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。     

结构声辐射计算的边界无网格法

研究函数有限维逼近插值形函数的一般要求,介绍采用移动最小二乘构建无网格插值形函数的方法与步骤;通过配点法将Kirchhoff-Helmholtz边界积分方程离散为受边界条件约束的线性方程组;最后通过分块矩阵法求解约束方程组,得到离散后的声辐射传输模型数值表达式。在计算实例中,分别用边界无网格法和边界元法建立声辐射传输模型进行声场计算,计算声场值与解析值相对比的结果表明,由于边界无网格法插值形函数根据求解情况自行构建,因此更灵活,具有更高的插值和计算精度。     

最小二乘网格的模型修补

最小二乘网格是在给定连接图和离散控制点集的基础上,通过求解线性系统对网格中的顶点重新定位而形成的网格.本文提出了一种最小二乘网格的模型修补算法,首先根据模型孔洞构造合适的连接图,然后根据网格连接图以及边界几何信息构造一个线性稀疏系统,最后求解连接网格中所有顶点的三维几何坐标.该算法计算速度快,能取得理想的效果.     

弹性地基圆形加肋板静力弯曲及弯曲自由振动分析的无网格法

基于一阶剪切理论和移动最小二乘法,提出分析弹性地基圆形加肋板线性弯曲的静力和自由振动问题的无网格方法.对弹性地基采用Winkler地基模型,对圆形加肋板则将平板和肋条分开考虑,通过位移协调条件建立两者的参数转换方程.平板和肋条均采用一系列点来离散,用移动最小二乘法建立的形函数来分别描述两者的位移场,再分别通过最小势能原...     

流体力学问题基于核重构思想的最小二乘配点法

将基于核重构思想的最小二乘配点法应用于流体力学问题,给出了离散二维不可压缩粘性流体非线性偏微分方程的最小二乘配点格式。为了检验该方法的有效性,以二维Stokes问题——Couette流动为典型算例,分别研究了正压与负压两种工况作用下Couette流动的速度分布。数值模拟结果表明,无论离散点是均匀分布还是随机分布,均给出了较准确的数值结果。     

基于欧拉插值的最小二乘混合配点法在弹性力学平面问题中的应用

由于常规配点型无网格法存在求解不稳定、精度差和求解高阶导数等问题,提出了基于欧拉插值的最小二乘混合配点法。该方法同时以位移和应变作为未知量,通过欧拉插值将未知变量的导数表达出来,同时在插值中引入高斯权函数,并代入微分方程,从而形成以位移和应变为未知数的超定方程组,然后形成最小二乘意义下的法方程,法方程和相应的位移边界条件、应力边界条件一起形成定解体系。该方法不需要域积分,是一种真正的无网格法。一些典型的弹性力学平面问题表明本文方法具有良好的精度。     

压电结构动力分析的无网格自然单元法

自然单元法是一种新兴的无网格数值计算方法,在本质边界条件的施加上较采用移动最小二乘法的无网格法具有明显的优势。将无网格自然单元法与精细积分法相结合,提出了压电结构动力响应分析的一条新途径。在空间域上采用自然单元法离散,并运用加权余量法推导了压电结构动力分析的离散控制方程。然后,采用精细积分法在时间域上进行求解。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。     

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。     

改进的移动最小二乘插值法研究

本文首先从内积的角度给出了移动最小二乘逼近法的新的推导方法,然后对Lancaster等提出的移动最小二乘插值法进行了重新推导,取在插值节点奇异的权函数,并对基函数进行部分正交化,建立了改进的移动最小二乘插值法,并证明了其形函数的插值性质。本文提出的改进移动最小二乘插值法的公式比Lancaster的公式更为简单,并可提高形函数的计算效率。本文为工程问题的插值型无网格方法提供了建立形函数的基本方法。     

随机动力学中FPK方程的径向基点插配点方法求解

应用径向基点插配点方法对随机动力学中的FPK方程进行了求解.所求未知函数的空间插值采用径向基点插近似,而时间导数离散采用差分格式,建立具有带宽特性的代数方程,采用逐次超松弛迭代法（SOR）有效地求解所得到的代数方程.针对线性振子和杜芬振子问题的FPK方程进行了具体的数值求解,计算结果表明了方法的有效性,尤其是散点模型的计算结果表明该方法具有比其它有网格数值方法对非规则离散模型适应性更强的优点.     

Hermite–Cloud: a novel true meshless method

In this paper, a novel true meshless numerical technique – the Hermite–Cloud method, is developed. This method uses the Hermite interpolation theorem for the construction of the interpolation functions, and the point collocation technique for discretization of the partial differential equations. This technique is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is employed instead. As a true meshless technique, the present method constructs the Hermite-type interpolation functions to directly compute the approximate solutions of both the unknown functions and the first-order derivatives. The necessary auxiliary conditions are also constructed to generate a complete set of partial differential equations with mixed Dirichlet and Neumann boundary conditions. The point collocation technique is then used for discretization of the governing partial differential equations. Numerical results show that the computational accuracy of the Hermite–Cloud method at scattered discrete points in the domain is much refined not only for approximate solutions, but also for the first-order derivative of these solutions.     

A gradient reproducing kernel collocation method for boundary value problems

The earlier work in the development of direct strong form collocation methods, such as the reproducing kernel collocation method (RKCM), addressed the domain integration issue in the Galerkin type meshfree method, such as the reproducing kernel particle method, but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of collocation points for optimal convergence. In this work, we intend to address the computational complexity in RKCM while achieving optimal convergence by introducing a gradient reproduction kernel approximation. The proposed gradient RKCM reduces the order of differentiation to the first order for solving second‐order PDEs with strong form collocation. We also show that, different from the typical strong form collocation method where a significantly large number of collocation points than the number of source points is needed for optimal convergence, the same number of collocation points and source points can be used in gradient RKCM. We also show that the same order of convergence rates in the primary unknown and its first‐order derivative is achieved, owing to the imposition of gradient reproducing conditions. The numerical examples are given to verify the analytical prediction. Copyright © 2012 John Wiley & Sons, Ltd.     

An adaptive isogeometric analysis meshfree collocation method for elasticity and frictional contact problems

A collocation method has been recently developed as a powerful alternative to Galerkin's method in the context of isogeometric analysis, characterized by significantly reduced computational cost, but still guaranteeing higher-order convergence rates. In this work, we propose a novel adaptive isogeometric analysis meshfree collocation (IGAM-C) for the two-dimensional (2D) elasticity and frictional contact problems. The concept of the IGAM-C method is based upon the correspondence between the isogeometric collocation and reproducing kernel meshfree approach, which facilitates the robust mesh adaptivity in isogeometric collocation. The proposed method reconciles collocation at the Greville points via the discretization of the strong form of the equilibrium equations. The adaptive refinement in collocation is guided by the gradient-based error estimator. Moreover, the resolution of the nonlinear equations governing the contact problem is derived from a strong form to avoid the disadvantages of numerical integration. Numerical examples are presented to demonstrate the effectiveness, robustness, and straightforward implementation of the present method for adaptive analysis.     

Reproducing kernel collocation method applied to the non-linear dynamics of pipe whip in a plane

A windowed collocation method, based on a moving least squares reproducing kernel particle approximation of functions, is explored for spatial discretization of the strongly non-linear system of partial differential equations governing large, planar whipping motion of a cantilever pipe subjected to a follower force pulse (the blow-down force) normal to the deflected centreline at its tip. This problem was discussed by Reid et al. [An elastic–plastic hardening–softening cantilever beam subjected to a force pulse at its tip: a model for pipe whip. Proc R Soc London A1998;454:997–1029] where a space–time finite difference discretization was employed to solve the governing partial differential equation of motion. It was shown that, despite the deflected shape predictions being accurate, numerical solutions of these equations might exhibit problematic (possibly spurious) steep localized gradients. The resolution of this problem in the context of structural mechanics is novel and is the subject of this paper. In particular, it is demonstrated that it is possible to reduce significantly such spurious and localized numerical instabilities through a windowed collocation approach with a suitable choice of the window size. The collocation procedure presently adopted is based on the moving least squares reproducing kernel particle method. Material and structural non-linearity in the beam (pipe) model is incorporated via an elastic–plastic-hardening–softening moment–curvature relationship. The projected ordinary differential equations are then integrated in time through a fifth order, explicit Runge–Kutta method with adaptive step sizes.     

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.     

Semi-Lagrangian reproducing kernel particle method for fragment-impact problems

Fragment-impact problems exhibit excessive material distortion and complex contact conditions that pose considerable challenges in mesh based numerical methods such as the finite element method (FEM). A semi-Lagrangian reproducing kernel particle method (RKPM) is proposed for fragment-impact modeling to alleviate mesh distortion difficulties associated with the Lagrangian FEM and to minimize the convective transport effect in the Eulerian or Arbitrary Lagrangian Eulerian FEM. A stabilized non-conforming nodal integration with boundary correction for the semi-Lagrangian RKPM is also proposed. Under the framework of semi-Lagrangian RKPM, a kernel contact algorithm is introduced to address multi-body contact. Stability analysis shows that temporal stability of the kernel contact algorithm is related to the velocity gradient between two contacting bodies. The performance of the proposed methods is examined by numerical simulation of penetration processes.     

The reproducing kernel particle method for two-dimensional unsteady heat conduction problems

The numerical solution to two-dimensional unsteady heat conduction problem is obtained using the reproducing kernel particle method (RKPM). A variational method is employed to furnish the discrete equations, and the essential boundary conditions are enforced by the penalty method. Convergence analysis and error estimation are discussed. Compared with the numerical methods based on mesh, the RKPM needs only the scattered nodes instead of meshing the domain of the problem. The effectiveness of the RKPM for two-dimensional unsteady heat conduction problems is examined by two numerical examples.     

A Lagrangian reproducing kernel particle method for metal forming analysis

A Meshless approach based on a Reproducing Kernel Particle Method is developed for metal forming analysis. In this approach, the displacement shape functions are constructed using the reproducing kernel approximation that satisfies consistency conditions. The variational equation of materials with loading-path dependent behavior and contact conditions is formulated with reference to the current configuration. A Lagrangian kernel function, and its corresponding reproducing kernel shape function, are constructed using material coordinates for the Lagrangian discretization of the variational equation. The spatial derivatives of the Lagrangian reproducing kernel shape functions involved in the stress computation of path-dependent materials are performed by an inverse mapping that requires the inversion of the deformation gradient. A collocation formulation is used in the discretization of the boundary integral of the contact constraint equations formulated by a penalty method. By the use of a transformation method, the contact constraints are imposed directly on the contact nodes, and consequently the contact forces and their associated stiffness matrices are formulated at the nodal coordinate. Numerical examples are given to verify the accuracy of the proposed meshless method for metal forming analysis.