有限元分析与应用课程大纲

Syllabus for Finite Element Analysis and

Application

课程介绍

Course Introduction

有限元方法(finite element method)：基于数学力学原理，采用计算机信息化

分析手段，完整获取复杂工程问题及科学研究中的定量化结果，也被称为一种基

于计算机信息化处理的“虚拟实验”。在数学上它是求取复杂微分方程近似解的有

效工具，是现代仿真技术的重要基础原理。有限元分析的力学基础是弹性力学，

方程求解的数学原理是加权残值法和泛函极值原理，实现的方法是数值化离散技

术，最终的载体是有限元分析软件。有限元方法已成为机械、航空航天、土木、

力学等专业学生的必备知识。

这门课程的主要内容包括：基本变量和力学方程、数学求解原理、离散结构和连

续体的有限元分析实现、各种应用领域、有限元分析的软件平台和建模技巧等。

在强调有限元理论的工程背景和物理概念的同时，通过经典的习题和典型的实例

来系统阐述有限元分析的基本原理。此外，课程基于 MATLAB 演示基于有限元

原理的编程方法，通过 ANSYS 来展示应用有限元方法的具体建模过程。

The finite element method has already become part of the

fundamental knowledge for people work in the field of machinery,

aerospace, civil engineering, mechanics and other professional fields.

The finite element method is based on elastic mechanics. Usually the

Galerkin method and functional maximum principles are used to solve

these equations. The most common finite element analysis software

packages such as ANSYS are based on the combination of

mathematics, mechanical methods, numerical discretization and

advanced computer technology.

The main content of this course includes: fundamental mathematics

and mechanics method, basic properties of the finite element method

and examples of typical applications for the finite element method.

课程语言

Course language

中文，英文字幕。

Mandarin and English subtitles

先修要求

Prerequisite

微积分、线性代数、材料力学

Calculus, Linear algebra, Mechanics

课程安排

Course format

课程持续 16 周，每周周一上传一讲内容，每周用时约 6 小时

This course lasts 16 weeks. The lectures will be uploaded on every

Monday. Approximately 6 hours per week are required.

学习内容

Course content

理论力学基本变量和方程；

有限元方法的数学和力学原理；

有限元方法基本性质；

有限元方法的典型单元；

有限元方法在多领域中的应用。

Basic variables and equations in the realm of theoretical mechanics

Fundamental mathematics and mechanics methods

Basic properties of the finite element method

Typical elements used in the finite element method

Applications for the finite element method

成绩组成及证书要求

Grading Policy and certificate requirement

每一讲（除第 0 讲和专题）都包含若干小节、讨论题和一套练习题（Exercise）。

每一小节包含一段视频和一套 Quiz，涵盖一个知识点。Quiz 侧重于对视频中知

识点的概念性考察；Exercise 侧重于对每一讲内容的考察，具有一定的难度；

讨论题会呈现在讨论区中供同学们探讨。课程的最后提供了期末考试（Final

Exam）供同学们检查最终的学习成果。

最终成绩将按照 Quiz（30%）、Exercise（40%）、Final Exam（30%）

的比例计算得到。成绩不低于 55%可获得证书。

Each lecture is divided into sections, which include a video and a quiz.

Each lecture also covers a discussion and an exercise. The quiz is

designed to check whether basic knowledge is acquired and the

exercise is designed to check the general comprehension of the

lecture. There will be a final exam at the end of this course.

The final grade will be calculate according to the following formula:

Quiz (30%), Exercise (40%), Final Exam (30%). To obtain the verified

certificate, the grade must higher than 55%.

课程章节

第 0 讲 有限的单元无限的能力

Lecture 0 Finite elements but infinite capabilities

第 1 讲 引论

Lecture 1 Introduction

1.1 力学的分类：质点、刚体、变形体的力学

1.1 Mechanical categories: mechanics of a mass point, a rigid body

and a deformable body

1.2 变形体力学的特点

1.2 Characteristics of the mechanics of deformation

1.3 微分方程求解的方法

1.3 Methods for solving differential equations

1.4 关于函数逼近的方法

1.4 Function approximation methods

1.5 针对复杂几何域上的函数表征及逼近

1.5 Function approximation for complex geometry

1.6 有限元的核心：针对复杂几何域的分片函数逼近

1.6 Sub-domain function approximations for complex geometry

1.7 有限元发展的历史和软件

1.7 History and development of finite element software

第 2 讲 基于直接刚度法的杆系有限元方法

Lecture 2 Finite element method for a bar based on the uniaxial

tension test

2.1 弹簧的力学分析原理

2.1 Principles for the mechanical analyses of springs

2.2 弹簧单元与杆单元的比较

2.2 Comparison of a spring and a bar element

2.3 杆单元的坐标变换

2.3 Coordinate transformation for a bar element

2.4 四杆结构的实例分析

2.4 An example of a structural analysis for a four bar system

2.5 四杆结构的 ANSYS 实例分析

2.5 An example of an ANSYS analysis for a four bar system

第 3 讲 针对复杂几何形状变形体的力学描述（1）

Lecture 3 Mechanical description of a deformable body with complex

geometry (1)

3.1 力学描述的基本思路及关于变形体材料的基本假设

3.1 Basic concept of mechanical descriptions and the basic hypothesis

of deformable material

3.2 指标记法

3.2 The index notation

3.3 关于三大变量及三大方程

3.3 Three categories of variables and three categories of equations

3.4 平面问题的平衡方程构建

3.4 Equilibrium equations for 2D problems

3.5 平面问题的几何方程构建

3.5 Strain-displacement relations (geometry equations) for 2D

problems

3.6 平面问题的物理方程构建

3.6 Stress-strain relations (physical equations) for 2D problems

3.7 两类边界条件

3.7 Two categories of boundary conditions

第 4 讲 针对复杂几何形状变形体的力学描述（2）

Lecture 4 Mechanical description of a deformable body with complex

geometry (2)

4.1 几种特殊情况的讨论

4.1 Discussion about several special cases

4.2 简单拉杆问题的完整弹性力学求解

4.2 Elastic mechanics solution of a simple bar problem

4.3 平面纯弯梁的描述及求解

4.3 Description and solution of a 2D bending beam

4.4 空间弹性问题的完整描述

4.4 A complete description for the 3D elasticity problem

4.5 关于张量的描述及理解

4.5 The description of a tensor and how to understand it

第 5 讲 变形体力学方程求解的试函数方法的原理

Lecture 5 The trial function method to solve the deformation

mechanics equations

5.1 变形体力学方程求解的主要方法分类及试函数方法

5.1 The main method to solve deformation mechanics equations and

the trial function method

5.2 平面弯曲梁求解的试函数方法-残值处理法

5.2 Trial function method for bending beam problems - residual

method

5.3 如何降低对试函数的高阶导数要求

5.3 How to reduce the requirements of higher derivatives for trial

functions

5.4 平面弯曲梁求解的虚功原理

5.4 The principle of virtual work and how to solve plane bending beam

problems

5.5 平面弯曲梁求解的最小势能原理的变分基础

5.5 The principle of the variation of the minimum potential energy for

a 2D beam bending

5.6 一般弹性问题的能量原理

5.6 Energy principle for general elastic problem

第 6 讲 基于试函数方法的经典实现及有限元实现

Lecture 6 The classical realization based on the trial function method

and the realization of finite elements

6.1 基于试函数的经典方法与有限元方法

6.1 The classical and finite element method of trial function method

6.2 有限元方法中的自然离散与逼近离散

6.2 Natural discretization and approximate discretization for the finite

element method

6.3 有限元方法中的基本步骤

6.3 Basic steps of the finite element method

6.4 经典方法及有限元方法的比较

6.4 Comparison of classical methods and the finite element method

第 7 讲 杆、梁结构的有限元分析

Lecture 7 Finite element analysis of bar and beam structures

7.1 局部坐标系中的杆单元构建及 MATLAB 编程

7.1 The MATLAB programming and design of a bar element in a local

coordinate system

7.2 局部坐标系中的平面纯弯梁单元构建及 MATLAB 编程

7.2 The MATLAB programming and design of a planar bending beam

in a local coordinate system

7.3 局部坐标系中的一般梁单元构建（组装）

7.3 Assembly of beam elements in local coordinate system

7.4 梁单元的坐标变化

7.4 Coordinate transformation for a beam element

7.5 分布力的处理

7.5 How to deal with a distributed force

7.6 门型框架结构的实例分析及 MATLAB 编程

7.6 Case Analysis and the MATLAB realization of a door type frame

structure

7.7 门型框架结构的 ANSYS 实例分析

7.7 ANSYS Analysis of a door type frame structure

第 8 讲 连续体结构的有限元分析（1）

Lecture 8 Finite Element Analysis of Continuum Structures (1)

8.1 平面 3 节点三角形单元及 MATLAB 编程

8.1 2D 3-node triangular element and its MATLAB realization

8.2 平面 4 节点矩形单元及 MATLAB 编程

8.2 2D 4-node rectangular element and its MATLAB realization

8.3 轴对称单元

8.3 Axisymmetric element

8.4 分布力的处理

8.4 How to deal with a distributed force

8.5 平面矩形薄板分析的 MATLAB 编程

8.5 The realization and analysis of a 2D rectangular sheet in MATLAB

8.6 平面矩形薄板的 ANSYS 实例分析

8.6 The realization and analysis of a 2D rectangular sheet in ANSYS

第 9 讲 连续体结构的有限元分析（2）

Lecture 9 Finite Element Analysis of Continuum Structures (2)

9.1 空间 4 节点四面体单元及 MATLAB 编程

9.1 3D 3-node tetrahedral elements and its MATLAB realization

9.2 空间 8 节点正六面体单元及 MATLAB 编程

9.2 3D 8-node hexahedral element and its MATLAB realization

9.3 参数单元的原理

9.3 The principle of a parametric element

9.4 数值积分

9.4 Numerical integration

9.5 典型空间问题的 MATLAB 编程

9.5 MATLAB realization for typical 3D problems

9.6 典型空间问题的 ANSYS 分析实例

9.6 ANSYS analysis for typical 3D problems

第 10 讲 有限元方法中的基本性质

Lecture 10 Fundamental properties of the finite element method

10.1 节点编号与储存带宽

10.1 Node numbering and bandwidth

10.2 形状函数矩阵与刚度矩阵的性质

10.2 Properties of the shape function matrix and stiffness matrix

10.3 边界条件的处理与支反力的计算

10.3 How to deal with boundary conditions and how to calculate

reaction forces

10.4 位移函数构造与收敛性要求

10.4 The construction of the displacement function and convergence

requirements

10.5 C0 单元与 C1 单元

10.5 The C0 element and the C1 element

10.6 单元的拼片试验

10.6 The element patch test

10.7 有限元分析数值解的精度与性质

10.7 The accuracy and properties of the numerical solution for the

finite element analysis

10.8 单元应力计算结果的误差与平均处理

10.8 The calculation error for a stress element and the average

processing

10.9 控制误差和提高精度的和方法和 p 方法

10.9 How to control errors and to improve the accuracy with the help

of different methods including the p-method

第 11 讲 高阶及复杂单元

Lecture 11 High order complex elements

11.1 1D 高阶单元

11.1 1D High order elements

11.2 2D 高阶单元

11.2 2D High order elements

11.3 3D 高阶单元

11.3 3D High order elements

11.4 基于薄板理论的弯曲板单元

11.4 Thin plate elements based on the theory of bended plate

elements

11.5 子结构与超级单元

11.5 Sub-structure and super elements

第 12 讲 有限元分析的应用领域引论（1）

Lecture 12 Introduction to the application of the finite element

analysis (1)

12.1 结构振动的有限元分析：基本原理

12.1 Finite element analysis of structural vibration: fundamental

principle

12.2 结构振动的有限元分析实例

12.2 Example of the finite element analysis of structural vibration

12.3 弹塑性问题的有限元分析：基本原理

12.3 Finite element analysis of elastoplastic problems: fundamental

principle

12.4 弹塑性问题的有限元分析：非线性问题求解

12.4 Finite element analysis of a elastoplastic problem: the solution of

a nonlinear problem

第 13 讲 有限元分析的应用领域引论（2）

Lecture 13 Introduction to the application of the finite element

analysis (2)

13.1 传热问题的有限元分析：基本原理

13.1 Finite element analysis of a heat transfer: fundamental principle

13.2 传热问题的有限元分析实例

13.2 Example of finite element analysis for a heat transfer problem

13.3 热应力问题的有限元分析：基本原理

13.3 Finite element analysis of thermal stress: fundamental principles

13.4 热应力问题的有限元分析实例

13.4 Example of finite element analysis for a thermal stress problem

专题 有限元分析的典型 project

Case study A small project with help of the finite element analysis