Boundary integral method for scattering problems with cracks buried in a piecewise homogeneous medium

We consider the scattering of time‐harmonic acoustic plane waves by a crack buried in a piecewise homogeneous medium. The integral representation for a solution is obtained in the form of potentials by using Green's formula. The density in potentials satisfies the uniquely solvable Fredholm integral equation. Then we obtain the existence and uniqueness of the solution. Copyright © 2011 John Wiley & Sons, Ltd.     

Hierarchical matrix techniques for low- and high-frequency Helmholtz problems

Wolfgang Hackbusch ^{In this paper, we discuss the application of hierarchical matrixtechniques to the solution of Helmholtz problems with largewave number in 2D. We consider the Brakhage–Werner integralformulation of the problem discretized by the Galerkin boundary-elementmethod. The dense n x n Galerkin matrix arising from this approachis represented by a sum of an -matrix and an 2-matrix, two different hierarchical matrix formats.A well-known multipole expansion is used to construct the 2-matrix. We present a new approach to dealingwith the numerical instability problems of this expansion: theparts of the matrix that can cause problems are approximatedin a stable way by an -matrix. Algebraic recompression methods are used to reducethe storage and the complexity of arithmetical operations ofthe -matrix.Further, an approximate LU decomposition of such a recompressed-matrix is aneffective preconditioner. We prove that the construction ofthe matrices as well as the matrix-vector product can be performedin almost linear time in the number of unknowns. Numerical experimentsfor scattering problems in 2D are presented, where the linearsystems are solved by a preconditioned iterative method.}     

A comparison of NRBCs for PUFEM in 2D Helmholtz problems at high wave numbers

In this work, exact and approximate non-reflecting boundary conditions (NRBCs) are implemented with the Partition of Unity Finite Element Method (PUFEM) to solve short wave scattering problems governed by the Helmholtz equation in two dimensions. By short wave problems, we mean situations in which the wavelength is a small fraction of the characteristic dimension of the scatterer. Various NRBCs are implemented and a comparison of their performance is carried out based on the accuracy of the results, ease of implementation and computational cost. The aim is to accurately model such problems in a reduced computational domain around the scatterer with fewer elements and without refining the mesh at each wave number.     

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     

An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium

Kozlov & Maz'ya (1989, Algebra Anal., 1, 144–170)proposed an alternating iterative method for solving Cauchyproblems for general strongly elliptic and formally self-adjointsystems. However, in many applied problems, operators appearthat do not satisfy these requirements, e.g. Helmholtz-typeoperators. Therefore, in this study, an alternating procedurefor solving Cauchy problems for self-adjoint non-coercive ellipticoperators of second order is presented. A convergence proofof this procedure is given.     

On radiation conditions for rough surface scattering problems

^** Email: arens{at}numathics.com^*** Email: hohage{at}math.uni-goettingen.de It is well known that Sommerfeld's radiation condition is nota valid characterization of outgoing waves for scattering problemsat rough surfaces. Instead, a radiation condition called upwardpropagating radiation condition (UPRC) is commonly used. Recently,a different radiation condition called the pole condition hasbeen investigated for scattering problems at bounded obstacles.In this paper we show the equivalence between the UPRC and thepole condition. In doing so, we give a rigorous interpretationof a formula called the angular spectrum representation forDirichlet data in the space of bounded continuous functions.     

双连通区域上的电磁波散射问题数值解

对于双连通区域上的电磁波散射问题,通过位势理论将其转化为边界积分方程组问题,然后采用Nystrom法和配置法对其离散求解,针对不同形状的障碍散射体,给出远场模式的数值解.     

Fast approximate computation of a time-harmonic scattered field using the on-surface radiation condition method

The numerical study of the on-surface radiation condition methodapplied to two- and three-dimensional time-harmonic scattering problems is examined. This approach allows us to quickly computean approximate solution to the initial exact boundary-valueproblem. A general background for the numerical treatment ofarbitrary convex-shaped objects is stated. New efficient on-surfaceradiation conditions leading in a natural way to a symmetricalboundary variational formulation are introduced. The approximationis based upon boundary finite-element methods. Moreover, thisstudy requires a specific numerical treatment of the curvatureoperator. To this end, a numerical procedure using some resultsabout the theory of local approximation of surfaces is described.Finally, the effectiveness and generality of the approach isnumerically tested for several scatterers.     

Wavelets collocation methods for the numerical solution of elliptic BV problems

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.     

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods appliedto boundary value problems for the positive definite Helmholtz-typeproblem –U + ²U = 0 in a bounded or unbounded domain,with the parameter real and possibly large. Applications arisein the implementation of space–time boundary integralmethods for the heat equation, where is proportional to 1/(t),and t is the time step. The corresponding layer potentials arisingfrom this problem depend nonlinearly on the parameter and havekernels which become highly peaked as , causing standard discretizationschemes to fail. We propose a new collocation method with arobust convergence rate as . Numerical experiments on a modelproblem verify the theoretical results.     

Comparing numerical methods for Helmholtz equation model problem

In this article, we implement a relatively new numerical technique, Adomian’s decomposition method for solving the linear Helmholtz partial differential equations. The method in applied mathematics can be an effective procedure to obtain for the analytic and approximate solutions. A new approach to a linear or nonlinear problems is particularly valuable as a tool for Scientists and Applied Mathematicians, because it provides immediate and visible symbolic terms of analytic solution as well as its numerical approximate solution to both linear and nonlinear problems without linearization [Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994; J. Math. Anal. Appl. 35 (1988) 501]. It does also not require discretization and consequently massive computation. In this scheme the solution is performed in the form of a convergent power series with easily computable components. This paper will present a numerical comparison with the Adomian decomposition and a conventional finite-difference method. The numerical results demonstrate that the new method is quite accurate and readily implemented.     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

一类带裂缝散射问题的边界积分方程方法

声波的散射问题中,如果散射体由不可穿透障碍物和可穿透裂缝两部分组成,障碍物表面分别满足第一类和第三类边界条件,裂缝两边满足不同的第二类边界条件,通过位势理论,可以将此混合问题转化为边界积分方程,通过Fredholm算子理论可以得到这个边界积分方程解的存在性和唯一性,从而获得原问题解的存在和唯一性.     

On surface radiation conditions for an ellipse

We compare several On Surface Radiation Boundary Conditions in two dimensions, for solving the Helmholtz equation exterior to an ellipse. We also introduce a new boundary condition for an ellipse based on a modal expansion in Mathieu functions. We compare the OSRC to a finite difference method.     

Analysis of the convected Helmholtz equation with a uniform mean flow in a waveguide with complete radiation boundary conditions

In this paper, we propose complete radiation boundary conditions (CRBCs) for solutions of the convected Helmholtz equation with a uniform mean flow in a waveguide. We first study CRBCs for the Helmholtz equation in a waveguide. Noting that the convected Helmholtz equation is associated with the Helmholtz equation via the Prandtl–Glauert transformation, CRBCs for the convected Helmholtz equation is derived from CRBCs for the Helmholtz equation. We analyse well-posedness and convergence of approximate solutions satisfying CRBCs for the convected Helmholtz equation. In addition, simple numerical experiments will be presented to confirm the theoretical results.     

Analysis of the coupling of BEM, FEM and mixed-FEM for a two-dimensional fluid–solid interaction problem

In this paper we consider a fluid–solid interaction problem posed in the plane. We employ a mixed variational formulation in the obstacle, in which the Cauchy stress tensor and the rotation are the only unknowns. This new mixed formulation is coupled, through suitable transmission conditions on the wet interface, with a Helmholtz equation satisfied by the pressure of the fluid in the unbounded domain. We use a traditional primal variational formulation in this part of the domain and incorporate the far field information through boundary integral equations. We approximate the resulting weak formulation by a Galerkin scheme based on PEERS in the solid and on a FEM-BEM approach in the fluid part. We show that our scheme is uniquely solvable and convergent, and then provide optimal error estimates. Finally, we illustrate our analysis with some computational experiments.     

The method of integral equation for scattering problem with a mixed crack

We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R² as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Coupling of mixed finite element and stabilized boundary element methods for a fluid–solid interaction problem in 3D

We introduce and analyze the coupling of a mixed finite element and a boundary element for a three‐dimensional time‐harmonic fluid–solid interaction problem. We consider a formulation in which the Cauchy stress tensor and the rotation are the main variables in the elastic structure and use the usual pressure formulation in the acoustic fluid. The mixed variational formulation in the solid is completed with boundary integral equations relating the Cauchy data of the acoustic problem on the coupling interface. A crucial point in our formulation is the stabilization technique introduced by Hiptmair and coworkers to avoid the well‐known instability issue appearing in the boundary element method treatment of the exterior Helmholtz problem. The main novelty of this formulation, with respect to a previous approach, consists in reducing the computational domain to the solid media and providing a more accurate treatment of the far field effect. We show that the continuous problem is well‐posed and propose a conforming Galerkin method based on the lowest‐order Arnold–Falk–Winther mixed finite element. Finally, we prove that the numerical scheme is convergent with optimal order.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1211–1233, 2014     

An adaptive solar radiation numerical model

A numerical model for the evaluation of solar radiation in different locations is presented. The solar radiation model is implemented taking into account the terrain surface using two-dimensional adaptive meshes of triangles that are constructed using a refinement/derefinement procedure in accordance with the variations of terrain surface and albedo. The selected methodology defines the terrain characteristics with a minimum number of points so that the computational cost is reduced for a given accuracy. The model can be used in atmospheric sciences as well as in other fields such as electrical engineering, since it allows the user to find the optimal location for maximum power generation in photovoltaic or solar thermal power plants. For this purpose, the effect of shadows is considered in each time step. The solar radiation is first computed for clear sky conditions considering the different components of the radiation. The real sky radiation is computed daily, starting from the results of clear sky radiation, in terms of the clear sky index. Maps for the clear sky index are obtained from a spatial interpolation of observational data that are available for each day at several points of the region under consideration. Finally, the solar radiation maps for a month are calculated from the daily results. The model can also be applied in solar radiation forecasting with the help of a forecasting meteorological model. This model takes into account the shadows cast, and allows the user to make a better estimation of the amount of solar power generation. Some numerical experiments related to the generation of solar radiation maps in Gran Canaria Island (Canary Islands, Spain) are presented.