基于径向基函数的无单元法的改进

基于径向基函数强形式的无单元(RBFS)法是真正意义上的无单元方法,但为了追求精度要求却未达到稀疏化。本文对RBFS进行了改进,通过构造具有δ函数性质的形函数,得到了具有稀疏带状性的系数矩阵,提高了计算效率,同时具有RBFS方法的优点。通过求解微分方程,得到节点均布时影响域半径与求解精度的关系曲线,验证了基函数中自由参数最佳取值的计算公式的适用性;并把节点均布下得到的影响域半径和自由参数的规律应用到节点任意排列的情况下,求解结果变化不大,均满足精度要求,由此得出这些规律仍然适用,这种无单元法对节点位置不敏感。     

基于逆复合二次径向基函数的对称复合材料层合板自由振动分析

利用逆复合二次径向基函数无网格配点法对Reddy的高阶剪切变形理论进行离散,预测了对称复合材料层合板的自由振动特性.将不同材料参数、几何尺寸和边界条件的层合板固有频率计算结果与相关文献中的结果进行对比,结果表明:逆复合二次径向基函数在对称复合材料层合板自由振动分析方面具有收敛性好及精度高等一系列优点.     

非均质中厚板的无网格LRPIM动力学分析

用局部加权残值法建立了非均质中厚板的局部径向点插值离散系统方程,采用无网格局部径向点插值法分析了非均质中厚板的自由振动和强迫振动问题。用径向基函数耦合多项式基函数来近似试函数,用四次样条函数做为加权残值法中的权函数。所构造的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件。该方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行。在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化。计算结果表明,利用该方法计算非均质中厚板的自由振动和强迫振动问题可以得到具有较高精度的解。     

用无网格局部径向点插值法分析非均质中厚板的弯曲问题

用无网格局部径向点插值法分析了非均质中厚板的弯曲问题.利用虚位移原理推导了中厚板的离散系统方程.采用径向基函数耦合多项式基函数来近似试函数,用四次样条函数作为加权残值公式中的权函数.所构造成的形函数具有Kronecker delta性质,可以很方便地施加本质边界条件.此方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,是一种真正的无网格方法.在计算过程中,取积分中的高斯点的材料参数来模拟问题域材料特性的变化.算例结果表明这种无网格方法具有效率高、精度高和易于实现等优点.     

THE MESHLESS SOLUTION FOR BENDING ANALYSIS OF FUNCTIONALLY GRADED BEAMS

利用全局薄板样条径向基配点法分析了功能梯度梁的弯曲问题,径向基函数的形状参数对近似精度有很大的影响,而薄板样条径向基函数的形状参数选取比其他径向基函数要容易. 利用高阶剪切变形理论推导了控制微分方程,将该文的计算结果与已有参考文献中的结果进行了对比,以验证该文方法的精度.     

基于Lasserre算法的自然单元法形函数计算

基于Lasserre体积算法推导了两种插值方案下自然单元法形函数及其导数 的具体计算方法,特别是对计算点处于某些特殊位置时可能造成计算失败的原因和处理方法 进行了较为深入的研究. 算例结果验证了Sibson与non-Sibson插值形函数在三角形外接圆 的圆周上具有不同的连续性,自然单元法形函数在凸区域的边界结点间是线性变化的,因而 可以方便地施加本质边界条件.     

双重点最小二乘配点无网格法

配点类无网格法需要计算近似函数的二阶导数,因而在移动最小二乘(MLS)近似中至少要采用二次基函数。本文利用Voronoi图对双重点移动最小二乘近似法进行了改进,建立了基于Voronoi图的双重点移动最小二乘近似(VDG),并利用加权最小二乘法离散微分方程,导出了双重点最小二乘配点无网格法(MD GLS)。该方法将求解域用节点离散,并以节点为生成点建立Voronoi图,取Voronoi多边形的顶点为辅助点。近似函数及其二阶导数的计算过程可分解为两个步骤:首先用场函数节点值拟合辅助点处近似函数的一阶导数,再以辅助点处近似函数的一阶导数值拟合节点处近似函数的二阶导数。由于在每一步中只需计算MLS形函数及其一阶导数,这种近似方法需要较少的影响点和较小的影响域。同时借助于Voronoi结构的优良几何性质,可以快速地搜索影响点。研究表明,与基于MLS的加权最小二乘无网格法(MWLS)相比,这种方法可以显著提高计算效率,并且在精度和收敛性方面也有所改善。     

中厚板弯曲分析的无网格弱-强式法(MWS)

基于局部弱式和强式配点相结合的无网格弱-强式法(meshfree weak-strong method,MWS)求解中厚板问题.MWS法对问题域使用整体离散节点表征和强形式配点法进行计算,在自然边界条件上或靠近自然边界条件的区域采用局部弱形式Petrov-Cralerkin法计算,用移动最小二乘法或径向点插值法来构造形函数,是一种理想的真正无网格法.采取MWS法,文中计算了中厚板的弯曲问题和能量误差.算例结果和对比分析表明,无网格弱-强式法(MWS)可以自然协调处理两类边界条件,计算效率高、数值结果稳定;对计算域采用规则节点布置,其解与弹性力学理论解以及有限元解都吻合很好.     

用无网格LRPIM分析中厚板弯曲问题

用径向基函数构造无网格点插值法的形函数,插值函数具有Kronecker delta函数性质,因此可以很方便地施加本质边界条件.利用无网格局部径向点插值方法分别对一个对边固支另对边简支中厚板和一个悬臂中厚板的弯曲进行了分析计算.该方法不需要任何形式的网格划分,所有的积分都在规则形状的子域及其边界上进行,是一种真正的无网格方法.算例表明:将无网格局部径向点插值法应用于计算中厚板的弯曲问题,所求得的位移场和应力场都是光滑的;在径向基函数的基础上,附加多项式大大提高了插值精度;所得结果与弹性力学理论解以及有限元解都十分吻合.     

基于分区径向基函数配点法的大变形分析

无网格法因为不需要划分网格, 可以避免网格畸变问题,使得其广泛应用于大变形和一些复杂问题. 径向基函数配点法是一种典型的强形式无网格法,这种方法具有完全不需要任何网格、求解过程简单、精度高、收敛性好以及易于扩展到高维空间等优点,但是由于其采用全域的形函数, 在求解高梯度问题时 存在精度较低和无法很好地反应局部特性的缺点. 针对这个问题,本文引入分区径向基函数配点法来求解局部存在高梯度的大变形问题. 基于完全拉格朗日格式,采用牛顿迭代法建立了分区径向基函数配点法在大变形分析中的增量求解模式.这种方法将求解域根据其几何特点划分成若干个子域, 在子域内构建径向基函数插值, 在界面上施加所有的界面连续条件,构建分块稀疏矩阵统一求解. 该方法仍然保持超收敛性, 且将原来的满阵转化成了稀疏矩阵, 降低了存储空间,提高了计算效率. 相比较于传统的径向基函数配点法和有限元法, 这种方法能够更好地反应局部特性和求解高梯度问题.数值分析表明该方法能够有效求解局部存在高梯度的大变形问题.      

NEW TREATMENT OF ESSENTIAL BOUNDARY CONDITIONS IN EFG METHOD BY COUPLING WITH RPIM

One of major difficulties in the implementation of meshfree methods using the moving least square (MLS) approximation, such as element-free Galerkin method (EFG), is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. Another class of meshfree methods based on the radial basis point interpolation can satisfy the essential boundary conditions exactly since its approximation function passes through each node in an influence domain and thus its shape functions possess the properties of delta function. In this paper, a coupled element-free Galerkin(EFG)-radial point interpolation method (RPIM) is proposed to enhance their advantages and avoid their disadvantages. Discretized equations of equilibrium are obtained in the RPIM region and the EFG region, respectively. Then a collocation approach is introduced to couple the RPIM and the EFG method. This method satisfies the linear consistency exactly and can maintain the stiffness matrix symmetric. Numerical tests show that this method gives reasonably accurate results consistent with the theory.     

Numerical analysis of Biot's consolidation process by radial point interpolation method

An algorithm is proposed to solve Biot's consolidation problem using meshless method called a radial point interpolation method (radial PIM). The radial PIM is advantageous over the meshless methods based on moving least-square (MLS) method in implementation of essential boundary condition and over the original PIM with polynomial basis in avoiding singularity when shape functions are constructed. Two variables in Biot's consolidation theory, displacement and excess pore water pressure, are spatially approximated by the same shape functions through the radial PIM technique. Fully implicit integration scheme is proposed in time domain to avoid spurious ripple effect. Some examples with structured and unstructured nodes are studied and compared with closed-form solution or finite element method solutions.     

Nonlocal continuum model for large deformation analysis of SLGSs using the kp-Ritz element-free method

A geometrically nonlinear large deformation analysis of SLGSs is presented using the element-free kp-Ritz method. Classical plate theory (CLP) is applied to describe the geometrically nonlinear behavior of SLGSs. Nonlocal elasticity theory is incorporated into CLP to take the small-scale effect into consideration. The system nonlinear equations are derived from the Ritz procedure based on the total Lagrangian formulation. The modified Newton–Raphson method and arc-length continuation are employed to solve the nonlinear equations. The efficiency of the element-free kp-Ritz method is verified through comparison with results reported in previous research. Numerical cases are studied to examine the influence of boundary conditions, aspect ratio, side length and nonlocal parameters on the nonlinear large deformation behavior of SLGSs. An interesting phenomenon is observed in that the nonlocal parameter effect is related to the mathematical expression of the transverse load.     

Conforming radial point interpolation method for spatial shell structures on the stress-resultant shell theory

The implementation of the conforming radial point interpolation method (CRPIM) for spatial thick shell structures is presented in this paper. The formulation of the discrete system equations is derived from a stress-resultant geometrically exact theory of shear flexible shells based on the Cosserat surface. A discrete singularity-free mapping between the five degrees of freedom of the Cosserat surface and the normal formulation with six degrees of freedom is constructed by exploiting the geometry connection between the orthogonal group and the unit sphere. A radial basis function is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometries. The major advantage of the CRPIM is that the shape functions possess a delta function property and the interpolation function obtained passes through all the scattered points in the influence domain. Thus, essential boundary conditions can be easily imposed, as in finite element method. A range of shape parameters is studied to examine the performance of CRPIM for shells, and optimal values are proposed. The phenomena of shear locking and membrane locking are illustrated by presenting the membrane and shear energies as fractions of the total energy. Several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the present CRPIM. The convergence rate of the results using a Gaussian (EXP) radial basis is relatively high compared to those using a multi-quadric (MQ) radial basis for the shell problems.     

有限覆盖径向点插值方法理论及其应用

数值流形方法能够统一地处理连续与非连续变形问题,有限覆盖技术是这种方法的核心。无网格方法前处理过程比较简单,径向点插值法是其中的一种计算格式。本文将有限覆盖技术与径向点插值方法相结合发展了有限覆盖径向点插值无网格方法,综合了数值流形方法与点插值方法的各自优点,能够有效地处理连续与非连续性问题,由此所构造的形函数具有Kronecker δ-函数属性,能够有效地处理位移边界条件。本文在阐述了这种方法基本原理的基础上,通过算例分析与数值计算论证了本文所建议方法的可靠性及其有效性。     

Numerical simulation of ground water flow via a new approach to the local radial point interpolation meshless method

A novel approach to local radial point interpolation meshless (LRPIM) method is introduced to investigate the influence of leakage on tidal response in a coastal leaky confined aquifer system, based on a local weighted residual method with the Heaviside step function as the weighting function over a local sub-domain. The present approach is a truly meshless method based only on a number of randomly located nodes. In this approach, neither global background integration mesh nor domain integration is needed. Radial basis functions (RBFs) interpolation is employed in shape function and its derivatives construction for evaluating the local weak form integrals. Due to satisfaction of kronecker delta property in RBF interpolation, no special treatment is needed to impose the essential boundary conditions. In order to obtain the optimum parameters, shape parameters of multiquadrics (MQ)-RBF are tuned and studied. The leakage has a significant impact on the tidal behaviour of the confined aquifer. The numerical results of this research indicate that both tidal amplitude of groundwater head in the aquifer and the distance over which the aquifer can be disturbed by the tide are considerably reduced by leakage. The novelty of the approach is the use of a local Heaviside weight function in the LRPIM which does not need local domain integration and only integrations on the boundary of the local domains are needed. Therefore, in this research a new local Heaviside weight function has been proposed. Numerical results are presented and compared with the results of analytical solution. It is observed that the obtained results agreed very well with the results of analytical solution. The numerical results show that the use of a local Heaviside weight function in the LRPIM is highly accurate, fast and robust. It is also noticed that this novel meshless approach using MQ radial basis is very stable.     

A nodal integration technique for meshfree radial point interpolation method (NI-RPIM)

A novel nodal integration technique for the meshfree radial point interpolation method (NI-RPIM) is presented for solid mechanics problems. In the NI-RPIM, radial basis functions (RBFs) augmented with polynomials are used to construct shape functions that possess the Delta function property. Galerkin weak form is adopted for creating discretized system equations, in which nodal integration is used to compute system matrices. A stable and simple nodal integration scheme is proposed to perform the nodal integration numerically. The NI-RPIM is examined using a number of example problems including stress analysis of an automobile mechanical component. The effect of shape parameters and dimension of local support domain on the results of the NI-RPIM is investigated in detail through these examples. The numerical solutions show that the present method is a robust, reliable, stable meshfree method and possesses better computational properties compared with traditional linear FEM and original RPIM using Gauss integration scheme.     

Local Heaviside‐weighted LRPIM meshless method and its application to two‐dimensional potential flows

In this paper, the local radial point interpolation meshless method (LRPIM) is used for the analysis of two‐dimensional potential flows, based on a local‐weighted residual method with the Heaviside step function as the weighting function over a local subdomain. Trial functions are constructed using radial basis functions. The present method is a truly meshless method based only on a number of randomly located nodes. Integration over the subdomains requires only a simple integration cell to obtain the solution. No element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions. The novelty of the paper is the use of a local Heaviside weight function in the LRPIM, which does not need local domain integration and integrations only on the boundary of the local domains are needed. Effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated. The behavior of shape parameters of multiquadrics has been systematically studied. Two numerical tests in groundwater and fluid flows are presented and compared with closed‐form solutions and finite element method. The results show that the use of a local Heaviside weight function in the LRPIM is highly accurate and possesses no numerical difficulties. Copyright © 2008 John Wiley & Sons, Ltd.     

粘弹性问题的插值型无单元伽辽金比例边界法

作为一种最近发展起来的半解析数值方法,插值型无单元伽辽金比例边界法不仅无需基本解,且在处理应力奇异性问题和无限域问题时十分有效．为了更有效地求解粘弹性问题,对插值型无单元伽辽金比例边界法应用于此类问题进行了研究,并发展了相应的算法. 通过时域分段展开,将时空耦合的初边值问题转化为一系列递推形式的边值问题,然后采用插值型无单元伽辽金比例边界法进行自适应计算．在径向保持解析特性的基础上,环向采用无单元伽辽金法离散可简化前处理和后处理工作量．此外,改进的插值型移动最小二乘法形函数具有插值性,有效地解决了本质边界条件不能直接施加的困难．最后给出了数值算例,并验证了所提方法的有效性和正确性.     

反平面断裂问题的无单元伽辽金比例边界法

将比例边界法与无单元伽辽金法相结合,建立了反平面断裂分析的无单元伽辽金比例边界法。这是一种边界型无网格法,在环向方向上采用无单元伽辽金法进行离散,因此计算时仅需要边界上的节点信息,不需要边界元所要求的基本解。为了便于施加本质边界条件,通过建立节点值和虚拟节点值之间的关系给出了修正的移动最小二乘形函数。在径向方向上,该方法利用解析的方法求解,因此是一种半解析的数值方法。最后,给出了数值算例,并验证了所提方法后处理简单和计算精度高的特点,适合于求解反平面断裂问题。