MESHLESS METHOD FOR 2D MIXED-MODE CRACK PROPAGATION BASED ON VORONOI CELL

A meshless method integrated with linear elastic fracture mechanics (LEFM) is presented for 2D mixed-mode crack propagation analysis. The domain is divided automatically into sub-domains based on Voronoi cells, which are used for quadrature for the potential energy. The continuous crack propagation is simulated with an incremental crack-extension method which assumes a piecewise linear discretization of the unknown crack path. For each increment of the crack extension, the meshless method is applied to carry out a stress analysis of the cracked structure. The J-integral, which can be decomposed into mode I and mode II for mixed-mode crack, is used for the evaluation of the stress intensity factors (SIFs). The crack-propagation direction, predicted on an incremental basis, is computed by a criterion defined in terms of the SIFs. The flowchart of the proposed procedure is presented and two numerical problems are analyzed with this method. The meshless results agree well with the experimental ones, which validates the accuracy and efficiency of the method.     

Collocated discrete least squares meshless (CDLSM) method for the solution of transient and steady‐state hyperbolic problems

A collocated discrete least squares meshless method for the solution of the transient and steady‐state hyperbolic problems is presented in this paper. The method is based on minimizing the sum of the squared residuals of the governing differential equation at some points chosen in the problem domain as collocation points. The collocation points are generally different from nodal points, which are used to discretize the problem domain. A moving least squares method is employed to construct the shape functions at nodal points. The coefficient matrix is symmetric and positive definite even for non‐symmetric hyperbolic differential equations and can be solved efficiently with iterative methods. The proposed method is a truly meshless method and does not require numerical integration. Advantages of the collocation points are shown to be threefold: First, the collocation points are shown to be responsible for stabilizing the method in particular when problems with shocked solution are attempted. Second, the collocation points are also shown to improve the accuracy of the solution even for problems with smooth solutions. Third, the collocation points are shown to contribute to the efficiency of the method when solving steady‐state problems via faster convergence of the resulting algorithm. The ability of the method and in particular the effect of collocation points are tested against a series of one‐dimensional transient and steady‐state benchmark examples from the literature and the results are presented. A sensitivity analysis is also carried out to investigate the effect of the base polynomials on the accuracy and convergence characteristics of the method in solving steady‐state problems. The results show the ability of the proposed method to accurately solve difficult hyperbolic problems considered. The method is also shown to be particularly stable for problems with shocked solution due to the inherent stabilizing mechanism of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

无网格介点法求解Helmholtz方程

作为一种配点型无网格法，无网格介点MIP法具有数值实施简单、计算精度高、运算高效和适用范围广等优点。Helmholtz方程是科学与工程问题中广泛应用的一类特殊方程，因此对MIP法求解此类方程的适用性进行了验证。利用MIP法的d适应性，给出了MIP法求解该方程的两种计算格式。在数值算例中，分别对平面规则域和不规则域上的一般Helmholtz方程，以及轴对称Helmholtz方程进行了数值分析。结果表明，MIP法完全适用于求解Helmholtz方程。而且，MIP法的计算精度和收敛性都优于普通配点法。此外，MIP法的两种计算格式中，L2C0型通常具有更好的计算效果，故建议将该计算格式作为MIP法求解该类方程的标准形式。     

加权最小二乘无网格法

在最小二乘法和移动最小二乘近似的基础上提出了加权最小二乘无网格法．该方法除节点外又引入了一些辅助点，控制方程在所有节点和辅助点处的残差用最小二乘法予以消除，边界条件用罚函数法引入．另外对移动最小二乘近似进行了改进，并给出了最小二乘法中泛函的简化格式，因而提高了计算效率．与配点法相比，新方法精度高，稳定性好，并且系数矩阵是对称正定矩阵．与Galerkin法相比，该方法不需要进行高斯积分，因而计算量小．算例表明该方法具有效率高、精度高和稳定性好等优点，并且易于实现．     

弹性静力问题的无网格弱-强形式结合法

基于改进的移动最小二乘(MLS)二阶导数近似,建立了一种求解弹性静力问题的无网格弱-强形式结合法(MLS-MWS)。该方法采用节点离散求解域,通过MLS构造形函数,将求解域划分为边界域和内部域,并分别使用控制方程的局部弱形式和强形式来建立离散系统方程。对强形式中涉及的近似函数二阶导数计算,提出了一种将其转化为求两次一阶导数的方法,与传统方法相比,该方法计算简单、精度高。MLS-MWS法结合了弱、强形式无网格法的优点,Neumann边界条件容易满足,并且只需在边界区域进行积分。文中应用该方法分析了两个弹性力学平面问题,分析结果表明本文方法具有良好的精度和收敛性。     

New cases of propagation of singularities along characteristic boundaries for model problems of shell theory

We continue a previous work [1] on propagation of singularities for model problems of thin shell theory in the parabolic and hyperbolic cases. The singularities along the characteristic boundaries are considered using extensions of the solutions out of the domain, adapted to either free or fixed boundaries. The corresponding transport equations are given except for the case of a characteristic fixed boundary for a hyperbolic shell, where the phenomenon is non local, but depends on the whole domain.     

h-ADAPTIVE ANALYSIS BASED ON MESHLESS LOCAL PETROV-G ALERKIN METHOD WITH B SPLINE WAVELET FOR PLATES AND SHELLS

Using the two-scale decomposition technique, the h-adaptive meshless local Petrov- Galerkin method for solving Mindlin plate and shell problems is presented. The scaling functions of B spline wavelet are employed as the basis of the moving least square method to construct the meshless interpolation function. Multi-resolution analysis is used to decompose the field variables into high and low scales and the high scale component can commonly represent the gradient of the solution according to inherent characteristics of wavelets. The high scale component in the present method can directly detect high gradient regions of the field variables. The developed adaptive refinement scheme has been applied to simulate actual examples, and the effectiveness of the present adaptive refinement scheme has been verified.     

Arching Action in Elastic-Plastic Plates

Abstract

The rate equation method is used to determine the large displacement static and dynamic response of viscoelastic shells of revolution. Among the results presented are those for which the effects of changes in the dimension of lineal elements through the thickness of the shell have been included. This effect has been incorporated within the confines of a structural theory, and full nonlinear shell equations are used to perform the analyses. Both the accuracy and economy of the rate equation method applied to problems of this nature are demonstrated through comparisons of solutions obtained by other methods.     

无网格局部Petrov-Galerkin法求解板壳弹塑性大变形

无网格局部Petrov-Galerkin法构造的高阶光滑的形函数非常适合建立板壳结构场函数的逼近函数,是一种比较理想的研究板壳问题的方法。基于Mindlin板壳理论,采用更新拉格朗日原理和大变形条件下场量的无网格表达形式,实现了率型无网格局部Petrov-Galerkin方法对板壳弹塑性大变形的求解,算例分析表明了方法的有效性和较高的分析精度。     

改进的奇异边界法模拟三维位势问题

奇异边界法是与基本解法相对应的一种边界型无网格数值离散方法. 该方法提出了源点强度因子的概念, 克服了传统基本解方法中最复杂最头疼的虚拟边界问题.基于边界元法中处理奇异积分的数值处理技术, 导出了源点强度因子的解析表达式, 提出了改进的无网格奇异边界法, 并进一步将该方法应用于三维位势问题. 该方法消除了传统方法中样本点的选取, 在不增加计算量的前提下, 极大地提高了奇异边界法的计算精度与稳定性.      

线性定常对流占优对流扩散问题的无网格解法

应用无网格Galerkin方法求解对流占优对流扩散问题时会出现非物理现象的数值伪振荡,本文将SUPG方法、GLS方法、SGS方法与无网格Galerkin方法相耦合,成功解决了对流扩散方程中对流项占优时的数值伪振荡问题。运用本文构造的方法,采用线性基和具有C2连续的权函数,应用移动最小二乘法可容易地构造高阶导数连续的形函数,从而避免了有限元方法中当采用线性元插值时,因忽略稳定项中二阶导数项而降低计算精度和稳定性的问题。数值实验表明:本文构造的方法具有计算精度高、稳定性好、计算算法实施简单、前后处理方便的优点,这些方法不仅能适用于对流项占优问题,而且也能很好地消除反应项占优时的数值伪振荡问题。     

基于局部Petrov-Galerkin离散方案的无网格法

基于局部Petrov-Galerkin离散方案,选用自然邻近插值构造试函数,用Shepard函数作为权函数,提出了一种无网格方法(MNNPG),这种方法充分发挥了局部Petrov-Galerkin法的优势,并且结合了自然邻近插值的特点,方便引入边界条件,由于以Shepard函数的圆形支集作为积分子域,用分片中点插值来完成区域积分,无需额外背景网格,是一种真正的无网格法。本文将该无网格方法用于求解二维弹性力学边值问题,算例结果很好地吻合了精确解,表明该方法具有良好的数值精度和稳定性。     

自适应一致性高阶无单元伽辽金法

近来提出的一致性高阶无单元伽辽金法通过导数修正技术大幅度减少了所需积分点数目,并能够精确地通过线性和二次分片试验,显著改善标准无单元伽辽金法的计算效率、精度和收敛性.本文在此基础之上,充分利用无单元法易于在局部区域添加节点的优势,发展了一致性高阶无单元伽辽金法的h型自适应分析方法.根据应变能密度梯度该方法自适应地确定需节点加密的区域,基于背景积分网格的局部多层细化要求生成新的计算节点,同时考虑了节点分布由密到疏渐进过渡的情形.采用相邻两次计算的应变能的相对误差作为自适应过程的停止准则,将所发展自适应无网格法应用于由几何外形、边界外载和体力等因素造成的应力集中问题的计算分析.数值结果表明,所发展方法能够自适应地对高应力梯度区域进行节点加密,自动给出合理的计算节点分布.与已有的标准无网格法的自适应分析相比,所发展方法在计算效率、精度和应力场光滑性等方面均展现出显著优势.与采用节点均匀分布的一致性高阶无单元伽辽金法相比,它大幅度地减少了计算节点数目,有效提高了一致性高阶无单元伽辽金法在分析应力集中等存在局部高梯度问题时的计算效率和求解精度.     

Analysis of 2-D thin structures by the meshless regular hybrid boundary node method

Thin structures are generally solved by the Finite Element Method (FEM), using plate or shell finite elements which have many limitations in applications, such as numerical locking, edge effects, length scaling and the envergence problem. Recently, by proposing a new approach to treating the nearly-singular integrals, Liu et al. developed a BEM to successfully solve thin structures with the thickness-to-length ratios in the micro- or nano-scales. On the other hand, the meshless Regular Hybrid Boundary Node Method (RHBNM), which is proposed by the current authors and based on a modified functional and the Moving Least-Square (MLS) approximation, has very promising applications for engineering problems owing to its meshless nature and dimension-reduction advantage, and not involving any singular or nearly-singular integrals. Test examples show that the RHBNM can also be applied readily to thin structures with high accuracy without any modification.     

Stabilization meshless method for convection-dominated problems

It is weN-known that the standard Galerkin is not ideally suited to deal with the spatial discretization of convection-dominated problems. In this paper, several techniques are proposed to overcome the instabilitY issues in convection-dominated problems in the simulation with a meshless method. These stable techniques included nodal refinement, enlargement of the nodal influence domain, full upwind meshless technique and adaptive upwind meshless technique. Numerical results for sample problems show that these techniques are effective in solving convection-dominated problems, and the adaptive upwind meshless technique is the most effective method of all.     

Meshless method based on collocation with consistent compactly supported radial basis functions

Based on our previously study, the accuracy of derivatives of interpolating functions are usually very poor near the boundary of domain when Compactly Supported Radial Basis Functions (CSRBFs) are used, so that it could result in significant error in solving partial differential equations with Neumann boundary conditions. To overcome this drawback, the Consistent Compactly Supported Radial Basis Functions (CCSRBFs) are developed, which satisfy the predetermined consistency conditions. Meshless method based on point collocation with CCSRBFs is developed for solving partial differential equations. Numerical studies show that the proposed method improves the accuracy of approximation significantly. The project supported by the National Natural Science Foundation of China (10172052)     

无网格MLPG法求解轴对称弹性体扭转问题

应用无网格局部彼得洛夫-伽辽金法(MLPG)研究轴对称弹性体扭转问题,给出了矩阵形式的控制方程,发展了MLPG求解轴对称体弹性扭转问题的数值计算方法。算例分析表明:此方法对求解此类问题具有良好的适应性,数值解能达到理想的计算精度。     

非均质问题中的无网格边界单元法

介绍了一种不需要内部网格计算非均匀介质问题的边界元算法.该算法是建立在一种能将任何区域积分转换成边界积分的径向积分转换法基础上,首先用对应各向同性问题的基本解来建立以正规化位移表示的非均质问题的积分方程,然后用径向积分转换法将出现在积分方程中的区域积分转换成边界积分,从而形成不需要使用内部网格来计算区域积分的纯边界元算法.与其它无网格法相比,此方法需要很少的内部点,有些问题甚至不需要内部点都能得到满意的结果,因此,可以计算大型的三维非均匀介质工程问题.由于此方法继承了边界元和无网格算法的优点,因而具有广阔的发展前景.     

Transient heat conduction analysis using the NURBS-enhanced scaled boundary finite element method and modified precise integration method

The Non-uniform rational B-spline(NURBS)enhanced scaled boundary finite element method in combination with the modified precise integration method is proposed for the transient heat conduction problems in this paper.The scaled boundary finite element method is a semi-analytical technique,which weakens the governing differential equations along the circumferential direction and solves those analytically in the radial direction.In this method,only the boundary is discretized in the finite element sense leading to a reduction of the spatial dimension by one with no fundamental solution required.Nevertheless,in case of the complex geometry,a huge number of elements are generally required to properly approximate the exact shape of the domain and distorted meshes are often unavoidable in the conventional finite element approach,which leads to huge computational efforts and loss of accuracy.NURBS are the most popular mathematical tool in CAD industry due to its flexibility to fit any free-form shape.In the proposed methodology,the arbitrary curved boundary of problem domain is exactly represented with NURBS basis functions,while the straight part of the boundary is discretized by the conventional Lagrange shape functions.Both the concepts of isogeometric analysis and scaled boundary finite element method are combined to form the governing equations of transient heat conduction analysis and the solution is obtained using the modified precise integration method.The stiffness matrix is obtained from a standard quadratic eigenvalue problem and the mass matrix is determined from the low-frequency expansion.Finally the governing equations become a system of first-order ordinary differential equations and the time domain response is solved numerically by the modified precise integration method.The accuracy and stability of the proposed method to deal with the transient heat conduction problems are demonstrated by numerical examples.     

Fracture mechanics analysis of curved stiffened panels using BEM

A dual boundary element method is developed for a analysis of reinforced cracked shallow shells. Boundary integral equations are derived from the Betti’s reciprocal theorem for a cracked shallow shell with transverse frames and longitudinal stiffeners. The effect of frames and stiffeners are treated as a distribution of line body forces. The radial basis function is used to transform domain integrals to boundary integrals. Stress intensity factors are evaluated from crack opening displacements. The effect of curvature on the stress intensity factors is illustrated by numerical examples. Three examples are presented to demonstrate the accuracy of this method compared with solutions obtained using the finite element method.