基于赫林格?赖斯纳变分原理的一致高效无网格本质边界条件施加方法

无网格法具有高阶连续光滑的形函数, 在结构分析中呈现出显著的精度优势. 但无网格形函数在节点处一般没有插值性, 导致伽辽金无网格法难以直接施加本质边界条件. 采用变分一致尼兹法施加边界条件的数值解具有良好的收敛性和稳定性, 因而得到了非常广泛的应用, 然而该方法仍然需要引入人工参数来保证算法的稳定性. 本文以赫林格?赖斯纳变分原理为基础, 建立了一种变分一致的本质边界条件施加方法. 该方法采用混合离散近似赫林格?赖斯纳变分原理弱形式中的位移和应力, 其中位移采用传统无网格形函数进行离散, 而应力则在背景积分单元中近似为相应阶次的多项式. 此时的无网格离散方程可视为一种新型的尼兹法施加本质边界条件, 其中修正变分项采用再生光滑梯度和无网格形函数进行混合离散, 稳定项则内嵌于赫林格?赖斯纳变分原理弱形式中, 无需额外增加稳定项, 消除了对人工参数的依赖性. 该方法无需计算复杂耗时的形函数导数, 并满足积分约束条件, 保证了数值求解的精度. 数值结果表明, 所提方法能够保证伽辽金无网格法的计算精度最优误差收敛率, 与传统的尼兹法相比明显提高了计算效率.      

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。     

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.     

局部彼得洛夫-伽辽金法分析各向异性板屈曲

基于Kirchhoff板理论和对挠度函数采用移动最小二乘近似函数进行插值，进一步研究无网格局部Petrov-Galerkin(MLPG)方法在各向异性板稳定问题中的应用．分析中，本质边界条件采用罚因子法施加，离散的特征值方程由板稳定控制方程的局部积分对称弱形式中得到．通过数值算例并与其他方法的结果进行比较，表明MLPG法求解各向异性薄板稳定问题具有收敛性好、精度高等一系列优点．     

带源参数热传导问题的基于滑动Kriging插值的MLPG法

利用基于滑动Kriging插值的无网格局部Petrov-Galerkin(MLPG)法来求解带源参数的二维热传导问题,推导了相应的离散方程。由于滑动Kriging插值法构造的形函数满足Kronecker Delta特性,因此可以直接施加本质边界条件。在离散过程中采用Heaviside分段函数作为局部弱形式的权函数,时间域则通过向后差分法进行离散,这一处理过程中刚度矩阵只涉及到边界积分,而没有涉及到区域积分。最后通过算例验证了本方法的有效性。     

用局部Petrov-Galerkin法分析薄板自由振动

利用薄板振型方程的等效积分弱形式和对振型函数采用移动最小二乘近似函数进行插值，本文进一步研究了无网格局部Petrov-Galerkin方法在薄板自由振动问题中的应用。它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。在插值近似时，采用虚拟-实际节点值变换方法直接引入本质边界条件。通过数值算例和与其他方法的结果进行比较，表明无网格局部Petrov-Galerkin法求解弹性薄板自由振动问题具有收敛性好、精度高等一系列优点。     

带源参数的二维热传导反问题的无网格方法

利用无网格有限点法求解带源参数的二维热传导反问题，推导了相应的离散方程. 与 其它基于网格的方法相比，有限点法采用移动最小二乘法构造形函数，只需要节点信息，不 需要划分网格，用配点法离散控制方程，可以直接施加边界条件，不需要在区域内部求积分. 用有限点法求解二维热传导反问题具有数值实现简单、计算量小、可以任意布置节点等优点. 最后通过算例验证了该方法的有效性.     

瞬态热传导问题的无网格法快速求解算法研究

为了提高基于Galerkin弱积分形式的无网格方法求解瞬态热传导问题的计算效率,提出了两种方案:第一种方案在空间离散上采用基于任意凸多边形节点影响域的无网格形函数,并通过选取适当的节点影响半径因子,使背景网格内的积分点仅对该背景网格内的无网格节点有贡献,从而避免了节点搜索问题,减少了系统刚度矩阵的带宽,且当节点影响半径因子为1.01时,无网格方法的形函数近似具有插值特性;第二种方案在求解线性方程组时,引入质量矩阵集中技术,从而避免了系统方程组的求解.二维矩形区域、二维圆形区域的瞬态热传导数值算例结果表明:在保证计算精度的同时,采用任意多边形节点影响域的无网格方法比传统无网格方法的计算时间至少节省44.09%,采用质量矩阵集中技术的无网格方法比传统无网格方法的计算时间至少节省76.15%,且当节点影响半径因子为1.01时,其本质边界条件的施加和有限元方法一样简单;由于采用质量矩阵集中技术的无网格方法比采用任意多边形节点影响域的无网格方法精度较低,因此如仅从计算效率考虑,对精度要求不是很高(误差在5%以内),建议采用质量矩阵集中技术,如同时考虑计算精度和效率,建议采用多边形节点影响域的技术.     

无网格局部强弱法求解不规则域问题

无网格局部彼得洛夫-伽辽金(meshless local Petrov-Galerkin,MLPG)法是一种具有代表性的无网格方法,在计算力学领域得到广泛应用.然而,这种方法在边界上需执行积分运算,通常很难处理不规则求解域问题.为了克服MLPG法的这种局限性,提出了无网格局部强弱(meshless local strong-weak,MLSW)法.MLSW法采用MLPG法离散内部求解域,采用无网格介点(meshless intervention-point,MIP)法施加自然边界条件,并采用配点法施加本质边界条件,避免执行边界积分运算,可适用于求解各类复杂的不规则域问题.从理论上讲,这种结合式方法,既保持了MLPG法稳定而精确计算的优势,同时兼备配点型方法在处理复杂结构问题时简洁而灵活的优势,实现了弱式法和强式法的优势互补.此外,MLSW法采用移动最小二乘核(moving least squares core,MLSc)近似法来构造形函数,是对传统移动最小二乘(moving least squares,MLS)近似法的一种改进.MLSc使用核基函数代替通常的基函数,有利于数值求解的精确性和稳定性,而且其导数近似计算变得更为简单.数值算例结果初步表明:这种新方法实施简单,求解稳定、精确,表现出适合工程运用的潜力.     

基于拉格朗日插值的无网格直接配点法和稳定配点法

由于无网格法中大多数近似函数均为有理式,不具有Kronecker delta性质,因此难以精确地施加本质边界条件.边界误差较大容易导致整个求解域求解结果精度低,甚至引起数值不稳定现象.文章在无网格直接配点法和稳定配点法中引入拉格朗日插值函数作为形函数,构建了拉格朗日插值配点法(LICM)和拉格朗日插值稳定配点法(SLICM).由于拉格朗日插值具有Kronecker delta性质,可以像有限元法一样简单而精确地施加本质边界条件,提高这两种方法的数值求解精度.稳定配点法基于子域对强形式方程进行积分,可以满足高阶积分约束,即可以保证形函数在积分形式下也满足高阶一致性条件,实现精确积分.同时,进行子域积分还可以减少离散矩阵的条件数,从而提高算法的稳定性.进一步提高拉格朗日插值稳定配点法的精度和稳定性.通过数值算例验证这两种方法的精度、收敛性和稳定性,结果表明基于拉格朗日插值的配点法的精度优于基于重构核近似的配点法,拉格朗日插值稳定配点法的精度和稳定性均优于拉格朗日插值配点法.     

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.     

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.     

一种新的数值方法——无网格伽辽金法（EFGM）

无网格伽辽金法（ＥＦＧＭ）是近几年发展起来的与有限元相似的一种数值算法,它采用移动的最小二乘法构造形函数,从能量泛函的弱变分形式中得到控制方程,并用拉氏乘子满足本征边界条件,从而得到偏微分方程的数值解中得到该法只需节点信息,不需将节点连成单元,此外,还有精度高,后处理方便等优点,本文介绍其基本原理及实现过程,并用算例表明,该法具有一定的发展前景。     

伽辽金型无网格法的数值积分方法

无网格法直接通过节点信息构造形函数,不依赖于节点之间的有序单元连接,能够建立任意高阶连续的整体协调形函数。与传统的有限元法相比,无网格法对大变形问题、移动边界问题和高阶问题的求解方面有比较明显的优势。伽辽金型无网格法是目前应用最为广泛的一类无网格法。虽然无网格形函数本身不依赖于单元,但伽辽金型无网格法需要采取合适的方法进行弱形式的数值积分。由于无网格形函数一般不是多项式,具有非插值性且影响域与背景积分网格通常不重合,伽辽金型无网格法通常需要采用高阶的高斯积分进行数值积分,导致了计算效率低下,难于求解大型实际问题。因此,如何通过建立高效积分方法提高无网格法的计算效率成为无网格法研究领域的一个核心问题。本文对无网格法中强制边界条件的施加方法进行了简要归纳,总结了伽辽金型无网格法中若干常用的数值积分方法,并对伽辽金型无网格法的数值积分方法领域存在的一些问题进行了探讨。     

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.     

A MESH-FREE ALGORITHM FOR DYNAMIC IMPACT ANALYSIS OF HYPERELASTICITY

A mesh-free method based on local Petrov-Galerkin formulation is presented to solve dynamic impact problems of hyperelastic material.In the present method,a simple Heaviside test function is chosen for simplifying domain integrals.Trial function is constructed by using a radial basis function(RBF)coupled with a polynomial basis function,in which the shape function possesses the kronecker delta function property.So,additional treatment is not required for imposing essential boundary conditions.Governing equations of impact problems are established and solved node by node by using an explicit time integration algorithm in a local domain,which is very similar to that of the collocation method except that numerical integration can be implemented over local domain in the present method.Numerical results for several examples show that the present method performs well in dealing with the dynamic impact problem of hyperelastic material.     

基于流形覆盖思想的无网格方法的研究

本语言基于流形思想,利用有限覆盖,单位分解等概念,引入建立在覆盖上的覆盖函数和具有紧支撑特性的单位分解函数,建立场逼近的近似表达,由弱形式的Galerkin变分得到数值分析模型,结合边界条件用于边值问题的求解,由此建立了一类新的无网格数值方法,论文采用这种方法分析了平面弹性问题,分析了体积闭锁现象,h、p型收敛性等,提出了一种选择覆盖大小的方案,且对狭长城采用了椭圆覆盖形式,取得了比较好的效果。     

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.     

二阶一致无网格与有限元耦合离散研究

为准确方便地施加本质边界条件,在连续掺混法（Continuous Blending Method, CBM）的框架下,通过增加一个边中节点,发展了采用二阶基底的无网格与二阶有限元的耦合离散方法。Galerkin弱形式的数值积分采用具二阶一致性的3点积分方法（Quadratically Consistent 3-point integration method,QC3）。与原本在QC3中采用的Nitsche法相比,所发展的耦合离散方法可像有限元法一样简单高效地施加本质边界条件,不向弱形式中引入额外项,也不依赖于任何人工参数。而且,数值结果还表明,QC3的计算精度也得到进一步提高。