求解二维多区域声波散射问题的边界元方法

对于多散射区域的声波散射问题的外Neumann边值问题,用单层位势来逼近每个散射域上的散射波,再利用位势理论的跳跃关系将问题转换为第二类边界积分方程组的求解问题,然后用Nystrom方法进行了求解.对多个随机散射区域的声波散射问题,数值例子体现了该求解方法的可行性和准确性.     

Finite element method to fluid–solid interaction problems with unbounded periodic interfaces

Consider a time‐harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This article is concerned with a variational approach to the fluid–solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasiperiodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet‐to‐Neumann mappings is proposed and the convergence analysis is performed. The Dirichlet‐to‐Neumann mappings are approximated by truncated Rayleigh series expansions. Finally, numerical tests in 2D are presented to confirm the convergence of solutions and the energy balance formula. In particular, the frequency spectrum of normally reflected signals is plotted for water–brass and water–brass–water interfaces. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 5–35, 2016     

Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems

We consider the approximation of the frequency domain three-dimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius . We also show exponential (in the parameter ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer.

Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.

     

Non-reflecting boundary conditions for waveguides

New non-reflecting boundary conditions are introduced for the solution of the Helmholtz equation in a waveguide. These boundary conditions are perfectly transparent for all propagating modes. They do not require the determination of these propagating modes but only their propagation constants. A quasi-local form of these boundary conditions is well suited as terminating boundary condition beyond finite element meshes. Related convergence properties to the exact solution and optimal error estimates are established.

     

Effect of ligand volume correction on PMF scoring

Recently, a knowledge‐based scoring function has been introduced that estimates the protein‐binding affinity based on the 3D structure of a protein–ligand complex (J Med Chem 1999, 42, 791). A ligand volume correction factor has been proposed and applied to filter out intraligand interactions in this simplified potential approach. Here we evaluate the effect of the ligand volume correction on the predictive power of the PMF scoring function. It is found that the effect of the ligand volume correction is significant on the derived potentials and large on the overall score. However, the effect of the ligand correction on the predictive power of the scoring function appears to be smaller. For a test set containing serine proteases the predictive power of the PMF scoring function does not change with the introduction of the volume correction. For a test set of metalloprotease complexes, the predictive power of the PMF scoring function improves only slightly when the volume correction is applied. For five test sets comprising a total of 225 diverse protein ligand complexes taken from the Brookhaven Protein Data Bank it is found, however, that the introduction of the ligand volume correction consistently improves the correlation between the PMF scores and the measured binding affinities. The effect of the correction factor on docking/scoring experiments is also analyzed using a test set of 61 biphenyl inhibitor‐stromelysin complexes. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 418–425, 2001     

Predicting the wind noise from the pantograph cover of a train

Finite element and boundary element calculations are combined to predict the flow noise radiated from a 1/10th-scale model of an aerodynamic cover used around the pantograph on a train at 250 km h⁻¹. The solutions of the unsteady air flow over the cover and the resulting sound propagation are divided into two parts in order to keep the problem tractable. First the unsteady fluid flow is solved using large-eddy simulation (LES). The pressure histories on the cover are then used to predict the radiated sound, using a boundary element method to solve the Helmholtz equation. The result thus leans heavily on assumptions about the coupling of the two solutions, the propagation of sound in a disturbed medium and the efficacy of LES. The predicted sound pressure levels are compared with experimental measurements made in an anechoic wind tunnel. © 1997 John Wiley & Sons, Ltd.     

Collocation discretization for an integral equation in ocean acoustics with depth‐dependent speed of sound

We analyze two collocation schemes for the Helmholtz equation with depth‐dependent sonic wave velocity, modeling time‐harmonic acoustic wave propagation in a three‐dimensional inhomogeneous ocean of finite height. Both discretization schemes are derived from a periodized version of the Lippmann‐Schwinger integral equation that equivalently describes the sound wave. The eigenfunctions of the corresponding periodized integral operator consist of trigonometric polynomials in the horizontal variables and eigenfunctions to some Sturm‐Liouville operator linked to the background profile of the sonic wave velocity in the vertical variable. Applying an interpolation projection onto a space spanned by finitely many of these eigenfunctions to either the unknown periodized wave field or the integral operator yields two different collocation schemes. A convergence estimate of Sloan [J. Approx. Theory, 39:97–117, 1983] on non‐polynomial interpolation allows to show converge of both schemes, together with algebraic convergence rates depending on the smoothness of the inhomogeneity and the source. Copyright © 2016 John Wiley & Sons, Ltd.     

Preconditioning spectral element schemes for definite and indefinite problems

Spectral element schemes for the solution of elliptic boundary value problems are considered. Preconditioning methods based on finite difference and finite element schemes are implemented. Numerical experiments show that inverting the preconditioner by a single multigrid iteration is most efficient and that the finite difference preconditioner is superior to the finite element one for both definite and indefinite problems. A multigrid preconditioner is also derived from the finite difference preconditioner and is found suitable for the CGS acceleration method. It is pointed out that, for the finite difference and finite element preconditioners, CGS does not always converge to the accurate algebraic solution. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 535–543, 1999     

Acoustic absorbers at low frequency based on split-tube metamaterials

The remarkable properties of acoustic metamaterials have attracted massive researches and applications, especially on low-frequency sound absorptions. Currently, most of the acoustic metamaterial absorbers employ resonances in plastic cavities, and their structural strengths are important in many circumstances, especially in harsh environment. However, studies of metamaterials including this point are very scarce. Here, we propose an acoustic metamaterial for low-frequency (<500 Hz) absorptions, composed of three nested square split tubes with inverted opening directions. The efficiency of the absorber is investigated both numerically and experimentally, and absorptions at the peeks are found to exceed 90% and the frequency can be effectively adjusted by tuning its geometric parameters. We further test its yield strength under compression and confirm its buckling behavior happens from the outmost layer. This tunable acoustic metamaterial with a fairly good mechanical strength may lead to broad applications in noise reduction.     

A hybrid numerical–asymptotic boundary element method for high frequency scattering by penetrable convex polygons

We present a novel hybrid numerical–asymptotic boundary element method for high frequency acoustic and electromagnetic scattering by penetrable (dielectric) convex polygons. Our method is based on a standard reformulation of the associated transmission boundary value problem as a direct boundary integral equation for the unknown Cauchy data, but with a nonstandard numerical discretization which efficiently captures the high frequency oscillatory behaviour. The Cauchy data is represented as a sum of the classical geometrical optics approximation, computed by a beam tracing algorithm, plus a contribution due to diffraction, computed by a Galerkin boundary element method using oscillatory basis functions chosen according to the principles of the Geometrical Theory of Diffraction. We demonstrate with a range of numerical experiments that our boundary element method can achieve a fixed accuracy of approximation using only a relatively small, frequency-independent number of degrees of freedom. Moreover, for the scattering scenarios we consider, the inclusion of the diffraction term provides an order of magnitude improvement in accuracy over the geometrical optics approximation alone.