Acoustic Wave Propagation Problems Research Articles

Numerical simulation of wave propagation by using a hybrid method with an arbitrary order accuracy in both spatial and temporal approximations

This paper introduces an innovative numerical methodology designed to achieve high precision solution of acoustic wave propagation problem in isotropic material. During the temporal discretization process, the Krylov deferred correction (KDC) technique is employed, wherein a new variable is introduced to handle the second-order time derivative in the governing equations. An improved strategy is adopted for precisely implementing boundary conditions. Following this, the arbitrary-order generalized finite difference method (GFDM) is employed to simulate the transformed boundary value problem at each time step, enabling our framework to select the Taylor series expansion order arbitrarily. Ultimately, a hybrid numerical approach for wave problems is developed to achieve arbitrary-order accuracy in both spatial and temporal approximations. The developed scheme undergoes validation through four different numerical experiments in two- or three-dimension, wherein the numerical solutions obtained are compared with either analytical solutions or results from COMSOL software.

A novel hybrid SBM-MFS methodology for acoustic wave propagation problems

In this paper, a novel hybrid meshless approach that combines the singular boundary method (SBM) and the method of fundamental solutions (MFS) to deal with two-dimensional (2D) exterior acoustic wave propagation problems is proposed and studied. The methodology is particularly devised to solve problems with complex boundary geometries containing geometric singularities such as corners and sharp edges. It employs the SBM to model intricate segments of these geometries and the MFS for the smooth ones. The proposed hybrid SBM-MFS method is studied in a 2D context in the framework of three benchmark examples involving acoustic radiation problems of circular-, square- and L-shaped objects in a full-space acoustic medium. In addition, the applicability of the proposed hybrid SBM-MFS methodology to predict the acoustic performance of a T-shaped thin barrier is also investigated. These examples are specifically designed to assess the feasibility, validity and accuracy of the hybrid SBM-MFS approach in comparison with the available analytical solutions and alternative numerical strategies such as the MFS, the SBM and the boundary element method (BEM). Numerical simulations demonstrate that the proposed method can match the level of accuracy of the MFS while keeps the robustness of the SBM when dealing with complex geometries, overcoming one the most important drawbacks of the traditional MFS. Moreover, the proposed hybrid SBM-MFS naturally avoids the non-uniqueness problem arising at the fictitious eigenfrequencies associated with the corresponding interior problems, a feature that neither the SBM nor the BEM inherently possesses.

An Efficient Method for Solving Problems of Acoustic Scattering on Three-Dimensional Transparent Structures

The article contains a study of methods for solving integral equations in the context of acoustic problems. The methodology considered is applied to describe acoustic wave propagation and scattering. Efficient discretization methods are used together with iterative methods to solve the operator equations, including an apparatus for fast multiplication of the resulting post-discretization Toeplitz matrices by a vector using the fast Fourier transform. The theoretical analysis of the proposed numerical algorithm demonstrates its efficiency in terms of the required number of arithmetic operations and the memory footprint of the computing system. The presented numerical simulation demonstrates the possibility of solving the problem of acoustic wave propagation in transparent media using the proposed methods. A visualization of the obtained solutions for a practical problem with a high level of discretization of the solution volume domain is also presented.

The edge-based smoothed FEM with [formula omitted]-Bathe implicit temporal discretization scheme for the analyses of underwater wave propagation problems

The underwater acoustic wave propagation problem is an important issue in ocean engineering field. The classical finite element approach with edge-based smoothed gradient smoothing technique is incorporated into the novel ρ∞-Bathe implicit direct time integration scheme in this paper for solving underwater transient acoustic wave propagation problems. This edge-based smoothed FEM (ES-FEM) and ρ∞-Bathe method are employed to perform the space and time domain discretization, respectively. Due to the appropriate softening effects from the ES-FEM, the space discretization errors can be markedly reduced. Meanwhile, the possible numerical error related to the temporal discretization also can be significantly suppressed by the novel ρ∞-Bathe method which possesses the controllable numerical damping effects. Through detailed analysis of the total dispersion error, we find that the numerical results from the proposed scheme can converge monotonically to the exact solutions when employing the decreasing temporal discretization intervals and the possible numerical anisotropy effects are also effectively suppressed. These excellent numerical performance clearly distinguish the present method from the conventional approaches and make it to be particularly suitable for complicated acoustic wave propagation analysis. Finally, various typical numerical experiments are performed to illustrate the capacities of the proposed scheme in tackling underwater transient acoustic wave propagation problems.

Level set-based shape optimization of deformable structures for achieving desired sound transmission and reflective responses

In this paper, we propose a level set-based shape optimization method for acoustic wave propagation problems with a deformable structure. First, we propose a mathematical model for acoustic wave propagation with a deformed structure based on coordinate transformation and the Eulerian approach. Next, we formulate the shape optimization problem and perform sensitivity analysis based on the shape derivative concept. We then construct an optimization algorithm based on the framework of a level set-based shape and topology optimization method. Finally, we provide two-dimensional optimization examples that demonstrate the effectiveness of our proposed method in providing optimized designs with desired functionality, while considering the structural deformation.

Analyzing the acoustic wave propagation characteristics in discontinuous bifurcated waveguide

This article examines the dispersion characteristics and energy distribution in different regions of the bifurcated waveguide with vertical step discontinuities. In this research study, we have considered mathematical model of noise reduction devices, specifically analyzing the acoustic wave propagation problem in a bifurcated waveguide with elastic membrane outer plates and rigid inner plates. Mode-matching (MM) technique is employed to match the eigenfunctions at the interface and analyzed the reflection and transmission amplitudes in all regions. Through MM scheme an infinite system of equations has been obtained for these boundary conditions by dividing the structure into three regions and finding dispersion relations and eigenfunctions using separation of variables and the corresponding orthogonality relations. Because of the flexible nature of the boundary surfaces, the acquired eigenfunctions and the dispersion relation do not satisfy the standard orthogonality relation. To resolve this difficulty, we have established a modified orthogonality relation to take into account the flexible nature of the boundary surfaces. With the implication of physical connections, the uniqueness of the problem has been ensured, and zero displacement and zero gradient edge conditions on bending surfaces have been considered to present results. Energy expressions are obtained showing how the energy is distributed in different sections of the waveguide, which is consistent with the energy conservation law. Finally, the numerical discussion illustrates the quantitative behavior of the physical parameters and summarizes the significant contribution of the study by presenting graphical data.

CABARET on rotating meshes

The Compact Accurately Boundary-Adjusting high-REsolution Technique (CABARET) method for 3D gas dynamics equations is generalised to rotating hexahedral mesh zones using the sliding interface approach, which extends applications of the original algorithm to a wide range of unsteady rotor flow problems. Important properties of the suggested extension of the CABARET scheme include the preservation of good linear dispersion and dissipation properties of the original CABARET method on fixed grids. To strictly preserve flux conservation at the sliding interface, the so-called supermesh approach is implemented that uses the projection of the contact faces on both sides of the sliding interface onto an intermediate virtual surface. The implementation is based on applying the sub-cell and sub-face solution reconstruction on the sliding interface of hexahedral meshes rather than deforming unstructured meshes. The sub-faces are defined by constructing projection overlap polygons using the Sutherland-Hodgman clipping algorithm. A local characteristic flux splitting is applied at each overlap polygon face, while updating the local stencil connectivity table at each time step. To further improve the dispersion properties of the CABARET grid at small Courant–Friedrichs–Lewy numbers, the dispersion-improved version of the CABARET method previously developed on rectangular Cartesian meshes is extended to non-uniform curvilinear meshes typical of open-rotor applications. Numerical examples are provided for the analytical problem of normal acoustic wave propagation through a rotating cylindrical grid zone embedded in an outer stationary grid. The error decay rates are calculated and compared with the theoretical order of CABARET convergence. In addition, the same algorithm accelerated on Graphics Processing Units (GPUs) is applied for simulations the flow around a generic two-bladed rotor in hover in the Wall Model Large Eddy Simulation framework. For the latter case, flow solutions are analysed, and their numerical grid sensitivity is discussed.

Partially pivoted ACA based acceleration of the energetic BEM for time-domain acoustic and elastic waves exterior problems

We consider acoustic and elastic wave propagation problems in 2D unbounded domains, reformulated in terms of space-time Boundary Integral Equations (BIEs). For their numerical solution, we employ a weak formulation related to the energy of the system and we discretize the weak problems by a Galerkin-type Boundary Element Method (BEM): this approach, called Energetic BEM, has revealed accurate and stable even on large time intervals of analysis. In particular, it results that, when standard Lagrangian basis functions are considered, the BEM matrices have Toeplitz lower triangular block structure, where blocks for growing time become fully populated; hence the overall memory cost of the energetic BEM is O(M2N), M and N being the number of the space degrees of freedom and the total number of performed time steps, respectively. This drawback prevents the application of such method to large scale problems. As a possible remedy, we propose a fast technique based on the Adaptive Cross Approximation (ACA). The core of this procedure is the approximation of sufficiently large time blocks of the energetic BEM matrix through the partially pivoted ACA algorithm, which allows to compute only few of the original entries. This leads to reduced assembly time, which for the energetic BEM is generally relevant, coupled with reduced memory storage requirements. Additionally, the consequent acceleration of the matrix/vector multiplication together with a marching on time procedure, leads to remarkable reduction of the computational solution time. The effectiveness of the proposed method is theoretically proved and several numerical results are presented and discussed.

Transient Acoustic Wave Propagation Problems in Multilayered Pavement Using a Time Discontinuous Galerkin Finite Element Method

This paper presents a modified time discontinuous Galerkin finite element method (MDGFEM) for transient acoustic wave propagation problem of multilayered pavement. The pavement consists of cement concrete pavement, semi-rigid base, and natural soil. The multilayered pavement is modeled as poroelastic mediums and assumed to be water-saturated. The well-known generalized Biot’s theory is employed to describe the wave propagation problem. The present MDGFEM, based on the artificial damping scheme, employs the Hermite (P3) functions and the linear (P1) shape functions to interpolate the global nodal vector and its temporal gradient respectively in a time step. Numerical results of 1D and 2D problems show that the MDGFEM can filter out spurious numerical oscillations before and after waves, boundaries of the hole, and the interface between the layers more effectively for the propagation of acoustic waves in multilayered pavement. Compared with widely used time-continuous methods such as the Newmark method, the method proposed in this paper presents better capabilities in the fluid–structure interaction behavior of multilayer pavements and provides a more accurate solution, which contributes to the further development of non-destructive testing of multilayer pavement structures.

Analysis of the interior acoustic wave propagation problems using the modified radial point interpolation method (M-RPIM)

Although the radial point interpolation method (RPIM), which is a typical meshless numerical technique, usually behaves much better than the conventional FEM in addressing the numerical dispersion error issue for acoustic computation and more accurate solutions can generally be yielded with the identical node distributions, the related numerical error still can not be completely removed and further improvements are still required. In this work, we proposed a modified RPIM (M-RPIM) to enhance the abilities of the original RPIM in suppressing the numerical dispersion error. In this M-RPIM, a simple and straightforward scheme is employed to ensure that the integrands in performing the numerical integration are continuously differentiable, while in the original RPIM the quadrature cells usually do not align with the shape function supports and then results in the discontinuously differentiable numerical approximation in the quadrature cells, hence considerable numerical integration error can be generated. Since the discontinuously differentiable entities in the system stiffness matrix can be completely avoided in the present M-RPIM, it is found that the numerical dispersion can be markedly suppressed and more accurate numerical solutions can be yielded than the original RPIM in solving acoustic problems. It should be pointed out that the numerical treatments and conclusions in this work are also applicable for most of other meshfree approximations which are similar to the RPIM.

Convergence analysis on seismic tomography for inverse problems of acoustic wave propagation

Convergence analysis on seismic tomography for inverse problems of acoustic wave propagation

A level-set approach based on reaction–diffusion equation applied to inversion problems in acoustic wave propagation

In this work, a topology optimization procedure based on the level-set method is applied to the solution of inverse problems for acoustic wave propagation in the time-domain. In this class of inverse problems the presence of obstacles in a background medium must be identified. Obstacles and background are defined by means of a level-set function that evolves by following the solution of a reaction–diffusion equation. Within this approach, no initial guess for the topology nor level-set reinitialization procedures are necessary, contrary to what is commonly observed when the Hamilton–Jacobi equation is used. The objective function is defined as the domain and time integration of the squared difference between experimental and simulation pressure signals. The finite element method is used for the spatial and level-set function discretizations and a time-marching procedure (Newmark scheme) is used to solve the wave propagation problem, as well as the adjoint problem for the sensitivity analysis. Both procedures provide the information needed to define the velocity field for the level set evolution. Results show that the proposed technique is capable to find the location and shape of obstacles within a background medium. Systematic tests show that, as expected, the distribution of sources and receivers shows to have a major influence on the final solution. Results also reproduce known difficulties; when the so called inverse crime is avoided, the identification procedure worsens its performance. Filters and smoothing are among different features that deserve further investigation. Although the formulation presented here focuses on the acoustic wave propagation problem, its extension to wave propagation in elastic media is straightforward.

The method of finite spheres in acoustic wave propagation through nonhomogeneous media: Inf-sup stability conditions

When the method of finite spheres is used for the solution of time-harmonic acoustic wave propagation problems in nonhomogeneous media, a mixed (or saddle-point) formulation is obtained in which the unknowns are the pressure fields and the Lagrange multiplier fields defined at the interfaces between the regions with distinct material properties. Then certain inf-sup conditions must be satisfied by the discretized spaces in order for the finite-dimensional problems to be well-posed. We discuss in this paper the analysis and use of these conditions. Since the conditions involve norms of functionals in fractional Sobolev spaces, we derive ‘stronger’ conditions that are simpler in form. These new conditions pave the way for the inf-sup testing, a tool for assessing the stability of the discretized problems.

Non-stiff boundary and interface penalties for narrow-stencil finite difference approximations of the Laplacian on curvilinear multiblock grids

The Laplacian appears in several partial differential equations used to model wave propagation. Summation-by-parts–simultaneous approximation term (SBP-SAT) finite difference methods are often used for such equations, as they combine computational efficiency with provable stability on curvilinear multiblock grids. However, the existing SBP-SAT discretization of the Laplacian quickly becomes prohibitively stiff as grid skewness increases. The stiffness stems from the SATs that impose inter-block couplings and Dirichlet boundary conditions. We resolve this issue by deriving stable SATs whose stiffness is almost insensitive to grid skewness. The new discretization thus allows for large time steps in explicit time integrators, even on very skewed grids. It also applies to the variable-coefficient generalization of the Laplacian. We demonstrate the efficacy and versatility of the new SATs by applying them to acoustic wave propagation problems inspired by marine seismic exploration and infrasound monitoring of volcanoes.

Analysis of acoustic wave propagation in composite laminates via a spectral element method

AbstractThe analysis of the acoustic or elastic wave propagation phenomena in the time‐domain is computationally expensive. An often‐cited rule of thumb is to use at least ten nodes per wavelength if using quadratic shape functions, and even more nodes are necessary in order to achieve highly accurate numerical results. For large‐scale problems, this results in a large system of linear algebraic equations, which must be solved for several thousands of time‐steps. This requires a vast amount of the computing time. In this paper, a spectral element method (SEM), which is a variant of the finite element method (FEM) by using high‐order shape functions with non‐equidistant node distributions, is presented in order to investigate the acoustic wave propagation problems taking the fluid‐structure interaction into account with a minimum of required degrees of freedoms (DOFs). Hence, the computing time for solving the system of linear algebraic equations is reduced while maintaining a high accuracy. The element matrices are computed using the Lobatto quadrature method, which results in a diagonal element mass matrix. For fluid‐structure interaction problems in acoustics, a non‐symmetric sparse mass matrix is obtained. By using an explicit time‐integration scheme only the mass matrix becomes the coefficient matrix of the linear system and, due to its high sparsity, the solution process is very efficient. In this work, the SEM is implemented and applied for the analysis of the transient behavior of the laminated composite structures excited by acoustic waves. Numerical examples are presented and discussed to show the efficiency of the SEM in the time‐domain for applications in acoustic wave propagation problems considering the fluid‐structure interaction.

Some dynamic features of ferroelectric thin films

The dynamic problem of the surface acoustic wave propagation in a thin-film structure is investigated. The propagation of plane waves in the film-substrate structure is considered. Thin film is made of barium titanate oxide, and solid substrate is made of magnesium oxide. Ferroelectric film is subjected to misfit strain because of the deposition. Assumed that the solid substrate retains its properties due to the large difference in the hardness of the materials. Built Green function, obtained phase velocities and electromechanical coupling coefficients in the C-phase of BaTiO3. The present investigation is of practical interest for developing high-frequencies smart ferroelectric devices for engineering applications.

Exact solution to the problem of acoustic wave propagation in a semi-infinite waveguide with flexible walls

Two boundary value problems for the Helmholtz equation in a semi-infinite strip are considered. The main feature of these problems is that, in addition to the function and its normal derivative on the boundary, the functionals of the boundary conditions possess tangential derivatives of the second and fourth orders. Also, the setting of the problems is complemented by certain edge conditions at the two vertices of the semi-strip. The problems model wave propagation in a semi-infinite waveguide with membrane and plate walls. A technique for the exact solution of these fluid–structure interaction problems is proposed. It requires application of two Laplace transforms with respect to both variables. The parameter of the second transform is a certain function of the first Laplace transform parameter. Ultimately, this method yields two scalar Riemann–Hilbert problems with the same coefficient and different right-hand sides. The dependence of the existence and uniqueness results of the physical model problems on the index of the Riemann–Hilbert problem is discussed, and numerical results for the membrane walls model are reported.

An Immersed Discontinuous Galerkin Method for Acoustic Wave Propagation in Inhomogeneous Media

We present an immersed discontinuous Galerkin finite element method on Cartesian meshes for two dimensional acoustic wave propagation problems in inhomogeneous media where elements are allowed to b...

On perfectly matched layers for discontinuous Petrov–Galerkin methods

In this article, several discontinuous Petrov–Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov–Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two- and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified.

Metric-based anisotropic mesh adaptation for 3D acoustic boundary element methods

This paper details the extension of a metric-based anisotropic mesh adaptation strategy to the boundary element method for problems of 3D acoustic wave propagation. Traditional mesh adaptation strategies for boundary element methods rely on Galerkin discretizations of the boundary integral equations, and the development of appropriate error indicators. They often require the solution of further integral equations. These methods utilize the error indicators to mark elements where the error is above a specified tolerance and then refine these elements. Such an approach cannot lead to anisotropic adaptation regardless of how these elements are refined, since the orientation and shape of current elements cannot be modified.In contrast, the method proposed here is independent of the discretization technique (e.g., collocation, Galerkin). Furthermore, it completely remeshes at each refinement step, altering the shape, size, and orientation of each element according to an optimal metric based on a numerically recovered Hessian of the boundary solution. The resulting adaptation procedure is truly anisotropic and independent of the complexity of the geometry. We show via a variety of numerical examples that it recovers optimal convergence rates for domains with geometric singularities. In particular, a faster convergence rate is recovered for scattering problems with complex geometries.