当前课程知识点：Finite Element Method (FEM) Analysis and Applications > 10、Basic properties in finite element method > 10.7 Accuracy and property of numerical solutions of finite element analysis > Video 10.7
返回《Finite Element Method (FEM) Analysis and Applications》慕课在线视频课程列表
返回《Finite Element Method (FEM) Analysis and Applications》慕课在线视频列表
我们先讨论一下有限元的求解精度问题
以平面问题为例，单元位移场可以展开为
如果单元的尺寸为h
也就是说上面这个式子里面Δx,Δy是h量级的
如果我们的位移场函数采用的是p阶完全多项式
那么它截断误差就是p+1
当然我们的Δx,Δy刚才提到了，它是h量级的
如果应变是位移的m阶导数
那么同样，要求m阶导数
所以它的误差阶次就要减一个m阶
而应变能是应变的平方项的表达
所以它误差的量级是
h的指数是2（p+1-m）
那么我们具体来用一下这个求解精度的估计
我们考虑一个两次网格划分的计算
假定第一次网格划分，我们网格尺寸是h
这个时候得到的结果是ui，这是第一次的
第二次我们进行网格的细分，细分的网格是细分一倍
也就是说网格的尺寸是h/2
这个时候得到的结果是ui，这个上面是2
假定准确解是ui
那么第一次得到的结果
也就是说有限元的结果减一个精确解
它的误差量级是h的p+1次
对于第二次也一样
由于这个时候是细分的，所以它的尺寸是h/2
同样它的阶次是p+1次
我们具体化
我们对一个平面3节点三角形单元
这个时候p=1，也就是取的完全的线性项
p+1=2
这样我们把刚才的估计式具体化
上面就得到了h的平方的量级的误差
下面是h/2的平方的量级的误差
我们大致估计一下，这两个一除
它就是4倍的关系
我们通过这个方程把ui解出来，它就等于
当然这只是一个初步的估计
真实的情况还非常复杂
因为它涉及到一些局部区域和整体区域的一些关系
也涉及到计算机的数值计算的误差
那么有限元分析是基于最小势能原理
我们看看，对象的总势能是可以表达成应变能减外力功
应变能是等于
由最小势能原理我们得到了刚度方程
我们考虑平衡时候的弹性系统的势能
我们把这个方程作一个代换
我们把刚才的刚度方程代到这里面来
然后前后两个一组合，我们就得到势能等于
这恰好就是等于负的应变能
也就是说等于负的U
它也等于-W/2，也就是说等于-1/2外力功
所以对于弹性问题来说
我们知道外力功，也知道应变能
就可以计算它的势能，互相是可以转换的
由最小势能原理的精确的解，或者精确的势能
它应该是最小的
所以我们把这个Πexact叫精确的势能，它是最小
所以我们近似的势能总是比它大
由刚才的势能公式，势能等于-U，也就是负的应变能
所以我们用应变能来表达就是
近似的应变能要小于精确的应变能
对于同一个问题
我们有近似的刚度矩阵，近似的位移
有精确的刚度矩阵，有精确的位移
因为在同一个问题中，载荷是一样的
我们来考查一下它们之间的关系
对于近似的，我们的应变能是按照这样来算的
对于精确的，也是按这样来算的
我们也知道，近似的要小于精确的应变能
我们把这两个公式代进来
再把刚才我们所得到的这一部分等于P代进去
这样我们就得到总体来说近似的位移
我们用模的方式来代表，它是小于精确的位移
所以我们有限元的解，它是近似的
它是具有下限性质
下限性质就是说它是偏小的
同样我们来看看
有限元模型Kq=P
实际上有限元它是用有限的节点和有限的单元
来逼近一个无限的自由度的对象
所以用有限的来逼近一个无限的
它总是使得刚度矩阵的数值变大了，也就是说变刚硬了
这样K就变大了
如果P外力不变
那这个时候q就要变小
刚才我们前面也分析了
我们用有限元算的位移值总体来说是偏小的
-Finite element, infinite capabilities
--Video
-1.1 Classification of mechanics：particle、rigid body、deformed body mechanics
--1.1 Test
-1.2 Main points for deformed body mechanics
--1.2 Test
-1.3 Methods to solve differential equation solving method
--1.3 Test
-1.4 Function approximation
--1.4 Test
-1.5 Function approximation defined on complex domains
--1.5 Test
-1.6 The core of finite element: subdomain function approximation for complex domains
--1.6 Test
-1.7 History and software of FEM development
--1.7 Test
-Discussion
-Homework
-2.1 Principles of mechanic analysis of springs
--2.1 Test
-2.2 Comparison between spring element and bar element
--2.2 Test
-2.3 Coordinate transformation of bar element
--2.3 Test
-2.4 An example of a four-bar structure
--2.4 Test
-2.5 ANSYS case analysis of four-bar structure
--ANSYS
-Discussion
-3.1 Mechanical description and basic assumptions for deformed body
--3.1 Test
-3.2 Index notation
--3.2 Test
-3.3 Thoughts on three major variables and three major equations
--3.3 Test
-3.4 Test
-3.4 Construction of equilibrium Equation of Plane Problem
-3.5 Test
-3.5 Construction of strain-displacement relations for plane problems
-3.6 Test
-3.6 Construction of constitutive relations for plane problems
-3.7 Test
-3.7 Two kinds of boundary conditions
- Discussion
-- Discussion
-4.1 Test
-4.1 Discussion of several special cases
-4.2 Test
-4.2 A complete solution of a simple bar under uniaxial tension based on elastic mechanics
-4.3 Test
-4.3 The description and solution of plane beam under pure bending
-4.4 Test
-4.4 Complete description of 3D elastic problem
-4.5 Test
-4.5 Description and understanding of tensor
-Discussion
-5.1 Test
-5.1Main method classification and trial function method for solving deformed body mechanics equation
-5.2 Test
-5.2 Trial function method for solving pure bending beam: residual value method
-5.3 Test
-5.3How to reduce the order of the derivative of trial function
-5.4 Test
-5.4 The principle of virtual work for solving plane bending beam
-5.5 Test
-5.5 The variational basis of the principle of minimum potential energy for solving the plane bending
-5.6 Test
-5.6 The general energy principle of elastic problem
-Discussion
-6.1Test
-6.1 Classic method and finite element method based on trial function
-6.2 Test
-6.2 Natural discretization and approximated discretization in finite element method
-6.3 Test
-6.3 Basic steps in the finite element method
-6.4 Test
-6.4 Comparison of classic method and finite element method
-Discussion
-7.1 Test
-7.1 Construction and MATLAB programming of bar element in local coordinate system
-7.2 Test
-7.2 Construction and MATLAB programming of plane pure bending beam element in local coordinate syste
-7.3 Construction of three-dimensional beam element in local coordinate system
-7.4 Test
-7.4 Beam element coordinate transformation
-7.5 Test
-7.5 Treatment of distributed force
-7.6 Case Analysis and MATLAB programming of portal frame structure
-7.7 ANSYS case analysis of portal frame structure
-8.1 Test
-8.1 Two-dimensional 3-node triangular element and MATLAB programming
-8.2 Test
-8.2 Two-dimensional 4-node rectangular element and MATLAB programming
-8.3 Test
-8.3 Axisymmetric element
-8.4 Test
-8.4 Treatment of distributed force
-8.5 MATLAB programming of 2D plane rectangular thin plate
-8.6 Finite element GUI operation and command flow of a plane rectangular thin plate on ANSYS softwar
-Discussion
-9.1 Three-dimensional 4-node tetrahedral element and MATLAB programming
-9.2 Three-dimensional 8-node hexahedral element and MATLAB programming
-9.3 Principle of the isoparametric element
-9.4Test
-9.4Numerical integration
-9.5 MATLAB programming for typical 2D problems
-9.6 ANSYS analysis case of typical 3Dl problem
-Discussion
-10.1Test
-10.1Node number and storage bandwidth
-10.2Test
-10.2 Properties of shape function matrix and stiffness matrix
-10.3Test
-10.3 Treatment of boundary conditions and calculation of reaction forces
-10.4Test
-10.4 Requirements for construction and convergence of displacement function
-10.5Test
-10.5C0 element and C1 element
-10.6 Test
-10.6 Patch test of element
-10.7 Test
-10.7 Accuracy and property of numerical solutions of finite element analysis
-10.8Test
-10.8 Error and average processing of element stress calculation result
-10.9 Test
-10.9 Error control and the accuracy improving method of h method and p method
-Discussion
-11.1 Test
-11.1 1D high-order element
-11.2 Test
-11.2 2D high-order element
-11.3 Test
-11.3 3D high-order element
-11.4 Test
-11.4 Bending plate element based on thin plate theory
-11.5 Test
-11.5 Sub-structure and super-element
-12.1Test
-12.1 Finite element analysis for structural vibration: basic principle
-12.2 Test
-12.2 Case of finite element analysis for structural vibration
-12.3 Test
-12.3 Finite element analysis for elastic-plastic problems: basic principle
-12.4 Test
-12.4 Finite element analysis for elastic-plastic problems: solving non-linear equations
-Discussion
-13.1 Test
-13.1 Finite element analysis for heat transfer: basic principle
-13.2 Test
-13.2 Case of finite element analysis for heat transfer
-13.3 Test
-13.3 Finite element analysis for thermal stress problems: basic principle
-13.4 Test
-13.4 Finite element analysis for thermal stress problems: solving non-linear equation
-Discussion
-2D problem: finite element analysis of a 2D perforated plate
-3D problem: meshing control of a flower-shaped chuck
-Modal analysis of vibration: Modal analysis of a cable-stayed bridge
-Elastic-plastic analysis: elastic-plastic analysis of a thick-walled cylinder under internal pressur
-Heat transfer analysis: transient problem of temperature field during steel cylinder cooling process
-Thermal stress analysis: temperature and assembly stress analysis of truss structure
-Probability of structure: Probabilistic design analysis of large hydraulic press frame
-Modeling and application of methods: Modeling and analysis of p-type elements for plane problem
