无网格方法.ppt
无网格法介绍
1
目
录
1
无网格法概述 无网格法分类 构造无网格形函数 导出无网格法公式
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3 4 5
无网格法研究主要进展及参考文献
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无网格法概述
无网格法定义
The meshfree method is used to establish a system of algebraic equations for the whole problem domain without the use of a predefined mesh, or uses easily generable meshes in a much more flexible or ‘‘freer’’ manner.(G R Liu,2009)
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导出无网格法公式
基于局部弱式的无网格法
MFree局部弱式法核心是对控制方程在每一个局部子域采用局部加权 残值法，使一个需要在全域求解的问题简化为在各个子域上对局部 伽辽金方程的求解问题,从而避免了全域的数值积分 代表方法：无网格局部伽辽金法(MLPG)，局部点插值法(LPIM) MLPG特点。优点：避免全局数值积分，减小了对背景积分网格依赖； 缺点： “刚度阵”带状但不对称，增加了计算难度，尽管在大部分 的边界积分可通过选用适当的权函数消除掉 ，但在问题域的边界上 或附近,边界积分是不可避免的，使其难以应用于复杂边界。 LPIM特点。优点：更易满足本质边界条件且有较好 的精度和收敛性； 不需满足全域相容性，更有应用前景；缺点：插值力矩阵容易发生 奇异，需要特殊处理。
在移动最小二乘近似（MLS）中，系数a(x)的选取使近 似函数 u h ( x) 在计算点x的邻域内待求函数u(x)在某种最 小二乘意义下的最佳近似。近似函数在节点 xi 处的误差 数。
无网格方法(刘欣著)PPT模板
5.6.1界面问 题的增强函 数
5.6.3数值 计算
07
ONE
第6章有限点方法
第6章有限点方法
6.1对流-扩散方程的有限点形式 6.2对流-扩散方程的有限点法求解 6.3Burgers方程的高阶时间格式有限点方法求解 6.4油藏数模的有限点法 6.5有限点方法在金融工程中的应用
第6章有限点 方法
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第9章流体-结构相互作用的无网 格方法研究进展
第9章流体-结构相互作用的无网 格方法研究进展
9.1流体-结构相互作用的计算研究 概述 9.2流体-结构相互作用模型描述 9.3FSI问题的扩展有限元方法求解 9.4浸入粒子方法 9.5气动弹性计算中的径向基函数法
第9章流体-结构相互作用的无网格方法研究进展
5.4增强型单位分解有限元方法
5.4.1增强 型覆盖函数 的实现
5.4.2数值 计算
第5章单位分解 有限元方法
5.5单位分解有限元在断裂力学中 的应用
1
5.5.1裂纹尖端附近的渐近解
2
5.5.2平面裂纹的单位分解有限 元计算
第5章单位分 解有限元方法
5.6单位分解有限元在界面问题中 的应用
5.6.2界面问 题的增强方 式
7.6.3数值求解
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ONE
第8章自适应无网格方法
第8章自适应无网 格方法
8.1自适应无网格Galerkin法 8.2结构动力问题的自适应无网 格计算 8.3hp自适应无网格方法
第8章自适应无网格方法
8.1自适应无网格Galerkin法
8.1.1后验误差估计
8.1.2背景网格重构 算法
8.1.3自适无网格静 力分析
无网格方法(刘欣著 )
无网格法的理论及应用
为了验证该方法的有效性和可行性，我们进行了一系列实验。实验过程中采 用了某稠油油田的实际数据集，包括地层压力、温度、渗透率等参数。同时，采 用了可视化评估指标，以便直观地评估计算结果的准确性。实验结果表明，该方 法在稠油热采数值模拟过程中具有较高的计算精度和计算效率，可为稠油热采技 术的优化提供有力支持。
1、算法开发：针对稠油热采的物理化学过程，开发相应的数值模拟算法， 如有限元法、有限差分法等。
2、软件架构：设计并实现数值模拟软件的架构，包括前后处理、求解器等 模块，以便用户进行快速高效的计算。
3、数据处理：针对稠油热采数值模拟过程中产生的大量数据，开发相应的 数据处理技术，如数据压缩、可视化等。
无网格法的数值积分采用移动最小二乘法（Moving Least Squares，MLS） 来实现。该方法通过对节点进行加权，构造一个局部近似函数来逼近真实的解。 数值积分通过在节点上建立局部近似函数，然后对该函数进行求导和积分来计算。 无网格法的数值积分具有高精度和高效性，同时避免了传统网格法中的网格生成 和数据处理问题。
1、结构分析
无网格法在结构分析中具有广泛的应用，可以处理各种复杂形状和材料属性 的结构。例如，桥梁、建筑物和飞机等结构分析中，无网格法能够适应复杂的几 何形状和非均匀的材料属性，同时提高计算效率和精度。此外，无网格法在疲劳 分析和振动分析中也得到了广泛应用。
2、流体分析
无网格法在流体分析中也有着广泛的应用，可以处理各种复杂的流体流动问 题。例如，无网格法可以应用于计算流体动力学（CFD）中的复杂流场模拟、燃 烧模拟以及噪声辐射模拟等。无网格法能够适应复杂的几何形状和流场特性，提 高计算精度和效率。
参考内容
稠油热采是一种重要的石油开采方法，具有提高采收率、降低开采成本等优 势。随着计算机技术的不断发展，数值模拟已成为稠油热采领域的重要工具。本 次演示旨在探讨稠油热采数值模拟自适应网格法计算软件的开发研究及实例应用。
无网格法的应用
无网格法的应用无网格方法的研究应用与进展引言有限元法（FEA）是随着电子计算机的发展而迅速发展起来的一种现代计算方法，但FEA 是基于网格的数值方法，在分析涉及特大变形（如加工成型、高速碰撞、流固耦合）、奇异性或裂纹动态扩展等问题时遇到了许多困难。
同时，复杂的三维结构的网格生成和重分也是相当困难和费时的。
近年来，无网格得到了迅速的发展，受到了国际力学界的高度重视。
与有限元的显著特点是无网格法不需要划分网格，只需要具体的节点信息，采用一种权函数（或核函数）有关的近似，用权函数表征节点信息。
克服了有限元对网格的依赖性，在涉及网格畸变、网格移动等问题中显示出明显的优势。
无网格方法的概述无网格方法（Meshless Method）是为有效解决有限元法在数值模拟分析时网格带来的重大问题而产生的，其基本思想是将有限元法中的网格结构去除，完全用一系列的节点排列来代之，摆脱了网格的初始化和网格重构对问题的束缚，保证了求解的精度[1]。
是一种很有发展的数值模拟分析方法。
目前发展的无网格方法有：光滑质点流体动力学法（SPH）、无网格枷辽金法（EFGM）、无网格局部枷辽金法（MLPGM）、扩散单元法（DEM）、Hp－clouds 无网格方法；有限点法（FPM）、无网格局部Petrov－Galerkin方法（MLPG）、多尺度重构核粒子方法（MRKP）、小波粒子方法（WPM）、径向基函数法（RBF）、无网格有限元法（MPFEM）、边界积分方程的无网格方法等。
这些方法的基本思想都是在问题域内布置一系列的离散节点，然后采用一种与权函数或核函数有关的近似，使得某个域上的节点可以影响研究对象上的任何一点的力学特性，进而求得问题的解。
无网格方法国内外研究的进展无网格法起源于20 世纪70 年代。
Perrone，Kao 最早采用任意网格技术将传统有限差分进行扩展，提出了有限差分法，这可看作无网格技术的最初萌芽。
1977年Lucy 和Monaghan 首次提出了基于拉格朗日公式的光滑质点流体动力法（Smoothed Particle Hydrocynamics：SPH），这是一种纯拉格朗日法，无需网格。
无网格法介绍
无网格法是在建立问题域的系统代数方程时，不需要利用预定义的 无网格法是在建立问题域的系统代数方程时， 网格信息，或者只利用更容易生成的更灵活、 网格信息，或者只利用更容易生成的更灵活、更自由的网格进行域 离散的方法。（刘桂荣，2009） 离散的方法。（刘桂荣，2009） 。（刘桂荣
无网格法概述
无网格法求解过程 FEM对比 对比） （与FEM对比）
导出无网格法公式
基于弱强式的无网格法
• MFree弱-强式法 弱 强式法 强式法(NWS)的核心思想是针对某一问题同时采用强式和 的核心思想是针对某一问题同时采用强式和 局部弱式建立起离散系统方程式，即对不同组别的节点根据其不同 局部弱式建立起离散系统方程式， 条件分别形成不同类型的方程，其中局部弱式被用于位于或接近导 条件分别形成不同类型的方程， 数边界条件的所有节点，强式被用于除此之外的其他节点。 数边界条件的所有节点，强式被用于除此之外的其他节点。 • 代表方法：MWS 代表方法： • MWS特点。MWS法使用最少数量的背景网格用于积分，对各类力学 特点。 法使用最少数量的背景网格用于积分， 特点 法使用最少数量的背景网格用于积分 问题均可得到稳定而精确的解，是目前近乎理想的无网格法。 问题均可得到稳定而精确的解，是目前近乎理想的无网格法。
构造无网格形函数
PIM形函数性质
• 一致性 如果单项式的完备阶数是p，则该形函数具有 C p 一致性 如果单项式的完备阶数是 ， • 再生性 PIM基函数可再生包含在其基函数当中的任意函数。 基函数可再生包含在其基函数当中的任意函数。 基函数可再生包含在其基函数当中的任意函数 • 线形独立性 PIM基函数在支持域上是线性独立的 基函数在支持域上是线性独立的 • δ 函数性
目
第六章 LS-DYNA无网格法
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创建了材料之后，选择 done 退出当前窗口，再选择 done 结束 材料的创建。 在sph的主窗口下，选择apply，可以看到我们创建的box形的 sph模型出现在窗口中。
ቤተ መጻሕፍቲ ባይዱ
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演示
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从计算角度看，该方法是将连续的物质表示为带有速 度的可运动的粒子的集合。每一个粒子均可以表示为其它 粒子的插值点。如粒子“I”与其相距一设定距离(通常为 2h)范围内的所有其它粒子“J”发生相互作用。它们间的 相互作用是由称为光滑函数(或核函数)的近似函数W(xx′,h )来衡量的，h称为光滑长度。因此，任意粒子“I”的 连续函数的值或其导数都可以利用周围粒子“J”的已知值 估计出来。这样，整个问题的解就转化为采用规则的插值 函数，对所有粒子进行插值计算的问题，守恒方程可以转 化为采用流量或内力表示的形式。
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临近搜索
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SPH计算步骤
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建模过程
在ls-prepost中第7页有sphgen 的功能，利用这个功能可以 建立一些简单的无网格模型。 选择其中的new，表示新建一 个无网格的命令。可以看到 下面有几个选项：box， sphere，cylinder，sketch
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选择box，出现如图2所示的菜单，即为创建一个具有长方体形 的无网格模型。
无网格讲义
令
和 是
得
上完全单调的。
CPD径向函数和多次单调函数
定理9.3
是正整数，设 不是至多m次的多项式，若 在 上 次单调且不是常数，那么 是 中m阶严格正定的且径向的。任意 s，使 得
类似前面，我们知道具有紧支集的径向 函数不是 (对所有s)中的严格正定函数, 同样，没有具有紧支集的条件正定径向 函数。 定理9.4： 若函数 有紧支集， 是m阶严格条件正定的，则m=0；也就 是， 是严格正定的。
其广义Fourier变换是 阶m=max(0, ) 其中 ：大于等于 的最小整数 ：修正的 阶第二类Bessel函数 注：应排除 ，因为这样的 会使多项式的 次数是偶数
由广义Fourier函数是正定的且在原点具m阶奇 异性，据TH7.3知 是 (or> ) 阶严格条件正定函数。 注：若 ，Fourier变换是一种古典的变换， 由第四章的广义逆multiquadrics，可知函数是 0阶严格条件正定的，也就是严格正定的。
特别的，可用薄板样条函数
和约束条件
求解 中的散乱数据插值问题，这样由线性 函数得到的插值结果是精确的
注：古典薄板样条（ ）是2阶严格条件正定的； 是 是3阶严格条件正定的 这些函数不是单调的 这两个图都包含负函数值的部分
径向幂和薄板样条通常统称为调和样条 注： 还没有理论可表明使用薄板样条插值的 解是适定的
定且迹非正，则 A有一个负的特征值及N-1个正的特征值。 证明：令A的特征值 由TH9.5知 所以A至少有N-1个正特征值 而 因此至少有一个负的特征值
定理9.7
假设 是一阶严格条件正定的且 (0) 0 对于任意不同的点 ，矩阵 A（ ）有N-1个正特征值和1个负特 征值，进一步，A是非奇异的。 证明： 显然A是一阶严格条件正定的； A的迹 ，由TH9.6立得结论。
计算固体力学(有限元以及无网格方法)全套教学【121P】PPT课件
i(xi , yi ) u i
um vj
uj j(xj , yj )
x O
三角形单元
将位移试函数代入上式，并求偏导数，得
xxyy222111 (((bcciiiuuviii
bjuj cjvj cjuj
bmum) cmvm) cmum)(bivi
bjvj
bmvm)
第二章 平面弹性力学的有限元法
反映了单元的位移形态，称为形函数
vm
m (xm, ym)
vi
i(xi , yi ) u i
um vj
uj j(xj , yj )
x
三角形单元
同理有 vN iv i N jvj N m v m N kv k
则位移向量可表示为
i,j,m
{ } e 单元节点位移向量
ui
vi
{f
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u v
Ni
0
0 Ni
求
L(u)0
解 域
u aiui
离 散
i
L'(ui) 0
AXB
各种数值方法
ui u(xi)离散节点的变量值
第一章 科学和工程中的数值方法
1.3 几个简单示例
(a) 开孔板力学模型
(b) 力学模型离散化
平面问题有限元法
第一章 科学和工程中的数值方法
BEM的变形
起重机吊钩
FEM的变形
第一章 科学和工程中的数值方法
2.2 三角形常应变单元
y
3 单元中的应变和应力
{}[B]{}e
由于[B]是常量，单元内各点应变分
量也都是常量，这是由于采用了线性位移 O 函数的缘故，这种单元称为三角形常应变 单元。
典型无网格法
无网格法典型无网格法V伽辽金型无网格法V配点型无网格法V基于局部弱形式和边界积分方程的无网格法V最小二乘无网格法V物质点法V EFG¾MLS¾Galerkin V RKPM¾RK¾Galerkin V PIM¾PIG l ki¾Galerkin等效积分弱形式（虚位移原理）MLS 近似：()()u =x N x dV 计算量大精度高稳定性V 精度高，稳定性好V 需要背景网格进行积分V 系数矩阵对称V 不易施加本质边界条件处理V背景网格积分,,d I i J j N N ΩΩ∫()()()()=ΔΩ+ΔΓP N x f x N x t x 11I I I I I II I ==∑∑零能模态V 单位分解积分11.函数ψk (x )只定义在子域Ωk 上；2.子域Ωk 相互重叠，且它们完全覆盖了域Ω；3l=3.函数ψk (x )满足单位分解条件1()d ()()d k k k f f ψΩΩ∩Ω=ΩΩ∑∫∫x x x配点型无网格法V FPM¾MLS¾CollocationV SPH¾KA¾Collocationp es ess c ouds V Hp meshless clouds ¾PUC ll ti¾Collocation基于局部弱形式和边界积分方程的无网格法V MLPG¾MLS¾LPGV LBIE¾MLS¾LBIENV BNM¾MLS¾BIEgn pn 1iI IJ iJJ p m v ==∑1IJ p Ip Jpp m m N N ==∑质量阵求逆iIp iJv?质量阵求逆！gpn n 11I IJ p IpJ p m m m N ====∑∑对角质量阵iI I iIp m v =V已知t k 时刻的物理量，求t k +1时刻的物理量11.更新网格结点数据kk Ip Ipm m N=∑ppn k k kiIip Ip p m v N=1p ppp =∑int,ext,kk k iI iIiIf ff=−2.在背景网格结点上积分动量方程并施加固定边界条件1k k k pp f t+=+ΔiIiI iI10,0k k iIiIp f +==在固定边界上6.更新密度，应力k k k 1/(1)p p iipρρε+=+Δ1k k kkij ij ij ij+=ijpijp ijp ijp r σσσ+Δ+Δkk k kk r σσΔ=ΔΩ−ΔΩ其中ijpijpijpijpijpk ijpσΔ根据弹塑性本构关系更新7.进行下一个时间步循环。
热传导无网格
Numerical solution of stokesflow in a circular cavityusing mesh-free local RBF-DQS.Soleimani Kutanaei a,Naeem Roshan b,A.Vosoughi c,S.Saghafic,A.Barari d,n,Soheil Soleimani ea Department of Civil Engineering,Babol University of Technology,Babol,Iranb Department of Mechanical,Materials,and Aerospace Engineering,Illinois Institute of Technology,Chicago,IL60616,USAc Department of Mechanical Engineering,Semnan University,Semnan,Irand Department of Civil Engineering,Aalborg University,Sohng˚a rdsholmsvej57,9000Aalborg,Aalborg,Denmarke Department of Mechanical Engineering,Babol University of Technology,Babol,Irana r t i c l e i n f oArticle history:Received3May2011Accepted16November2011Available online11January2012Keywords:Mesh-free methodStokesflowcircular cavityLocal radial basis function-baseddifferential quadraturea b s t r a c tThis work reports the results of a numerical investigation of Stokesflow problem in a circular cavity asan irregular geometry using mesh-free local radial basis function-based differential quadrature(RBF-DQ)method.This method is the combination of differential quadrature approximation of derivativesand function approximation of Radial Basis Function.As a result,the method can be used to directlyapproximate the derivatives of dependent variables on a scattered set of knots.In this study knots weredistributed irregularly in the solution domain using the Halton sequences.The method is applied on atwo-dimensional geometry.The obtained results from the numerical simulations are compared withthose gained by previous works.Outcomes prove that the current technique is in very good agreementwith previous investigations and this fact that RBF-DQ method is an accurate andflexible method insolution of partial differential equations(PDEs).&2011Elsevier Ltd.All rights reserved.1.IntroductionStudy of slow and viscousflow in a cavity with rotating bound-aries provides useful information onflow behavior and distributivemixing,which has many industrial applications.The Stokesflow canbe considered for a range of engineering processes and naturalphenomena.Flow contains very small objects moving in afluid,suchas the settling of dust particles and the swimming of microorganismscan be considered as an example of Stokesflow.Other examplesinclude theflow offluid through small channels or cracks(groundwater or oil),such as in hydrodynamic lubrication or the seepage insand or rock formations.Theflow of high-viscosityfluids may also bedescribed by creepingflow,such as the extrusion of melts or thetransport of paints,heavy oils or food-processing materials[1–3].Though theflow is dominated only by shear force,thepresence of obstacles and the rotation of cylinders produce eddyzones and dead zones,which affect thefluidflow and mixingprocess.For simpleflow configurations involving only a movingwall with a stationary cylinder the Stokes equations could besolved directly by analytical method[4].Mixing operations,as used in e.g.polymer,chemical and foodprocessing,have been the subject of considerable study andresearch for several decades,for instance,review articles byIrving and Saxton[5],Edwards[6],Ottino[7],Young et.al[8],Krasnopolskaya et.al[1]and separate chapters in the textbooksby Brodkey[9],Middleman[10],Ottino[11]and Baird and Collias[12].Sometimes,experimental methods have also been used tounderstand the complexflow structures created by chaoticmixing produced by discontinuous and sinusoidalflow[13]andshearflow past a circular cylinder with arbitrary rotational speeds[14].Finn and Cox[2]described a slow-flow mixing device thatmimics a natural mixing technique.Their analytical,numericaland experimental investigations showed that applying bothtranslating and rotating movement of inner cylinder can improvethe mixing behavior.Analytical solutions of Stokes equationsforflow domains such as a rectangular cavity with rotatingcylinders involve complex solution procedure[15].Hence Hellouand Coutanceau[15]supported their analytical and numericalpredictions with experimental results.In a recent work,Galaktionovet al.[16]solved the bi-harmonic form of the Stokes equations usingthe method of superposition to study the distributive mixingproduced by a circular cylinder placed at the center of a rectangularcavity.Traditional numerical techniques such asfinite difference,finite volume andfinite element methods(FDM,FVM and FEM)are routinely used to solve complex problems.It is well knownthat these methods are strongly dependent on mesh properties.However,in solution of partial differential equations(PDEs)byContents lists available at SciVerse ScienceDirectjournal homepage:/locate/enganaboundEngineering Analysis with Boundary Elements0955-7997/$-see front matter&2011Elsevier Ltd.All rights reserved.doi:10.1016/j.enganabound.2011.11.009n Corresponding author.E-mail addresses:ab@civil.aau.dk,amin78404@(A.Barari).Engineering Analysis with Boundary Elements36(2012)633–638these methods for complex geometries,mesh generation takes up very much time and is the most expensive part of the simulation.In light of difficulties of the meshing-related issues various mesh-free methods have been developed.Among them are the smooth particle hydrodynamics (SPH)[17],meshless local Petrov–Galerkin approach (MLPG)[18,19],least-squares mesh-free method (LSMFM)[20,21],etc.Recently,a new mesh-free method is proposed based on the so-called radial basis functions (RBF)[22–24].Kansa [23–24]introduced the direct collocation method using RBFs.It is found that RBFs are able to construct an interpolation scheme with favorable properties such as high efficiency,good quality and capability of dealing with scattered data,especially for higher dimension problems.The ‘‘truly’’mesh-free nature of RBFs motivated researchers to use them to deal with partial differential equations.It should be noted that the Kansa’s RBF method is actually based on the function approximation.To approximate derivatives by using RBFs,Shu and co-workers [25–27]proposed the RBF-DQ method,which combines the differential quadrature (DQ)approximation [28,29]of derivatives and function approximation of RBF.Previous applications [25–27,30–34]showed that RBF-DQ is an efficient method to solve fluid flow,heat transfer,linear and nonlinear PDEs and proved that the local RBF-DQ method is very flexible,simple in code writing and it can be easily applied to linear and nonlinear problems.In this method the problem of ill-conditioned global matrix has been removed by replacement of global solvers by block partitioning schemes for large simulation problems as shown in Fig.1.In this study the mesh-free local RBF-DQ method is applied to obtain the flow characteristics in a circular rotating cavity.Outcomes reveal that current technique is in good agreement with previous works have been done using other numerical methods,namely finite element method (FEM)and boundary element method (BEM).1.1.Radial basis functionsA radial basis function,denoted by j ð:x Àx j :2Þ,is a continuous spline,which depends on the separation distances of a subset of scattered points x A R d ,in which d denotes the spatial dimension.The most commonly used RBF isMultiquadrics ðMQ Þ:j ðr Þ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffir 2þc 2p ,c 40ð1Þwhere r ¼:x Àx j :2.For scattered points the approximation of afunction f (x )can be written as a linear combination of N RBFs:f ðx ÞffiX N j ¼1l j j ð:x Àx j :2Þþc ðx Þ,ð2Þwhere N is the number of centers or knots x ,x ¼(x 1,x 2,...,x d ),d isthe dimension of the problem,l ’s are coefficients to be determined and j is the RBF.If C d q denotes the space of d -variate polynomials of order not exceeding q ,and letting the polynomials letting thepolynomials P 1,...,P m be the basis of C d q in R d,then the polynomial c (x ),in Eq.(2),is usually written in the following form:c ðx Þ¼X m 1z i P i ðx Þ,ð3Þwhere m ¼(q À1þd )!/(d !(q À1)!).To determine the coefficients(l 1,y ,l N )and (z 1,y ,z N ),extra m equations are required in addition to the N equations resulting from the collocating Eq.(5)at the N knots.This is insured by the m conditions for Eq.(5),that isX N j ¼1l j P i ðx j Þ¼0,i ¼1,...,m :ð4Þ1.2.Differential quadrature method (DQ)The essence of the DQ method is that the partial derivative of an unknown function with respect to an independent variable can be approximated by a linear weighted sum of functional values at all mesh points in that direction.Suppose that a function f (x )is sufficiently smooth.Then its m th order derivative with respect to x at a point x i can be approximated by DQ as f ðm Þx ðx i Þ¼X N j ¼1w m ij f ðx j Þ,i ¼1,2,...,N ,ð5Þwhere x j are the discrete points in the domain;f (x j )and w ðm Þijare the function values at these points and the related weighting coefficients,respectively.One of the most important advantages of the DQ method as a global (high order)approach,in compar-ison with the low order methods such as FV or FD,is its ability to generate numerical results with high order of accuracy by using a considerably small number of mesh points.1.3.Local MQ-DQ method formulationSuppose that the solution of a partial differential equation iscontinuous,which can be approximated by MQ RBFs,and only a constant is included in the polynomial term c (x ).Then,the function in the domain can be approximated by MQ RBFs as f ðx ,y Þ¼X N j ¼1l j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx Àx j Þ2þðy Ày j Þ2þc 2j q þl N þ1:ð6ÞTo make the problem be well-posed,one more equation isrequired.From Eq.(4),we haveX N j ¼1l j ¼0)l i ¼ÀX N j ¼1,j a il j :ð7ÞSubstituting Eq.(7)into Eq.(6)gives f ðx ,y Þ¼X N j ¼1,j a il j g j ðx ,y Þþl N þ1,ð8Þwhereg j ðx ,y Þ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx Àx j Þ2þðy Ày j Þ2þc 2j q Àffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx Àx i Þ2þðy Ày i Þ2þc 2iq ,ð9Þl N þ1can be replaced by l i and Eq.(8)can be written asf ðx ,y Þ¼X N j ¼1,j a il j g j ðx ,y Þþl i :ð10Þf (x ,y )in Eq.(10)constitutes N -dimensional linear vector space V Nwith respect to the operation of addition andmultiplication.Fig.1.Supporting knots around a centered knot.S.Soleimani Kutanaei et al./Engineering Analysis with Boundary Elements 36(2012)633–638634Substituting all the base functions into Eq.(5),we have0¼X Nk¼1w mðÞi,k,ð11Þ@m g jðx i,y iÞ@x m ¼X Nk¼1wðnÞi,kg jðx k,y kÞ,j¼1,2,...,N,but j a i:ð12ÞFor the given i,equation system(11)–(12)has N unknowns with N equations.Therefore,solving this equation system canobtain the weighting coefficients wðmÞik .From Eq.(9),one can easilyobtain thefirst order derivative of g j(x,y)as@g jðx i,y iÞ@x ¼xÀx jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxÀx jÞ2þðyÀy jÞ2þc2jqÀxÀx iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxÀx iÞ2þðyÀy iÞ2þc2iqð13ÞIn the matrix form,the weighting coefficient matrix of the x-derivative can then be determined by½G ½W n T¼f G x g,ð14Þwhere[W n]T is the transpose of the weighting coefficient matrix [W n],and½W m ¼wðmÞ1,1wðmÞ1,2:::wðmÞ1,NwðmÞ2,1wðmÞ2,2:::w mðÞ2,N ^^&^wðmÞN,1wðmÞN,2:::wðmÞN,N26666643777775,½G ¼11 (1)g1ðx1,y1Þg1ðx2,y2Þ...g1ðx N,y NÞ^^&g Nðx1,y1Þg Nðx2,y2Þ...g Nðx N,y NÞ266664377775,½G x ¼00ÁÁÁ0g mxð1,1Þg mxð1,2ÞÁÁÁg mxð1,NÞ^^&g mxðN,1Þg mxðN,2Þ...g mxðN,NÞ266664377775:ð15ÞWith the known matrices[G]and[G x],the weighting coeffi-cient matrix of x-derivative[W m]can be obtained by using a direct or iterative method such as LU decomposition or SOR.The weighting coefficient matrix of the y-derivative can be obtained in a similar ing these weighting coefficients,we can discretize the spatial derivatives and transform the governing equations into a system of algebraic equations,which can be solved by iterative or direct method.1.4.Shape parameter(c)in local MQ-DQ methodAs mentioned before,the MQ approximation of the function contains a shape parameter C that could be knot-dependent and must be determined by the user.In our study,c is chosen as a constant,and is taken as0.12,and the number of supporting knots is taken as16based on the previous work of Ding et al.[26].2.Mathematical modelingConsidered geometry of the problem with related boundary condition is depicted in Fig.2.The continuity and dimensionless form of the Navier–Stokes equations for a Newtonian viscousfluid of constant density and constant viscosity are as follows:r u¼0ð16ÞSt @u@tþðu U rÞu¼ÀEu r Pþ1Rer2uþ1Fr2gð17Þwhere u¼(u,v,w)is the velocity vector,P is the pressure,g is theexternal body force,St is Strouhal number,Eu is Euler number,Reis Reynolds number and Fr is Froude number.In theflow offluids,a Reynolds number(densityÂlengthÂvelocity/viscosity)describes the relative importance of inertiaeffects to viscous effects.Forflows with very small(less than1)Reynolds number the inertia effects can be ignored in comparisonto the viscous resistance.These kinds offlows are known asStokesflow or creepingflow.The general form of momentumequation for stokesflow while Fr51is shown below:St@u@t¼ÀEu r Pþ1Rer2uð18ÞFor the cases in which the rate of change in the velocityfielddue to geometrical changes is much smaller than viscous diffu-sion(StÂRe51),the momentum equation can be reduced tosteady form asm r2u¼r Pð19Þsubjected to the following no-slip boundary conditions:u¼Uð20Þwhere m is the dynamic viscosity and U is known a prioriboundary velocity.Eqs.(16),(19)and(20)constitute the so calledprimitive variables of velocity–pressure formulation.By defini-tion,the vorticity vector X¼(x,Z,z)is expressed asX¼rÂuð21ÞTaking the curl to Eq.(19)with constant m,and using Eq.(21),we obtain the steady-state vorticity transport equation for Stokesflows as follows:r2X¼0ð22ÞTaking the curl to Eq.(19)and using Eq.(16),we getr2u¼ÀrÂXð23ÞSubstituting Eq.(23)in Eq.(19)with taking the divergenceoperator,we can obtainr2P¼0ð24ÞEqs.(22)–(24)are the governing equations for steady stateStokesflows in another form known as the velocity–vorticityformulation.Fig.2.Geometry and boundary conditions.S.Soleimani Kutanaei et al./Engineering Analysis with Boundary Elements36(2012)633–638635In two dimensional if (u ,v )are the velocity vectors and o is the associated vorticity vector,then we haver 2o ¼0ð25Þr 2u ¼À@o @yð26Þr 2v ¼@o @xð27Þ3.Results and discussionAn iterative procedure is applied to the velocity–vorticity formulation of Stokes flow.In the present work,the velocity Poisson equations are first solved to obtain the vorticity boundary conditions that are used in the solution of the vorticity Laplace equation.The solution procedure is described in the following steps:1.Solve the discretized form of velocity Poisson Eqs.(26)and (27)using the SOR.Discritizing of space derivation for Eqs.(26)and (27)using local RBF-DQ is gained as follows:X n i k ¼1w 2x i ,k u k i þX n i k ¼1w 2y i ,k u k i¼ÀX n i k ¼1w 1y i ,k o kið28ÞX n i k ¼1w 2x i ,k v ki þX n ik ¼1w 2y i ,k v ki¼X n ik ¼1w 1x i ,k v k ið29Þ2.Calculate the velocity distribution and velocity derivatives atall nodal points.3.Determine new vorticity boundary values using the definition of vorticity in Eq.(21).4.Solve the vorticity Laplace Eq.(25)using iterative local RBF-DQX n i k ¼1w 2xi ,koki þX n i k ¼1w 2y i ,k o ki¼0ð30Þ5.Calculate the unknown vorticity values throughout the domain.6.Calculate the derivatives of the vorticity vectors to be used in the velocity Poisson equations.7.Check for the convergence of the velocity and vorticity components (the convergence criterion is set to 10À6).8.If the convergence criterion is satisfied,then stop;otherwise go to Step 1.Point generation in the present method is consequently easier in complex domains as compared with mesh-based methods.In general,there are three grid generation algorithms that can be selected according to the practical applications [35]:(1)Using the grid generation algorithm of conventional finite-difference schemes for simple geometries.(2)Using the random point generation algorithm.(3)Using algebraic formulation to produce several layers oflocally orthogonal grids near the boundary,and using the random point generation algorithm to generate other nodes.A comparison on the CPU time of the solution procedure employing the points generated by the second and the third algorithms can be found in [36].In the present study the second algorithm is applied for the problem.RBF-DQ nodes used for the current problems are illu-strated in Fig.3.In the current investigation,knots were distributed in the domain using the Halton sequences [37].In the present code we have used the all-pair search approach [17]for identifying the nearest supporting knots.The increase of knots number was stopped since an appropriate accuracy obtained.We selected 841knots totally,with 107knots on the external boundary.The results obtained by local MQ-DQ are shown as the vorticity,u-and v-velocity contours in Figs.4,5and 6,respec-tively,which exhibit the symmetric characteristics of the flow variables as expected for the Stokes flow.In order to comparison the results in details,u-velocity variation was calculated along the vertical line x ¼1as shown in Fig.7.Outomes gained by RBF-DQ is in excellent agreement with the previous work [38].4.ConclusionsTwo dimensional Stokes flow in an irregular geometry applic-able in engineering are analyzed numerically in terms of velocity and vorticity contour by mesh-free local RBF-DQ method.Results are compared with solutions achieved by FE andBEM.Fig.3.Irregular nodes for RBF-DQ methodsolution.Fig.4.Vorticity contour for the Stokes flow in a circular cavity.S.Soleimani Kutanaei et al./Engineering Analysis with Boundary Elements 36(2012)633–638636Comparison of the present outcomes with pervious works demonstrates that the local RBF-DQ method is an attractive approach in terms of accuracy,capability and flexibility in programming,even for complex irregular boundaries.References[1]Krasnopolskaya TS,Meleshko VV,Peters GWM,Meijer HEH.Mixing in Stokesflow in an annular wedge cavity.Eur J Mech B/Fluids 1999;18:793–822.[2]Finn Matthew D,Cox Stephen M.Stokes flow in a mixer with changinggeometry.J Eng Math 2001;41:75–99.[3]Lefevre A,Mota JPB,Rodrigo AJS,Saatdjian E.Chaotic advection and heattransfer enhancement in Stokes flows.Int J Heat Fluid Flow 2003;24:310–21.[4]Jeffrey DJ,Sherwood JD.Streamline patterns and eddies in low-Reynolds-number flow.J Fluid Mech 1980;96:315–34.[5]Irving HF,Saxton RL.Mixing of high viscosity materials.In:Uhl VW,Gray JB,editors.Mixing,theory and practice,vol. 2.London:Academic;1967.p.169–224.[6]Edwards minar flow and distributive mixing.In:Hardy N,Edwards MF,Nienow AW,editors.Mixing in the process industries.London:Butterworth;1985.p.202–25.[7]Ottino JM.Mixing,chaotic advection,and turbulence.Annu Rev Fluid Mech1990;22:207–53.[8]Young DL,Chen CW,Fan CM,Murugesan K,Tsai CC.The method offundamental solutions for Stokes flow in a rectangular cavity with cylinders.Eur J Mech B/Fluids 2005;24:703–16.[9]Brodkey RS.The phenomena of fluid motions.London:Addison-Wesley;1967.[10]Middleman S.Fundamentals of polymer processing.New York:McGraw-Hill;1977.[11]Ottino JM.The kinematics of mixing:stretching,chaos and transport.Cam-bridge:Cambridge University Press;1989.[12]Baird DG,Collias DI.Polymer processing:principles and design.London:Butterworth;1995.[13]Leong CW,Ottino JM.Experiments on mixing due to chaotic advection in acavity.J Fluid Mech 1989;209:463–99.[14]Robertson CR,Acrivos A.Low Reynolds number shear flow past a rotatingcircular cylinder.Part 1.Momentum transfer.J Fluid Mech 1970;40:685–704.[15]Hellou M,Coutanceau M.Cellular Stokes flow induced by rotation of acylinder in a closed channel.J Fluid Mech 1992;236:557–77.[16]Galaktionov OS,Meleshko VV,Peters GWM,Meijer HEH.Stokes flow in arectangular cavity with a cylinder.Fluid Dyn Res 1999;24:81–102.[17]Liu GR,Liu MB.Smoothed particle hydrodynamics:a meshfree particlemethod.Singapore:World Scientific Publishing Co.Pte.Ltd.;2003.[18]Sladek V,Sladek J,Tanaka M,Zhang CH.Transient heat conduction inanisotropic and functionally grade media by local integral equations.Eng Anal Boundary Elem 2005;29:1047–65.[19]Sladek J,Sladek V,Hellmich C,Eberhardsteiner J.Heat conduction analysis of3-D axisymmetric and anisotropic FGM bodies by meshless local Petrov–Galerkin put Mech 2007;39:323–33.[20]Xuan ZC,Zhang XK.Simulation of Stokes flow over microelectrodes withleast-squares meshfree method.Simulation Modelling Pract Theory 2008;16:294–314.[21]Liu WK,Jun S,Zhang YF.Reproducing kernel particle methods.Int J NumerMethods Fluids 1995;20:1081–106.[22]Franke R.Scattered data interpolation:tests of some methods.Math Comp1982;38:181–99.[23]Kansa EJ.Multiquadrics —a scattered data approximation scheme withapplications to computational fluid dynamics —I.Surface approximations and partial derivative put Math Appl 1990;19:127–45.[24]Kansa EJ.Multiquadrics —a scattered data approximation scheme withapplications to computational fluid dynamics —II.Solutions to parabolic,hyperbolic,and elliptic partial differential put Math Appl 1990;19(6–8):147–61.[25]Shu C,Ding H,Yeo KS.Local radial basis function-based differential quad-rature method and its application to solve two dimensional incompressible Navier–Stokes put Methods Appl Mech Eng 2003;192:941–54.[26]Ding H,Shu C,Tang DB.Error estimates of local multiquadric-baseddifferential quadrature (LMQDQ)method through numerical experiments.Int J Numer Methods Eng 2005;63:1513–29.[27]Shu C,Ding H,Chen HQ,Wang TG.An upwind local RBF-DQ method forsimulation of inviscid compressible flput Methods Appl Mech Eng 2005;194:2001–17.[28]Bellman RE,Kashef BG,Casti J.Differential quadrature:a technique for therapid solution of nonlinear partial differential equations.J Comput Phys 1972;10:40–52.[29]Shu C.Differential quadrature and its application in engineering.London:Springer-Verlag;2000[340pp.].[30]Soleimani S,Qajarjazi A,Bararnia H,Barari A,Domairry G.Entropy generationdue to natural convection in a partially heated cavity by local RBF-DQ method.Meccanica2011;46:1023–33.Fig.5.u-velocity contour for the Stokes flow in a circularcavity.Fig.6.v-velocity contour for the Stokes flow in a circularcavity.parison of u-velocity profile at x ¼1for a circular cavity obtained by local MQ-DQ and previous work [38].S.Soleimani Kutanaei et al./Engineering Analysis with Boundary Elements 36(2012)633–638637[31]Jalaal M,Soheil Soleimani,Domairry G,Ghasemi E,Bararnia H,MohammadiF,et al.Numerical simulation of electricfield in complex geometries for different electrode arrangements using meshless local MQ-DQ method.J Electrostat2011;69:168–75.[32]Soleimani S,Jalaal M,Bararnia H,Ghasemi E,Ganji DD,Mohammadi F.LocalRBF-DQ method for two-dimensional transient heat conduction problems.Int Commun Heat Mass Transfer2010;37:1411–8.[33]Bararnia H,Jalaal M,Ghasemi E,Soheil Soleimani DD,Ganji F.Mohammadi,Numerical simulation of joule heating phenomenon using meshless RBF-DQ method.Int J Therm Sci2010;49:2117–27.[34]Soleimani S,Ganji DD,Ghasemi E,Jalaal M,Bararnia H.Meshless local RBF-DQfor2-D heat conduction:a comparative study.Therm Sci2011;15:117–21.[35]Ding H,Shu C,Yeo KS,Xu D.Development of least-square-based twodimensionalfinite-difference schemes and their application to simulate natural convection in a put Fluids2004;33:137–54.[36]Ding H,Shu C,Yeo KS,Lu ZL.Simulation of natural convection in eccentricannuli between a square outer cylinder and a circular inner cylinder using local MQ-DQ method.Numer Heat Trans Part A2005;47:291–313.[37]Fasshauer GE.Meshfree approximation methods with MATLAB.Singapore:World Scientific Publisher;2007.[38]Young DL,Chiu CL,Fan CM,Tsai CC,Lin YC.Method of fundamental solutionsfor multidimensional Stokes equations by the dual-potential formulation.Eur J Mech B/Fluids2006;25:877–93.S.Soleimani Kutanaei et al./Engineering Analysis with Boundary Elements36(2012)633–638 638。
10_无网格方法
a( x) A1 ( x) B( x)u (8)
把式(8)代入式(1)得近似函数为
u ( x, x) pT ( x) A1 ( x) B( x)u N T ( x, x)u (9)
U T RA T a d V T RB T a d 0
式中 RA a A a 和 RB a B a 为余量。 虽然 U 和V
T T T Tຫໍສະໝຸດ
为任意权函数，但在实际应用时，不可能也不需要取无穷多个权函数。 与试函数表达式类似，可以把权函数也写成已知基函数的组合，即
pT ( x) [1, x, y, x 2 , xy, y 2 ]
m6
m 10
基函数的个数 m 、 基函数中包含的完备多项式的最高阶数 n 和问题的维数 nDim 之间的关系为
m
(n 1)(n 2) (n nDim ) nDim !
(3)
在移动最小二乘近似的（MLS）中，坐标 ai ( x ) 的选取应该使近似位移
p ( x) [1, x, y, r cos , r sin , r sin sin , r cos sin ] 2 2 2 2
T
式中：r 为某点距裂纹尖端的距离， 为该点与裂纹尖端连线与裂纹线的夹角。
若把式(1)作为有限单元容许位移函数，则 p1 ( x) 1 可以保证单元容许位
求解方法。只要试函数是利用离散点来建立的，则由紧支试函
数加权余量法导出的各种近似方法都称为无网格方法。
紧支近似函数 紧支近似函数是定义在局部区域(支撑区域)中的函数，它只 在支撑域中有定义，而在支撑域外为零。在二维问题中，支撑 域一般为圆形或矩形，与求解域相比，支撑域是一个很小的区 域，并且是可以互相重叠。有限元网格表示的区域是不能彼此 重叠的。
固体力学中的无网格方法
• • • • • 无网格方法的概述 无网格方法的近似方案 不连续性近似 离散化实现 基本边界条件的实现
无网格方法的概述
无网格法是在建立问题域的系统代 数方程时，不需要利用预定义的网 格信息，或者只利用更容易生成的 更灵活、更自由的网格进行域离散 的方法。（刘桂荣，2009）
无网格方法的概述
最近几年，Duarte和Oden等人提出了单位分解法， 并且认识到基于移动最小二乘法的近似方法实际上是 单位分解法的一种特例，从而将这类近似方法加以扩 展；Liu等人也对此类方法做了大量的研究工作，并对 其收敛性给以证明。
无网格方法的概述
无网格法求解过 程（与FEM对比）
无网格方法的概述
无网格方法模拟裂纹扩展
无网格方法的近似方案
• 核函数近似方法 • 移动最小二乘近似(MLS) • 单位分解法
无网格方法的近似方案
核函数近似方法
核函数近似方法最初主要用于SPH方 法。它对函数u（x）利用核函数进行近 似 u ( x ) x y , h u ( y ) d x y , h 被称为核函数或权函数，h是紧 支集尺寸的一个度量。
无网格方法的概述
一条构造无网格方法的途径是采用 移动最小二乘法（moving least square approximation method，简记为MLS）进 行近似。Nayroles等人最早将移动最小二 乘近似用于Galerkin方法，并将之称为扩 散单元法（difflnse elernent methods，简 称DEM）。Belytschko等人提出了无单元 的Galerkin法（element free galerkin method，简称EFG）。这类方法具有较 好的协调性及稳定性。
计算固体力学(有限元以及无网格方法)
σz ≠ 0
E 1− µ2
平面应变问题的弹性矩阵只需将上页中的 E 换成
µ 换成 1 − µ 即可。 即可。
µ
1 E (1 − µ ) µ [D] = (1 + µ )(1 − 2µ ) 1 − µ 0
µ
1− µ 1 0
0 1 − 2µ 2(1 − µ ) 0
第二章 平面弹性力学的有限元法
2.2 三角形常应变单元 3 单元中的应变和应力
{ε } = [ B ]{δ }e
i ( xi , yi )
y
vm
Hale Waihona Puke m ( xm , y m )
vi
um
vj
uj
ui
j(x j , y j )
由于[ 是常量， 由于[B]是常量，单元内各点应变分 量也都是常量， 量也都是常量，这是由于采用了线性位 移函数的缘故， 移函数的缘故，这种单元称为三角形常 应变单元。 应变单元。
2.2 三角形常应变单元 2 位移试函数
由于位移函数适用于单元中的任意 一点，所以代入3个节点的坐标后， 一点，所以代入3个节点的坐标后，得 出节点处位移函数为
ui = α1 + α 2 xi + α 3 yi
u j = α1 + α 2 x j + α 3 y j u m = α 1 + α 2 xm + α 3 y m
y
vm
m ( xm , y m )
vi
um
vj
uj
i ( xi , yi )
ui
j(x j , y j )
O
x
三角形单元
1 xi 1 ∆ = 1 xj 2 1 xm yi