Transparent boundary conditions for the Helmholtz equation in some ramified domains with a fractal boundary

The paper addresses a class of boundary value problems in some self-similar ramified domains, with the Laplace or Helmholtz equations. Much stress is placed on transparent boundary conditions which allow the solutions to be computed in subdomains. A self similar finite element method is proposed and tested. It can be used for numerically computing the spectrum of the Laplace operator with Neumann boundary conditions, as well as the eigenmodes. The eigenmodes are normalized by means of a perturbation method and the spectral decomposition of a compactly supported function is carried out. Finally, a numerical method for the wave equation is addressed.     

Nonreflecting boundary conditions for discrete waves

We introduce a new class of nonreflecting boundary conditions for lattice models, which minimizes reflections at artificial boundaries. Exact integrodifferential boundary conditions for finite chains and half-spaces are obtained by means of Green’s functions for initial value problems. Truncating the resulting integrals in time, we obtain absorbing boundary conditions. Numerical tests illustrate the ability of these conditions to suppress reflections.     

Computation of bound states for 2D potentials using discrete basis

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate in closed form the matrix representation of the associated Hamiltonian for two exactly solvable 2D potentials. This enabled us to treat analytically the full Hamiltonian and compute the associated bound states spectrum as the eigenvalues of the associated analytical matrix representing their Hamiltonians. Finally we compared our results satisfactorily with those obtained using the Gauss quadrature numerical integration approach.

PACS numbers: 03.65.Ge, 34.20.Cf, 03.65.Nk, 34.20.Gj     

Helmholtz边界积分方程的多频计算

提出了利用无穷级数展开的方法,将波数从Helmholtz边界积分方程的特解中分离出来,使随波数变化的系统矩阵变为波数的矩阵级数形式,同时证明了级数截断时的收敛性。数值结果表明,结合CHIEF方法,用级数展开的方法不仅能有效地克服频域内非唯一现象,节省计算时间;而且当频率较高时,在单元粗剖分下也能得到满意的结果。     

Fast multipole accelerated boundary element method for the Helmholtz equation in acoustic scattering problems

We apply the fast multipole method (FMM) accelerated boundary element method (BEM) for the three-dimensional (3D) Helmholtz equation, and as a result, large-scale acoustic scattering problems involving 400000 elements are solved efficiently. This is an extension of the fast multipole BEM for two-dimensional (2D) acoustic problems developed by authors recently. Some new improvements are obtained. In this new technique, the improved Burton-Miller formulation is employed to over-come non-uniqueness difficultie...     

Fast integral equation methods for the modified Helmholtz equation

We present integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(x) ? α²Δu(x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of our methods on several numerical examples, and we show that they have both the ability to handle highly complex geometry and the potential to solve large-scale problems.     

Two transparent boundary conditions for the electromagnetic scattering from two-dimensional overfilled cavities

We consider electromagnetic scattering from two-dimensional (2D) overfilled cavities embedded in an infinite ground plane. The unbounded computational domain is truncated to a bounded one by using a transparent boundary condition (TBC) proposed on a semi-ellipse. For overfilled rectangular cavities with homogeneous media, another TBC is introduced on the cavity apertures, which produces a smaller computational domain. The existence and uniqueness of the solutions of the variational formulations for the transverse magnetic and transverse electric polarizations are established. In the exterior domain, the 2D scattering problem is solved in the elliptic coordinate system using the Mathieu functions. In the interior domain, the problem is solved by a finite element method. Numerical experiments show the efficiency and accuracy of the new boundary conditions.     

Helmholtz equation and non-singular boundary elements applied to multi-disciplinary physical problems

The famous scientist Hermann von Helmholtz was born 200 years ago. Many complex physical wave phenomena in engineering can effectively be described using one or a set of equations named after him: the Helmholtz equation. Although this has been known for a long time, from a theoretical point of view, the actual numerical implementation has often been hindered by divergence-free and/or curl-free constraints. There is further a need for a numerical method that is accurate, reliable and takes into account radiation conditions at infinity. The classical boundary element method satisfies the last condition, yet one has to deal with singularities in the implementation. We review here how a recently developed singularity-free three-dimensional boundary element framework with superior accuracy can be used to tackle such problems only using one or a few Helmholtz equations with higher order (quadratic) elements which can tackle complex curved shapes. Examples are given for acoustics (a Helmholtz resonator among others) and electromagnetic scattering.     

Thomas-Fermi equation with non-spherical boundary conditions

The confined atom Thomas-Fermi equation with non-spherical boundary conditions is considered. A 2-D finite element code for solving the Thomas-Fermi equation with general boundary conditions is demonstrated. Results for both Dirichlet and Neumann boundary conditions for ellipsoids of revolution are presented.     

新的对角形式快速多极边界元法求解声学Helmholtz方程

传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势, 具有良好的工程应用前景.     

新的对角形式快速多极边界元法求解声学Helmholtz方程

传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势,     

Exact solution for the fractional cable equation with nonlocal boundary conditions

The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.     

Some remarkableR-separable coordinate systems for the Helmholtz equation

We present examples to show that the phenomenon ofR-separation occurs nontrivially for the Helmholtz equation on a pseudo-Riemannian manifold.R-separable coordinate systems can be both orthogonal and non-orthogonal and a given coordinate system mayR-separate in more than one way. A satisfactory theory of variable separation for the Helmholtz equation must incorporateR-separation.Supported in part by NSF Grant MCS 78-26216.     

High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces

This work overcomes the difficulty of the previous matched interface and boundary (MIB) method in dealing with interfaces with non-constant curvatures for optical waveguide analysis. This difficulty is essentially bypassed by avoiding the use of local cylindrical coordinates in the improved MIB method. Instead, novel jump conditions are derived along global Cartesian directions for the transverse magnetic field components. Effective interface treatments are proposed to rigorously impose jump conditions across arbitrarily curved interfaces based on a simple Cartesian grid. Even though each field component satisfies the scalar Helmholtz equation, the enforcement of jump conditions couples two transverse magnetic field components, so that the resulting MIB method is a full-vectorial approach for the modal analysis of optical waveguides. The numerical performance of the proposed MIB method is investigated by considering interface problems with both constant and general curvatures. The MIB method is shown to be able to deliver a fourth order of accuracy in all cases, even when a high frequency solution is involved.     

Improved transmission conditions for a one-dimensional domain decomposition method applied to the solution of the Helmholtz equation

The scattering problem of a time-harmonic electromagnetic wave from a perfect electric conductor (PEC) coated with materials is considered, and solved by coupling a finite element method with an integral equation prescribed on the outer boundary of the computational domain. To reduce the numerical complexity, a one-dimensional domain decomposition method (DDM) is employed: the computational domain is partitioned into concentric subdomains (SDs), and Robin transmission conditions (TCs) are prescribed on the interfaces. For some configurations and/or materials, the convergence of the corresponding DDM algorithm happens to be slow. A possible remedy is to enhance the efficiency of the TCs by approximating the exact ones more accurately. To this end, we first consider the simplified 2D problem of a circular PEC cylinder with an homogeneous coating and up to four SDs with circular interfaces, thus allowing to obtain the exact TCs in closed-form. Approximate local or non-local TCs are derived from these exact ones, and numerical examples demonstrate their superiority over the standard Robin TCs. Then, the case of an elliptical PEC cylinder with one interface in free-space is investigated. Also, the issues pertaining to the uniqueness of the solutions and convergence of the algorithm are addressed.     

A dispersion minimizing finite difference scheme and preconditioned solver for the 3D Helmholtz equation

In this paper, a new 27-point finite difference method is presented for solving the 3D Helmholtz equation with perfectly matched layer (PML), which is a second order scheme and pointwise consistent with the equation. An error analysis is made between the numerical wavenumber and the exact wavenumber, and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters. A full-coarsening multigrid-based preconditioned Bi-CGSTAB method is developed for solving the linear system stemming from the Helmholtz equation with PML by the finite difference scheme. The shifted-Laplacian is extended to precondition the 3D Helmholtz equation, and a spectral analysis is given. The discrete preconditioned system is solved by the Bi-CGSTAB method, with a multigrid method used to invert the preconditioner approximately. Full-coarsening multigrid is employed, and a new matrix-based prolongation operator is constructed accordingly. Numerical results are presented to demonstrate the efficiency of both the new 27-point finite difference scheme with refined parameters, and the preconditioned Bi-CGSTAB method with the 3D full-coarsening multigrid.     

Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation

Rarefied gas flow behavior is usually described by the Boltzmann equation, the Navier-Stokes system being valid when the gas is less rarefied. Slip boundary conditions for the Navier-Stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level (Boltzmann equation). These slip conditions are explicitly written in terms of asymptotic behavior of some linear half-space problems. The validity of this analysis is established in the simple case of the Couette flow, for which it is proved that the right boundary conditions are obtained.