当前课程知识点：Finite Element Method (FEM) Analysis and Applications > 5、Principle of trial function method for solving mechanical equations of deformed body > 5.2 Trial function method for solving pure bending beam: residual value method > Video 5.2
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那么下面我们具体给一个例子
对于受均布外载的简支梁
我们用Galerkin加权残值法来进行求解
我们来看一下梁的长度是l，高度是h
受一个均布载荷p0
那么控制方程就是[方程]
边界条件呢就是左端点挠度为0
右端点挠度为0
关于力的边界条件呢是
左端点的弯矩为0，右端点的弯矩为0
当然弯矩呢我们都表达成挠度的二阶导数了
试函数我们取为一项
也就是说我们这个φ1取为
那么这个待定系数呢是c1
那么我们可以验证所设定的这个试函数
它是满足BC(u)和BC(p)的
那么我们把它代入到控制方程中间去
那么我们就得到这么一个残差的函数
那么我们对残差函数乘上基底函数
也就是说φ1，我们这个时候的φ1是
那么我们在整个定义域里面
也就是说0到l上面让它为0
这样我们就可以求解，求出c1等于这么一个系数
最后得到的结果，把c1代到试函数里面去
也就是说c1乘上一个基底函数
这就是这个问题的解
这个解是在取定
这个试函数的基础上得到的真实的解
那么同样我们为了提高计算精度
我们可以取多项基底函数
比如我们可以取两项
其中φ1还是和原来一样
φ2呢我们取一个
这样的话我们线性组合就是
那同样我们可以验证这个试函数它也是
满足所有的边界条件BC(u)和BC(p)
我们代入控制方程中间，这个残差函数是这样的
那么我们对残差函数分别做两次加权以后
在定义域作积分，让它等于0
那么就得到这两个方程
那么我们把φ1，φ2，也就是基底函数分别代进去
这样我们就得到关于c1和c2的两个方程
这个方程组我们求解以后就可以求出c1和c2
我们把刚才的两种计算
也就是说取一项和取两项的情况进行一个精度的比较
我们可以看一下
我们以简支梁在中点也就是l/2的位置上
我们看看它的挠度和精确解进行一个对照
我们发现取一项的误差是0.3861%
取两项的时候误差就可以达到-0.027%
我们可以看一看，多取一项，它的精度大幅提高
那么同样我们用最小二乘法来作一下这个简支梁
那么同样，基底函数我们也是取两项
我们把取的试函数代到控制方程里面去
我们就得到一个残差函数
我们把这个残差函数R平方以后对整个域进行积分
我们就定义一个误差指标叫Err
那么我们现在需要调整待定系数c1,c2
来使得我们定义的误差指标取最小
那么我们分别对c1,c2
对我们这个问题里取的两项作一阶导数
一阶导数分别为0
那就是关于c1,c2的线性方程组
同样我们也可以求出c1,c2的值
那么我们可以看见，所算出来的c1和c2
和前面Galerkin加权残值法得到的结果一样
其原因就是我们取的基底函数是一样的
-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
