Two-dimensional Acoustic Wave Equation Research Articles

Physics-informed neural networks for transcranial ultrasound wave propagation

Transcranial ultrasound imaging has been playing an increasingly important role in the non-invasive treatment of brain disorders. However, the conventional mesh-based numerical wave solvers, which are an integral part of imaging algorithms, suffer from limitations such as high computational cost and discretization error in predicting the wavefield passing through the skull. In this paper, we explore the use of physics-informed neural networks (PINNs) for predicting the transcranial ultrasound wave propagation. The wave equation, two sets of time snapshots data and a boundary condition (BC) are embedded as physical constraints in the loss function during training. The proposed approach has been validated by solving the two-dimensional (2D) acoustic wave equation under three increasingly complex spatially varying velocity models. Our cases demonstrate that due to the meshless nature of PINNs, they can be flexibly applied to different wave equations and types of BCs. By adding physics constraints to the loss function, PINNs can predict wavefields far outside the training data, providing ideas for improving the generalization capability of existing deep learning methods. The proposed approach offers exciting perspectives because of the powerful framework and simple implementation. We conclude with a summary of the strengths, limitations and further research directions of this work.

Over-reflection of acoustic waves by supersonic exponential boundary layer flows

The two-dimensional acoustic wave equation for inviscid compressible boundary layer flows, i.e. the Pridmore-Brown equation with an exponential velocity profile for homentropic flows, is studied for the reflection and over-reflection of acoustic waves based on the exact solution in terms of the confluent Heun function. The reflection coefficient $R$ , which is the ratio of the amplitude of the reflected to that of the incoming acoustic wave, is determined as a function of the streamwise wavenumber $\alpha$ , the Mach number $M$ and the incident angle $\phi$ of the acoustic waves. Over-reflection refers to $R>1$ , i.e. the reflected wave has a larger amplitude than the incident wave. We prove that, in the supersonic context, energy is always transferred from the base flow to the reflected wave, i.e. $R<1$ does not exist. Meanwhile, this fact is intimately linked to the critical layer. We show that the presence of the critical layer leads to an optimal energy exchange from the base flow into the acoustic wave, i.e. the critical layer ensures $R>1$ . In our analysis, we observe a special phenomenon, resonant over-reflection, which is proven to be closely related to resonant frequencies $\omega _r$ of unstable modes of the temporal stability of the base flow. At resonant frequencies of the first unstable mode, the over-reflection coefficient exhibits an unusual peak in an extremely narrow frequency interval. The maximum values of these peaks are largely synchronized with the variation of the growth rate $\omega _i$ of the unstable modes. In addition, resonant over-reflection appears also at resonant frequencies of other higher unstable modes, but their peaks of the over-reflection coefficient are always smaller than that induced by the first unstable mode.

A parallel hybrid implementation of the 2D acoustic wave equation

Abstract In this paper, we propose a hybrid parallel programming approach for a numerical solution of a two-dimensional acoustic wave equation using an implicit difference scheme for a single computer. The calculations are carried out in an implicit finite difference scheme. First, we transform the differential equation into an implicit finite-difference equation and then using the alternating direction implicit (ADI) method, we split the equation into two sub-equations. Using the cyclic reduction algorithm, we calculate an approximate solution. Finally, we change this algorithm to parallelize on graphics processing unit (GPU), GPU + Open Multi-Processing (OpenMP), and Hybrid (GPU + OpenMP + message passing interface (MPI)) computing platforms. The special focus is on improving the performance of the parallel algorithms to calculate the acceleration based on the execution time. We show that the code that runs on the hybrid approach gives the expected results by comparing our results to those obtained by running the same simulation on a classical processor core, Compute Unified Device Architecture (CUDA), and CUDA + OpenMP implementations.

Study of transmission loss model using the phase of the source

MOE(Measure Of Effectiveness analysis) is needed to predict the sonar detection performance and maneuvering tactics in the ocean. To effect analysis, the detection probability is calculated by sonar characteristics and sound velocity distribution. The detection probability considering the sonar characteristics and the sound velocity distribution can be derived by using the FOM (figure of merit) and the transmission loss. Generally, the transmission loss used to calculate the detection probability is obtained by calculating the path of ray and the attention for the frequency of the source, sound velocity distribution. Although these models have advantages of being efficient and fast, it cannot consider the effect of constructive and destructive interference for the source. In this study, the propagation phenomena for source were calculated by solving the two-dimensional acoustic wave equation to obtain transmission loss. For the computational efficiency, the proposed method uses the geometric spreading correction for the amplitude correction of the three dimensions. The calculated transmission loss is compared with transmission loss using BELLHOP, which is a sound transmission model. As a result, it is confirmed that the proposed method can consider the phase of the source.

A Modified PML Acoustic Wave Equation

In this paper, we consider a two-dimensional acoustic wave equation in an unbounded domain and introduce a modified model of the classical un-split perfectly matched layer (PML). We apply a regularization technique to a lower order regularity term employed in the auxiliary variable in the classical PML model. In addition, we propose a staggered finite difference method for discretizing the regularized system. The regularized system and numerical solution are analyzed in terms of the well-posedness and stability with the standard Galerkin method and von Neumann stability analysis, respectively. In particular, the existence and uniqueness of the solution for the regularized system are proved and the Courant-Friedrichs-Lewy (CFL) condition of the staggered finite difference method is determined. To support the theoretical results, we demonstrate a non-reflection property of acoustic waves in the layers.

Fourth Order Finite Difference Methods for the Wave Equation with Mesh Refinement Interfaces

We analyze two types of summation-by-parts finite difference operators for approximating the second derivative with variable coefficient. The first type uses ghost points, while the second type does not use any ghost points. A previously unexplored relation between the two types of summation-by-parts operators is investigated. By combining them we develop a new fourth order accurate finite difference discretization with hanging nodes on the mesh refinement interface. We take the model problem as the two-dimensional acoustic wave equation in second order form in terms of acoustic pressure, and prove energy stability for the proposed method. Compared to previous approaches using ghost points, the proposed method leads to a smaller system of linear equations that needs to be solved for the ghost point values. Another attractive feature of the proposed method is that the explicit time step does not need to be reduced relative to the corresponding periodic problem. Numerical experiments, both for smoothly varying and discontinuous material properties, demonstrate that the proposed method converges to fourth order accuracy. A detailed comparison of the accuracy and the time-step restriction with the simultaneous-approximation-term penalty method is also presented.

Multiple band gaps of a ferroelectric based 2D-phononic crystal slab

In this work, we study the band gap properties of a two-dimensional phononic crystal composed of a hexagonal cavity inside a circular enclosure embedded in a homogeneous matrix. A simple two-dimensional structure with a cavity that is modeled with an opening in one side that allows air to enter was studied theoretically using the finite element method (FEM). The properties of acoustic wave propagation in a phononic crystal consisting of periodic inclusions with a cavity placed in an air host material are analyzed. The phononic crystal dispersion relations, transmission loss spectrum, and displacement fields of the eigenmodes of this phononic crystal are investigated. The pressure field distribution inside the domain is solved parametrically through a two-dimensional acoustic wave equation in the horizontal plane as a function of frequency. We have found that several complete band gaps with a variable bandwidth exist for the acoustic waves of longitudinal polarization.

Error Analysis of Chebyshev Spectral Element Methods for the Acoustic Wave Equation in Heterogeneous Media

This work concerns the error analysis of the spectral element method with Gauss–Lobatto–Chebyshev collocation points with the implicit Newmark average acceleration scheme for the two-dimensional acoustic wave equation. The analysis is restricted to homogeneous Dirichlet boundary conditions, constant compressibility and variable density. The proposed error estimates are optimal with respect to the mesh parameter although suboptimal on the polynomial degree. Numerical examples illustrate the theoretical results.

Error analysis of the spectral element method with Gauss–Lobatto–Legendre points for the acoustic wave equation in heterogeneous media

We present the error analysis of a high-order method for the two-dimensional acoustic wave equation in the particular case of constant compressibility and variable density. The domain discretization is based on the spectral element method with Gauss–Lobatto–Legendre (GLL) collocation points, whereas the time discretization is based on the explicit leapfrog scheme. As usual, GLL points are also employed in the numerical quadrature, so that the mass matrix is diagonal and the resulting algebraic scheme is explicit in time. The analysis provides an a priori estimate which depends on the time step, the element length, and the polynomial degree, generalizing several known results for the wave equation in homogeneous media. Numerical examples illustrate the validity of the estimate under certain regularity assumptions and provide expected error estimates when the medium is discontinuous.

A fourth order accuracy summation-by-parts finite difference scheme for acoustic reverse time migration in boundary-conforming grids

The fourth order accuracy finite difference scheme is known advantageous in reducing memory and improving efficiency. Summation-by-parts finite difference operator is a natural way for wavefield simulation in complicated domains containing surface topography and irregular interfaces. The application of summation-by-parts method guarantees the stability of numerical approximation for heterogeneous media on curvilinear grids. This paper extends the second order summation-by-parts finite difference method to the fourth order case for the discretization of acoustic wave equation and perfect matched layer in boundary-conforming grids. In particular, the implementation of the fourth order method for wavefield simulation and reverse time migration in complicated domains can significantly improve the efficiency and decrease the storage. The elliptic method is applied for boundary-conforming grid generation in complicated domains. Under such grids, the two-dimensional acoustic wave equation in second order displacement formulation is compactly reformulated for forward modeling and reverse time migration, and the symmetric and compact form of perfectly matched layers expressed in a curvilinear coordinate system are applied to suppress artificial reflections. The discretizations of the acoustic wave equation and perfectly matched layer formula are fourth and second order accuracy in space and time respectively, where the spatial discretization satisfies the principle of summation-by-parts and is stable. Numerical experiments are presented to compare the accuracy of the second with fourth order summation-by-parts finite difference methods and to evaluate the efficiency of reverse time migration by using these two methods. As well, comparisons are performed between the fourth order accuracy summation-by-parts finite difference method and central finite difference method to illustrate the stability superiority of summation-by-parts operators.

Acoustic reverse time migration and perfectly matched layer in boundary-conforming grids by elliptic method

Traditionally, finite difference method is chosen as a fast and accurate solution method for numerical simulation of wave equation. However, finite difference method faces obstacles when surface topography and irregular interfaces exist. Boundary-conforming grids by the elliptic method provide an optimal choice for finite difference wavefield simulation in complicated domains containing not only surface topography but also irregular interfaces. By such grids, the calculations of spatial derivatives are transformed by a chain rule into those in the regular computational space, where traditional finite difference schemes are still applicable. Boundary-conforming grids are superior to other irregular grid methods, such as interpolation method, mapping method and unstructured grids, on the aspects of generality, in accuracy and stability. This paper comprehensively applies the elliptic method and acoustic wave equation simulation, reverse time migration, perfectly matched layers in such boundary-conforming grids. The two-dimensional acoustic wave equation is compactly reformulated in boundary-conforming grids by the elliptic method for forward modeling and reverse time migration, and the symmetric and compact form of perfectly matched layers expressed in curvilinear coordinate system are applied to suppress artificial reflections. A stable and explicit second order accuracy finite difference method is used for discretization. Two models are presented to evaluate the ability of boundary-conforming grids to deal with surface topography and complex interfaces, and to demonstrate the feasibility of wavefield propagation and reverse time migration. Comparisons between the numerical simulations with and without the perfectly matched layers are performed to show the effect of the reformulated perfectly matched layer.

A nonlinear multigrid method for inversion of two-dimensional acoustic wave equation

Abstract. We investigate the problem of estimating the velocity in a two-dimensional acoustic wave equation, which plays an important role in geological survey. The forward problem is discretized using finite-difference methods and the estimation is formulated as a least-square minimization problem with a regularization term. To reduce the computational burden, a nonlinear multigrid method is applied to solve this inverse problem. In the multigrid inversion process, in order to make the objective functionals at different scales compatible, they are dynamically adjusted. In this way, the necessary condition of “the optimal solution should be the fixed point of multigrid inversion” can be met. The stable and fast regularized Gauss–Newton method is applied to each grid. The results of numerical simulations indicate that the proposed method can effectively reduce the required computation, improve the inversion results, and have the anti-noise ability.

An efficient fourth-order low dispersive finite difference scheme for a 2-D acoustic wave equation

In this paper, we propose an efficient fourth-order compact finite difference scheme with low numerical dispersion to solve the two-dimensional acoustic wave equation. Combined with the alternating direction implicit (ADI) technique and Padé approximation, the standard second-order finite difference scheme can be improved to fourth-order and solved as a sequence of one-dimensional problems with high computational efficiency. However such compact higher-order methods suffer from high numerical dispersion. To suppress numerical dispersion, the compact and non-compact stages are interlinked to produce a hybrid scheme, in which the compact stage is based on Padé approximation in both y and temporal dimensions while the non-compact stage is based on Padé approximation in y dimension only. Stability analysis shows that the new scheme is conditionally stable and superior to some existing methods in terms of the Courant–Friedrichs–Lewy (CFL) condition. The dispersion analysis shows that the new scheme has lower numerical dispersion in comparison to the existing compact ADI scheme and the higher-order locally one-dimensional (LOD) scheme. Three numerical examples are solved to demonstrate the accuracy and efficiency of the new method.

High accuracy wave simulation – Revised derivation, numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation

The nearly analytic integration discrete (NAID) method for solving the two-dimensional acoustic wave equation has been fully mathematically revised, analyzed and tested. The NAID method is an alternative numerical modeling method for generating synthetic seismograms. The acoustic wave equation is first transformed into a system of first-order ordinary differential equations (ODEs) with respect to time variable t, and then directly integrated at a small time interval of [tn,tn+1] to obtain semi-discrete ordinary differential equations. The integral kernel is expanded into a truncated Taylor series, to which the integration operator is explicitly applied. The high-order temporal derivatives involved in the integral kernel are replaced by high-order spatial derivatives, which then are approximately calculated as a weighted linear combination of the displacement, the particle-velocity, and their spatial gradients. In this article, we investigate the theoretical properties of the revised NAID method, including the discrete error and the stability criteria. Numerical results for constant and layered velocity models show that, comparing to the Lax–Wendroff correction (LWC) scheme and the staggered-grid finite difference method, the NAID method can effectively suppress the numerical dispersion and source-noises caused by the discretization of the acoustic wave equation with too-coarse spatial grids or when models have strong velocity contrasts between adjacent layers. The proposed NAID method has been applied in computing the acoustic wavefields for two heterogeneous models – the corner edge model and the Marmousi model. Promising numerical results illustrate that the NAID method provides an encouraging tool for large-scale and complex wave simulation and inversion problems based on the acoustic equation.

A wavelet multiscale–homotopy method for the inverse problem of two-dimensional acoustic wave equation

By introducing the wavelet multiscale method and the homotopy method to the inversion process for the velocity estimation problem of two-dimensional acoustic wave equation in seismic prospecting, a joint inversion method—wavelet multiscale–homotopy method is designed, which is stable, globally convergent, and has the ability of noise suppression. The results of numerical simulations and noise suppression tests indicate our method’s effectiveness.

A homotopy method for the inversion of a two-dimensional acoustic wave equation

This article considers the problem of estimating the velocity in a two-dimensional acoustic wave equation. In order to overcome the defects of local convergence of conventional methods, a general homotopy inversion strategy is proposed for the continuous inverse model. On the basis of this strategy, a practical algorithm is constructed by introducing Tikhonov regularization method for the nonlinear operator equation which is obtained from the discretization of the continuous inverse model. Numerical simulations show that the algorithm is a fast and widely convergent inversion method.

A widely convergent generalized pulse-spectrum technique for the inversion of two-dimensional acoustic wave equation

An algorithm of the generalized pulse-spectrum technique (GPST) is used for the inverse problems of two-dimensional acoustic wave equation, and a skill of saving computational cost is designed in choosing regularizing parameters. To overcome the defect of the local convergence of the GPST, a widely convergent GPST (WCGPST) is constructed. Numerical simulations are carried out to test the feasibility and to study the general characteristics of the WCGPST. It is found that the WCGPST is not only as robust as the standard GPST but also possessing widely convergent region.

A wavelet multiscale method for the inverse problems of a two-dimensional wave equation

This article considers the problem of estimating the velocity in a two-dimensional acoustic wave equation. The wavelet analysis is introduced and a wavelet multiscale method is constructed, based on the idea of hierarchical approximation. The inverse problem is decomposed into a sequence of inverse problems which rely on the scale variables and are solved successively according to the size of scale from the longest to the shortest. And in every inversion, at different scales, the regularized Gauss–Newton method is used, which is stable and fast, until the parameter of the primary inverse problem is found. The results of numerical simulations indicate that the method is a widely convergent optimization method (in some cases it may be global), and exhibits the advantages of conventional methods on computational efficiency and precision.

Determination of source parameters by wavefield extrapolation

The extrapolation of an observed (seismogram) wavefield backward in time produces an image, in both time and space, of the source. This concept is implemented through a finite difference solution of the two-dimensional acoustic wave equation. The seismograms themselves are used as time-dependent boundary values that drive the finite difference mesh, Numerical examples illustrate the determination of hypocentre location and origin time for point sources and of source extent, orientation and rupture velocity for finite sources in both homogeneous and heterogeneous media. Spatial resolution of source parameters is typically one-half wavelength of the dominant frequency generated by the source.

On the added mass and radiation damping of rod bundles oscillating in compressible fluids

This paper presents a theoretical analysis as well as numerical results for the added mass and radiation damping coefficients of a group of two-dimensional circular cylinders oscillating harmonically in an infinite compressible fluid. The fluid reaction force on these vibrating cylinders is obtained by solving the two-dimensional acoustic wave equation with Neumann conditions on the cylinders and the radiation condition at infinity. Numerical results show that when the acoustic wavelength is large compared with the cylinder radius, the added mass predominates over the radiation damping, and both are independent of the dimensionless wavenumber. On the other hand, when the acoustic wavelength is small compared with the cylinder radius, the radiation damping predominates over the added mass, and both are small.