks. In this study, we focus on the development of novel routing algorithms to tackle acoustic transmission problems in underwater sensor networks. The proposed scheme is based on reinforcement learning and game theory and is designed as a routing game model to provide an effective packet-forwarding mechanism. In particular, our Q-learning game paradigm captures the dynamics of the underwater sensor networks system in a decentralized, distributed manner. The results of a performance simulation analysis show that the proposed scheme can outperform existing schemes while displaying balanced system performance in terms of energy efficiency and underwater sensor networks throughput.

Acoustic wave incidence on a multilayer medium containing a bubbly fluid layer

The problem of acoustic wave reflection and transmission through a multilayer medium containing a bubbly fluid layer is considered. For the water-water with air bubbles-water model the wave reflection and transmission coefficients are calculated and compared with the experimental data. The problem parameters, at which these coefficients take extremum values, are determined. The influence of vapor within the bubbles on the acoustic wave transmission through a layer of a fluid with the vapor-gas bubbles is shown.

A priori error estimates of the DtN-FEM for the transmission problem in acoustics

This paper is concerned with a variational approach solving the two-dimensional acoustic transmission problems. The original problem is reduced to an equivalent nonlocal boundary value problem by introducing an exact Dirichlet-to-Neumann (DtN) mapping in terms of Fourier expansion series on an artificial boundary. Uniqueness and existence of solutions in appropriate Sobolev spaces are established for the corresponding variational problem and its modification due to the truncation of DtN mapping. A priori error estimates containing the effects of both element meshsize and truncation order of series for the finite element approximation are derived. Numerical experiments are also presented to illustrate the efficiency and accuracy of the numerical scheme.

Characterization of compressed earth blocks using low frequency guided acoustic waves.

The objective of this work was to analyze the influence of compaction pressure on the intrinsic acoustic parameters (porosity, tortuosity, air-flow resistivity, viscous, and thermal characteristic lengths) of compressed earth blocks through their identification by solving an inverse acoustic wave transmission problem. A low frequency acoustic pipe (60-6000 Hz of length 22 m, internal diameter 3.4 cm) was used for the experimental characterization of the samples. The parameters were identified by the minimization of the difference between the transmissions coefficients data obtained in the pipe with that from an analytical interaction model in which the compressed earth blocks were considered as having rigid frames. The viscous and thermal effects in the pores were accounted for by employing the Johnson-Champoux-Allard-Lafarge model. The results obtained by inversion for high-density compressed earth blocks showed some discordance between the model and experiment especially for the high frequency limit of the acoustic characteristics studied. This was as a consequence of applying high compaction pressure rendering them very highly resistive therefore degrading the signal-to-noise ratios of the transmitted waves. The results showed that the airflow resistivity was very sensitive to the degree of the applied compaction pressure used to form the blocks.

A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems

The present work is concerned with the shape reconstruction problem of isotropic elastic inclusions from far-field data obtained by the scattering of a finite number of time-harmonic incident plane waves. This paper aims at completing the theoretical framework which is necessary for the application of geometric optimization tools to the inverse transmission problem in elastodynamics. The forward problem is reduced to systems of boundary integral equations following the direct and indirect methods initially developed for solving acoustic transmission problems. We establish the Fréchet differentiability of the boundary to far-field operator and give a characterization of the first Fréchet derivative and its adjoint operator. Using these results we propose an inverse scattering algorithm based on the iteratively regularized Gauß–Newton method and show numerical experiments in the special case of star-shaped obstacles.

Regularized Combined Field Integral Equations for Acoustic Transmission Problems

We present a new class of well-conditioned integral equations for the solution of two and three dimensional scattering problems by homogeneous penetrable scatterers. Our novel boundary integral equations result from using regularizing operators which are suitable approximations of the admittance operators that map the transmission boundary conditions to the exterior and, respectively, interior Cauchy data on the interface between the media. We refer to these regularized boundary integral equations as generalized combined source integral equations (GCSIE). The admittance operators can be expressed in terms of Dirichlet-to-Neumann operators and their inverses. We construct our regularizing operators in terms of simple boundary layer operators with complex wavenumbers. The choice of complex wavenumbers in the definition of the regularizing operators ensures the unique solvability of the GCSIE. The GCSIE are shown to be integral equations of the second kind for regular enough interfaces of material discontinu...

The factorization method for the acoustic transmission problem

In this work, the shape reconstruction problem of acoustically penetrable bodies from the far-field data corresponding to time-harmonic plane wave incidence is investigated within the framework of the factorization method. Although the latter technique has received considerable attention in inverse scattering problems dealing with impenetrable scatterers and it has not been elaborated for inverse transmission problems with the only exception being a work by the first two authors and co-workers. We aim to bridge this gap in the field of acoustic scattering; the paper on one hand focuses on establishing rigorously the necessary theoretical framework for the application of the factorization method to the inverse acoustic transmission problem. The main outcome of the investigation undertaken is the derivation of an explicit formula for the scatterer's characteristic function, which depends solely on the far-field data feeding the inverse scattering scheme. Extended numerical examples in three dimensions are also presented, where a variety of different surfaces are successfully reconstructed by the factorization method, thus, complementing the method's validation from the computational point of view.

COMPARATIVE STUDY OF TWO DIFFERENT FMM–BEM METHODS IN SOLVING 2-D ACOUSTIC TRANSMISSION PROBLEMS WITH A MULTILAYERED OBSTACLE

The fast multipole method (FMM) is an effective approach for accelerating the computation efficiency of the boundary element method (BEM) in solving problems that are computationally intensive. This paper presents two different BEMs, i.e., Kress' and Seydou's methods, for solving two-dimensional (2D) acoustic transmission problems with a multilayered obstacle, along with application of the FMM to solution of the related boundary integral equations. Conventional BEM requires O(MN2) operations to compute the equations for this problem. By using the FMM, both the amount of computation and the memory requirement of the BEM are reduced to order O(MN), where M is the number of layers of the obstacle. The efficiency and accuracy of this approach in dealing with the acoustic transmission problems containing a multilayered obstacle are demonstrated in the numerical examples. It is confirmed that this approach can be applied to solving the acoustic transmission problems for an obstacle with multilayers.