A dispersion minimizing scheme for the 3-D Helmholtz equation based on ray theory

We develop a new dispersion minimizing compact finite difference scheme for the Helmholtz equation in 2 and 3 dimensions. The scheme is based on a newly developed ray theory for difference equations. A discrete Helmholtz operator and a discrete operator to be applied to the source and the wavefields are constructed. Their coefficients are piecewise polynomial functions of $hk$, chosen such that phase and amplitude errors are minimal. The phase errors of the scheme are very small, approximately as small as those of the 2-D quasi-stabilized FEM method and substantially smaller than those of alternatives in 3-D, assuming the same number of gridpoints per wavelength is used. In numerical experiments, accurate solutions are obtained in constant and smoothly varying media using meshes with only five to six points per wavelength and wave propagation over hundreds of wavelengths. When used as a coarse level discretization in a multigrid method the scheme can even be used with downto three points per wavelength. Tests on 3-D examples with up to $10^8$ degrees of freedom show that with a recently developed hybrid solver, the use of coarser meshes can lead to corresponding savings in computation time, resulting in good simulation times compared to the literature.Comment: 33 pages, 12 figures, 6 table

A multigrid method for the Helmholtz equation with optimized coarse grid corrections

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let G_c denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement G_c > 2. However, in examples the requirement is more like G_c >= 10, in a trade-off involving also the amount of damping present and the number of multigrid iterations. We conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators. Standard 5-point finite differences in 2-D are our first example. A new coarse scale 9-point operator is constructed to match the fine scale phase speeds. We then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator. The required G_c is reduced from about 10 to about 3.5, with less damping present so that waves propagate over > 100 wavelengths in the new scheme. Next we consider extensions of the method to more general cases. In 3-D comparable results are obtained with standard 7-point differences and optimized 27-point coarse grid operators, leading to an order of magnitude reduction in the number of unknowns for the coarsest scale linear system. Finally we show how to include PML boundary layers, using a regular grid finite element method. Matching coarse scale operators can easily be constructed for other discretizations. The method is therefore potentially useful for a large class of discretized high-frequency Helmholtz equations.Comment: Coarse scale operators are simplified and only standard smoothers used in v3; 5 figures, 12 table

A Scale-selective Multilevel Method for Long-Wave Linear Acoustics

A new method for the numerical integration of the equations for one-dimensional linear acoustics with large time steps is presented. While it is capable of computing the "slaved" dynamics of short-wave solution components induced by slow forcing, it eliminates freely propagating compressible short-wave modes, which are under-resolved in time. Scale-wise decomposition of the data based on geometric multigrid ideas enables a scale-dependent blending of time integrators with different principal features. To guide the selection of these integrators, the discrete-dispersion relations of some standard second-order schemes are analyzed, and their response to high wave number low frequency source terms are discussed. The performance of the new method is illustrated on a test case with "multiscale" initial data and a problem with a slowly varying high wave number source term

Gaussian Process Regression for Estimating EM Ducting Within the Marine Atmospheric Boundary Layer

We show that Gaussian process regression (GPR) can be used to infer the electromagnetic (EM) duct height within the marine atmospheric boundary layer (MABL) from sparsely sampled propagation factors within the context of bistatic radars. We use GPR to calculate the posterior predictive distribution on the labels (i.e. duct height) from both noise-free and noise-contaminated array of propagation factors. For duct height inference from noise-contaminated propagation factors, we compare a naive approach, utilizing one random sample from the input distribution (i.e. disregarding the input noise), with an inverse-variance weighted approach, utilizing a few random samples to estimate the true predictive distribution. The resulting posterior predictive distributions from these two approaches are compared to a "ground truth" distribution, which is approximated using a large number of Monte-Carlo samples. The ability of GPR to yield accurate and fast duct height predictions using a few training examples indicates the suitability of the proposed method for real-time applications.Comment: 15 pages, 6 figure

Computational Modeling of Airborne Noise Demonstrated Via Benchmarks, Supersonic Jet, and Railway Barrier

In the last several years, there has been a growing demand for mobility to cope with the increasing population. All kinds of transportation have responded to this demand by expanding their networks and introducing new ideas. Rail transportation introduced the idea of high-speed trains and air transportation introduced the idea of high-speed civil transport (HSCT). In this expanding world, the noise legislation is felt to inhibit these plans. Accurate computational methods for noise prediction are in great demand. In the current research, two computational methods are developed to predict noise propagation in air. The first method is based on the finite differencing technique on generalized curvilinear coordinates and it is used to solve linear and nonlinear Euler equations. The dispersion-relation-preserving scheme is adopted for spatial discretization. For temporal integration, either the dispersion-relation-preserving scheme or the low-dispersion-and-dissipation Runge-Kutta scheme is used. Both characteristic and asymptotic nonreflective boundary conditions are studied. Ghost points are employed to satisfy the wall boundary condition. A number of benchmark problems are solved to validate different components of the present method. These include initial pulse in free space, initial pulse reflected from a flat or curved wall, time-periodic train of waves reflected from a flat wall, and oscillatory sink flow. The computed results are compared with the analytical solutions and good agreements are obtained. Using the method developed, the noise of Mach 2.1, perfectly expanded, two-dimensional supersonic jet is computed. The Reynolds-averaged Navier-Stokes equations are solved for the jet mean flow. The instability waves, which are used to excite the jet, are obtained from the solution of the compressible Rayleigh equation. Then, the linearized Euler equations are solved for jet noise. To improve computational efficiency, flow-adapted grid and a multi-block time integration technique are developed. The computations are compared with the experimental results for both the mean flow and the jet noise. Good agreement is obtained. The method proved to be fast and efficient. The second computational method is based on the boundary element technique. The Helmholtz equation is solved for the sound field around a railway noise barrier. Linear elements are used to discretize the barrier surface. Frequency-dependent grids are employed for efficiency. The train noise is represented by a point source located above the nearest rail. The source parameters are estimated from a typical field measurement of train noise spectrum. Both elevated and ground-level train decks are considered. The performance of the noise barrier at low and high frequencies is investigated. Moreover, A-weighted sound pressure levels are calculated. The computed results are successfully compared with field measurements

Analysis of high-order finite elements for convected wave propagation

In this paper, we examine the performance of high-order finite element methods (FEM) for aeroacoustic propagation, based on the convected Helmholtz equation. A methodology is presented to measure the dispersion and amplitude errors of the p-FEM, including non-interpolating shape functions, such as ‘bubble’ shape functions. A series of simple test cases are also presented to support the results of the dispersion analysis. The main conclusion is that the properties of p-FEM that make its strength for standard acoustics (e.g., exponential p-convergence, low dispersion error) remain present for flow acoustics as well. However, the flow has a noticeable effect on the accuracy of the numerical solution, even when the change in wavelength due to the mean flow is accounted for, and an approximation of the dispersion error is proposed to describe the influence of the mean flow. Also discussed is the so-called aliasing effect, which can reduce the accuracy of the solution in the case of downstream propagation. This can be avoided by an appropriate choice of mesh resolution