THE NUMERICAL SOLUTIONS OF CUT-OFF FREQUENCIES IN TWO-DIELECTRIC LAYERED WAVEGUIDE BY USING BOUNDARY ELEMENT METHOD

This paper diseusses the general principle of finding the cut-off frequencies in two-dielectric lay-ered waveguides by using the boundary element method,based on the fundamental solution of a twodimensional Helmholtz equation.In terms of the formulae given in the paper,some numerical resultsare obtained for two commonly used configurations.The fimal results show that the method is of an appre-ciable precision.

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 difficulties in the conventional BEM for exterior acoustic problems. The computational efficiency is further improved by adopting the FMM and the block diagonal preconditioner used in the generalized minimum residual method (GMRES) iterative solver to solve the system matrix equation. Numerical results clearly demonstrate the complete reliability and efficiency of the proposed algorithm. It is potentially useful for solving large-scale engineering acoustic scattering problems.

Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications

This paper describes formulation and implementation of the fast multipole boundary element method （FMBEM） for 2D acoustic problems. The kernel function expansion theory is summarized, and four building blocks of the FMBEM are described in details. They are moment calculation, moment to moment translation, moment to local translation, and local to local translation. A data structure for the quad-tree construction is proposed which can facilitate implementation. An analytical moment expression is derived, which is more accurate, stable, and efficient than direct numerical computation. Numerical examples are presented to demonstrate the accuracy and efficiency of the FMBEM, and radiation of a 2D vibration rail mode is simulated using the FMBEM.

A simplified two-dimensional boundary element method with arbitrary uniform mean flow

To reduce computational costs, an improved form of the frequency domain boundary element method（BEM） is proposed for two-dimensional radiation and propagation acoustic problems in a subsonic uniform flow with arbitrary orientation. The boundary integral equation（BIE） representation solves the two-dimensional convected Helmholtz equation（CHE） and its fundamental solution, which must satisfy a new Sommerfeld radiation condition（SRC） in the physical space. In order to facilitate conventional formulations, the variables of the advanced form are expressed only in terms of the acoustic pressure as well as its normal and tangential derivatives, and their multiplication operators are based on the convected Green＇s kernel and its modified derivative. The proposed approach significantly reduces the CPU times of classical computational codes for modeling acoustic domains with arbitrary mean flow. It is validated by a comparison with the analytical solutions for the sound radiation problems of monopole,dipole and quadrupole sources in the presence of a subsonic uniform flow with arbitrary orientation.

Analysis of Numerical Integration Error for Bessel Integral Identity in Fast Multipole Method for 2D Helmholtz Equation

In 2D fast multipole method for scattering problems,square quadrature rule is used to discretize the Bessel integral identity for diagonal expansion of 2D Helmholtz kernel,and numerical integration error is introduced. Taking advantage of the relationship between Euler-Maclaurin formula and trapezoidal quadrature rule,and the relationship between trapezoidal and square quadrature rule,sharp computable bound with analytical form on the error of numerical integration of Bessel integral identity by square quadrature rule is derived in this paper. Numerical experiments are presented at the end to demonstrate the accuracy of the sharp computable bound on the numerical integration error.

A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems

This paper presents a novel wideband fast multipole boundary element approach to 3D half-space/planesymmetric acoustic wave problems.The half-space fundamental solution is employed in the boundary integral equations so that the tree structure required in the fast multipole algorithm is constructed for the boundary elements in the real domain only.Moreover,a set of symmetric relations between the multipole expansion coefficients of the real and image domains are derived,and the half-space fundamental solution is modified for the purpose of applying such relations to avoid calculating,translating and saving the multipole/local expansion coefficients of the image domain.The wideband adaptive multilevel fast multipole algorithm associated with the iterative solver GMRES is employed so that the present method is accurate and efficient for both lowand high-frequency acoustic wave problems.As for exterior acoustic problems,the Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method.Details on the implementation of the present method are described,and numerical examples are given to demonstrate its accuracy and efficiency.

A Modified Helmholtz Equation with Impedance Boundary Conditions

Here considered is the problem of transient electromagnetic scattering from overfilled cavities embedded in an impedance ground plane.An artificial boundary condition is introduced on a semicircle enclosing the cavity that couples the fields from the infinite exterior domain to those fields inside.A Green’s function solution is obtained for the exterior domain,while the interior problem is solved using finite element method.Well-posedness of the associated variational formulation is achieved and convergence and stability of the numerical scheme confirmed.Numerical experiments show the accuracy and robustness of the method.

Boundary Element Modeling of Multiconnected Ocean Basin in Visakhapatnam Port Under the Resonance Conditions

A mathematical model has been developed to analyze the influence of extreme water waves over multiconnected regions in Visakhapatnam Port,India by considering an average water depth in each multiconnected regions.In addition,partial reflection of incident waves on coastal boundary is also considered.The domain of interest is divided mainly into two regions,i.e.,open sea region and harbor region namely as Region-I and Region-II,respectively.Further,Region-II is divided into multiple connected regions.The 2-D boundary element method(BEM)including the Chebyshev point discretization is utilized to solve the Helmholtz equation in each region separately to determine the wave amplification.The numerical convergence is performed to obtain the optimum numerical accuracy and the validation of the current numerical approach is also conducted by comparing the simulation results with existing studies.The four key spots based on the moored ship locations in Visakhapatnam Port are identified to perform the numerical simulation.The wave amplification at these locations is estimated for monochromatic incident waves,considering approximate water depth and different reflection coefficients on the wall of port under the resonance conditions.In addition,wave field analysis inside the Visakhapatnam Port is also conducted to understand resonance conditions.The current numerical model provides an efficient tool to analyze the amplification on any realistic ports or harbors.

Directional H^(2) Compression Algorithm: Optimisations and Application to a Discontinuous Galerkin BEM for the Helmholtz Equation

This study aimed to specialise a directional H^(2)(DH^(2))compression to matrices arising from the discontinuous Galerkin(DG)discretisation of the hypersingular equation in acoustics.The significantfinding is an algorithm that takes a DG stiffness matrix andfinds a near-optimal DH^(2) approximation for low and high-frequency problems.We introduced the necessary special optimisations to make this algorithm more efficient in the case of a DG stiffness matrix.Moreover,an automatic parameter tuning strategy makes it easy to use and versatile.Numerical comparisons with a classical Boundary Element Method(BEM)show that a DG scheme combined with a DH^(2) gives better computational efficiency than a classical BEM in the case of high-order finite elements and hp heterogeneous meshes.The results indicate that DG is suitable for an auto-adaptive context in integral equations.

耦合边界元法和等效源法的稳健CHIEF法

针对声学边界元法中解的非唯一性和奇异积分问题,基于组合亥姆霍兹积分方程公式(combined helmholtz integral equation formulation,CHIEF)法思想,将常规边界元方程和等效源方程进行联立,并利用两者方程系数矩阵间的耦合等价关系,间接替换计算常规边界元法中的奇异系数矩阵,进而提出一种具有全频域唯一解、高计算精度和高稳定性的耦合CHIEF法。该方法将等效源方程作为补充方程,不仅解决了传统CHIEF法内点补充方程失效的问题,而且矩阵的间接替换计算避免了直接计算奇异积分,显著提高了计算效率和精度。通过声辐射和声散射的典型算例对比了所提方法、常规边界元法、常规Burton-Miller法和等效源法的计算效果。结果表明,所提方法不仅在全波数域内均能获得唯一解,且其计算精度和效率均优于常规边界元法和常规Burton-Miller方法,其系数矩阵条件数远低于等效源法。

伸缩虚拟边界元法解二维Helmholtz外问题

以位势理论为基础,提出了求解Helmholtz外问题的伸缩虚拟边界元法.给出了该方法在全波数域内获得唯一解的严格数学证明,其核心是通过伸缩虚拟边界使对偶内问题的特征频率(本征值)避开与波数重合,从而保证了解的唯一性,同以往前人提出的几种解法途径相比,该法简单得多;通过诸多边界曲线形状和不同边界量的声辐射算例,从计算精度、稳定性以及克服解的非唯一性等方面,对该方法进行了检验.计算结果表明:对远场或近场辐射声压,该方法都具有非常高的效率和精度.

三维Helmholtz方程外边值问题的虚边界元法

基于位势的延拓,推导出三维虚边界积分方程.通过选择不同的虚边界,避免相应内问题的特征值与波数重合,从而保证解的唯一性.数值算例验证了该方法求解任意波数三维Helmholtz方程外边值问题的有效性.

求解三维Helmholtz方程外边值问题的一种新的边界积分方程法

本文应用多重互反法（ｔｈｅｍｕｌｔｉｐｌｅｒｅｃｉｐｒｏｃｉｔｙｍｅｔｈｏｄ）给出了求解三维Ｈｅｌｍｈｏｌｔｚ外边值问题的一种新的边界积分方程法。首先，在限制解在无穷远处性态的Ｄｉｒｉｃｈｌｅｔ条件下，导出了解在外区域及边界上的积分表达式，其特点在于积分核是由Ｌａｐｌａｃｅ方程的常规基本解衍生出来的无穷级数且与波数无关。在此基础上，对Ｄｉｒｉｃｈｌｅｔ问题和Ｎｅｕｍａｎｎ问题导出了边界积分方程，并对数值求解这些方程所涉及的一些问题进行了评述，最后，总结了这一方法与传统边界元法相比较所具有的优点。

二维Helmholtz方程的边界元法

在已有位势问题工作的基础上,建立求解二维Helmholtz方程边值问题的间接变量规则化边界积分方程,它不包含CPV强奇异积分和HFP超奇异积分的计算。数值算例表明:本文方法在低频率下可取得较好的精度和效率。

求解二维非齐次Helmholtz方程的边界元法

基于Laplace方程的基本解讨论了二维非齐次Helmholtz方程的直接边界元解法.通过将Helmholtz方程变形之后加权Laplace方程的基本解和应用Green公式得到相应的直接积分方程,针对积分方程中同时存在域积分项和边界积分项,在应用边界元法分析求解时采用了耦合关于内点和边界点的积分方程求解,最后,通过数值算例验证方法的有效性.

论Helmholtz方程的一类边界积分方程的合理性

本文导出了Helmholtz 方程超定边值问题有解的一个充要条件,和用非解析开拓法证明了文[1]中的Helmholtz 方程在外域中的解的边界积分表示式的合理性,并将此类边界积分表示式推广用于带空洞的有限域。这样就比较严密而又浅近地证明了基于该表示式建立起来的间接变量和直接变量边界积分方程的合理性。

三维Helmholtz问题的间接规则化边界积分方程

进行了三维Helmholtz方程边值问题的间接变量规则化边界元法研究。使用位势问题的基本解逼近Helmholtz问题的基本解,将Helmholtz问题的边界积分方程的规则化问题转化为位势问题边界积分方程的规则化问题。应用已有的三维位势问题的基本解积分恒等式及规则化边界积分方程的极限定理,建立三维Helmholtz方程边值问题的间接变量规则化边界积分方程。此外,本文提出一般参数表示的几何边界上的精确单元的数值实施方案,即边界几何采用精确单元描述,单元上的边界量用二次8节点不连续插值函数逼近。数值算例表明:本文方法在低频下可获得很高的计算精度和效率。

快速多极边界元方法在大规模声学问题中的应用

为克服传统边界元方法不适合进行大规模声学问题仿真的困难,将快速多极算法应用到传统边界元方法中,对大规模声学问题进行数值计算。由于在快速多极边界元方法中引入基本解的多极扩展,并应用预处理后的广义极小残差法迭代求解器求解线性方程系统,使得快速多极边界元方法的计算效率与传统边界元方法相比显著提高,计算量和存储量减少到O(N)量级(N为问题的自由度数)。对于传统边界元方法求解外部声学问题时的非唯一解现象,在快速多极边界元方法中采用改进的Burton-Miller方法获得全频段的唯一解。数值算例验证了快速多极边界元方法的准确性,表明快速多极边界元方法的计算效率与传统边界元方法相比有数量级的提高,能够有效求解大规模声学问题。

脊的位置变化对双脊波导单模带宽的影响

采用亥姆霍兹方程有限元方法分析了非对称双脊波导脊的变化对其主模单模带宽的影响,结果表明,可以通过改变脊的尺寸和位置来调节单模带宽。与已有文献报道的结果进行了对比,结果显示利用该计算方法所得数据精确有效。

脊的变化对对称双脊波导主模截止波长的影响

关于双脊波导的研究报道很多,但对双脊波导的研究都集中于脊中心对称情况,对于脊偏离中心位置的情况则没有涉及。采用对亥姆霍兹方程有限元方法分析了脊的变化对双脊波导主模截止波长的影响,数值结果与文献结果对比,计算方法精确有效,从而可以通过改变脊尺寸和位置调节截止波长。