scholarly journals Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies

2017 ◽  
Vol 473 (2207) ◽  
pp. 20170242 ◽  
Author(s):  
Oscar P. Bruno ◽  
Stephen P. Shipman ◽  
Catalin Turc ◽  
Venakides Stephanos
Keyword(s):  
Green Function ◽  
Blow Up ◽  
Three Dimensional ◽  
Acoustic Scattering ◽  
Three Dimensions ◽  
Scattering Problems ◽  
Periodic Surfaces ◽  
Lattice Sum ◽  
The Green Function ◽  
First Time

This work, part II in a series, presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called ‘Wood anomaly frequencies’. At these frequencies, there is a grazing Rayleigh wave, and the quasi-periodic Green function ceases to exist. We present a modification of the Green function by adding two types of terms to its lattice sum. The first type are transversely shifted Green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence. The second type are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on this new Green function for scattering by doubly periodic surfaces at and around Wood frequencies. We believe this is the first solver able to handle Wood frequencies for doubly periodic scattering problems in three dimensions. We demonstrate the method by applying it to acoustic scattering.

Adaptivity and Three Dimensional Hierarchical Boundary Elements for Rigid Acoustic Scattering Problems

1995 ◽  
Author(s):  
Steven J. Newhouse ◽  
Ian C. Mathews
Keyword(s):  
Boundary Element ◽  
Acoustic Analysis ◽  
Three Dimensional ◽  
Acoustic Scattering ◽  
Error Estimator ◽  
Infinite Domain ◽  
Scattering Problems ◽  
Effective Form ◽  
Mesh Density ◽  
Acoustic Scattering Problems

Abstract The boundary element method is an established numerical tool for the analysis of acoustic pressure fields in an infinite domain. There is currently no well established method of estimating the surface pressure error distribution for an arbitrary three dimensional body. Hierarchical shape functions have been used as a highly effective form of p refinement in many finite and boundary element applications. Their ability to be used as an error estimator in acoustic analysis has never been fully exploited. This paper studies the influence of mesh density and interpolation order on several acoustic scattering problems. A hierarchical error estimator is implemented and its effectiveness verified against the spherical problem. A coarse cylindrical mesh is then refined using the new error estimator until the solution has converged. The effectiveness of this analysis is shown by comparing the error indicators derived during the analysis to the solution generated from a very fine cylindrical mesh.

INVERSE ACOUSTIC SCATTERING PROBLEMS IN OCEAN ENVIRONMENTS

1999 ◽  
Vol 07 (02) ◽  
pp. 111-132 ◽  
Author(s):  
YONGZHI XU
Keyword(s):  
Acoustic Waves ◽  
Three Dimensional ◽  
Acoustic Scattering ◽  
Stratified Medium ◽  
Computational Results ◽  
Scattering Problems ◽  
Numerical Examples ◽  
Layered Waveguide ◽  
Shape Determination

This paper presents theoretical and computational results from our research on inverse scattering problems for acoustic waves in ocean environments. In particular, we discuss the determination of a three-dimensional (3-D) distributed inhomogeneity in a two-layered waveguide from scattered sound and the shape determination of an object in a stratified medium. Numerical examples are presented.

On the Numerical Radiation Condition in the Steady-State Ship Wave Problem

1987 ◽  
Vol 31 (01) ◽  
pp. 14-22
Author(s):  
Peter Schjeldahl Jensen
Keyword(s):  
Green Function ◽  
Three Dimensional ◽  
Radiation Condition ◽  
Function Technique ◽  
Wave Problem ◽  
Ship Wave ◽  
Rankine Source ◽  
Wigley Hull ◽  
The Green Function ◽  
The Waves

The waves created by a thin ship sailing in calm water are examined. The velocity potential of the ship in the zero Froude number case is known and the additional potential due to the waves is calculated by the Green function technique. The simple Green function corresponding to the Rankine source potential is used here. Two major problems exist with this method. In the Neumann-Poisson boundary-value problem- probably the first iteration toward a full nonlinear solution to the ship wave problem _it is necessary to impose a radiation condition in order to get uniqueness. This problem is related to the second one, which arises due to the existence of eigensolutions. The two-dimensional situation is here analyzed first, thereby easing the three-dimensional analysis. A numerical scheme is constructed and results for the twodimensional waves generated by a submerged vortex and for the three-dimensional waves due to the Wigley hull are presented.

scholarly journals Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160255 ◽  
Author(s):  
Oscar P. Bruno ◽  
Stephen P. Shipman ◽  
Catalin Turc ◽  
Stephanos Venakides
Keyword(s):  
Green Function ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Green Functions ◽  
Lattice Sums ◽  
Periodic Arrays ◽  
The Green Function ◽  
Three Dimensional Space

This work, part I in a two-part series, presents: (i) a simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) an associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain ‘Wood frequencies’ at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function—that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) This paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.

Three-dimensional interaction between uniform current and a submerged horizontal cylinder in an ice-covered channel

2021 ◽  
Vol 928 ◽  
Author(s):  
Y.F. Yang ◽  
G.X. Wu ◽  
K. Ren
Keyword(s):  
Green Function ◽  
Circular Cylinder ◽  
Body Surface ◽  
Velocity Potential ◽  
Three Dimensional ◽  
Ice Sheet ◽  
The Body ◽  
Thin Elastic Plate ◽  
Uniform Current ◽  
The Green Function

The problem of interaction of a uniform current with a submerged horizontal circular cylinder in an ice-covered channel is considered. The fluid flow is described by linearized velocity potential theory and the ice sheet is treated as a thin elastic plate. The potential due to a source or the Green function satisfying all boundary conditions apart from that on the body surface is first derived. This can be used to derive the boundary integral equation for a body of arbitrary shape. It can also be used to obtain the solution due to multipoles by differentiating the Green function with its position directly. For a transverse circular cylinder, through distributing multipoles along its centre line, the velocity potential can be written in an infinite series with unknown coefficients, which can be determined from the impermeable condition on a body surface. A major feature here is that different from the free surface problem, or a channel without the ice sheet cover, this problem is fully three-dimensional because of the constraints along the intersection of the ice sheet with the channel wall. It has been also confirmed that there is an infinite number of critical speeds. Whenever the current speed passes a critical value, the force on the body and wave pattern change rapidly, and two more wave components are generated at the far-field. Extensive results are provided for hydroelastic waves and hydrodynamic forces when the ice sheet is under different edge conditions, and the insight of their physical features is discussed.

An isogeometric formulation of locally-conformal perfectly matched layer for acoustic scattering problems

2021 ◽  
Vol 263 (6) ◽  
pp. 829-833
Author(s):  
Yongzhen Mi ◽  
Xiang Yu
Keyword(s):  
Acoustic Scattering ◽  
Perfectly Matched Layer ◽  
Real Space ◽  
Three Dimensions ◽  
Sound Waves ◽  
Computational Accuracy ◽  
Scattering Problems ◽  
New Formulation ◽  
Locally Conformal ◽  
Acoustic Scattering Problems

This paper presents an isogeometric formulation of the locally-conformal perfectly matched layer (PML) for time-harmonic acoustic scattering problems. The new formulation is a generalization of the conventional locally-conformal PML, in which the NURBS patch supporting the PML domain is transformed from real space to complex space. This is achieved by complex coordinate stretching, based on a stretching vector field indicating the directions in which incident sound waves are absorbed. The performance of the isogeometric PML formulation is discussed through several acoustic scattering problems, spanning from one to three dimensions. It is found that the proposed method presents superior computational accuracy, high geometric adaptivity, and good robustness against challenging geometric features. The geometry-preserving ability inherent in the isogeometric framework could bring extra benefits by eliminating geometric errors that are unavoidable in the conventional PML. Meanwhile, these properties are not sensitive to the location of the sound source or the depth of the PML domain.

FICTITIOUS DOMAIN METHODS FOR THE NUMERICAL SOLUTION OF THREE-DIMENSIONAL ACOUSTIC SCATTERING PROBLEMS

1999 ◽  
Vol 07 (03) ◽  
pp. 161-183 ◽  
Author(s):  
ERKKI HEIKKOLA ◽  
YURI A. KUZNETSOV ◽  
KONSTANTIN N. LIPNIKOV
Keyword(s):  
Boundary Value Problem ◽  
Numerical Solution ◽  
Boundary Value ◽  
Three Dimensional ◽  
Acoustic Scattering ◽  
Fictitious Domain ◽  
Scattering Problems ◽  
Element Discretization ◽  
Fictitious Domain Methods ◽  
Acoustic Scattering Problems

Efficient iterative methods for the numerical solution of three-dimensional acoustic scattering problems are considered. The underlying exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a non-reflecting boundary condition on the artificial boundary. The finite element discretization of the approximate boundary value problem is performed using locally fitted meshes, and algebraic fictitious domain methods with separable preconditioners are applied to the solution of the resultant mesh equations. These methods are based on imbedding the original domain into a larger one with a simple geometry (for example, a sphere or a parallelepiped). The iterative solution method is realized in a low-dimensional subspace, and partial solution methods are applied to the linear systems with the preconditioner. The results of numerical experiments demonstrate the efficiency and accuracy of the approach.

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