伽辽金多极边界元法及其在声学中的应用
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摘要
随着计算机技术的飞速发展，声学数值计算方法已成为交通运输、航空航天、机械、国防等诸多领域中噪声预测与噪声控制的有效手段。其中，边界元法以其降维、计算精度高、自动满足远场辐射条件等方面的优点，被广泛用于声学问题的数值计算中。但是，传统边界元法从发展之初就存在一个明显的弱点，即由边界积分方程离散生成系统线性方程组的系数矩阵为满秩矩阵。求解该类型线性方程组所需要的高计算量以及高存储量，限制了传统边界元法求解声学问题的规模。因此，在传统边界元法的基础上，发展一种可高效求解声学问题的数值方法具有重要的意义。本文在深入研究声学伽辽金边界元法以及快速多极算法的基础上，发展了一种在宽频带范围内具有较高计算效率的伽辽金多极边界元法，并对二维和三维算法实现过程中的关键问题进行了详细分析。本文主要的研究工作如下：
(1)将快速多极算法引入伽辽金边界元法中，提出了一种求解二维声学问题的宽频带伽辽金多极边界元法。首先，在求解无限域声学问题时，使用Burton-Miller方程来保证数值结果在全频率段上的唯一性。在处理由Burton-Miller方程中引入的奇异算子时，利用Laplace方程基本解相关性质以及奇异减方法得到一类可快速求解的积分形式；其次，在分析边界积分方程基本解的分波扩展以及平面波展开形式的基础上，构造了一种适合求解宽频带声学问题的伽辽金多极边界元法，并结合快速多极边界元法求解声学问题一般步骤，给出了宽频带伽辽金多极边界元法的实现过程。同时，在构造二维四叉树结构时，使用改进的相互作用列表划分方式，以提高快速多极算法自身的计算效率。在使用广义极小残差方法求解线性方程组时，使用近似求逆方法处理系统线性方程组的系数矩阵，以加速迭代法的收敛。使用二维刚性圆柱面声散射的标准算例验证了二维宽频带伽辽金多极边界元法的计算效率和计算精度。对系统方程组采用的预处理方法，可显著减少求解方程组所需迭代步数，提高线性方程组的计算效率。最后，对含有34个刚性圆柱面的大规模多体散射模型进行数值计算，清楚的表明二维宽频带伽辽金多极边界元法在求解大规模声学问题的潜力。
(2)基于快速多极算法中的低频和高频展开形式，结合伽辽金格式的边界积分方程,提出了一种求解三维声学问题的宽频带伽辽金多极边界元法。首先，对求解三维声学的低频和高频快速多极算法进行了详细分析，并基于Burton-Miller方程推导了三维宽频带声学伽辽金多极边界元法的相关公式。其次，在求解超奇异积分算子时，采用广义函数的正则化方法，将边界积分方程中含有基本解的两阶偏导数转化为边界未知函数的旋度，将该积分的超奇异性降为弱奇异性，以提高超奇异积分的计算效率。而后，在三维宽频带伽辽金多极边界元法中截断项的选取时，结合低频和高频算法中的选取方式，给出了适合宽频带算法的展开系数截断项的确定方法。最有，通过L形箱体，三维刚性球声辐射和声散射等多个算例验证了三维宽频带伽辽金多极边界元法的准确性与有效性。
(3)利用半空间形式的基本解，提出了求解半空间声学问题的伽辽金多极边界元法。首先，在二维和三维全空间算法的基础上，构造半空间形式的基本解，推导了求解半空间声学问题的伽辽金多极边界元法的计算公式。与全空间镜像方法相比，半空间方法略去了由建立镜像模型带来的计算量及存储量问题；其次，结合三维树结构，推导了半空间格林函数中源点与镜像点之间局部系数的转移公式，给出了求解半空间声学问题的伽辽金多极边界元法的实现过程；最后，通过二维和三维半空间数值算例，验证了半空间伽辽金多极边界元法的准确性，高效性。
(4)利用半空间宽频带伽辽金多极边界元法进行了三维有限长声屏障的降噪性能分析,证明了该方法的有效性。首先，建立了直立形声屏障的模型，通过与文献结果对比，验证了由半空间宽频带伽辽金多极边界元法得到的数值结果的正确性；其次，对比了不同频率时，直立形声屏障和T形声屏障的降噪效果；最后，以三维有限长T形声屏障为分析对象，对影响降噪效果的T形声屏障几何参数以及不同多孔介质材料吸声处理等因素对声屏障降噪性能进行了初步分析。相关工作为三维声屏障的设计提供了一种有效的数值预测方法。
With the rapid development of computational technology, numerical method iswidely used to solve acoustic problems in many areas of engineering, such astransportation, aerospace, mechanical and military fields. For the acoustic problems,the boundary element method (BEM) is one of the most developed and acceptednumerical methods. Since it process many advantages, for instances, reduce thedimensionalities of the problems by one, high accuracy and suitable to solveunbounded domain problems. However, one disadvantage makes the BEM falling tosolve large scale acoustic problems, that is, it leads to linear system equations withfully populated, sometime ill conditioned coefficient matrices. The storagerequirements and computational time to solve this kind of system equations arerelatively high. As a result, it is necessary to develop a new method, which inherits allthe advantages of the BEM and at the same time processes higher computationalefficiency. In this dissertation, the fast multipole method is introduced into Galerkinboundary element method, and a wideband multipole Galerkin boundary elementmethod is developed to solve large scale acoustic problems. The implementation ofthe wideband multipole Galerkin boundary element is investigated in detail. The mainresearch works and results of this dissertation are listed as follows:
(1) The fast multipole method is introduced into the Galerkin BEM, a newwideband multipole Galerkin BEM is presented for solving two-dimensional acousticproblems. In order to remove the non-uniqueness problems associated withconventional BEM when solving exterior acoustic problems, the Burton-Millerformulation–a linear combination of the boundary integral equation and its normalderivative equation is employed. The hyper-singular integral is desingularized usingsome properties of Laplace equation and the singular subtraction approach, the newformulation of the hypersingular integral can be calculated efficiency. Based on thepartial wave expansion method and the plane wave expansion method, twoformulations of the fast method are developed for the low and high frequency acousticproblems repectively. In order to obtain overall computational efficiency in widebandfrequencies, a wideband multipole Galerkin BEM is proposed. It unifies previousexisting fast multipole method for low and high frequencies into an algorithm, which is accurate and efficient for any frequency. This wideband multipole Galerkin BEMhaving a CPU time of O(n) if low frequency computations dominate, or O(nlog2n) ifthe high frequency computations dominate. A brief procedure of this widebandmultipole Galerkin BEM is presented. During the process of construction of the quadtree, a modified definition of the interaction list, which can reduce the multipoleexpansion coefficient to local expansion coefficient translations is suitably adopted.Furthermore, an efficient preconditioning technique–spares approximate inverse pre-conditioner is employed to improve the convergence of the generalized minimalresidual (GMRES) solver. Finally, the numerical result of the rigid cylinder scatteringproblem demonstrates the accuracy and efficiency of the wideband multipole GalerkinBEM for two-dimensional acoustic problems. The sparse approximate inversepreconditioning technique dramatically reduces the iteration steps required by theGREMS solver, the total computational efficiency is further improved. A multi-bodyscattering problem with34cylinders is solved effectively. This example clearly showsthe great potential of the wideband multipole Galerkin BEM for engineeringapplications.
(2) Based on the Burton-Miller formulation, a wideband multipole Galerkin BEMcombining with the low frequency and high frequency method is proposed to solve thethree-dimensional acoustic problems. The method evaluating the hyper-singularintegral in three dimension problem is differ from the method used in chapter two. Byusing the idea of regularization in sense of distributions, the double normalderivatives for hyper-singular integral are shifted to the boundary rotations of trialfunction and test function, then the hyper-singular integral is transformed to a weaklysingular form. To determine the number of terms in multipole and local expansions,an improved empirical formula is applied. Finally, several examples including an Lshaped box radiation problem, a sphere radiation problem and a sphere scatteringproblem are studied to investigate the accuracy and efficiency of the threedimensional wideband multipole Galerkin BEM.
(3) The half space fundamental solution of the boundary integral equation isemployed, a new multipole Galerkin BEM for solving half space acoustic problems isproposed. Compared with full space method, a tree structure of the boundary elementsonly for the boundaries of the real domain need to be constructed. The implementationof the half space multipole Galerkin BEM is simplified, only the local expansion isdifferent from the full space method. Finally, the two-dimensional and three-dimensional examples validate the accuracy and efficiency of the half spacewideband multipole Galerkin BEM. By using the half space method, the half spaceacoustic problems can be solved with less CPU time.
(4) The half space multipole Galerkin BEM is employed to predict the acousticperformance of three dimensional finite noise barrier, and the accuracy of the methodis validated. Firstly, a three dimensional upright noise barrier with finite length isestablished. The simulation result is coincidence with the result given in the referencebook, and the accuracy of the half space method is validated again. Secondly, underdifferent frequencies, the performance of T shape noise barrier is considered.Examination of the source and receiver location, the geometry parameters andtreatment of the T top surface is investigated. Finally, more engineering applicationsare expected to be simulated by the wideband multipole Galerkin BEM algorithm inthe future.
(1)将快速多极算法引入伽辽金边界元法中，提出了一种求解二维声学问题的宽频带伽辽金多极边界元法。首先，在求解无限域声学问题时，使用Burton-Miller方程来保证数值结果在全频率段上的唯一性。在处理由Burton-Miller方程中引入的奇异算子时，利用Laplace方程基本解相关性质以及奇异减方法得到一类可快速求解的积分形式；其次，在分析边界积分方程基本解的分波扩展以及平面波展开形式的基础上，构造了一种适合求解宽频带声学问题的伽辽金多极边界元法，并结合快速多极边界元法求解声学问题一般步骤，给出了宽频带伽辽金多极边界元法的实现过程。同时，在构造二维四叉树结构时，使用改进的相互作用列表划分方式，以提高快速多极算法自身的计算效率。在使用广义极小残差方法求解线性方程组时，使用近似求逆方法处理系统线性方程组的系数矩阵，以加速迭代法的收敛。使用二维刚性圆柱面声散射的标准算例验证了二维宽频带伽辽金多极边界元法的计算效率和计算精度。对系统方程组采用的预处理方法，可显著减少求解方程组所需迭代步数，提高线性方程组的计算效率。最后，对含有34个刚性圆柱面的大规模多体散射模型进行数值计算，清楚的表明二维宽频带伽辽金多极边界元法在求解大规模声学问题的潜力。
(2)基于快速多极算法中的低频和高频展开形式，结合伽辽金格式的边界积分方程,提出了一种求解三维声学问题的宽频带伽辽金多极边界元法。首先，对求解三维声学的低频和高频快速多极算法进行了详细分析，并基于Burton-Miller方程推导了三维宽频带声学伽辽金多极边界元法的相关公式。其次，在求解超奇异积分算子时，采用广义函数的正则化方法，将边界积分方程中含有基本解的两阶偏导数转化为边界未知函数的旋度，将该积分的超奇异性降为弱奇异性，以提高超奇异积分的计算效率。而后，在三维宽频带伽辽金多极边界元法中截断项的选取时，结合低频和高频算法中的选取方式，给出了适合宽频带算法的展开系数截断项的确定方法。最有，通过L形箱体，三维刚性球声辐射和声散射等多个算例验证了三维宽频带伽辽金多极边界元法的准确性与有效性。
(3)利用半空间形式的基本解，提出了求解半空间声学问题的伽辽金多极边界元法。首先，在二维和三维全空间算法的基础上，构造半空间形式的基本解，推导了求解半空间声学问题的伽辽金多极边界元法的计算公式。与全空间镜像方法相比，半空间方法略去了由建立镜像模型带来的计算量及存储量问题；其次，结合三维树结构，推导了半空间格林函数中源点与镜像点之间局部系数的转移公式，给出了求解半空间声学问题的伽辽金多极边界元法的实现过程；最后，通过二维和三维半空间数值算例，验证了半空间伽辽金多极边界元法的准确性，高效性。
(4)利用半空间宽频带伽辽金多极边界元法进行了三维有限长声屏障的降噪性能分析,证明了该方法的有效性。首先，建立了直立形声屏障的模型，通过与文献结果对比，验证了由半空间宽频带伽辽金多极边界元法得到的数值结果的正确性；其次，对比了不同频率时，直立形声屏障和T形声屏障的降噪效果；最后，以三维有限长T形声屏障为分析对象，对影响降噪效果的T形声屏障几何参数以及不同多孔介质材料吸声处理等因素对声屏障降噪性能进行了初步分析。相关工作为三维声屏障的设计提供了一种有效的数值预测方法。
With the rapid development of computational technology, numerical method iswidely used to solve acoustic problems in many areas of engineering, such astransportation, aerospace, mechanical and military fields. For the acoustic problems,the boundary element method (BEM) is one of the most developed and acceptednumerical methods. Since it process many advantages, for instances, reduce thedimensionalities of the problems by one, high accuracy and suitable to solveunbounded domain problems. However, one disadvantage makes the BEM falling tosolve large scale acoustic problems, that is, it leads to linear system equations withfully populated, sometime ill conditioned coefficient matrices. The storagerequirements and computational time to solve this kind of system equations arerelatively high. As a result, it is necessary to develop a new method, which inherits allthe advantages of the BEM and at the same time processes higher computationalefficiency. In this dissertation, the fast multipole method is introduced into Galerkinboundary element method, and a wideband multipole Galerkin boundary elementmethod is developed to solve large scale acoustic problems. The implementation ofthe wideband multipole Galerkin boundary element is investigated in detail. The mainresearch works and results of this dissertation are listed as follows:
(1) The fast multipole method is introduced into the Galerkin BEM, a newwideband multipole Galerkin BEM is presented for solving two-dimensional acousticproblems. In order to remove the non-uniqueness problems associated withconventional BEM when solving exterior acoustic problems, the Burton-Millerformulation–a linear combination of the boundary integral equation and its normalderivative equation is employed. The hyper-singular integral is desingularized usingsome properties of Laplace equation and the singular subtraction approach, the newformulation of the hypersingular integral can be calculated efficiency. Based on thepartial wave expansion method and the plane wave expansion method, twoformulations of the fast method are developed for the low and high frequency acousticproblems repectively. In order to obtain overall computational efficiency in widebandfrequencies, a wideband multipole Galerkin BEM is proposed. It unifies previousexisting fast multipole method for low and high frequencies into an algorithm, which is accurate and efficient for any frequency. This wideband multipole Galerkin BEMhaving a CPU time of O(n) if low frequency computations dominate, or O(nlog2n) ifthe high frequency computations dominate. A brief procedure of this widebandmultipole Galerkin BEM is presented. During the process of construction of the quadtree, a modified definition of the interaction list, which can reduce the multipoleexpansion coefficient to local expansion coefficient translations is suitably adopted.Furthermore, an efficient preconditioning technique–spares approximate inverse pre-conditioner is employed to improve the convergence of the generalized minimalresidual (GMRES) solver. Finally, the numerical result of the rigid cylinder scatteringproblem demonstrates the accuracy and efficiency of the wideband multipole GalerkinBEM for two-dimensional acoustic problems. The sparse approximate inversepreconditioning technique dramatically reduces the iteration steps required by theGREMS solver, the total computational efficiency is further improved. A multi-bodyscattering problem with34cylinders is solved effectively. This example clearly showsthe great potential of the wideband multipole Galerkin BEM for engineeringapplications.
(2) Based on the Burton-Miller formulation, a wideband multipole Galerkin BEMcombining with the low frequency and high frequency method is proposed to solve thethree-dimensional acoustic problems. The method evaluating the hyper-singularintegral in three dimension problem is differ from the method used in chapter two. Byusing the idea of regularization in sense of distributions, the double normalderivatives for hyper-singular integral are shifted to the boundary rotations of trialfunction and test function, then the hyper-singular integral is transformed to a weaklysingular form. To determine the number of terms in multipole and local expansions,an improved empirical formula is applied. Finally, several examples including an Lshaped box radiation problem, a sphere radiation problem and a sphere scatteringproblem are studied to investigate the accuracy and efficiency of the threedimensional wideband multipole Galerkin BEM.
(3) The half space fundamental solution of the boundary integral equation isemployed, a new multipole Galerkin BEM for solving half space acoustic problems isproposed. Compared with full space method, a tree structure of the boundary elementsonly for the boundaries of the real domain need to be constructed. The implementationof the half space multipole Galerkin BEM is simplified, only the local expansion isdifferent from the full space method. Finally, the two-dimensional and three-dimensional examples validate the accuracy and efficiency of the half spacewideband multipole Galerkin BEM. By using the half space method, the half spaceacoustic problems can be solved with less CPU time.
(4) The half space multipole Galerkin BEM is employed to predict the acousticperformance of three dimensional finite noise barrier, and the accuracy of the methodis validated. Firstly, a three dimensional upright noise barrier with finite length isestablished. The simulation result is coincidence with the result given in the referencebook, and the accuracy of the half space method is validated again. Secondly, underdifferent frequencies, the performance of T shape noise barrier is considered.Examination of the source and receiver location, the geometry parameters andtreatment of the T top surface is investigated. Finally, more engineering applicationsare expected to be simulated by the wideband multipole Galerkin BEM algorithm inthe future.
引文
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