边界无网格技术在噪声源识别中的应用

根据非共形面声全息对声辐射传播建模的高精确度要求,将边界无网格法引入声辐射传播建模过程中,实现对三维空间任意封闭曲面上Kirchhoff-Helmholtz边界积分方程高精度离散和求解计算。进一步研究非共形面声全息逆向重构问题的基本原理及其不适定性,采用了Landweber迭代正则化和L曲线正则化参数选取方法,从而确立有效的声场逆向重构求解方法。最后,还进行减速电机噪声源表面振动重构实验,验证研究的基于边界无网格的非共形面声全息的相关理论和方法可行。     

应用三次B样条函数插值的边界元法计算结构振动声辐射问题

本文通过在边界元方法中采用三次Ｂ样条函数作为插值形函数，对结构振动声辐射的计算进行了研究。并以脉动球作为算例，对其辐射声场中的有关声场参数进行了计算。通过将计算结果与理论解进行比较，结果表明：即使在边界剖分比较粗的情况下，利用该方法计算结构振动声辐射问题在较宽的振动频率范围内，也能给出良好的计算精度。     

求解远场声辐射问题的一种快速边界元法

针对传统边界元法奇异性的固有缺陷,研究一种求解结构辐射声场的间接边界元法。通过在内域虚拟曲面上配置一系列单偶极子源,等效模拟结构外辐射声场,并根据单元体积速度匹配法的边界条件方案,给出数值计算辐射声场的间接边界元模型。结合球冠形辐射器的仿真算例,表明该间接边界元法在求解远场声特性时具有较高的计算效率和精度,为分析设计具有大空间指向性的辐射器提供参考。     

基于分布源边界点法的近场声全息技术

分布源边界点法是一种新型的声辐射数值计算方法,通过分布在振动体内部一系列的特解源间接地构造出声源与声场之间的传递矩阵,从而有效地避开了采用边界元法求解声辐射问题时所存在的复杂的变量插值、奇异积分的处理、特征波数处解的非唯一性等问题.本文首先建立了分布源边界点法实现近场声全息变换的模型,并提出采用Tikhonov正则化方法稳定重建过程,抑制了重建误差的影响,随后通过实验研究进一步验证了文中方法的可行性和正确性.     

由水中结构物表面振速场计算辐射声场的理论及数值方法

一、计算原理 描述三维水介质中单频、稳态声场的波动方程有如下形式:式中:φ──速度势,c──水中声速 将波动方程与格林公式结合可推导出关于三维空间中结构体的边界积分方程:式中:C1──与表面形状有关的系数 S──包围结构体的曲面 u=e～jkr/4πr点源函数 u　=　n/　n;n──S面的内法向 φ=　φ/　n 边界积分方程中含有已知和未知的边界值,是计算结构体辐射声场的原始方程。 二、数值方法 采用边界元法计算辐射声场,首先将壳体表面边界离散成数个面单元,可采用四边形或三角形单元。单元内任一点的值可由节点处相应的值来表示,将离散单…     

基于滑动Kriging插值的无网格MLPG法求解结构动力问题

利用基于滑动Kriging插值的无网格局部Petrov-Galerkin (MLPG) 法来求解二维结构动力问题,Heaviside分段函数作为局部弱形式的权函数并采用精细积分法来离散时间域。基于滑动Kriging插值构造的形函数满足Kronecker Delta性质,因此可以直接施加本质边界条件。刚度矩阵形成过程中只涉及到边界积分,而没有涉及到区域积分和奇异积分。计算结果表明：基于滑动Kriging插值的MLPG法具有模拟简单、计算精度高等优点。     

电厂球磨机筒体辐射声场的计算

本文将噪声预测理论应用到球磨机的降噪工程上,针对球磨机建立了结构和激励简化模型,采用半解析的组合梁函数法求解出振动响应后,结合边界元方法计算了辐射声场分布。计算结果正确预测出近场中低频段的指向性特点和远场声压级水平,与实测谱值有较好的一致性。     

分阶拟合直接配点无网格法

在子域插值的基础上提出了分阶拟合直接配点无网格方法。该方法通过分阶拟合使近似函数在节点的残差达到最小,边界条件直接引入,然后使用直接配点法求解方程。与其它插值或拟合方法相比,分阶拟合避免了矩阵奇异产生的困难;与最小移动二乘法(MLS)相比,分阶拟合只需用六个点来构造二次基近似函数,减小了计算量;而与其它基于Galerkin法的无网格法相比,分阶拟合直接配点无网格法计算量小。     

电子球磨机筒体辐射声场的计算

本文将噪声预测理论应用到球磨机的降噪工程上，针对球磨机建立了结构和激励简化模型，采用半解析的组合梁函数法求解出振动响应后，结合边界元方法计算了辐射声场分布，计算结果正确预测出近场中低频段的指向性特点和远场声压级水平，与实测谱值有较好的一致性。     

房间隔声的声辐射度模拟计算

为计算房间之间的空气声隔声,提出一种基于声辐射度的隔声计算模型。通过房间之间的声辐射度传递,模拟房间隔声中的声传播,建立房间的声辐射度系统方程。通过辐射度方程的数值求解实现隔声房间的声场模拟,由此计算获得房间之间的隔声量。通过房间隔声的实测对比验证该隔声计算模型的正确性。对于不同条件的房间隔声模拟表明,当房间中声场扩散度较高时该隔声模型与统计声学模型的结果符合良好;当房间中的声场偏离扩散场时,该模型仍可准确计算房间隔声量,因而具有广泛的适用性。     

Isoparametric finite point method in computational mechanics

In this paper, a new meshless method, the isoparametric finite point method (IFPM) in computational mechanics is presented. The present IFPM is a truly meshless method and developed based on the concepts of meshless discretization and local isoparametric interpolation. In IFPM, the unknown functions, their derivatives, and the sub-domain and its boundaries of an arbitrary point are described by the same shape functions. Two kinds of shape functions that satisfy the Kronecker-Delta property are developed for the scattered points in the domain and on the boundaries, respectively. Conventional point collocation method is employed for the discretization of the governing equation and the boundary conditions. The essential (Dirichlet) and natural (Neumann) boundary conditions can be directly enforced at the boundary points. Several numerical examples are presented together with the results obtained by the exact solution and the finite element method. The numerical results show that the present IFPM is a simple and efficient method in computational mechanics.     

A meshless local boundary integral equation method for two-dimensional steady elliptic problems

A novel meshless local boundary integral equation (LBIE) method is proposed for the numerical solution of two-dimensional steady elliptic problems, such as heat conduction, electrostatics or linear elasticity. The domain is discretized by a distribution of boundary and internal nodes. From this nodal points’ cloud a “background” mesh is created by a triangulation algorithm. A local form of the singular boundary integral equation of the conventional boundary elements method is adopted. Its local form is derived by considering a local domain of each node, comprising by the union of neighboring “background” triangles. Therefore, the boundary shape of this local domain is a polygonal closed line. A combination of interpolation schemes is taken into account. Interpolation of boundary unknown field variables is accomplished through boundary elements’ shape functions. On the other hand, the Radial Basis Point Interpolation Functions method is employed for interpolating the unknown interior fields. Essential boundary conditions are imposed directly due to the Kronecker delta-function property of the boundary elements’ interpolation functions. After the numerical evaluation of all boundary integrals, a banded stiffness matrix is constructed, as in the finite elements method. Several potential and elastostatic benchmark problems in two dimensions are solved numerically. The proposed meshless LBIE method is also compared with other numerical methods, in order to demonstrate its efficiency, accuracy and convergence.     

A boundary meshless method with shape functions computed from the PDE

This paper presents a new meshless method using high degree polynomial shape functions. These shape functions are approximated solutions of the partial differential equation (PDE) and the discretization concerns only the boundary. If the domain is split into several subdomains, one has also to discretize the interfaces. To get a true meshless integration-free method, the boundary and interface conditions are accounted by collocation procedures. It is well known that a pure collocation technique induces numerical instabilities. That is why the collocation will be coupled with the least-squares method. The numerical technique will be applied to various second order PDE's in 2D domains. Because there is no integration and the number of shape functions does not increase very much with the degree, high degree polynomials can be considered without a huge computational cost. As for instance the p-version of finite elements or some well established meshless methods, the present method permits to get very accurate solutions.     

多联通封闭空间声场响应的基于核重构的最小二乘无网格解法

针对多通域封闭空间声场响应的亥姆霍兹方程的求解问题,本文基于核重构思想,应用无网格配点法构造近似函数,并利用最小二乘方法的原理解决边界问题,离散控制微分方程,建立求解的代数方程。边界问题以及稳定性问题一直是无网格法的难点,该方法的系数矩阵是对称正定的,因此结果具有较好的稳定性。通过数值算例分析多联通域二维问题中配点均匀分布与随机分布时此方法的精确性以及稳定性,利用典型算例对比无网格方法数值解与解析解,结果证明此方法不需要进行网格划分,节点可随机分布,精度较高且具有良好的收敛性。     

Comparison of local weak and strong form meshless methods for 2-D diffusion equation

A comparison between weak form meshless local Petrov-Galerkin method (MLPG) and strong form meshless diffuse approximate method (DAM) is performed for the diffusion equation in two dimensions. The shape functions are in both methods obtained by moving least squares (MLS) approximation with the polynomial weight function of the fourth order on the local support domain with 13 closest nodes. The weak form test functions are similar to the MLS weight functions but defined over the square quadrature domain. Implicit timestepping is used. The methods are tested in terms of average and maximum error norms on uniform and non-uniform node arrangements on a square without and with a hole for a Dirichlet jump problem and involvement of Dirichlet and Neumann boundary conditions. The results are compared also to the results of the finite difference and finite element method. It has been found that both meshless methods provide a similar accuracy and the same convergence rate. The advantage of DAM is in simpler numerical implementation and lower computational cost.     

A point interpolation meshless method based on radial basis functions

A point interpolation meshless method is proposed based on combining radial and polynomial basis functions. Involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis. This non‐singularity is useful in constructing well‐performed shape functions. Furthermore, the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property. This makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least‐squares approximation. In addition, the partial derivatives of shape functions are easily obtained, thus improving computational efficiency. Examples on curve/surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high. Copyright © 2002 John Wiley & Sons, Ltd.     

Virtual boundary meshless least square integral method with moving least squares approximation for 2D elastic problem

Moving least squares approximation (MLSA) has been widely used in the meshless method. The singularity should appear in some special arrangements of nodes, such as the data nodes lie along straight lines and the distances between several nodes and calculation point are almost equal. The local weighted orthogonal basis functions (LWOBF) obtained by the orthogonalization of Gramm–Schmidt are employed to take the place of the general polynomial basis functions in MLSA. In this paper, MLSA with LWOBF is introduced into the virtual boundary meshless least square integral method to construct the shape function of the virtual source functions. The calculation format of virtual boundary meshless least square integral method with MLSA is deduced. The Gauss integration is adopted both on the virtual and real boundary elements. Some numerical examples are calculated by the proposed method. The non-singularity of MLSA with LWOBF is verified. The number of nodes constructing the shape function can be less than the number of LWOBF and the accuracy of numerical result varies little.     

热弹性动力学耦合问题的插值型移动最小二乘无网格法研究

该文基于插值型移动最小二乘法,将无网格局部Petrov-Galerkin（MLPG）法用于二维耦合热弹性动力学问题的求解。修正的Fourier热传导方程和弹性动力控制方程通过加权余量法来离散,Heaviside分段函数作为局部弱形式的权函数,从而得到描述热耦合问题的二阶常微分方程组。然后利用微分代数方法,温度和位移作为辅助变量,将上述二阶常微分方程组转换成常微分代数系统,采用Newmark逐步积分法进行求解。该方法无需Laplace变换可直接得到温度场和位移场数值结果,同时插值型移动最小二乘法构造的形函数由于满足Kroneckerdelta特性,因此能直接施加本质边界条件。最后通过两个数值算例来验证该方法的有效性。     

Reproducing polynomial particle methods for boundary integral equations

Since meshless methods have been introduced to alleviate the difficulties arising in conventional finite element method, many papers on applications of meshless methods to boundary element method have been published. However, most of these papers use moving least squares approximation functions that have difficulties in prescribing essential boundary conditions. Recently, in order to strengthen the effectiveness of meshless methods, Oh et al. developed meshfree reproducing polynomial particle (RPP) shape functions, patchwise RPP and reproducing singularity particle (RSP) shape functions with use of flat-top partition of unity. All of these approximation functions satisfy the Kronecker delta property. In this paper, we report that meshfree RPP shape functions, patchwise RPP shape functions, and patchwise RSP shape functions effectively handle boundary integral equations with (or without) domain singularities.