1 Introduction of Finite Element Method Computational ...

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Computational Rock Mechanics Research GroupSchool of Mechanical and Civil EngineeringChina University of Mining and Technology (BJ)

Monday, May 10, 2021

Computational Mechanics(Bilingual Course)

Yongliang Wang

Computational Mechanics(Bilingual Course)

Yongliang Wang

1.1 Development process of finite element method

1.2 Computation procedure of finite element method

1.3 Main contents of the book

1.4 Exercises

1 Introduction of Finite Element Method1 Introduction of Finite Element Method

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(a) The continuum is divided into a finite number of parts (elements), the behavior of which is specified by a finite number of parameters, and

(b) the solution of the complete system as an assembly of its elements follows precisely the same rules as those applicable to standard discrete problems.

Continuous problems to discrete problems


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Develops

在这个信息时代，像许多其它方

面的进步一样，在有限元分析中，

应用程序和软件往往比书籍、文

献更好地代表了新的进展。


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在20世纪60年代，由于Ed Wilson发布了他的第一个程序，

这种激情终于被点燃了。这些程序的第一代没有名字。在

遍布世界的许多实验室里，通过改进和扩展这些早期在

Berkeley开发的软件，工程师们扩展了新的用途，带来了

对工程分析的巨大冲击和有限元软件的随之发展。

SAP：在Berkeley开发的第二代线性程序称

之为SAP(Structural Analysis Program)，之

后发展的第一个非线性程序是NONSAP，它具有隐式积分进行平衡求解和瞬时问题求

解的功能。北京大学Prof. Mingwu YUAN引

入中国并进行开发、使用。 Prof. Mingwu YUAN


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MARC：1969年，在Brown大学任教的Pedro Marcal，为

了第一个非线性商业有限元程序进入市场，于建立了一个

公司；程序命名为MARC，目前它仍然是主要软件，1999年被MSC公司兼并，MSC/MARC。

ANSYS：大约在同期，John Swanson为了核能应用在

Westinghouse发展了一个非线性有限元程序。为了使

ANSYS程序进入市场，他于1969年离开Westinghouse。ANSYS尽管主要是关注非线性材料而非求解完全的非线性

问题，它多年来仍垄断了商业非线性有限元软件的舞台。


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ABAQUS：David Hibbitt，他与Pedro Marcal合作到

了1972，1978建立了HKS公司，使ABAQUS商用软件

进入市场。因为该程序是能够引导研究人员增加用户

单元和材料模型，对软件行业带来了实质性的冲击。

UEL: User subroutine to define an ELement.UMAT: User subroutine to define an element with access to Abaqus Materials.

Prof. Zhuo ZHUANG

清华大学Prof. Zhuo ZHUANG引入中国

并进行开发、使用、推广。

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ADINA ： Jürgen Bathe 是在 Ed Wilson 的指导下在

Berkeley获得博士学位的，不久之后开始在MIT任教，这

期间他便发布了他的程序。这是NONSAP软件的派生产

品，称为ADINA。

NASTRAN ：大型通用有限元软件。 The MacHeal-Schwendler Corporation (MSC)，1963年创立，主要得到

美国航空界赞助，如NASA和FAA，为飞行器验证软件。

前处理为PATRAN。


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DYNA：显式有限元程序发展的里程碑来自于LawrenceLivermore实验室的John Hallquist的工作。1975年，John开始他的工作，1976年，他首先发布DYNA程序。他慧眼

吸取了前面许多人的成果，并且与Berkeley的研究人员紧

密交流合作，包括 Jerry Goudreau，Bob Taylor，TomHughes和Juan Simo。他之所以成功的部分关键因素是与

Dave Benson合作发展了接触－冲击相互作用，和他的令人

敬畏的编程效率，以及计算程序DYNA-2D和DYNA-3D的

广泛传播。


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OPENSEES: http://opensees.berkeley.edu/ FEAP: http://www.ce.berkeley.edu/projects/feap/ …… …

Open source programs





1.4 Exercises

1 Introduction of Finite Element Method1 Introduction of Finite Element Method 1.2 Computation procedure of finite element method

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Computational mechanics is the discipline concerned with the useof computational methods to study phenomena governed by the principles ofmechanics. Before the emergence of computational science (also calledscientific computing) as a “third way” besides theoretical andexperimental sciences, computational mechanics was widely considered tobe a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.

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Numerical methods in computational mechanics

作为力学分支的计算力学，发展了有限元 (finite element

method, FEM)、离散元(discrete element method, DEM) 、有限

差分法 (finite difference method, FDM)、无网格法 (mesh-less

method, MLM)、扩展有限元法(extended finite element method,

XFEM)、边界元(boundary element method, BEM)、半解析方法

(semi-analytic methods）等理论和方法，为虚拟仿真提供了工具。

有限元分析是虚拟设计的基本组成部分，它提供了更快捷和

低成本的方式评估设计的概念和细节，使得人们越来越多地应用

数值试验代替样品原型的物理试验。


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(a) The continuum is divided into a finite number of parts (elements), the behavior of which is specified by a finite number of parameters, and

(b) the solution of the complete system as an assembly of its elements follows precisely the same rules as those applicable to standard discrete problems.

Continuous problems to discrete problems


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1. Define the problem to be solved in terms of differential equations. Construct the integral form for the problem as a virtual work, variational, or weak formulation.

2. Select the type and order of finite elements to be used in the analysis.

3. Define the mesh for the problem. This involves the description of the node and element layout, as well as the specification of boundary conditions and parameters for the formulation used.

4. Compute and assemble the element arrays. The particular virtual work, variational, or weak form provides the basis for computing the specific relationships of each element.

5. Solve the resulting set of linear algebraic equations for the unknown parameters. For static problems this requires the solution of linear algebraic equations.

6. Output the results for the nodal and element variables. Graphical outputs also are useful for this step.


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Mining engineering——coal mining

3D numerical model Evolution of Y-stress Tunnel excavation

Y-stress field Y-displacement field


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Geotechnical engineering——slope excavation

Y-stress field Y-displacement field

2D numerical model 2D structural mesh Slope excavation


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Rock mechanics——crack initiation and propagation

Displacement field

2D numerical model Crack propagation Rock disc model


Petroleum Engineering——hydraulic fracturing

1 : 1 1 : 1.5

σH σh

Hydraulic fracturing

of tight shale reservoir

Numerical simulation Multiple wells fracturing


Petroleum Engineering——hydraulic fracturing

Hydraulic fracturing

of tight shale reservoir

Numerical simulation Multiple wells fracturing


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Civil engineering——deformation of buildings

Deformation of high-rise building Deformation of single-story

industrial building


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Civil engineering——vibration of cable-stayed bridge

施工过程仿真和分析

结构动力响应仿真和分析

地震响应时程仿真和分析

荆州长江大桥：全长4397.6m ，接线长4444m ，桥面宽24.5米，其中斜拉桥部分宽度26.5米，双向四车道。大桥由北岸的

引桥、荆江大堤桥、北岸滩桥、北汊通航孔桥、三八洲桥、南汊通航孔桥、南岸滩桥、荆南干堤桥和南岸引桥等九个部分主组成，总投资13.73亿元。北汊桥主桥为双

塔双索面混凝土梁斜拉桥，是目前中国主大跨径的混凝土梁斜拉桥，居同类桥梁世界第二。


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Civil engineering——vibration of cable-stayed bridge

1 0.18459(纵漂)2 0.28296(一阶对称竖弯)3 0.36452(一阶对称侧弯)4 0.47058(一阶反对称竖弯)5 0.51384(塔反对称侧弯)6 0.58846(对称扭转)13 0.66620(竖弯)32 0.77994(塔对称侧弯)46 0.82465(竖弯)50 0.84960(反对称扭转)

Frequencies

1st-oder mode

6th-oder mode


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Structure optimization——Zhaozhou Bridge

Zhaozhou Bridge(赵州桥) Model and load

Bending moment

该桥是一座空腹式的圆弧形石拱桥，是中国

现存早、保存好的巨大石拱桥。赵州桥

是入选世界纪录协会世界早的敞肩石拱桥

，创造了世界之。桥长50.82米，跨径37.02米，券高7.23米，两端宽9.6米，桥的设计完

全合乎科学原理，施工技术更是巧妙绝伦。

唐朝的张嘉贞说它“制造奇特，人不知其所

以为”。


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Mechanical Engineering - vehicle crash simulation

在汽车设计领域中，对初期设计概

念和终设计细节的评估，碰撞的

仿真正代替整车的试验如布置判定

气囊释放的加速计，内部的缓冲装

置，以及选择材料和满足碰撞准则

的构件截面。

Mesh Numerical model

Flow field simulation


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Aerospace——airplane and aerocraft

Aerocraft simulation

点阵材料翼型结构的拓扑优化 Airplane simulation


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Aerospace——bird strike

鸟撞试验: 1814g bird, 200m/s velocity

1 msec(兆秒) 5 msec(兆秒)


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心脏起搏器，心脏肌肉材料

Biomechanics——organ


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Model and mesh

Biomechanics——bone


Biomechanics——shoes

穿高跟鞋脚骨应力更高、

位移（变形）更大！


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Simulation of desk

Physical model

Displacement Von-Mises stress

假设桌子的四只脚同地面完全固定，桌子上存放物品，物品产生的均匀分布压力作用在桌面，模拟分析桌子的变形和应力。


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Numerical model Axial force Bending moment Physical model

Simulation of chair


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Simulation of dressing

3D virtual fitting system Von-Mises stress





1.4 Exercises

1 Introduction of Finite Element Method1 Introduction of Finite Element Method

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Structural chart of main contents and chapters 36

Chapter 1. Introduction of Finite Element Method

Chapter 2. Fundamentals of Elastic Mechanics

Chapter 3. Weak Form of Equivalent Integration

Chapter 4. Elements and Shape Functions

Chapter 5. Isoparametric Element and Numerical Integration

Chapter 6. Finite Element Computation Scheme of Elasticity Problems

Chapter 7. Solutions of Linear Algebraic Equations

Chapter 8. Error Estimation and Adaptive Analysis

Chapter 9. Programs of Finite Element Method


Contents

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