大规模声学问题的快速多极边界元方法研究
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摘要
边界元方法是一种有效的数值计算方法,它伴随着有限元方法发展起来。由于具有只在边界离散、计算精度高、适合处理无限域问题等优点,边界元方法被广泛用于声学问题研究。然而,边界元方法形成的线性系统方程组的系数矩阵通常是稠密、非对称满阵,使用常规的求解方法将消耗高昂的计算机资源,所以传统的边界元方法只适合分析中小规模的声学问题,无法求解大规模问题。计算能力成为制约边界元方法在大规模工程领域发展与应用的瓶颈。因此,发展一种新型的快速边界元方法求解大规模声学问题至关重要。本文主要是对快速多极方法和边界元方法进行深入研究,提出一种新型的快速多极边界元方法有效求解大规模声学问题。
针对传统边界元方法求解外部Helmholtz方程时存在的非唯一解问题,提出了采用Burton-Miller公式解决这个问题。对于Burton-Miller公式引入的超奇异及强奇异积分困难,利用新的奇异性减少技术及Laplace方程相关性质,推导了只包含完全弱奇异积分的Burton-Miller公式改进形式。此外,提出了一种简单有效的块对角预处理技术改善线性系统方程组的性态,加速求解的迭代收敛效率。数值结果证明了改进的边界元方法求解声学问题的准确性和有效性,同时也揭示出传统边界元方法的计算效率为O(N2)量级(N为问题的自由度数),不适合求解大规模声学问题。
基于改进的Burton-Miller公式,提出了一种新型的快速多极边界元方法求解大规模二维(2D)声学问题。利用多极扩展理论,推导了快速多极边界元方法的计算公式及数值程序处理流程。此外,根据树结构叶子结点的特点,提出了一种有效的稀疏近似求逆预处理技术全面提高快速多极边界元方法的迭代求解效率,并从理论上证明了新算法各求解步骤具有O(N)量级的计算复杂性。数值算例验证了快速多极边界元方法求解2D声学问题的准确性和有效性,进一步的数值测试表明新算法具有O(N)量级的计算效率,比传统边界元方法的计算效率有数量级的提高。包含24万个自由度的多体声散射算例有效证明了快速多极边界元方法可以在个人计算机上快速求解大规模2D声学问题,成功克服了传统边界元方法不能求解大规模声学问题的技术瓶颈,并且表现出潜在的工程应用价值。
将2D问题延伸到3D(三维)问题,基于改进的Burton-Miller公式,提出了一种新型的宽频快速多极边界元方法求解大规模全空间声学问题。分别利用分波扩展方法及平面波扩展方法,推导了快速多极边界元方法的低、高频问题计算公式,并提出一种可自适应实现求解低、高频问题的宽频无缝连接形式。此外,根据现有的经验方法,提出了适合宽频计算的截断项数确定公式。包含52万个自由度的数值算例验证了快速多极边界元方法求解3D大规模声学问题的准确性、有效性及工程适用性。
进一步将全空间问题延伸到半空间问题。基于3D全空间算法,提出了一种新型的快速多极边界元方法求解大规模半空间声学问题。利用半空间Green函数,推导了半空间快速多极边界元方法的计算公式。新的半空间算法只需要在实域建立树结构,而不需要建立包含实域和影像域的更大的树结构,这明显减少了新算法的计算时间和存储量要求。数值结果验证了半空间快速多极边界元方法求解大规模半空间声学问题的准确性、有效性,建筑物及声障分析的算例表明新算法可以处理大规模工程声学问题。
本文对声学快速多极边界元方法的基础理论及应用进行了深入的研究,结果表明发展的方法能够有效求解大规模声学问题,具有重要的学术研究价值及广阔的工程应用前景。
The boundary element method (BEM) is a numerical method along with the development of the finite element method (FEM). The BEM is widely used to solve acoustic problems, since it has attractive advantages of boundary discretization, high accuracy and is especially suitable to handle infinite domain problems. However, the most serious problem is that the BEM leads to linear system of equations with general dense, non-symmetrical coefficient matrices. Solving the BEM system of equations needs expensive computational costs, when traditional solution techniques are used. As a result, the BEM has been limited to solve relatively small- and moderate-size problems, and is not available for large-scale problems. The computational ability of the BEM becomes a bottleneck problem. This restricts the large-scale engineering development and application of the BEM. Thus, it is crucial to develop a new fast BEM for solving large-scale acoustic problems. This dissertation focuses on the research of the fast multipole method (FMM) and BEM, and develops a new fast multipole BEM (FMBEM) for solving large-scale acoustic problems.
The Burton-Miller formulation is employed to successfully remove the non-uniqueness problem associated with the conventional BEM for exterior Helmholtz equation. The major difficulty of the Burton-Miller formulation is that it includes a hypersingular integral. This dissertation proposes an improved form of the Burton-Miller formulation which it only contains weakly singular integrals, and avoids the difficulty of the hypersingular integral evaluation. Furthermore, the iterative efficiency of the presented method is significantly improved by adopting a simple and effective block diagonal preconditioner to improve the condition of the system matrix equations. The block diagonal preconditioner is very efficient and results in a large reduction of required iteration steps. Numerical results demonstrate the accuracy and efficiency of the improved BEM for acoustic problems, and show the conventional BEM needs O(N2) computational time and computer memory, where N is number of degrees of freedom (DOFs). Thus, the BEM is prohibitively expensive for solving large-scale acoustic problems.
A new fast multipole BEM based on the improved Burton-Miller formulation is presented for solving large-scale two-dimensional (2D) acoustic problems. According to the theories of multipole expansions, the formulations and algorithms of the fast multipole BEM are developed. Furthermore, for overall improving the computational efficiency of the presented method, an effective sparse approximate inverse preconditioner is constructed based on the leaves of tree structure. Then the O(N) complexity of the fast multipole BEM is verified using the theoretical analysis. Numerical results demonstrate the accuracy and efficiency of the fast multipole BEM for solving 2D acoustic problems. Further numerical tests show that the presented method has O(N) computational efficiency and provides an order of magnitude increase in efficiency compared to the conventional BEM. A multiple scattering model with 240000 DOFs is solved effectively on a personal computer. The results demonstrate that the fast multipole BEM has the advantage for large-scale acoustic problems, and successfully solves the bottleneck problem of the BEM. This example shows the great potential of the presented method for large-scale engineering applications.
The fast multipole BEM is extended from 2D to 3D acoustic problems. Based on the improved Burton-Miller formulation, a new wideband fast multipole BEM is presented for solving large-scale 3D full-space acoustic problems. According to the partial wave expansion method and plane wave expansion method, the formulations of the fast multipole BEM are developed for the low- and high-frequency problems, respectively. In order to further obtain overall computational efficiency in all frequencies, a seamless framework for adaptively combining the low- and high-frequency formulation is proposed. Furthermore, the practical formulation is presented to determine the number of truncation terms based on the empirical methods. The numerical examples including a model with 520000 DOFs clearly demonstrate the accuracy and efficiency of the fast multipole BEM for solving large-scale 3D acoustic problems in a wide frequency range, and show the potentially useful engineering applications.
The fast multipole BEM is further extended from full-space to half-space acoustic problems. Based on the full-space algorithm, a new fast multipole BEM for solving large-scale 3D half-space acoustic problems is presented. Using the half-space Green's function, the formulations of the half-space fast multipole BEM are developed. In the new half-space algorithm, a tree structure of boundary elements can be constructed in the real domain only, instead of using a larger tree structure that contains both the real domain and its mirror image, which greatly simplifies the implementation of the half-space fast multipole BEM and reduces the computational time and memory storage. The numerical examples validate the accuracy and efficiency of the fast multipole BEM for solving large-scale 3D half-space acoustic problems. The analysis of the building and sound barrier noise further illustrates the potential of the presented method for solving large-scale practical problems.
This dissertation mainly studies the fundamental theories and applications of the fast multipole BEM in acoustics, the research results demonstrate that the fast multipole BEM is efficient for solving large-scale acoustic problem, and show important academic value and extensive engineering prospect.
针对传统边界元方法求解外部Helmholtz方程时存在的非唯一解问题,提出了采用Burton-Miller公式解决这个问题。对于Burton-Miller公式引入的超奇异及强奇异积分困难,利用新的奇异性减少技术及Laplace方程相关性质,推导了只包含完全弱奇异积分的Burton-Miller公式改进形式。此外,提出了一种简单有效的块对角预处理技术改善线性系统方程组的性态,加速求解的迭代收敛效率。数值结果证明了改进的边界元方法求解声学问题的准确性和有效性,同时也揭示出传统边界元方法的计算效率为O(N2)量级(N为问题的自由度数),不适合求解大规模声学问题。
基于改进的Burton-Miller公式,提出了一种新型的快速多极边界元方法求解大规模二维(2D)声学问题。利用多极扩展理论,推导了快速多极边界元方法的计算公式及数值程序处理流程。此外,根据树结构叶子结点的特点,提出了一种有效的稀疏近似求逆预处理技术全面提高快速多极边界元方法的迭代求解效率,并从理论上证明了新算法各求解步骤具有O(N)量级的计算复杂性。数值算例验证了快速多极边界元方法求解2D声学问题的准确性和有效性,进一步的数值测试表明新算法具有O(N)量级的计算效率,比传统边界元方法的计算效率有数量级的提高。包含24万个自由度的多体声散射算例有效证明了快速多极边界元方法可以在个人计算机上快速求解大规模2D声学问题,成功克服了传统边界元方法不能求解大规模声学问题的技术瓶颈,并且表现出潜在的工程应用价值。
将2D问题延伸到3D(三维)问题,基于改进的Burton-Miller公式,提出了一种新型的宽频快速多极边界元方法求解大规模全空间声学问题。分别利用分波扩展方法及平面波扩展方法,推导了快速多极边界元方法的低、高频问题计算公式,并提出一种可自适应实现求解低、高频问题的宽频无缝连接形式。此外,根据现有的经验方法,提出了适合宽频计算的截断项数确定公式。包含52万个自由度的数值算例验证了快速多极边界元方法求解3D大规模声学问题的准确性、有效性及工程适用性。
进一步将全空间问题延伸到半空间问题。基于3D全空间算法,提出了一种新型的快速多极边界元方法求解大规模半空间声学问题。利用半空间Green函数,推导了半空间快速多极边界元方法的计算公式。新的半空间算法只需要在实域建立树结构,而不需要建立包含实域和影像域的更大的树结构,这明显减少了新算法的计算时间和存储量要求。数值结果验证了半空间快速多极边界元方法求解大规模半空间声学问题的准确性、有效性,建筑物及声障分析的算例表明新算法可以处理大规模工程声学问题。
本文对声学快速多极边界元方法的基础理论及应用进行了深入的研究,结果表明发展的方法能够有效求解大规模声学问题,具有重要的学术研究价值及广阔的工程应用前景。
The boundary element method (BEM) is a numerical method along with the development of the finite element method (FEM). The BEM is widely used to solve acoustic problems, since it has attractive advantages of boundary discretization, high accuracy and is especially suitable to handle infinite domain problems. However, the most serious problem is that the BEM leads to linear system of equations with general dense, non-symmetrical coefficient matrices. Solving the BEM system of equations needs expensive computational costs, when traditional solution techniques are used. As a result, the BEM has been limited to solve relatively small- and moderate-size problems, and is not available for large-scale problems. The computational ability of the BEM becomes a bottleneck problem. This restricts the large-scale engineering development and application of the BEM. Thus, it is crucial to develop a new fast BEM for solving large-scale acoustic problems. This dissertation focuses on the research of the fast multipole method (FMM) and BEM, and develops a new fast multipole BEM (FMBEM) for solving large-scale acoustic problems.
The Burton-Miller formulation is employed to successfully remove the non-uniqueness problem associated with the conventional BEM for exterior Helmholtz equation. The major difficulty of the Burton-Miller formulation is that it includes a hypersingular integral. This dissertation proposes an improved form of the Burton-Miller formulation which it only contains weakly singular integrals, and avoids the difficulty of the hypersingular integral evaluation. Furthermore, the iterative efficiency of the presented method is significantly improved by adopting a simple and effective block diagonal preconditioner to improve the condition of the system matrix equations. The block diagonal preconditioner is very efficient and results in a large reduction of required iteration steps. Numerical results demonstrate the accuracy and efficiency of the improved BEM for acoustic problems, and show the conventional BEM needs O(N2) computational time and computer memory, where N is number of degrees of freedom (DOFs). Thus, the BEM is prohibitively expensive for solving large-scale acoustic problems.
A new fast multipole BEM based on the improved Burton-Miller formulation is presented for solving large-scale two-dimensional (2D) acoustic problems. According to the theories of multipole expansions, the formulations and algorithms of the fast multipole BEM are developed. Furthermore, for overall improving the computational efficiency of the presented method, an effective sparse approximate inverse preconditioner is constructed based on the leaves of tree structure. Then the O(N) complexity of the fast multipole BEM is verified using the theoretical analysis. Numerical results demonstrate the accuracy and efficiency of the fast multipole BEM for solving 2D acoustic problems. Further numerical tests show that the presented method has O(N) computational efficiency and provides an order of magnitude increase in efficiency compared to the conventional BEM. A multiple scattering model with 240000 DOFs is solved effectively on a personal computer. The results demonstrate that the fast multipole BEM has the advantage for large-scale acoustic problems, and successfully solves the bottleneck problem of the BEM. This example shows the great potential of the presented method for large-scale engineering applications.
The fast multipole BEM is extended from 2D to 3D acoustic problems. Based on the improved Burton-Miller formulation, a new wideband fast multipole BEM is presented for solving large-scale 3D full-space acoustic problems. According to the partial wave expansion method and plane wave expansion method, the formulations of the fast multipole BEM are developed for the low- and high-frequency problems, respectively. In order to further obtain overall computational efficiency in all frequencies, a seamless framework for adaptively combining the low- and high-frequency formulation is proposed. Furthermore, the practical formulation is presented to determine the number of truncation terms based on the empirical methods. The numerical examples including a model with 520000 DOFs clearly demonstrate the accuracy and efficiency of the fast multipole BEM for solving large-scale 3D acoustic problems in a wide frequency range, and show the potentially useful engineering applications.
The fast multipole BEM is further extended from full-space to half-space acoustic problems. Based on the full-space algorithm, a new fast multipole BEM for solving large-scale 3D half-space acoustic problems is presented. Using the half-space Green's function, the formulations of the half-space fast multipole BEM are developed. In the new half-space algorithm, a tree structure of boundary elements can be constructed in the real domain only, instead of using a larger tree structure that contains both the real domain and its mirror image, which greatly simplifies the implementation of the half-space fast multipole BEM and reduces the computational time and memory storage. The numerical examples validate the accuracy and efficiency of the fast multipole BEM for solving large-scale 3D half-space acoustic problems. The analysis of the building and sound barrier noise further illustrates the potential of the presented method for solving large-scale practical problems.
This dissertation mainly studies the fundamental theories and applications of the fast multipole BEM in acoustics, the research results demonstrate that the fast multipole BEM is efficient for solving large-scale acoustic problem, and show important academic value and extensive engineering prospect.
引文
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