基于径向基函数的自适应网格方法

本文给出了一种基于径向基函数的自适应网格方法．该方法利用网格依赖方法的解与径向基函数插值解的信息来细化或粗化网格，充分利用了径向基函数计算格式简单、节点配置灵活的优点与网格依赖方法的稳健性．提出的算法很容易编程实现．数值算例表明该算法可以在解变化剧烈的区域加密网格，在解变化平缓的地方粗化网格，从而在保证相同数值求解精度的情况下，能够极大地节省计算量．     

Hermite径向基函数点插值配点法求解消声器横向模态

为了避免划分网格,应用Hermite径向基函数点插值配点法(HRPIC)求解消声器横向本征方程,应用该方法计算的圆形和跑道圆横截面本征波数分别与解析结果和有限元计算结果吻合较好。进而分析影响域尺寸,问题域内计算点数目以及径向基函数的形状参数对本征波数计算误差的影响。结果表明,本征波数的计算误差在一定范围内会随着影响域尺寸和问题域内节点数目的增大而减小,但是不会一直减小,存在最优的数值,无量纲的形状参数直接影响本征波数的计算精度。最后比较Hermite径向基函数点插值配点法与有限元法的计算速度。     

基于径向基函数的高效网格变形算法研究

在流固耦合时域仿真与气动外形优化中,网格变形技术得到了普遍应用。基于径向基函数的网格变形技术以其诸多优良的特性,在近年来得到了广泛的关注。其基本原理是采用物面网格节点的位移构造一个径向基函数序列,再利用此序列将物面的位移光滑的插值到空间网格上。其计算耗时与物面插值节点数与空间等插值节点数的乘积成正比,为了减少其计算量,目前多数文献集中于使用数据精简算法减少物面插值节点数。本文通过引入子空间逐级逼近思想,构造了一种精简空间待插值节点数的方案,该方案主要思想是采用多次插值,每一次插值的对象为上一次插值在物面产生的误差,并且通过限制每一次插值的插值区域来实现减缩空间节点的目的。计算结果表明此方案可以支持大变形运动,同时显著的减少了计算时间。     

径向基函数应用于流固耦合分析初探

该文在径向基函数(radial basis function, RBF)及其插值函数研究基础上,构建了基于RBF插值的流固耦合信息传递模型,进一步通过程序编制,在同济大学空间结构自动分析与设计软件—AADS系统中实现了对任一时间步上的流固交互作用分析,并得到与ADINA软件非常吻合的计算结果。算例分析表明:径向基函数可作为一种有效的界面信息传递方法应用于对三维流固耦合问题的研究。     

基于遗传算法和复合二次径向基函数的复合材料层合板自由振动分析

为了提高复合材料层合板自由振动分析的精度,采用无网格径向基配点法分析复合材料材料层合板的自由振动问题,径向基函数的形状参数对计算精度有很大影响。利用遗传算法对复合二次径向基函数的形状参数进行优化,用优化后的形状参数的复合二次径向基函数计算复合材料层合板的固有频率,计算结果与文献中的结果具有较好的一致性。遗传算法在形状参数优化方面具有很大的潜力,所提出的方法具有较高的计算精度。     

正交各向异性和各向同性方板自由振动的三维无网格解

推导了正交各向异性和各向同性弹性体自由振动的三维控制微分方程,利用基于逆复合二次径向基函数的无网格配点法对三维控制微分方程和边界条件进行离散,通过数值算例选取了逆复合二次径向基函数的形状参数,结果表明:形状参数(是x方向的节点数)时计算结果收敛最快。计算了不同边界条件的正交各向异性和各向同性板的固有频率,该文中的结果与文献中的结果具有较好的一致性。     

径向基函数、散乱数据拟合与无网格偏微分方程数值解

介绍了近年来国际上有关散乱数据拟合研究中的径向基函数方法，及其在散乱线性泛函信息插值，无网格偏微分方程数值解中应用的主要内容。     

基于径向基函数的点云岛屿孔洞自动修复

研究了利用点云获得的模型的孔洞修复,针对目前主要通过人工修复带有岛屿面片的孔洞耗时较长的问题,提出了一种基于径向基函数(RBF)自动修复岛屿孔洞的方法。该方法首先利用最小权重三角化法修复模型主体上的孔洞,其次计算模型主体上孔洞与岛屿面片的相关性,利用模型主体上孔洞和与其相关岛屿面片周围点来计算径向基函数,最后将粗修复后细分的点调整到径向基函数描述的曲面上。实验表明,与其他方法相比,该方法能快速、准确地修复缺陷模型。     

一种局部形式的流固耦合界面插值方法

摘 要：提出了一种局部形式的流固耦合问题界面插值方法,并应用于机翼颤振计算的界面插值。用全局径向基函数和单项式基函数构造插值函数,插值定义域选择局部形式,使得气动节点的插值仅与其相邻的结构节点相关。给出局部形式定义域的设置方法,并确定出定义域的下界。按照插值过程的能量守恒原则推导出气动力的插值矩阵。三维变形球面的插值结果表明方法具有良好的精度和效率。将此方法应用于AGARD 445.6机翼跨音速颤振的界面插值,得到的颤振结果与实验值吻合。     

利用径向基函数网络修正机枪有限元模型

在对机枪进行结构设计时,建立有限元模型并在此基础上进行动力学仿真,对了解其结构动态特性如何影响整个武器射击精度十分重要.提出了一种利用径向基函数网络进行机枪有限元模型修正的方法,将模型修正转化为正问题进行研究.根据实测模态数据对所建立的有限元模型进行修正,以径向基函数网络反映机枪结构参数与其动态特性之间的非线性映射关系,利用神经网络的泛化特性求解设计参数的目标值,不需迭代求解,并且避开了反问题面临的非线性优化计算.反演仿真数据代入的有限元模型计算结果与实测结果较为吻合,证明了该方法的有效性.     

Dual reciprocity hybrid boundary node method for acoustic eigenvalue problems

The hybrid boundary node method (HBNM) is a truly meshless method, and elements are not required for either interpolation or integration. The method, however, can only be used for solving homogeneous problems. For the inhomogeneous problem, the domain integration is inevitable. This paper applied the dual reciprocity hybrid boundary node method (DRHBNM), which is composed by the HBNM and the dual reciprocity method (DRM) for solving acoustic eigenvalue problems. In this method, the solution is composed of two parts, i.e. the complementary solution and the particular solution. The complementary solution is solved by HBNM and the particular one is obtained by DRM. The modified variational formulation is applied to form the discrete equations of HBNM. The moving least squares (MLS) is employed to approximate the boundary variables, while the domain variables are interpolated by the fundamental solutions. The domain integration is interpolated by radial basis function (RBF). The Q–R algorithm and Householder algorithm are applied for solving the eigenvalues of the transformed matrix. The parameters that influence the performance of DRHBNM are studied through numerical examples. Numerical results show that high convergence rates and high accuracy are achievable.     

Eigenvalue analysis for acoustic problem in 3D by boundary element method with the block Sakurai–Sugiura method

This paper presents accurate numerical solutions for nonlinear eigenvalue analysis of three-dimensional acoustic cavities by boundary element method (BEM). To solve the nonlinear eigenvalue problem (NEP) formulated by BEM, we employ a contour integral method, called block Sakurai–Sugiura (SS) method, by which the NEP is converted to a standard linear eigenvalue problem and the dimension of eigenspace is reduced. The block version adopted in present work can also extract eigenvalues whose multiplicity is larger than one, but for the complex connected region which includes a internal closed boundary, the methodology yields fictitious eigenvalues. The application of the technique is demonstrated through the eigenvalue calculation of sphere with unique homogenous boundary conditions, cube with mixed boundary conditions and a complex connected region formed by cubic boundary and spherical boundary, however, the fictitious eigenvalues can be identified by Burton–Miller's method. These numerical results are supported by appropriate convergence study and comparisons with close form.     

特征值问题迭代伽略金法与Rayleigh商加速

该文讨论特征值问题非协调有限元和混合有限元的加速计算方法。基于迭代伽略金法和Rayleigh商加速技巧,我们建立了特征值问题Wilson非协调有限元和Ciarlet-Raviart混合有限元的加速计算方案。这些新方案把在细网格上解一个特征值问题简化为在粗网格上解一个特征值问题和在细网格上解一个线性方程。文中证明了新方案的计算结果仍然保持了渐近最优精度阶,并用数值实验验证了理论结果。     

Node‐by‐node meshless approach and its applications to structural analyses

The meshless method is expected to become an effective procedure for realizing a CAD/CAE seamless system for analyses ranging from modelling to computation, because time‐consuming mesh generation processes are not required. In the present study, a new meshless approach, referred to as the Node‐By‐Node Meshless method is proposed, in which only nodal data is utilized to discretize the governing equations, which are derived using either the principle of virtual work or the Galerkin method. In this method, three key methodologies are utilized: (i) nodal integration using stabilization terms, (ii) interpolation by the Moving Least‐Squares Method, and (iii) a node‐by‐node iterative solver. This paper presents the formulation of the proposed method along with numerical results obtained for two‐dimensional elastostatic and eigenvalue problems. Copyright © 1999 John Wiley & Sons, Ltd.     

A meshless solution for two dimensional density-driven groundwater flow

Density-driven groundwater flow is a complicated nonlinear problem in groundwater hydraulics. The local boundary integral method is a promising meshless scheme that is used for solving several difficult problems in different areas. This method applies the boundary integral equations to the local domain around every node. The nodes can be randomly distributed in the domain and on the global boundary. Therefore, this method is characterised as meshless. The unknown potentials and concentrations in all of the nodes are approximated by interpolation to obtain a system of linear equations. Solving this system of equations leads to the numerical solution for the main problem. In this paper, a combination of the radial basis function interpolation and the local boundary element method is used to solve groundwater flow problem combined with the transport of pollution, which also influences the density of groundwater.     

C1连续曲面重构与光顺的有限元算法

提出了一种基于离散的测量数据重建光顺自由曲面的有限元新方法。根据最佳逼近与能量光顺原理，建立正定的目标泛函，采用18自由度三角形板单元对泛函离散，进行极小化，求得最优解。根据有限元插值计算，重新构造出全场C^1连续的自由曲面。这种方法结合了能量光顺技术，有效地抑制了输入数据上误差噪声的影响，曲面重建的精度高、光顺性好，而且能给出合理的一阶导数。该方法计算简单、便于应用，所需的输入数据点少，并可用于处理曲线边界区域的问题。     

Stability of parallel flows by the finite element method

This paper presents and results obtained from the stabilitystudies of plane Poiseuille flow and magnetohydrodynamic flow by the finite element method. Applying Galerkin's weighted residual method and introducing the interpolation function in the exponetial form with respect to time, the governing flow equations are reduced to an eigenvalue problem. This formulation is much simpler than that of the asymptotic expansion method. Solutions are obtained directly in terms of velocity and pressure. Results of the critical Reynolds number obtained by this method compare well with those of other methods for plane Poiseuille flow and magnetohydrodynamic flow.     

用有限元亏量校正求特征值下界

对解2阶椭圆特征值问题的线性有限元法,本文考虑了一种计算简单的有限元亏量校正方案。基于插值校正和Rayleigh商给出了新的校正特征值。理论分析表明该校正特征值或者达到二次元的精度阶或者当网格直径充分小时下逼近准确特征值,并用数值实验验证了理论结果。     

A meshless collocation method based on the differential reproducing kernel interpolation

A differential reproducing kernel (DRK) interpolation-based collocation method is developed for solving partial differential equations governing a certain physical problem. The novelty of this method is that we construct a set of differential reproducing conditions to determine the shape functions of derivatives of the DRK interpolation function, without directly differentiating the DRK interpolation function. In addition, the shape function of the DRK interpolation function at each sampling node is separated into a primitive function processing Kronecker delta properties and an enrichment function constituting reproducing conditions, so that the nodal interpolation properties are satisfied. A point collocation method based on the present DRK interpolation is developed for the analysis of one-dimensional bar problems, two-dimensional potential problems, and plane problems of elastic solids. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach, with excellent accuracy and fast convergence rate.     

A successive bounding method to find the exact eigenvalues of transcendental stiffness matrix formulations

An alternative algorithm for finding exact natural frequencies or buckling loads from the transcendental, e.g. dynamic, stiffness matrix method is presented in this paper and evaluated by using the plate assembly testbed program VICONOPT. The method is based on the bounding properties of the eigenvalues provided by either linear or quadratic matrix pencils on the exact solutions of the transcendental eigenvalue problem. The procedure presented has five stages, including two accuracy checking stages which prevent unnecessary calculations. Numerical tests on buckling of general anisotropic plate assemblies show that significant time savings can be achieved compared with an earlier multiple determinant parabolic interpolation method.