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我们在工程中间遇到很多结构具有重复性
我们把重复性分为几何空间上的重复
我们叫采用子结构
在计算时间上,比如一些非线性计算,还有一些动态的
我们充分利用重复性,这个时候就用超级单元来进行处理
对于子结构我们看一下
这是一个带孔的多排的横梁结构
我们看看它有两排,每一排里还带有几个孔
首先这个两排它是重复的
我们把其中重复的一排取出来
这叫一级子结构
在一级子结构里面,它里面带有多个孔
如果它的孔的位置和大小都是一致的
我们把带孔重复的这部分把它分出来
这样我们就得到二级子结构
这个二级子结构带有一个孔
我们看看,对于一个也的这么一个形状
它同样也满足对称性
它的对称性的话我们可以取出1/4
这叫三级子结构
那么子结构就是说把系统中具有相同特征
和性质的局部结构取出来,它就是子结构
那子结构的应用方法
第一就是把具有重复性的结构作为子结构
可以有多级子结构,把它划分出来
然后对底层子结构进行分析
逐级形成各级的刚度方程并进行缩聚
假定我们设第k级子结构的刚度方程是这么一个表达
我们把它分块
我们把它分别表达成基于和外部节点
发生连接关系的自由度qout
还有只是在子结构里面,只是内部自由度的qinn
那么我们通过这么一个刚度方程把qinn代换掉
就得到这么一个仅表达成qout的显式的刚度方程
这个带波浪的K和p就写成这么一个表达式
然后把子结构进行拼装形成上一级子结构
对多级子结构全部处理以后
最终得到整体刚度方程,进行求解
当求解以后,再把结果进行回代
再求出各级子结构内部的节点位移和其它物理量
超级单元主要是用于比如接触问题
还有一些和时间相关的迭代计算
超级单元它实际上就是一种广义的特定单元
是经过缩聚内部自由度过后的子结构
它只有与外部有连接关系的节点自由度
它的目的是为了减少计算量
特别是在需要多次迭代的复杂计算中有明显的优势
它的效果是大大减小每次生成刚度矩阵的计算量
同时也减小了计算的规模
可以获得较高的计算效率
我们看看,这就是接触问题中间超级单元的应用
我们把这部分作为超级单元
它和外部相连接的地方我们叫主自由度
它内部这个我们叫从自由度
我们可以看看,超级单元应用以后
它只有主自由度和接触的节点相关联
同样我们也是把这个问题用刚度方程进行描述
我们把位移分成两块
一块是主节点的节点位移,我们叫主自由度
另外还有一个是从节点的节点位移
我们叫从自由度
分成两个部分以后
我们把这个分块的刚度矩阵和刚度方程
进行一个缩聚,我们就可以得到这么一个刚度方程
qm就是我们的主自由度对应的节点位移
我们所得到的这个方程就是超级单元的刚度方程
这个带波浪的K就是超级单元的刚度矩阵
带波浪的P就是超级单元的外载节点列阵
同学们,这一讲的内容就是这些
我们下一讲再见
-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