基于无网格方法的声学问题数值模拟研究
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摘要
无网格法是近年来发展比较迅速的一种数值方法,它可以克服传统数值分析方法对网格的依赖性,彻底或部分的消除网格,抛开网格的初始划分和网格重构,成为科学和工程数值计算研究的热点。径向基函数(Radial Basis Function, RBF)在散乱数据插值中已经得到了广泛的应用,基于RBF配点的无网格方法是一种真正的无网格方法,在求解偏微分方程过程中具有不需要任何网格、易于实施、计算精度高等诸多优点,近年来逐步发展成为一种比较成熟的无网格方法。
论文将基于RBF配点的无网格方法较为完整的引入到声学数值计算领域,针对RBF配点型无网格方法的自身特点和其应用于声学计算时所面临的特殊问题进行了深入的研究,以期利用RBF方法不需要网格划分等优点突破基于网格的数值方法在声学领域应用中的局限性,建立了适用于声学问题求解的无网格方法。
论文将RBF配点点型无网格方法引入计算气动声学(Computational aeroacoustics,CAA)领域,提出了适用于CAA问题求解的局部RBF配点法,克服了RBF全域配点方法所带来的病态密集矩阵难以求解问题,同时仅在计算数据点周围灵活配点的处理方式,也为求解中采用矢通量分裂方法引入迎风机制提供了保证。
论文提出了适用于声学特征值问题求解的边界型RBF无网格方法。该方法将修正变分原理和移动最小二乘相结合,利用RBF插值实现方程中非齐次项的求解。该方法的实施过程中只用在求解域的边界上布点,域内少量布点仅是为了实现径向基函数插值,这就使得该方法既具有边界元方法的半解析半数值方法的优点,求解精度比较高,同时又具有无网格方法的特点,不用在求解域内以及边界上划分网格,前处理非常简单。
论文研究了不同气流流动、不同壁面条件、不同截面形状的管道中声传播的基本规律和管道声模态的传播特点。利用RBF无网格方法对具有不同壁面边界条件的单扩张腔消声器的声传播模态及截止频率等进行了数值模拟,验证了RBF无网格方法应用于简单消声器结构声学性能计算的有效性。最后论文将基于RBF的无网格数值方法拓展至实际工程领域中消声器的声学性能计算。由单室扩张式、内插管单室扩张式和穿孔管共振式三种简单消声器结构的计算拓展至基于这三种基本结构组合的复合型消声器结构。通过数值计算与消声器实验结果对比表明,RBF无网格方法可以有效应用于工程实际中消声器的设计和计算。相对于传统的基于一维结构的计算方法而言,该数值计算中应用的是消声器结构的三维模型,得到的计算结果对于消声器结构的设计和改进更有指导意义。
本文对RBF无网格方法应用于声学问题计算的系统研究表明：基于RBF配点的无网格方法作为一种真正的无网格方法,摆脱了求解中对网格剖分的依赖,具有精度高、易于扩展到高维问题、中心点和配点设置灵活、编程简单等优点。可以预期RBF配点无网格方法必将会成为一种重要的声学数值模拟方法。
Meshless method is a new kind of numerical methods which has rapidly developed in the recent years. This method is independent of the concept of element which traditional element-type method depends on, and has the advantages that no elements are needed totally or partly, avoiding the onerous mesh generation and re-meshing, etc. The method is paid very much attention by scientists and engineers in various computational researches because of its greatly theoretical and applicable value. Radial basis function (RBF) has been used successfully in scattered data interpolation, the introduction of the RBF to the collocation method for solving partial differential equations has many advantages, such as the method is a truly meshless method, does not require any meshes, easy to implement, high accuracy, and gradually developed into a mature meshless method.
The main purpose of this dissertation is to introduce the basis system of the RBF collocation meshless method into acoustics researches and develop new types of meshless methods which are suitable for the acoustics numerical simulation. Therefore, two parts of important contents are studied in details, including researches on the common theoretical problems of meshless methods themselves and the special applicable background of acoustics, especially for the analysis of computational aeroacoustics (CAA) problems.
A localized RBF collocation method was proposed to solve CAA problems, the method yields the generation of a small interpolation matrix for each data center and hence circumvent the highly ill-conditioned dense matrix dute to the globally supported RBF. An upwind implementation is further introduced to contain the hyperbolic property of the governing equations by using flux vector splitting method.
A boundary type RBF meshless method is presented for acoustic eigenvalue problems, in which the modified variational formulation is combined with the moving least squares (MLS), while the domain integration is interpolated by RBF. This method is a truly meshless method, and elements are not required for either interpolation or integration, only discrete nodes are constructed on the boundary of a domain, several nodes in the domain are needed just for the RBF interpolation. The numerical examples show that the accuracy and convergence of the methodare high, and the pre-processing is very easy.
The propagation of acoustic waves along a duct with different air flow, different wall conditions, and different shapes of the duct cross-section is considered in this dissertation. The duct cut-off frequency and the transmission characteristics of acoustic mode are dicussed deeply. The numerical simulation on the acoustics perforces of a single cavity expansion muffler are carried by the localized RBF collocation method, including the rigid and impedance wall boundary conditions. Some studies on the transmission characteristics of acoustic mode and duct cut-off frequency are carried by the boundary-type RBF collocation method. The meshless method is finally extended to the calculation on acoustic performance of mufflers in practical engineering field. The comparision between the results obtained from the numerical method and which from the expermential shows that the meshless method can be applied to engineering pratice in the muffler design and calculation. As opposed to the traditional one-dimensional models, the use of RBF meshless method which based on three-dimensional models is more guidance for the muffler structure design.
The systematic studies on the numerical simulation of acoustic problems show that, as a truly meshless method, the RBF collocation method has several advantages over the traditional methods:such as it does not need any mesh, it is an independent spatial dimension which can be easily extended to high-dimensional problems, it can also be extended to solve the partial differentional equations with higher order derivatives whithout any difficulties, and it is easy in implementation and programming. RBF collocation method will become an important method in acoustic numerical simulation because of its feasibility, effectiveness and outstanding advantages in the simulation of acoustics.
论文将基于RBF配点的无网格方法较为完整的引入到声学数值计算领域,针对RBF配点型无网格方法的自身特点和其应用于声学计算时所面临的特殊问题进行了深入的研究,以期利用RBF方法不需要网格划分等优点突破基于网格的数值方法在声学领域应用中的局限性,建立了适用于声学问题求解的无网格方法。
论文将RBF配点点型无网格方法引入计算气动声学(Computational aeroacoustics,CAA)领域,提出了适用于CAA问题求解的局部RBF配点法,克服了RBF全域配点方法所带来的病态密集矩阵难以求解问题,同时仅在计算数据点周围灵活配点的处理方式,也为求解中采用矢通量分裂方法引入迎风机制提供了保证。
论文提出了适用于声学特征值问题求解的边界型RBF无网格方法。该方法将修正变分原理和移动最小二乘相结合,利用RBF插值实现方程中非齐次项的求解。该方法的实施过程中只用在求解域的边界上布点,域内少量布点仅是为了实现径向基函数插值,这就使得该方法既具有边界元方法的半解析半数值方法的优点,求解精度比较高,同时又具有无网格方法的特点,不用在求解域内以及边界上划分网格,前处理非常简单。
论文研究了不同气流流动、不同壁面条件、不同截面形状的管道中声传播的基本规律和管道声模态的传播特点。利用RBF无网格方法对具有不同壁面边界条件的单扩张腔消声器的声传播模态及截止频率等进行了数值模拟,验证了RBF无网格方法应用于简单消声器结构声学性能计算的有效性。最后论文将基于RBF的无网格数值方法拓展至实际工程领域中消声器的声学性能计算。由单室扩张式、内插管单室扩张式和穿孔管共振式三种简单消声器结构的计算拓展至基于这三种基本结构组合的复合型消声器结构。通过数值计算与消声器实验结果对比表明,RBF无网格方法可以有效应用于工程实际中消声器的设计和计算。相对于传统的基于一维结构的计算方法而言,该数值计算中应用的是消声器结构的三维模型,得到的计算结果对于消声器结构的设计和改进更有指导意义。
本文对RBF无网格方法应用于声学问题计算的系统研究表明：基于RBF配点的无网格方法作为一种真正的无网格方法,摆脱了求解中对网格剖分的依赖,具有精度高、易于扩展到高维问题、中心点和配点设置灵活、编程简单等优点。可以预期RBF配点无网格方法必将会成为一种重要的声学数值模拟方法。
Meshless method is a new kind of numerical methods which has rapidly developed in the recent years. This method is independent of the concept of element which traditional element-type method depends on, and has the advantages that no elements are needed totally or partly, avoiding the onerous mesh generation and re-meshing, etc. The method is paid very much attention by scientists and engineers in various computational researches because of its greatly theoretical and applicable value. Radial basis function (RBF) has been used successfully in scattered data interpolation, the introduction of the RBF to the collocation method for solving partial differential equations has many advantages, such as the method is a truly meshless method, does not require any meshes, easy to implement, high accuracy, and gradually developed into a mature meshless method.
The main purpose of this dissertation is to introduce the basis system of the RBF collocation meshless method into acoustics researches and develop new types of meshless methods which are suitable for the acoustics numerical simulation. Therefore, two parts of important contents are studied in details, including researches on the common theoretical problems of meshless methods themselves and the special applicable background of acoustics, especially for the analysis of computational aeroacoustics (CAA) problems.
A localized RBF collocation method was proposed to solve CAA problems, the method yields the generation of a small interpolation matrix for each data center and hence circumvent the highly ill-conditioned dense matrix dute to the globally supported RBF. An upwind implementation is further introduced to contain the hyperbolic property of the governing equations by using flux vector splitting method.
A boundary type RBF meshless method is presented for acoustic eigenvalue problems, in which the modified variational formulation is combined with the moving least squares (MLS), while the domain integration is interpolated by RBF. This method is a truly meshless method, and elements are not required for either interpolation or integration, only discrete nodes are constructed on the boundary of a domain, several nodes in the domain are needed just for the RBF interpolation. The numerical examples show that the accuracy and convergence of the methodare high, and the pre-processing is very easy.
The propagation of acoustic waves along a duct with different air flow, different wall conditions, and different shapes of the duct cross-section is considered in this dissertation. The duct cut-off frequency and the transmission characteristics of acoustic mode are dicussed deeply. The numerical simulation on the acoustics perforces of a single cavity expansion muffler are carried by the localized RBF collocation method, including the rigid and impedance wall boundary conditions. Some studies on the transmission characteristics of acoustic mode and duct cut-off frequency are carried by the boundary-type RBF collocation method. The meshless method is finally extended to the calculation on acoustic performance of mufflers in practical engineering field. The comparision between the results obtained from the numerical method and which from the expermential shows that the meshless method can be applied to engineering pratice in the muffler design and calculation. As opposed to the traditional one-dimensional models, the use of RBF meshless method which based on three-dimensional models is more guidance for the muffler structure design.
The systematic studies on the numerical simulation of acoustic problems show that, as a truly meshless method, the RBF collocation method has several advantages over the traditional methods:such as it does not need any mesh, it is an independent spatial dimension which can be easily extended to high-dimensional problems, it can also be extended to solve the partial differentional equations with higher order derivatives whithout any difficulties, and it is easy in implementation and programming. RBF collocation method will become an important method in acoustic numerical simulation because of its feasibility, effectiveness and outstanding advantages in the simulation of acoustics.
引文
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