轴对称动力学问题的无网格自然邻接点Petrov-Galerkin法

基于无网格自然邻接点Petrov-Galerkin法,提出了复杂轴对称动力学问题求解的一条新途径。几何形状和边界条件的轴对称特点,将原来的空间问题转化为平面问题求解。计算时仅仅需要横截面上离散节点的信息,无论积分还是插值都不需要网格。自然邻接点插值构造的试函数具有Kronecker delta函数性质,因此能够直接准确地施加本质边界条件。有限元三节点三角形单元的形函数作为权函数,可以减少域积分中被积函数的阶次,提高了计算效率。数值算例结果表明,本文提出的方法对求解轴对称动力学问题是行之有效的。     

材料非线性分析的自然单元法

自然单元法主要是基于给定结点的Voronoi图,利用自然相邻插值进行形函数的构造,其形函数满足Kronecker delta性质,便于施加本质边界条件,这使得自然单元法同时兼有有限单元法和无网格法的优点。在材料非线性本构关系的基础上,推导了考虑材料非线性问题的自然单元法模型。算例表明:该模型在处理材料非线性问题时,具有一定的合理性和可行性,是一种有效的数值方法。     

基于C~1自然邻近插值的曲面拟合

以C~0连续non-Sibsonian插值作为三次单纯形Bernstein-Bézier多项式的基坐标,构造C1连续自然邻近插值函数。介绍了高阶连续函数的构建原理和性质。将C1连续自然邻近插值函数应用于曲面拟合场合,由于Voronoi图能够自动调整数据点分布不规则和密度不均匀在空间上的差异,即使对于散乱数据点,也能获得较好的拟合结果。     

自然元与无限元耦合方法在岩土工程粘弹性分析中的应用

自然单元法是一种新的偏微分方程数值解法,由于其位移插值函数采用无网格的方式构造且形函数满足插值性质,从而克服传统有限元方法对单元网格信息的依赖,大大简化数值计算的前处理过程,同时又能像有限元那样准确施加边界条件,在岩土工程中具有广阔的应用前景;介绍了自然元与无限元的基本原理,针对在处理岩土工程无限域或半无限域问题时需要人为确定边界条件而带来计算误差的问题,引入无限元模拟无穷远处边界条件,与自然元相结合形成耦合分析方法;并根据粘弹性理论,采用Laplace插值,编制了基于自然元与无限元耦合方法的二维粘弹性分析程序,通过算例验证了算法的正确性,结果也表明相对于纯自然单元法,耦合方法能够显著提高分析结果的精度,在此基础上拓展了自然单元法在岩土工程中的应用范围.     

线弹性断裂力学问题的扩展自然单元法

为了更加有效地求解线弹性断裂问题,提出了扩展自然单元法。该方法基于单位分解的思想,在自然单元法的位移模式中加入扩展项表征不连续位移场和裂纹尖端奇异场。通过水平集方法确定裂纹面和裂纹尖端区域,并基于虚位移原理推导了平衡方程的离散线性方程。由于自然单元法的形函数满足Kronecker delta函数性质,本质边界条件易于施加。混合模式裂纹的应力强度因子由相互作用能量积分方法计算。数值算例结果表明扩展自然单元法可以方便地求解线弹性断裂力学问题。     

基于代理模型的指数型功能梯度板固有频率优化

针对指数型功能梯度板的固有频率优化问题,提出一种基于代理模型的优化方法。采用一种基于自然邻近插值的无网格数值方法——自然单元法,建立功能梯度板的一阶剪切变形理论的自由振动分析格式。选取样本点,用自然单元法计算相应固有频率,然后基于此建立梯度指数-频率代理模型,采用Nelder-Mead单纯形法优化梯度指数使板的固有频率达到预设值。通过算例验证算法的计算效率和精度。结果表明,基于分段三次Hermite插值构造的代理模型具有较高的精度;基于代理模型的优化结果避免了多次调用自然单元法计算频率,提高了计算效率。      

轴对称结构动力弹塑性分析的无网格自然邻接点Petrov-Galerkin法

将无网格自然邻接点Petrov-Galerkin法与预校正形式的Newmark法相结合,建立了一种轴对称结构动力弹塑性分析的新方法。由于几何形状和边界条件的轴对称特点,三维的轴对称问题可转化为二维问题。此外,计算时仅需要轴对称面上的一组离散节点,有效地避免了复杂的网格划分和网格畸变的影响。在轴对称面上的局部多边形子域上采用局部加权余量法推导了轴对称结构动力弹塑性分析的离散化控制方程,并采用预校正形式的Newmark法在时间域上进行求解。为了克服本质边界条件不能直接施加的缺点,试函数采用自然邻接点插值进行构造。数值算例结果表明,该研究所提出的轴对称结构动力弹塑性分析方法是行之有效的。     

单位分解增强自然单元法计算应力强度因子

自然单元法是一种新兴的无网格数值计算方法,但应用于裂纹问题计算时,其近似函数并不能准确反映裂纹尖端附近应力场的奇异性,需要在缝尖附近增大结点布置密度以获得一定的计算精度。在单位分解框架下将缝尖渐近位移场函数嵌入到自然单元法近似函数中,应用伽辽金过程获得平衡方程的离散线性方程,用相互作用能量积分方法计算了混合模式裂纹的应力强度因子。算例分析表明:单位分解增强自然单元法可以方便地处理裂纹问题,在不增加结点布置密度的情况下可有效提高应力强度因子的计算精度。     

梁杆结构二阶效应分析的一种新型梁单元

推导了一种计及梁杆二阶效应的新型两结点梁单元。首先依据插值理论构造了三结点Euler-Bernoulli梁单元的位移场:使用五次Hermite插值函数建立梁单元的侧向位移场,二次Lagrange插值函数建立梁单元的轴向位移场,进而由非线性有限元理论推导了单元的线性刚度矩阵和几何刚度矩阵,然后使用静力凝聚方法消除三结点梁单元中间结点的自由度,从而得到一种考虑轴力效应的新型两结点梁单元。实例分析表明,此新型梁单元具有很高的计算精度,使用此单元进行梁杆结构分析可获得相当准确的二阶位移和内力。     

空间独立插值下的节点ICM拓扑优化方法

为了克服连续体结构拓扑优化中的数值不稳定性现象,在独立连续映射法中,采用节点拓扑变量描述单元的有无。单元内任意一点的弹性模量和单元体积建模基于空间独立的拓扑变量场,基于低阶单元形函数和改进过滤方式下的插值函数,得到了各类不同空间插值下的ICM法。通过二维拓扑优化数值算例比较了各类方法下的优化结果。结果表明,混合插值模式下的ICM法能得到较为理想的拓扑优化结果。     

Combination of Newmark method and natural element method for elastodynamics

The natural element method (NEM) is a meshless method. The trial and test functions of the NEM are constructed using natural neighbor interpolations which are based on the Voronoi tessellation of a set of nodes. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which makes imposition of essential boundary conditions easy to implement. We investigate the performance of the NEM combined with the Newmark method for problems of elastodynamics in this article. Applications are considered for a cantilever beam with different initial load conditions. The NEM numerical results are compared with the finite element method. NEM shows promise for these applications.     

Polytope finite elements

In this paper, finite elements based on arbitrary convex and non‐convex polytopes are introduced. Polytopes in combination with natural element coordinates (NECs) permit a uniform element formulation of interpolation functions that are independent of the dimension of space, localization and the number of vertices. NECs based on the natural neighbor interpolation are restricted to the polytope and can be understood as an extension of the barycentric coordinates on simplexes. The differentiation and integration of these interpolation functions on the basis of NECs is essential for finite element approximations. The accuracy of the finite element interpolation or approximation can be controlled by either applying the h‐version or by utilizing the p‐version of the finite element method (FEM). Advantages in the handling of hanging nodes are discussed. Furthermore, we present construction methods for Lagrangian as well as for hierarchical interpolation functions based on NECs. Numerical experiments on different convex and non‐convex decompositions will show the usability, accuracy and convergence of the developed polytope FEM. Copyright © 2007 John Wiley & Sons, Ltd.     

Calculation of the derivative of interpolation shape function for three-dimensional natural element method

The natural element method (NEM) is a special meshless method. Its shape functions are constructed using natural neighbor node interpolations based on the concepts of Voronoi tessellation. The NEM interpolation is linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. However, for a three-dimensional problem, the computation of shape function derivative of NEM is still very complicated even with the non-Sibson interpolation function, which makes the NEM an unpopular numerical method. In this paper, we adopt the direct mathematical derivative technique, and after some rigorous deduction, finally obtain the shape function derivative expression of three-dimensional NEM. Compared with the Lasserre algorithm, this algorithm is more intuitionistic and can be conveniently programmed. The NEM numerical results for cantilever beams verify the correctness of the shape function derivative expression of NEM derived in this paper.     

Numerical analysis of dental implants using a new advanced discretization technique

In this work an advanced discretization meshless technique is applied to analyze the elastostatic behavior of implants inserted in the mandible bone. The natural neighbor radial point interpolation method (NNRPIM) is a flexible and efficient meshless method capable of predicting smooth and accurate stress fields. One of the advantages of meshless methods over the finite element method (FEM) is the complete freedom of the domain computational discretization, allowing to discretize biological structures with higher accuracy. In order to show the NNRPIM efficiency, two implant systems' numerical models are analyzed with the NNRPIM and the obtained results are compared with the FEM.     

A natural neighbour-based moving least-squares approach for the element-free Galerkin method

The element-free Galerkin method (EFG) and the natural element method (NEM) are two well known and widely used meshless methods. Whereas the EFG method can represent moving boundaries like cracks only by modifying the weighting functions the NEM requires an adaptation of the nodal set-up. But on the other hand the NEM is computationally more efficient than EFG. In this paper a new concept for the automatic adjustment of nodal influence domains in the EFG method is presented in order to obtain an efficiency similar to the NEM. This concept is based on the definition of natural neighbours for each meshless node which can be determined from a Voronoi diagram of the nodal set-up. In this approach adapted nodal influence domains are obtained by interpolating the distances to the natural neighbours depending on the direction. In the paper we show that this concept leads, especially for problems with grading node density, to a reduced number of influencing nodes at the interpolation points and consequently a significant reduction of the numerical effort. Copyright © 2006 John Wiley & Sons, Ltd.     

A study on the performance of natural neighbour‐based Galerkin methods

In this paper we address the problem of improving natural element simulations in terms of computational cost. Several problems are discussed, that include an efficient natural neighbour search algorithm and a comparison of different natural neighbour‐based interpolation algorithms. In particular, we review the so‐called pseudo‐NEM, a moving least squares‐like approximation scheme that employs natural neighbours, and compare it with traditional Sibson and Laplace interpolation schemes in terms of both accuracy and computational cost. Some examples in linear Elasticity and visco‐plasticity are analysed in order to test the proposed schemes in engineering problems. Copyright © 2007 John Wiley & Sons, Ltd.