当前课程知识点：Finite Element Method (FEM) Analysis and Applications > 12、Introduction to the application of finite element analysis (1) > 12.4 Finite element analysis for elastic-plastic problems: solving non-linear equations > Video 12.4
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在弹塑性问题有限元分析问题里面要涉及到非线性方程求解
我们最常用的方法就是Newton-Raphson方法
我们也叫N-R方法
我们看看，有限元分析的非线性方程组是这么一个情况
非线性方程求解有直接的迭代法
有Newton-Raphson方法
还有改进的Newton-Raphson方法
那么下面重点考查一下Newton-Raphson方法
它的主要思想就是进行分步的逼近迭代
它将总载荷分成一系列的载荷段
在每一个载荷段里面进行非线性方程的迭代
它的特点是在迭代中
非线性方程都变为线性方程进行求解
我们看看N-R方法的计算流程
首先，把总载荷P分成一系列的载荷段
在第k个载荷段进行多步迭代
要保证它收敛
然后再进入下一步的载荷段
在第k个载荷步，我们假定里面第i次迭代是这么一个表达
每次表达都需要计算弹塑性的切线刚度矩阵
就是把这个Kep求出来
然后再施加
这个时候可以求出
同样在这个载荷段，第k段里面
要进行多步的计算
我们看看这其中的子步
首先我们对第一次施加的外载荷的增量第k步的ΔP1
也就是说这个大步
再用这一点的切线刚度
我们可以把Δq1算出来
这是第一个子步
第二个子步呢，我们把k这一点的q2代进去
再算这一点的切线
再把这一点的差ΔP2代进去，可以算出Δq2
这样我们可以把后面每一段算出来
一直到Δqi它的模小于εR，也就是说小于一个误差
这样我们就停止迭代
我们把每一段的Δq加起来就得到整个载荷段的Δq
刚才我们在第k个载荷段里面进行了多次迭代
并且保证它最后收敛
最后得到了相应的Δq
这是对一个载荷段
那么对每一个载荷段进行同样的迭代计算
我们把每一个载荷段的节点位移的增量全部加起来
这样就可以得到最后的结果
我们可以看出来，常规的N-R方法
每一次都需要形成切线刚度矩阵
也就是说都要对K矩阵重新形成，并且要进行求解求逆
计算量很大
而改进的N-R方法就是
我们在一个范围里面，比如在一个载荷段里面
我们就用初始的切线刚度矩阵
也就是说只算一次
后面迭代的话都用它，保持不变
这样就省得每次去求逆
这样我们就可以大大减少计算量
我们称这种采用初始的切线刚度矩阵的方法叫改进的N-R方法
当然我们可以把多个载荷步里面
可以分成几个切线刚度矩阵不变
这样既保证了我们收敛的效率
同时又减少了大量的计算
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
