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好，我们构造10节点的四面体二次单元
我们有10个节点，它是四面体
它有4个角节点
每一条边上面还有一个中节点
一共是6条边，所以一共是10个节点
每个节点3个自由度
所以它是30个自由度
我们由Pascal三角形可知
一个完全的二次项，当然是3D问题了
它有10项，所以正好我们是10个节点
我们就可以构造出完全的二次项的插值关系
从a1一直取到a10
我们同样基于体的自然坐标来进行描述
对应的角节点的形状函数写出来是这样的
对于棱边上的中间节点，同样有6项
每一个形状函数我们写出来是这么一个关系
对于四面体的三次单元
如果我们要构造一个完全的三次函数
它需要20项
那么我们就需要20个节点
我们来构建一下就是
首先4个角节点
12个分布在6条棱边上的三等分的节点
再加上4个面心节点
这样我们就由20个节点组成四面体的三次单元
它的自由度是60
在每个方向可以取20项，也就是20个自由度
这样我们就可以完全地从a1取到a20
这样就是完全的三次项
对应的角节点的形状函数
我们基于自然坐标描述出来是这样的
对于棱边上的三等分的点
对应出来的形状函数是这么一个关系
对于面心节点得到的形状函数是这样的
前面是四面体的单元
我们看看正六面体的高阶单元
首先正六面体的单元用Lagrange插值来进行
和前面矩形的Lagrange插值情况类似
我们可以通过三个坐标方向的Lagrange多项式
相乘来获得
这样我们假定在ξ,η,ζ方向
分别取m+1,n+1,p+1个节点
这样我们就可以得到从低阶到高阶的，直达
这么一个高阶项
相应的形状函数的描述
我们对于i节点
这个i节点分别对应着I,J,K
这么一个行列式的3个方向的编号
它对应的Lagrange一维的插值函数是
每个方向的插值函数我们把它乘起来
就可以得到Lagrange的正六面体的高次单元的形状函数
与构造Serendipity矩形单元的情况类似
我们同样也可能构造各种节点的Serendipity正六面体单元
也就是说我们把增加的节点尽量放在边上
而不放在这个体单元的中间
比如我们有Serendipity二次单元
它有20个节点
它对应的角节点的形状函数是这么一个表达
对于棱边上的内节点，它的形状函数就是这样的
对于三次的Serendipity单元，它有32个节点
对应着角节点的形状函数是这样的
对于棱边上的内节点，其形状函数为这样的
我们看看，这是一个20节点的Serendipity单元
它在一个侧面上
我们看看它的二维节点是这么一个划分的情况
我们所增加的节点都是在棱边上
在这个面上，我们再看看
取出一个边，同样这个边也是把节点取在这条线上
32节点情况也是类似的
这是面上的二维节点
这是面上的一维方向的节点情况
-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
