Acoustic Wave Equation Research Articles

Acoustic emission of pulsating bubbles in viscous media

<sec>The classical single bubble’s acoustic emission equation has been used to describe the sound filed radiated by bubble for a long time. Because this formula does not consider the influence of the medium viscosity in the process of sound wave propagation, it is more reasonable to modify it in some special cases.</sec><sec>Based on the boundary condition of the bubbles, i.e. the vibration velocity of the bubble wall is equal to the particle vibration velocity of the external medium at the bubble boundary, the acoustic wave equation in spherical coordinate system in viscous medium is solved, and the modified acoustic emission formula of the bubble in the viscous medium is given.</sec><sec>The bubble radius <i>R</i>(<i>t</i>) is obtained numerically from the bubble dynamics equation by using the fourth-fifth order Runge-Kutta method. Then the bubble's radiation sound field is obtained by using the direct substitution method and the finite element (The pressure acoustics module; two-dimensional (2D) axisymmetric geometric model) method, respectively. The modified expression <i>p</i><sub>present</sub> given in this work is more accurate to describe the bubble’s radiation than the classical expression <i>p</i><sub>classical</sub> in the cases of high-viscosity, high-frequency and long-distance. In these cases, continuing to measure the acoustic emission of bubbles by using the classical expression may have an influence on the characteristics of cavitation, such as the inaccurate descriptions of parameters such as cavitation intensity and cavitation threshold.</sec>

A Lattice-adaptive Model for Solving Acoustic Wave Equations Based on Lattice Boltzmann Method

A Lattice-adaptive Model for Solving Acoustic Wave Equations Based on Lattice Boltzmann Method

Joint Migration Inversion Based on a Full-Wavefield Acoustic Wave Equation With Vector Reflectivity

Joint Migration Inversion Based on a Full-Wavefield Acoustic Wave Equation With Vector Reflectivity

Numerical Simulation and Parallel Computing of Acoustic Wave Equation in Isotropic-Heterogeneous Media

Numerical Simulation and Parallel Computing of Acoustic Wave Equation in Isotropic-Heterogeneous Media

A method to analyze the radiation characteristics of a liquid column resonance transducer based on fluid motion.

Liquid column resonance (LCR) transducers have been widely used in deep-sea acoustic applications because of their fluid-filled structures. Until now, studies of pipe resonance have generally been based on the plane acoustic wave equation, but for a vibrating object, the velocity is the primary focus instead of the pressure. Thus, the motion equation of a pipe resonance mode can be deduced based on the Navier-Stokes (N-S) equations. In this work, the velocity of an LCR transducer is obtained using the finite element model, and the velocity distribution inside the liquid column is examined. In addition, the radiating surface of the LCR transducer is identified and a simplified model of the radiation that consists of concave pistons and ring sources is proposed and verified. The theory behind the high mechanical quality (Q) value of the LCR transducer is explained through the radiation of the LCR transducer and the low viscosity of the water. This is also verified through a finite element model and measurements. Due to the high mechanical Q value and the low frequency of the LCR transducer, such measurements should be carried out in open-field water and the pulse should be long enough to achieve a steady state.

Full-wave Image Reconstruction in Transcranial Photoacoustic Computed Tomography using a Finite Element Method.

Transcranial photoacoustic computed tomography presents challenges in human brain imaging due to skull-induced acoustic aberration. Existing full-wave image reconstruction methods rely on a unified elastic wave equation for skull shear and longitudinal wave propagation, therefore demanding substantial computational resources. We propose an efficient discrete imaging model based on finite element discretization. The elastic wave equation for solids is solely applied to the hard-tissue skull region, while the soft-tissue or coupling-medium region that dominates the simulation domain is modeled with the simpler acoustic wave equation for liquids. The solid-liquid interfaces are explicitly modeled with elastic-acoustic coupling. Furthermore, finite element discretization allows coarser, irregular meshes to conform to object geometry. These factors significantly reduce the linear system size by 20 times to facilitate accurate whole-brain simulations with improved speed. We derive a matched forward-adjoint operator pair based on the model to enable integration with various optimization algorithms. We validate the reconstruction framework through numerical simulations and phantom experiments.

Overcoming the Spectral Bias Problem of Physics-Informed Neural Networks in Solving the Frequency-Domain Acoustic Wave Equation

Overcoming the Spectral Bias Problem of Physics-Informed Neural Networks in Solving the Frequency-Domain Acoustic Wave Equation

A model-based simulation framework for coupled acoustics, elastodynamics, and damage with application to nano-pulse lithotripsy

We develop a model for solid objects surrounded by a fluid that accounts for the possibility of acoustic pressures giving rise to damage on the surface of the solid. The propagation of an acoustic pressure in the fluid domain is modeled by the acoustic wave equation. On the other hand, the response of the solid is described by linear elastodynamics coupled with a gradient damage model, one that is based on a cohesive-type phase-field description of fracture. The interaction between the acoustic pressure and the deformation and damage of the solid are represented by transmission conditions at the fluid–solid interface. The resulting governing equations are discretized using a finite-element/finite-difference method that pays particular attention to the spatial and temporal scales that need to be resolved. Results from model-based simulations are provided for a benchmark problem as well as for recent experiments in nano-pulse lithotripsy. A parametric study is performed to illustrate how damage develops in response to the driving force (magnitude and location of the acoustic source) as a function of the fracture resistance of the solid. The results are shown to be qualitatively consistent with experimental observations for the location and size of the damage fields on the solid surface. A study of limiting cases also suggests that both the threshold for damage and the critical fracture energy are important to consider in order to capture the transition from damage initiation to complete localization. A low-cycle fatigue model is proposed that degrades the fracture resistance of the solid as a function of accumulated tensile strain energy, and it is shown to be capable of capturing damage localization in simulations of multi-pulse nano-pulse lithotripsy.

Numerical Simulation of Wave Equation Using the Finite Difference Method with Variable Refined Grid in Complex Near-surface Condition

The horizontal layered refined grid is adopted in the conventional finite difference method (FDM) of variable refined grid for the numerical simulation of wave equation. But, the method cannot well adapt to the characteristics of undulated topography and the variation of velocity structure in near-surface region. To solve this problem, a numerical simulation of wave equation using FDM with undulated variable refined grid is proposed in this paper. The grids are generated according to the undulated topography and the spatial distribution of velocity in the near-surface region from low-velocity to the high-velocity layer. For ensuring the accuracy while taking into account the computational efficiency, the FDM with variable coefficient in above grids is used to discretize the acoustic wave equation. The numerical evaluation and example show that the same accuracy as the method using fine grid can be obtained by the new method. However, its computational time is only 43.6% of the one spent by the fine grid. Furthermore, the new method can also stably adapt to the real complex loess plateau near-surface model in a working area of Ordos Basins.

Complex Source Distributions in Wave-based Virtual Acoustics

In the typical scenario in time-domain wave-based acoustics, the solution to the acoustic wave equation is approximated over a three-dimensional volumetric grid using a time stepping method. In the setting of very large simulations, the grid is normally assumed to be regular (e.g. Cartesian) so that massive parallelism may be exploited. One difficulty has been in representing source distributions that do not conform neatly to a regular grid. Using a Fourier-based optimisation procedure in the wave vector domain, it is possible to represent arbitrary source distributions in a flexible way over a pre-defined collection of grid points. Such a methodology is independent of the particular choice of simulation method and depends only on the regularity of the grid. In this paper, approximations to various simple distributions, including the line source and piston, are examined, with regard to accuracy, rotation of the distribution relative to the grid, and the size of the point cloud used to represent the source. Numerical results are presented.

Massively parallel nodal discontinous Galerkin finite element method simulator for room acoustics

We present a massively parallel and scalable nodal discontinuous Galerkin finite element method (DGFEM) solver for the time-domain linearized acoustic wave equations. The solver is implemented using the libParanumal finite element framework with extensions to handle curvilinear geometries and frequency dependent boundary conditions of relevance in practical room acoustics. The implementation is benchmarked on heterogeneous multi-device many-core computing architectures, and high performance and scalability are demonstrated for a problem that is considered expensive to solve in practical applications. In a benchmark study, scaling tests show that multi-GPU support gives the ability to simulate large rooms, over a broad frequency range, with realistic boundary conditions, both in terms of computing time and memory requirements. Furthermore, numerical simulations on two non-trivial geometries are presented, a star-shaped room with a dome and an auditorium. Overall, this shows the viability of using a multi-device accelerated DGFEM solver to enable realistic large-scale wave-based room acoustics simulations.

Robust data driven discovery of a seismic wave equation

SUMMARY Despite the fact that our physical observations can often be described by derived physical laws, such as the wave equation, in many cases, we observe data that do not match the laws or have not been described physically yet. Therefore recently, a branch of machine learning has been devoted to the discovery of physical laws from data. We test this approach for discovering the wave equation from the observed spatial-temporal wavefields. The algorithm first pre-trains a neural network (NN) in a supervised fashion to establish the mapping between the spatial-temporal locations (x, y, z, t) and the observation displacement wavefield function u(x, y, z, t). The trained NN serves to generate metadata and provide the time and spatial derivatives of the wavefield (e.g. utt and uxx) by automatic differentiation. Then, a preliminary library of potential terms for the wave equation is optimized from an overcomplete library by using a genetic algorithm. We, then, use a physics-informed information criterion to evaluate the precision and parsimony of potential equations in the preliminary library and determine the best structure of the wave equation. Finally, we train the ‘physics-informed’ neural network to identify the corresponding coefficients of each functional term. Examples in discovering the 2-D acoustic wave equation validate the feasibility and effectiveness of our implementation. We also verify the robustness of this method by testing it on noisy and sparsely acquired wavefield data.

Estimating the Location of Acoustic Sources with Oceanic Internal Gravity Waves via Frequency-Difference Source Localization

Frequency-difference source localization (FDSL) is a relatively new method for locating sources emitting acoustic or electromagnetic waves from recorded data. Estimates of location are produced by matching a nonlinear function of the received data with the same nonlinear function of modeled data. This so-called matched-field approach depends on the accuracy of the modeled field. FDSL exhibits less sensitivity to model errors in some cases of interest. Fluctuations of the speed of sound in the ocean are affected by a wide variety of phenomena and internal gravity waves is one where their spectrum is usually well known. There is no analytical theory for predicting how these fluctuations of sound speed affect the accuracy of FDSL, leaving a numerical evaluation as the alternative for the focus of this paper. Experimental application of FDSL was previously made for an acoustic source of 100 Hz bandwidth surrounding 250 Hz for a 129 km section in the Philippine Sea with data recorded on a long vertical array [Geroski and Dowling, Long-range frequency-difference source localization in the Philippine Sea, J. Acoust. Soc. Am. 146 (2019) 4727–4739.]. The role of internal waves is quantified with a realistic FDSL simulation using models based on the parabolic and ray approximations of the linear acoustic wave equation. The modeled vertical variation of FDSL error is similar to observed while the modeled horizontal error is too small. The complicated software model is automatic but is computationally intensive, with our version needing about 1.5 mo to compute on a PC with 16 cpu cores.

Full-waveform inversion using level set and cut elements for sharp-interface problems

This paper proposes a topology optimization procedure to reconstruct sharp interfaces of salt bodies in a background medium with known properties. The reconstruction technique uses the time-domain acoustic wave equation that provides the pressure field to minimize a data misfit objective function. The salt-background distinction is described by the level set-cut element approach proposed by Andreasen et al. (2020) [1]. This approach preserves the abrupt transition of properties across interfaces and provides accurate solutions to the problem. The nodal values of the level set function are parameterized as an explicit function of the design variables, and a “discretize (space)-then-derive”approach is implemented consistently with the discretized optimization problem. This setting allows for solving the minimization problem with mathematical programming methods; we use the conjugate gradient to update the level set shape-based reconstruction. A set of numerical examples demonstrate that the optimization algorithm is quite robust and is able to identify complex geometries of salt bodies.

Mixed Virtual Element approximation of linear acoustic wave equation

Abstract We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In the absence of external loads, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical $\theta $-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing an optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering applications of the method.

An efficient method for transcranial ultrasound focus correction based on the coupling of boundary integrals and finite elements

Transcranial focused ultrasound is a novel technique for the noninvasive treatment of brain diseases. The success of the treatment greatly depends on achieving precise and efficient intraoperative focus. However, compensating for aberrated ultrasound waves caused by the skull through numerical simulation-based phase corrections is a challenging task due to the significant computational burden involved in solving the acoustic wave equation. In this article, we propose a promising strategy using the coupling of the boundary integral equation method (BIEM) and the finite element method (FEM) to overcome the above limitation. Specifically, we adopt the BIEM to obtain the Robin-to-Dirichlet maps on the boundaries of the skull and then couple the maps to the FEM matrices via a dual interpolation technique, resulting in a computational domain including only the skull. Three simulation experiments were conducted to evaluate the effectiveness of the proposed method, including a convergence test and two skull-induced aberration corrections in 2D and 3D ultrasound. The results show that the method’s convergence is guaranteed as the element size decreases, leading to a decrease in pressure error. The computation times for simulating a 500 kHz ultrasound field on a regular desktop computer were found to be 0.47 ± 0.01 s in the 2D case and 43.72 ± 1.49 s in the 3D case, provided that lower–upper decomposition (approximately 13 s in 2D and 2.5 h in 3D) was implemented in advance. We also demonstrated that more accurate transcranial focusing can be achieved by phase correction compared to the noncorrected results (with errors of 1.02 mm vs. 6.45 mm in 2D and 0.28 mm vs. 3.07 mm in 3D). The proposed strategy is valuable for enabling online ultrasound simulations during treatment, facilitating real-time adjustments and interventions.

Nonlinear aeroelastic fluid-structure-acoustic interaction analysis of a coupled composite panel with an acoustic cavity in supersonic flow

This paper is concerned with numerical modeling and nonlinear dynamics analyses of fluid-structure-acoustic interactions of an elastic composite laminated panel with a backed cavity in supersonic flow. To achieve strongly coupled solutions to the aeroelastic fluid-structure-acoustic interaction problem, an implicit partitioned arbitrary Lagrangian-Eulerian solution method is developed based on a finite volume model for the supersonic flow and a monolithic nonlinear finite element model for the composite panel-cavity system. A nonlinear higher-order shear deformation zig-zag theory is adopted to accommodate the large-deformable vibration of the composite panel, and the acoustic wave equation is employed for modeling the compressible fluid in the cavity. The method is verified by comparisons with benchmark results. Physical insights into the effects of the dynamic pressure, cavity depth, and the ply-angle of composite material on the aeroelastic behaviors of the panel-cavity system are provided. Some new phenomena are revealed. A special flutter behavior dominated by acoustic resonance is found for the composite panel with a backed cavity in supersonic flow, which is due to local mode coalescence as the mode frequencies of the acoustic cavity and the composite panel are close to each other. The transition between the aeroelastic flutters of the composite panel induced by the lower-order structural mode coalescence and the acoustic resonance of the cavity can be progressively adjusted by alerting the cavity depth and the ply-angle of the composite panel.

A Splitting Nearly-Analytic Symplectic Partitioned Runge–Kutta Method for Solving 2D Elastic Wave Equations

In this paper, we present a new splitting nearly-analytic symplectic partitioned Runge–Kutta (SNSPRK) method for the two-dimensional (2D) elastic wave equations. It is an extension to elastic wave equation of our recent work on the locally one-dimensional nearly-analytic symplectic partitioned Runge–Kutta (LOD-NSPRK) method for the 2D acoustic wave equations. The method is based on the spatial differential operator-split technique, in which the resulting spatial discrete matrices are symmetric unlike the conventional nearly-analytic symplectic partitioned Runge–Kutta (NSPRK) method. The stability condition is given, which has more relax restriction for the time step than the NSPRK schemes. To show the performance of the new method, numerical experiments are given. Numerical results illustrate that the SNSPRK method has better long-term calculation capability as time proceeds than the NSPRK method.

Physics-informed neural network assisted spherical microphone array signal processing

Thanks to their rotational symmetry that facilitates three-dimensional signal processing, spherical microphone arrays are the common array apertures used for spatial audio and acoustic applications. However, practical implementations of spherical microphone arrays suffer from two issues. First, at high frequency range, a large number of sensors are needed to accurately capture a sound field. Second, the accompanying signal processing algorithm, i.e., the spherical harmonic decomposition method, requires a variable radius array or a rigid surface array to circumvent the spherical Bessel function nulls. Such arrays are hard to design and introduce a scattering field. To address these issues, this paper proposes to assist a spherical microphone array with a physics-informed neural network (PINN) for three-dimensional signal processing. The PINN models the sound field around the array based on the sensor measurements and the acoustic wave equation, augmenting the sound field information captured by the array through prediction. This makes it possible to analyze a high frequency sound field with a reduced number of sensors and avoid the spherical Bessel function nulls with a simple single radius open-sphere microphone array.

Enhancement of brain hyperthermia via transcranial magnetic resonance imaging-guided focused ultrasound and microbubbles—Heating mechanism investigation using COMSOL

Noninvasive methods for enhancing the brain drug delivery has been pursued for years. Previously we developed a new MR-guided focused ultrasound (FUS)-based technique, which can achieve targeted brain hyperthermia for heat-triggered drug release and simultaneously open the blood–brain barrier safely for drug penetration. However, the underline mechanisms were unclear. This study aimed to explore the mechanisms for the enhanced FUS brain tissue hyperthermia with microbubbles via numerical modeling in COMSOL. The acoustic wave equation was employed to describe the FUS propagation. A bubble dynamics equation was adopted for calculating the stable bubble oscillations under FUS exposures. A modified bioheat transfer equation was utilized to compute the heating, with various heating sources including FUS, microbubble acoustic emission (MAE), and viscous dissipation (VD). The microbubbles were randomly distributed within the focal region. The sonication time was 6s with an initial temperature of 41°C. The average temperature in the focal region were 41.65°C, 42.24°C, 42.98°C, and 43.59°C for FUS alone, FUS + MAE, FUS + VD, and FUS + MAE + VD, respectively. Compared with the FUS alone, both MAE and VD made significant contributions to the heating with additional temperature increases of 47.6% and 67.2%, respectively.