基于分区光滑理论与无网格法的声学数值方法研究
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摘要
随着数值计算技术的发展和计算机性能的快速提高,工程师们可以在设计和试制阶段利用数值方法对工业产品的噪声性能进行预测,并提出改进降噪措施。一些声学领域中原本不可能预测的工程实际问题可以借助数值方法来进行仿真。声学数值计算己成为噪声控制和预测的关键技术。针对Helmholtz方程的数值计算方法的改进是近年来声学数值计算研究的一个热点问题,其重点在于如何提高声学数值计算的精度、效率以及算法的适用性。
众所周知,数值色散误差是求解Helmholtz方程时的一个难点,它使得计算精度随波数的增加而变差,并且计算精度受波数、网格模型质量的影响严重。为了改善数值计算结果,本文对声学光滑有限元法、声学分区光滑径向点插值法、声学有限元法。最小二乘点插值法和声学有限元-径向点插值法等进行了深入系统的研究,并将这些方法应用到一些工程问题的求解中,包括“中气专项”轿车车内声场分析和微车车内的结构-声场耦合分析。
论文主要研究工作和创新性成果有：
(1)声学有限元法(Finite Element Method, FEM)模型偏硬,导致数值色散误差明显且计算精度易受网格质量和波数的影响。针对这一问题,将分区光滑理论推广至声学数值计算领域,研究提出了声压梯度分区光滑处理技术,推导了声学光滑有限元法(Smoothed Finite Element Method, SFEM)的基本公式。以二维矩形域声学问题和“中气专项”轿车车内声腔为研究对象,深入研究了声学SFEM的计算精度与网格质量、波数等因素之间的关系,分析结果表明：与声学FEM相比,声学SFEM具有更高的计算精度,且数值结果不易受网格质量、波数的影响。因此,声学SFEM比FEM更适合求解单元扭曲严重、波数高的工程问题。
(2)由于FEM模型存在数值色散误差,导致应用FEM分析结构-声场耦合问题时计算结果易受分析频率和单元尺寸的影响。针对这一问题,将光滑有限元法应用于结构-声场耦合问题分析中,研究提出了光滑有限元法/边界元法(Smoothed Finite Element Method/Boundary Element Method, SFEM/BEM)和光滑有限元/有限元法(SFEM/FEM),推导了SFEM/BEM和SFEM/FEM分析板、壳结构-声场耦合系统的计算公式。算例分析和工程应用研究表明：与SFEM/BEM和SFEM/FEM比较,SFEM/BEM和SFEM/FEM在分析结构-声场耦合问题时的计算精度和效率均有所提高,具有良好的工程应用前景。
(3)为了降低无网格法的数值色散误差,将声压梯度分区光滑处理技术与径向点插值法相结合,研究提出了声学数值分析的分区光滑径向点插值法(Cell-based Smoothed Radial Point Interpolation Method,CS-RPIM),推导了CS-RPIM分析声学问题的基本公式。将该方法的数值仿真结果与SFEM及FEM的数值解进行对比验证了该方法的有效性,同时证明了该方法在网格质量要求、计算精度和高波数问题分析上都有较大的优势。
(4)针对FEM模型过硬、而无网格法计算复杂、积分不稳定的问题,将有限元法与无网格法相结合的思想推广到声学波动方程计算中,研究提出了声学数值分析的有限元-最小二乘点插值法(Finite Element-least Square Point Interpolation Method,FE-LSPIM),推导了FE-LSPIM求解声学波动方程的公式。数值算例分析表明：与声学FEM和EFGM相比,FE-LSPIM分析声学问题时计算误差更小,能更好地适用于求解网格质量较差的工程问题。
(5)根据FE-LSPIM的基本思路,研究并提出了声学有限元-径向点插值法(Finite Element-Radial Point Interpolation,FE-RPIM)。该方法的构造的形函数继承了有限元法的单元兼容性和径向点插值法(The Radial Point Interpolation,RPIM)的克罗内克尔性质。与FE-LSPIM不同的是,FE-RPIM的局部近似采用RPIM而非LSPIM。数值算例表明：声学FE-RPIM比FEM和(?)FE-LSPIM具有更高的精度和更好的收敛性。
(6)以“中气专项”轿车车内声学模型和微车产品的结构-声场耦合模型为工程应用对象,验证了本文所提方法的有效性和优越性。
本文在声学数值计算方法方面做了深入的探讨,研究提出了声学光滑有限元法、声学分区光滑径向点插值法、声学有限元-最小二乘点插值法和声学有限元-径向点插值法。研究成果能有效应用于声学问题数值计算中,具有广阔的工程实际应用前景。
本文研究工作受到国家“跃升计划”专项一中国高水平汽车自主创新能力建设(简称“中气专项”)与教育部“长江学者和创新团队发展计划”项目(5311050050037)的资助。
With the development of the numerical techniques and the computer performance, engineers can predict the acoustic performance of industrial products and proposes schemes to improve the acoustic performance by using some numerical methods. Some acoustic engineering problems which can not be predicted previously can be simulated currently by using the numerical methods, and the computational acoustics has become a key technology of forecasting and controlling of noise. In the past decades, there has been an increasing interest in the simulation of noise, either to satisfy more and more stringent national and international standards or to improve end-user's comfort. The numerical simulation of the elastic and acoustic wave propagation, addressed by the Helmholtz equation, is a field of intense developments. The search for simple, efficient and accurate numerical methods applicable to acoustic problem, has been receiving much interest.
It is well-known that one key issue of solving the Helmholtz equation using numerical method is the accuracy deterioration in the solution with increasing wave number due to the "numerical dispersion error". In order to improve the accuracy of numerical solution, this dissertation makes an intensive and systemic study of the acoustic smoothed finite element method, the cell-based smoothed radial point interpolation method, the finite element-least square point interpolation method and the finite element-radial point interpolation method. These methods are applied to solve some engineering problems, including the acoustic simulation of "Zhongqi Specific Projects" car and the coupled analysis of mirco-car structural-acoustic models.
The main research work and innovative achievements in this dissertation are:
(1) The "overly-stiff" property of acoustic finite element method leads to biggish numerical dispersion error, and the accuracy is easily affected by the mesh quality and the wave number. Aim at this problem, the cell-based smoothed theory is extended to the field of acoustic numerical simulation and the acoustic gradient cell-based smoothing operation is proposed, and then the basic formulation of the acoustic smoothed finite element method (SFEM) is induced. Numerical examples, such as the two-dimensional acoustic square domain model and the cavity model of "Zhongqi Specific Projects" car, are analyzed intensively. The relationship, between the accuracy and the factors, such as the mesh quality and the wave number, is obtained. The results show that, compared to the corresponding FEM, the error of the acoustic SFEM model is smaller, and the numerical solution is more insusceptible to the mesh quality and the wave number. Hence, the acoustic SFEM model is more suitable than FEM model to analyze engineering problem with severe element distortion and high wave number.
(2) The results of structural-acoustic problems using the FEM are susceptible to the element size and the analytical frequency because of the numerical dispersion error. Aim at this problem, the smoothed finite element method is extended to the coupled analysis of structural-acoustic problems, and then the SFEM/FEM and SFEM/BEM are further derived for the analysis of structural-acoustic problems. The basic formulations of SFEM/FEM and SFEM/BEM are induced, respectively. The investigation of numerical examples and engineering applications show that, the accuracy and efficiency of the SFEM/BEM and SFEM/FEM is higher than that of the corresponding SFEM/BEM and SFEM/FEM, respectively. Hence, the SFEM/BEM and SFEM/FEM have great potential in the practical analysis of engineering problems.
(3) In order to reduce the numerical dispersion error of meshless methods, a cell-based smoothed radial point interpolation method (CS-RPIM) is extended to the field of acoustic numerical simulation by incorporating the cell-based acoustic gradient smoothing operation into the radial point interpolation method, and then the basic formulation of cell-based smoothed radial point interpolation method for acoustic problems is induced. At the same time, the great advantage of the method in demand of mesh quality, calculation precision and wave number are proved.
(4) The FEM model is in general overly-stiff; the meshless methods are complex and suffer from numerical instability. In order to overcome these problems, the strategy, which is presented by incorporating the meshless method and FEM, is extended to the field of acoustic numerical simulation, and the finite element-least square point interpolation method is investigated as well as the basic formulations are induced. The results of numerical examples show that, the error of acoustic FE-LSPIM model is smaller and the approximate solution is more insusceptible to the mesh quality and the wave number as compared to the FEM and EFGM. Hence, the acoustic FE-LSPIM model is more suitable than the corresponding FEM and EFGM model to analyze engineering problems with low quality mesh.
(5) According to the basic ideas of FE-LSPIM, the finite element-the radial point interpolation method (FE-RPIM) is proposed and extended to the field of acoustic numerical simulation. The shape functions of the FE-RPIM inherit the compatibility properties of finite element method and the Kronecker-delta property of the radial point interpolation method (RPIM). Unlike the FE-LSPIM, the local approximation in FE-RPIM doesn't use the least-square point interpolation method but uses the radial point interpolation method. Numerical tests demonstrate that the accuracy and convergence property of the acoustic FE-RPIM are more excellent than the FEM and FE-LSPIM.
(6) This dissertation analyzes some engineering problems, such as the "Zhongqi Specific Projects" car's cavity model and the mirco-car structural-acoustic coupled model. The analysis results show the effectiveness and superiority of proposed methods.
A detail study for the acoustic numerical computation method is implemented in this dissertation. Some acoustic numerical methods such as the smoothed finite element method, the cell-based smoothed radial point interpolation method, the finite element-the least square point interpolation method and the finite element-the radial point interpolation method, are proposed to the acoustic problems. The Research findings can be applied to solving the acoustic problems and has more engineering application foreground.
The research of this dissertation is supported by the national major projects called "Yueshen program"-the high level of automotive innovation capacity building (referred to as "Zhongqi Specific Projects") and the program for Changjiang Scholar and Innovative Research Team in University (5311050050037).
众所周知,数值色散误差是求解Helmholtz方程时的一个难点,它使得计算精度随波数的增加而变差,并且计算精度受波数、网格模型质量的影响严重。为了改善数值计算结果,本文对声学光滑有限元法、声学分区光滑径向点插值法、声学有限元法。最小二乘点插值法和声学有限元-径向点插值法等进行了深入系统的研究,并将这些方法应用到一些工程问题的求解中,包括“中气专项”轿车车内声场分析和微车车内的结构-声场耦合分析。
论文主要研究工作和创新性成果有：
(1)声学有限元法(Finite Element Method, FEM)模型偏硬,导致数值色散误差明显且计算精度易受网格质量和波数的影响。针对这一问题,将分区光滑理论推广至声学数值计算领域,研究提出了声压梯度分区光滑处理技术,推导了声学光滑有限元法(Smoothed Finite Element Method, SFEM)的基本公式。以二维矩形域声学问题和“中气专项”轿车车内声腔为研究对象,深入研究了声学SFEM的计算精度与网格质量、波数等因素之间的关系,分析结果表明：与声学FEM相比,声学SFEM具有更高的计算精度,且数值结果不易受网格质量、波数的影响。因此,声学SFEM比FEM更适合求解单元扭曲严重、波数高的工程问题。
(2)由于FEM模型存在数值色散误差,导致应用FEM分析结构-声场耦合问题时计算结果易受分析频率和单元尺寸的影响。针对这一问题,将光滑有限元法应用于结构-声场耦合问题分析中,研究提出了光滑有限元法/边界元法(Smoothed Finite Element Method/Boundary Element Method, SFEM/BEM)和光滑有限元/有限元法(SFEM/FEM),推导了SFEM/BEM和SFEM/FEM分析板、壳结构-声场耦合系统的计算公式。算例分析和工程应用研究表明：与SFEM/BEM和SFEM/FEM比较,SFEM/BEM和SFEM/FEM在分析结构-声场耦合问题时的计算精度和效率均有所提高,具有良好的工程应用前景。
(3)为了降低无网格法的数值色散误差,将声压梯度分区光滑处理技术与径向点插值法相结合,研究提出了声学数值分析的分区光滑径向点插值法(Cell-based Smoothed Radial Point Interpolation Method,CS-RPIM),推导了CS-RPIM分析声学问题的基本公式。将该方法的数值仿真结果与SFEM及FEM的数值解进行对比验证了该方法的有效性,同时证明了该方法在网格质量要求、计算精度和高波数问题分析上都有较大的优势。
(4)针对FEM模型过硬、而无网格法计算复杂、积分不稳定的问题,将有限元法与无网格法相结合的思想推广到声学波动方程计算中,研究提出了声学数值分析的有限元-最小二乘点插值法(Finite Element-least Square Point Interpolation Method,FE-LSPIM),推导了FE-LSPIM求解声学波动方程的公式。数值算例分析表明：与声学FEM和EFGM相比,FE-LSPIM分析声学问题时计算误差更小,能更好地适用于求解网格质量较差的工程问题。
(5)根据FE-LSPIM的基本思路,研究并提出了声学有限元-径向点插值法(Finite Element-Radial Point Interpolation,FE-RPIM)。该方法的构造的形函数继承了有限元法的单元兼容性和径向点插值法(The Radial Point Interpolation,RPIM)的克罗内克尔性质。与FE-LSPIM不同的是,FE-RPIM的局部近似采用RPIM而非LSPIM。数值算例表明：声学FE-RPIM比FEM和(?)FE-LSPIM具有更高的精度和更好的收敛性。
(6)以“中气专项”轿车车内声学模型和微车产品的结构-声场耦合模型为工程应用对象,验证了本文所提方法的有效性和优越性。
本文在声学数值计算方法方面做了深入的探讨,研究提出了声学光滑有限元法、声学分区光滑径向点插值法、声学有限元-最小二乘点插值法和声学有限元-径向点插值法。研究成果能有效应用于声学问题数值计算中,具有广阔的工程实际应用前景。
本文研究工作受到国家“跃升计划”专项一中国高水平汽车自主创新能力建设(简称“中气专项”)与教育部“长江学者和创新团队发展计划”项目(5311050050037)的资助。
With the development of the numerical techniques and the computer performance, engineers can predict the acoustic performance of industrial products and proposes schemes to improve the acoustic performance by using some numerical methods. Some acoustic engineering problems which can not be predicted previously can be simulated currently by using the numerical methods, and the computational acoustics has become a key technology of forecasting and controlling of noise. In the past decades, there has been an increasing interest in the simulation of noise, either to satisfy more and more stringent national and international standards or to improve end-user's comfort. The numerical simulation of the elastic and acoustic wave propagation, addressed by the Helmholtz equation, is a field of intense developments. The search for simple, efficient and accurate numerical methods applicable to acoustic problem, has been receiving much interest.
It is well-known that one key issue of solving the Helmholtz equation using numerical method is the accuracy deterioration in the solution with increasing wave number due to the "numerical dispersion error". In order to improve the accuracy of numerical solution, this dissertation makes an intensive and systemic study of the acoustic smoothed finite element method, the cell-based smoothed radial point interpolation method, the finite element-least square point interpolation method and the finite element-radial point interpolation method. These methods are applied to solve some engineering problems, including the acoustic simulation of "Zhongqi Specific Projects" car and the coupled analysis of mirco-car structural-acoustic models.
The main research work and innovative achievements in this dissertation are:
(1) The "overly-stiff" property of acoustic finite element method leads to biggish numerical dispersion error, and the accuracy is easily affected by the mesh quality and the wave number. Aim at this problem, the cell-based smoothed theory is extended to the field of acoustic numerical simulation and the acoustic gradient cell-based smoothing operation is proposed, and then the basic formulation of the acoustic smoothed finite element method (SFEM) is induced. Numerical examples, such as the two-dimensional acoustic square domain model and the cavity model of "Zhongqi Specific Projects" car, are analyzed intensively. The relationship, between the accuracy and the factors, such as the mesh quality and the wave number, is obtained. The results show that, compared to the corresponding FEM, the error of the acoustic SFEM model is smaller, and the numerical solution is more insusceptible to the mesh quality and the wave number. Hence, the acoustic SFEM model is more suitable than FEM model to analyze engineering problem with severe element distortion and high wave number.
(2) The results of structural-acoustic problems using the FEM are susceptible to the element size and the analytical frequency because of the numerical dispersion error. Aim at this problem, the smoothed finite element method is extended to the coupled analysis of structural-acoustic problems, and then the SFEM/FEM and SFEM/BEM are further derived for the analysis of structural-acoustic problems. The basic formulations of SFEM/FEM and SFEM/BEM are induced, respectively. The investigation of numerical examples and engineering applications show that, the accuracy and efficiency of the SFEM/BEM and SFEM/FEM is higher than that of the corresponding SFEM/BEM and SFEM/FEM, respectively. Hence, the SFEM/BEM and SFEM/FEM have great potential in the practical analysis of engineering problems.
(3) In order to reduce the numerical dispersion error of meshless methods, a cell-based smoothed radial point interpolation method (CS-RPIM) is extended to the field of acoustic numerical simulation by incorporating the cell-based acoustic gradient smoothing operation into the radial point interpolation method, and then the basic formulation of cell-based smoothed radial point interpolation method for acoustic problems is induced. At the same time, the great advantage of the method in demand of mesh quality, calculation precision and wave number are proved.
(4) The FEM model is in general overly-stiff; the meshless methods are complex and suffer from numerical instability. In order to overcome these problems, the strategy, which is presented by incorporating the meshless method and FEM, is extended to the field of acoustic numerical simulation, and the finite element-least square point interpolation method is investigated as well as the basic formulations are induced. The results of numerical examples show that, the error of acoustic FE-LSPIM model is smaller and the approximate solution is more insusceptible to the mesh quality and the wave number as compared to the FEM and EFGM. Hence, the acoustic FE-LSPIM model is more suitable than the corresponding FEM and EFGM model to analyze engineering problems with low quality mesh.
(5) According to the basic ideas of FE-LSPIM, the finite element-the radial point interpolation method (FE-RPIM) is proposed and extended to the field of acoustic numerical simulation. The shape functions of the FE-RPIM inherit the compatibility properties of finite element method and the Kronecker-delta property of the radial point interpolation method (RPIM). Unlike the FE-LSPIM, the local approximation in FE-RPIM doesn't use the least-square point interpolation method but uses the radial point interpolation method. Numerical tests demonstrate that the accuracy and convergence property of the acoustic FE-RPIM are more excellent than the FEM and FE-LSPIM.
(6) This dissertation analyzes some engineering problems, such as the "Zhongqi Specific Projects" car's cavity model and the mirco-car structural-acoustic coupled model. The analysis results show the effectiveness and superiority of proposed methods.
A detail study for the acoustic numerical computation method is implemented in this dissertation. Some acoustic numerical methods such as the smoothed finite element method, the cell-based smoothed radial point interpolation method, the finite element-the least square point interpolation method and the finite element-the radial point interpolation method, are proposed to the acoustic problems. The Research findings can be applied to solving the acoustic problems and has more engineering application foreground.
The research of this dissertation is supported by the national major projects called "Yueshen program"-the high level of automotive innovation capacity building (referred to as "Zhongqi Specific Projects") and the program for Changjiang Scholar and Innovative Research Team in University (5311050050037).
引文
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