本讲内容 5.1 概述 5.2 FEM 5.3 FDM 5.4 DEM. 5.5 Case study

Size: px

Start display at page:

Download "本讲内容 5.1 概述 5.2 FEM 5.3 FDM 5.4 DEM. 5.5 Case study"

Brice Hunter
6 years ago
Views:

1 5 地下工程数值计算方法

2 本讲内容 5.1 概述 5.2 FEM 5.3 FDM 5.4 DEM 5.5 Case study

3 Agenda 4.1 Introduction 4.2 Finite Element Method (FEM) 4.3 Finite Difference Method (FDM) 4.4 Boundary Element Method (BEM) 4.5 Discrete Element Method (DEM) 4.6 Comparison of Numerical Methods 4.7 How Does The Finite Element Method Work 4.8 Phased Excavation of A Shield Tunnel 3

4 Introduction What? Numerical methods are often divided into elementary ones such as finding the root of an equation, integrating a function or solving a linear system of equations to intensive ones like the finite element method. Why? To simulate physical phenomena to solve science or engineering. How? Computer technology, Intensive methods, simplifications, approximation. 4

5 Introduction Numerical Analysis To study the behavior of numerical methods. It is a mathematical subject that considers the modeling of the error in the processing of numerical methods and the subsequent re-design of methods. The objective by numerical methods To give an overview of what can be done. To give insight into how it can be done. To give the confidence to tackle numerical solutions in underground engineering. 5

6 Agenda 4.1 Introduction 4.2 Finite Element Method (FEM) 4.3 Finite Difference Method (FDM) 4.4 Boundary Element Method (BEM) 4.5 Discrete Element Method (DEM) 4.6 Comparison of Numerical Methods 4.7 How Does The Finite Element Method Work 4.8 Phased Excavation of A Shield Tunnel 6

7 Finite Element Method (FEM) History The finite element method originated from the need for solving complex elasticity and structural analysis problems in civil and aeronautical engineering. 7

8 Finite Element Method (FEM) History Its development can be traced back to the work by Alexander Hrennikoff (1941) and Richard Courant (1942). Hrennikoff Courant Discretizing the domain by using a lattice analogy Dividing the domain into finite triangular sub-regions for solution of second order elliptic partial differential equations that arise from the problem of torsion of a cylinder. While, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete subdomains(elements) 8

9 Finite Element Method (FEM) Definition A numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. Approach Eliminate the differential equation completely (steady state problems). Render the PDE into an approximating system of ordinary differential equations which can be solved by using standard techniques (Euler's method, Runge-Kutta). Generally, it is an implicit approach. 9

10 Finite Element Method (FEM) Solving Process A complex global stiffness matrix (relates unknown quantities to known quantities) would be established. An equation should be created that approximates the equation to be studied, but is numerically stable. When to use? When the domain changes; When the desired precision varies over the entire domain; When the solution lacks smoothness. 10

11 Finite Element Method (FEM) Examples In a tunnel excavation simulation it is possible to increase prediction accuracy in "important" areas; 11

12 Finite Element Method (FEM) Examples In the simulation of the slope failure patterns, it is more important to have accurate predictions over slope itself than over the wideopen mountain. 12

13 Finite Element Method (FEM) Strengths increased accuracy virtual prototyping enhanced design increased revenue better insight into critical design parameters fewer hardware prototypes a faster and less expensive design cycle increased productivity 13

14 Finite Element Method (FEM) Weaknesses Requirement of large computer processing and storage capacity desired level of accuracy required and associated computational time requirements a pre- and postprocessing program is indispensable to facilitate data handling comprehension of the results are difficult 14

15 Agenda 4.1 Introduction 4.2 Finite Element Method (FEM) 4.3 Finite Difference Method (FDM) 4.4 Boundary Element Method (BEM) 4.5 Discrete Element Method (DEM) 4.6 Comparison of Numerical Methods 4.7 How Does The Finite Element Method Work 4.8 Phased Excavation of A Shield Tunnel 15

16 Finite Difference Method (FDM) What? In mathematics, finite difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives. How? The subsurface is also modeled as a continuum that is divided into a number of elements which are interconnected at their nodes. Explicit approach is used to solve unknown parameters. 16

17 Finite Difference Method (FDM) Examples 17

18 *Discussion on explicit and implicit methods Implicit method Explicit method Common Ground (1)Both used in computer simulations of physical processes; (2)Numerical methods for solving time-variable ordinary and partial differential equations. Differences It have to solve an equation involving both the current state and the later one: G(Y(t), Y(t+ t)) = 0. An extra computation is required. In solving stiff problem, it takes much less computational time, although with larger time steps. It should calculate the state of a system at a later time: Y(t + t) = F(Y(t)). In solving stiff problem, the use of an explicit method requires impractically small time steps t to keep the error in the result bounded 18

19 Finite Difference Method (FDM) Strengths Weaknesses 19

20 Agenda 4.1 Introduction 4.2 Finite Element Method (FEM) 4.3 Finite Difference Method (FDM) 4.4 Boundary Element Method (BEM) 4.5 Discrete Element Method (DEM) 4.6 Comparison of Numerical Methods 4.7 How Does The Finite Element Method Work 4.8 Phased Excavation of A Shield Tunnel 20

21 Boundary Element Method (BEM) What? The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in boundary integral form). How? The boundary element method attempts to use the given boundary conditions to fit boundary values into the integral equation, rather than values throughout the space defined by a partial differential equation. Once this is done, in the post-processing stage, the integral equation can then be used again to calculate numerically the solution directly at any desired point in the interior of the solution domain. 21

22 Boundary Element Method (BEM) The BEM Model for a Furnace Lining 22

23 Boundary Element Method (BEM) Strengths Weaknesses Small computer capacity and small system of equations The method is confined to these boundary elements Efficient and economical for 2/3D problems when the defined boundaries are of greatest concern. Difficult to model complex construction procedures and time dependency of material characteristics 23

24 Agenda 4.1 Introduction 4.2 Finite Element Method (FEM) 4.3 Finite Difference Method (FDM) 4.4 Boundary Element Method (BEM) 4.5 Discrete Element Method (DEM) 4.6 Comparison of Numerical Methods 4.7 How Does The Finite Element Method Work 4.8 Phased Excavation of A Shield Tunnel 24

25 Discrete Element Method (DEM) What? The term Discrete Element Method (DEM) is a family of numerical methods for computing the motion of a large number of particles like molecules or grains of sand. History 1971, Peter Cundall; 1985, Williams, Hocking, and Mustoe, generalized discrete element method ; 1988, Genhua Shi, DDA; 2004, Munjiza and Owen, finite-discrete element method. 25

26 How? Put all particles in certain positions and give them an initial velocity Compute the forces which act on each particle Add the forces to find the total force on each particle Compute the new positions Compute the change in the position and the velocity of each particle during a certain time step from Newton's laws of motion 26

27 Discrete Element Method (DEM) Strengths Weaknesses simulate a wide variety of granular flow situations allows a more detailed study of the microdynamics of powder flows especially useful for kinematic studies of large block systems The maximum number of particles and duration of a virtual simulation is limited by computational power simulations are generally limited to spherical particles Parameters input are always difficult to determine 27

28 Discrete Element Method (DEM) Examples Used in rock mechanics 28

29 Discrete Element Method (DEM) Examples Used in soil mechanics 29

30 Agenda 4.1 Introduction 4.2 Finite Element Method (FEM) 4.3 Finite Difference Method (FDM) 4.4 Boundary Element Method (BEM) 4.5 Discrete Element Method (DEM) 4.6 Comparison of Numerical Methods 4.7 How Does The Finite Element Method Work 4.8 Phased Excavation of A Shield Tunnel 30

31 Comparison of Numerical Methods Comparison between FDM and FEM FDM An approximation to the differential equation Restricted to handle rectangular shapes and simple alterations FEM An approximation to its solution Able to handle complex geometries (and boundaries) with relative ease Easy to implement It defines the function on a discrete domain The quality of the approximation between grid points is poor Not easy to implement The approximations are defined on the entire domain The mathematical foundation is more sound Suitable for all types of analysis in structural mechanics Should be used in solving computational fluid dynamics 31

32 Comparison of Numerical Methods DEM should be applied only if it Allows finite displacements and rotations of discrete bodies, including complete detachment. Recognizes new interactions (contact) automatically as the calculation progresses. A discrete element code will embody an efficient algorithm for detecting and classifying contacts. It will maintain a data structure and memory allocation scheme that can handle many hundreds or thousands of discontinuities or contacts. 32

34 文件名格式 : 班级学号姓名简略实验名称邮件标题同文件名 Any questions please 发送至 xingzhengwu@163.com

CHAPTER 1. Introduction

CHAPTER 1. Introduction ME 475: Computer-Aided Design of Structures 1-1 CHAPTER 1 Introduction 1.1 Analysis versus Design 1.2 Basic Steps in Analysis 1.3 What is the Finite Element Method? 1.4 Geometrical Representation, Discretization