Finite-difference time-domain simulation of acoustic propagation in dispersive medium: An application to bubble clouds in the ocean

Accurate modeling of pulse propagation and scattering is a problem in many disciplines (i.e. electromagnetics and acoustics). For the case of an acoustic wave propagating in a two-dimensional non-dispersive medium, a routine 2nd order in time and space Finite-Difference Time-Domain (FDTD) scheme representation of the linear wave equation can be used to solve for the acoustic pressure. However when the medium is dispersive, one is required to take into account the frequency dependent attenuation and phase speed. Until recently to include the dispersive effects one typically solved the problem in the frequency domain and not in the time domain. The frequency domain solutions were Fourier transformed into the time domain. However by using a theory first proposed by Blackstock [D.T. Blackstock, J. Acoust. Soc. Am. 77 (1985) 2050. [1]], the linear wave equation has been modified by adding an additional term (the derivative of the convolution between the causal time-domain propagation factor and the acoustic pressure) that takes into account the dispersive nature of the medium. In the case of acoustic propagation through water, the water environment becomes strongly dispersive due to the presence of air bubbles that are present below the air-water interface.     

An augmented method for free boundary problems with moving contact lines

An augmented immersed interface method (IIM) is proposed for simulating one-phase moving contact line problems in which a liquid drop spreads or recoils on a solid substrate. While the present two-dimensional mathematical model is a free boundary problem, in our new numerical method, the fluid domain enclosed by the free boundary is embedded into a rectangular one so that the problem can be solved by a regular Cartesian grid method. We introduce an augmented variable along the free boundary so that the stress balancing boundary condition is satisfied. A hybrid time discretization is used in the projection method for better stability. The resultant Helmholtz/Poisson equations with interfaces then are solved by the IIM in an efficient way. Several numerical tests including an accuracy check, and the spreading and recoiling processes of a liquid drop are presented in detail.     

Unstructured Grid-Based Discontinuous Galerkin Method for Broadband Electromagnetic Simulations

A parallel, unstructured, high-order discontinuous Galerkin method is developed for the time-dependent Maxwell's equations, using simple monomial polynomials for spatial discretization and a fourth-order Runge–Kutta scheme for time marching. Scattering results for a number of validation cases are computed employing polynomials of up to third order. Accurate solutions are obtained on coarse meshes and grid convergence is achieved, demonstrating the capabilities of the scheme for time-domain electromagnetic wave scattering simulations.     

An effective algorithm for simulating acoustical wave propagation

In this paper, the acoustical wave propagator scheme is implemented in Fortran for predicting sound propagation in a one-dimensional duct. Example calculations are performed for a semi-infinite duct and a duct with a solid blockage. Numerical accuracy of our results is examined and compared with the finite-difference time-domain method. This scheme is found to be highly accurate and computationally effective for describing the time-domain evolution of acoustic waves. Multiple reflections within the solid blockage and phase changes of the transmitting wave from solid back into air are illustrated through the implementation of this scheme.     

Comparison of homogeneous and heterogeneous modeling of transient scattering from dispersive media directly in the time domain

Accurate modeling of pulse propagation and scattering in dispersive medium is a problem in many disciplines (i.e. electromagnetics and acoustics). The inclusion of an additional term in the wave equation (the derivative of the convolution between the causal time-domain propagation factor and the acoustic pressure) that takes into account the dispersive nature of the medium is utilized to make these problems tractable. The resulting modified wave equation (either homogeneous or heterogeneous) is applicable to either linear or non-linear propagation. For the case of an acoustic wave propagating in a two-dimensional heterogeneous dispersive medium, a finite-difference time-domain (FDTD) representation of the modified linear wave equation can been used to solve for the acoustic pressure. The method is applied to the case of scattering from and propagating through a 2D infinitely long cylinder with real world material properties. It is found that ignoring the heterogeneity in the medium can lead to significant error in the propagated/scattered field.     

Hexagonal grid methods with applications to partial differential equations

We consider in this work hexagonal grids for two-dimensional applications. A finite volume-based finite difference approach to solving Laplacian-related differential equations on hexagonal grids is developed. Both ordinary and compact hexagonal seven-point schemes are investigated. Theoretical properties of the associated linear algebraic systems are determined. These methods are applied to solve PDEs on both regular and curved domains, successfully exhibiting linear and spiral wave propagations in regular domains and curved wave in a reversed C-type domain.     

基于分布参数模型的高炉炉温预测

高炉炉温的高低直接影响了高炉的生产稳定性以及炼铁过程中的低能耗、高质量等性能指标.本文采用分布参数系统思想建立了偏微分方程预测控制模型.首先,使用普通最小二乘方法分两种不同方案对模型进行参数估计并对他们的仿真结果进行比较分析,然后,使用易适应的参数估计方法(flexibleleast squaremethod,FLS)对模型进行变系数估计处理.此方法能够体现系数的时变性,提高模型的准确度.最后,从仿真效果图中展示了本方法对硅含量预报的结果无论在命中率上还是精确度上的优势和有效性.     

Dispersion analysis of numerical wave propagation and its computational consequences

We present in this paper a comparison of the dispersion properties for several finite-difference approximations of the acoustic wave equation. We investigate the compact and staggered schemes of fourth order accuracy in space and of second order or fourth order accuracy in time. We derive the computational cost of the simulation implied by a precision criterion on the numerical simulation (maximum allowed error in phase or group velocity). We conclude that for moderate accuracy the staggered scheme of second order in time is more efficient, whereas for very precise simulation the compact scheme of fourth order in time is a better choice. The comparison increasingly favors the lower order staggered scheme as the dimension increases. In three dimensional simulation, the cost of extremely precise simulation with any of the schemes is very large, whereas for simulation of moderate precision the staggered scheme is the least expensive.     

Dispersive and Dissipative Properties of Discontinuous Galerkin Finite Element Methods for the Second-Order Wave Equation

Discontinuous Galerkin finite element methods (DGFEM) offer certain advantages over standard continuous finite element methods when applied to the spatial discretisation of the acoustic wave equation. For instance, the mass matrix has a block diagonal structure which, used in conjunction with an explicit time stepping scheme, gives an extremely economical scheme for time domain simulation. This feature is ubiquitous and extends to other time-dependent wave problems such as Maxwell’s equations. An important consideration in computational wave propagation is the dispersive and dissipative properties of the discretisation scheme in comparison with those of the original system. We investigate these properties for two popular DGFEM schemes: the interior penalty discontinuous Galerkin finite element method applied to the second-order wave equation and a more general family of schemes applied to the corresponding first order system. We show how the analysis of the multi-dimensional case may be reduced to consideration of one-dimensional problems. We derive the dispersion error for various schemes and conjecture on the generalisation to higher order approximation in space     

A simple implementation of the immersed interface methods for Stokes flows with singular forces

In this paper, we introduce a simple version of the immersed interface method (IIM) for Stokes flows with singular forces along an interface. The numerical method is based on applying the Taylor’s expansions along the normal direction and incorporating the solution and its normal derivative jumps into the finite difference approximations. The fluid variables are solved in a staggered grid, and a new accurate interpolating scheme for the non-smooth velocity has been developed. The numerical results show that the scheme is second-order accurate.     

A Framework for Moving Least Squares Method with Total Variation Minimizing Regularization

In this paper, we propose a computational framework to incorporate regularization terms used in regularity based variational methods into least squares based methods. In the regularity based variational approach, the image is a result of the competition between the fidelity term and a regularity term, while in the least squares based approach the image is computed as a minimizer to a constrained least squares problem. The total variation minimizing denoising scheme is an exemplary scheme of the former approach with the total variation term as the regularity term, while the moving least squares method is an exemplary scheme of the latter approach. Both approaches have appeared in the literature of image processing independently. By putting schemes from both approaches into a single framework, the resulting scheme benefits from the advantageous properties of both parties. As an example, in this paper, we propose a new denoising scheme, where the total variation minimizing term is adopted by the moving least squares method. The proposed scheme is based on splitting methods, since they make it possible to express the minimization problem as a linear system. In this paper, we employed the split Bregman scheme for its simplicity. The resulting denoising scheme overcomes the drawbacks of both schemes, i.e., the staircase artifact in the total variation minimizing based denoising and the noisy artifact in the moving least squares based denoising method. The proposed computational framework can be utilized to put various combinations of both approaches with different properties together.     

Mathematical modeling of plasma opening switches

This article investigates Plasma Opening Switches with the help of an implicit code. The plasma is described using a collisional bifluid model, where the low ratio of electronic and ionic masses allows us to drop inertial terms from the electronic momentum equation. This asymptotic model filters plasma oscillations from the equations and the scheme is stable for time steps exceeding the electron plasma period. However, an implicit coupling of fluid and Maxwell's equations is considered because of the presence of a damped mode, whose characteristic time scale restricts the time steps of explicit schemes to small values compared to the simulation requirements. The stability of the scheme is illustrated by simulating certain operating phases of the Plasma Erosion Opening Switch.     

A High-Order Accurate Method for Frequency Domain Maxwell Equations with Discontinuous Coefficients

Maxwell equations contain a dielectric coefficient ɛ that describes the particular media. For homogeneous materials the dielectric coefficient is constant. There is a jump in this coefficient across the interface between differing media. This discontinuity can significantly reduce the order of accuracy of the numerical scheme. We present an analysis and implementation of a fourth order accurate algorithm for the solution of Maxwell equations with an interface between two media and so the dielectric coefficient is discontinuous. We approximate the discontinuous function by a continuous one either locally or in the entire domain. We study the one-dimensional system in frequency space. We only consider schemes that can be implemented for multidimensional problems both in the frequency and time domains.     

A family of fourth-order difference schemes on rotated grid for two-dimensional convection–diffusion equation

We derive a family of fourth-order finite difference schemes on the rotated grid for the two-dimensional convection–diffusion equation with variable coefficients. In the case of constant convection coefficients, we present an analytic bound on the spectral radius of the line Jacobi’s iteration matrix in terms of the cell Reynolds numbers. Our analysis and numerical experiments show that the proposed schemes are stable and produce highly accurate solutions. Classical iterative methods with these schemes are convergent with large values of the convection coefficients. We also compare the fourth-order schemes with the nine point scheme obtained from the second-order central difference scheme after one step of cyclic reduction.     

ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D

The ADER scheme for solving systems of linear, hyperbolic partial differential equations in two-dimensions is presented in this paper. It is a finite-volume scheme of high order in space and time. The scheme is explicit, fully discrete and advances the solution in one single step. Several numerical tests have been performed. In the first test case the dissipation and dispersion behaviour of the schemes are studied in one space dimension. Dispersion as well as dissipation effects strongly influence the discrete wave propagation over long distances and are very important for, e.g., aeroacoustical calculations. The next test, the so-called co-rotating vortex pair, is a demonstration of the ideas of the two-dimensional ADER approach. The linearised Euler equations are used for the simulation of the sound emitted by a co-rotating vortex pair.     

A revisit to block and recursive least squares for parameter estimation

In this paper, the classical least squares (LS) and recursive least squares (RLS) for parameter estimation have been re-examined in the light of the present day computing capabilities. It has been demonstrated that for linear time-invariant systems, the performance of blockwise least squares (BLS) is always superior to that of RLS. In the context of parameter estimation for dynamic systems, the current computational capability of personal computers are more than adequate for BLS. However, for time-varying systems with abrupt parameter changes, standard blockwise LS may no longer be suitable due to its inefficiency in discarding “old” data. To deal with this limitation, a novel sliding window blockwise least squares approach with automatically adjustable window length triggered by a change detection scheme is proposed. Two types of sliding windows, rectangular and exponential, have been investigated. The performance of the proposed algorithm has been illustrated by comparing with the standard RLS and an exponentially weighted RLS (EWRLS) using two examples. The simulation results have conclusively shown that: (1) BLS has better performance than RLS; (2) the proposed variable-length sliding window blockwise least squares (VLSWBLS) algorithm can outperform RLS with forgetting factors; (3) the scheme has both good tracking ability for abrupt parameter changes and can ensure the high accuracy of parameter estimate at the steady-state; and (4) the computational burden of VLSWBLS is completely manageable with the current computer technology. Even though the idea presented here is straightforward, it has significant implications to virtually all areas of application where RLS schemes are used.     

薄膜体声波谐振器的FDTD数值分析

提出了基于时域有限差分方法对薄膜体声波谐振器进行数值分析的新方法。利用时域有限差分法理论对压电材料的控制方程,牛顿方程和电学方程在空间和时间进行了离散化,通过得到的差分方程直接得出了声场传播的时域数值解。使用该数值方法对薄膜体声波谐振器的电学特性阻抗进行了分析,并将结果与一维Mason模型的解析解进行了比较验证。     

Wave-splitting and absorbing boundary condition for Maxwell's equations on a curved surface

Wave-splitting of Maxwell's equations on a curved surface in three-dimension is derived from the conditions for out-going and in-coming waves. The condition for out-going wave is used to derive an absorbing boundary condition on a non-planar surface. A local approximation of the absorbing boundary condition is given. As a special case, a spherical absorbing boundary which truncates the computational domain is also considered in detail.     

Efficient and Accurate Sound Propagation Using Adaptive Rectangular Decomposition

Accurate sound rendering can add significant realism to complement visual display in interactive applications, as well as facilitate acoustic predictions for many engineering applications, like accurate acoustic analysis for architectural design (Monks et al., 2000). Numerical simulation can provide this realism most naturally by modeling the underlying physics of wave propagation. However, wave simulation has traditionally posed a tough computational challenge. In this paper, we present a technique which relies on an adaptive rectangular decomposition of 3D scenes to enable efficient and accurate simulation of sound propagation in complex virtual environments. It exploits the known analytical solution of the wave equation in rectangular domains, and utilizes an efficient implementation of the discrete cosine transform on graphics processors (GPU) to achieve at least a 100-fold performance gain compared to a standard finite-difference time-domain (FDTD) implementation with comparable accuracy, while also being 10-fold more memory efficient. Consequently, we are able to perform accurate numerical acoustic simulation on large, complex scenes in the kilohertz range. To the best of our knowledge, it was not previously possible to perform such simulations on a desktop computer. Our work thus enables acoustic analysis on large scenes and auditory display for complex virtual environments on commodity hardware.     

Some results on energy-conserving lattice Boltzmann models

We consider the problem of “energy conserving” lattice Boltzmann models. A major difficulty observed in previous studies is the coupling between the viscous and thermal waves even at moderate wave numbers. We propose a theoretical framework based on the knowledge of the partial equivalent equations of the lattice Boltzmann scheme at several orders of precision. With the help of linearized models (inviscid and dissipative advective acoustics and classical acoustics), we suggest natural sets of relations for the parameters of lattice Boltzmann schemes. The application is proposed for three two-dimensional schemes. Numerical test cases for simple linear and nonlinear waves establish that the main difficulty in the previous contributions can now be overcome.