Finite Element Method - PowerPoint PPT Presentation

Loading...

PPT – Finite Element Method PowerPoint presentation | free to download - id: 3cd6cc-OGVkZ



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

Finite Element Method

Description:

Finite Element Method for readers of all backgrounds G. R. Liu and S. S. Quek CHAPTER 1: COMPUTATIONAL MODELLING CONTENTS INTRODUCTION PHYSICAL PROBLEMS IN ... – PowerPoint PPT presentation

Number of Views:139
Avg rating:3.0/5.0
Slides: 28
Provided by: elsevierd
Learn more at: http://www.elsevierdirect.com
Category:
Tags: element | finite | method

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: Finite Element Method


1
Finite Element Method
for readers of all backgrounds
G. R. Liu and S. S. Quek
  • CHAPTER 1
  • COMPUTATIONAL MODELLING

2
CONTENTS
  • INTRODUCTION
  • PHYSICAL PROBLEMS IN ENGINEERING
  • COMPUTATIONAL MODELLING USING FEM
  • Geometry modelling
  • Meshing
  • Material properties specification
  • Boundary, initial and loading conditions
    specification
  • SIMULATION
  • Discrete system equations
  • Equation solvers
  • VISUALIZATION

3
INTRODUCTION
  • Design process for an engineering system
  • Major steps include computational modelling,
    simulation and analysis of results.
  • Process is iterative.
  • Aided by good knowledge of computational
    modelling and simulation.
  • FEM an indispensable tool

4

C
onceptual design

Modelling

Physical
,
mathematical
,
computational
, and

operational, economical


Simulation

Experimental, analytical, and
computational

Virtual prototyping
Analysis

Photography, visual
-
tape, and

computer graphics, visual reality

Design

Prototyping

Testing

Fabrication

5
PHYSICAL PROBLEMS IN ENGINEERING
  • Mechanics for solids and structures
  • Heat transfer
  • Acoustics
  • Fluid mechanics
  • Others

6
COMPUTATIONAL MODELLING USING FEM
  • Four major aspects
  • Modelling of geometry
  • Meshing (discretization)
  • Defining material properties
  • Defining boundary, initial and loading conditions

7
Modelling of geometry
  • Points can be created simply by keying in the
    coordinates.
  • Lines/curves can be created by connecting
    points/nodes.
  • Surfaces can be created by connecting/rotating/
    translating the existing lines/curves.
  • Solids can be created by connecting/
    rotating/translating the existing surfaces.
  • Points, lines/curves, surfaces and solids can be
    translated/rotated/reflected to form new ones.

8
Modelling of geometry
  • Use of graphic software and preprocessors to aid
    the modelling of geometry
  • Can be imported into software for discretization
    and analysis
  • Simplification of complex geometry usually
    required

9
Modelling of geometry
  • Eventually represented by discretized elements
  • Note that curved lines/surfaces may not be well
    represented if elements with linear edges are
    used.

10
Meshing (Discretization)
  • Why do we discretize?
  • Solutions to most complex, real life problems are
    unsolvable analytically
  • Dividing domain into small, regularly shaped
    elements/cells enables the solution within a
    single element to be approximated easily
  • Solutions for all elements in the domain then
    approximate the solutions of the complex problem
    itself (see analogy of approximating a complex
    function with linear functions)

11
A complex function is represented by piecewise
linear functions
12
Meshing (Discretization)
  • Part of preprocessing
  • Automatic mesh generators an ideal
  • Semi-automatic mesh generators in practice
  • Shapes (types) of elements
  • Triangular (2D)
  • Quadrilateral (2D)
  • Tetrahedral (3D)
  • Hexahedral (3D)
  • Etc.

13
Mesh for the design of scaled model of aircraft
for dynamic analysis
14
Mesh for a boom showing the stress distribution
(Picture used by courtesy of EDS PLM Solutions)
15
Mesh of a hinge joint
16
Axisymmetric mesh of part of a dental implant
(The CeraOne? abutment system, Nobel Biocare)
17
Property of material or media
  • Type of material property depends upon problem
  • Usually involves simple keying in of data of
    material property in preprocessor
  • Use of material database (commercially available)
  • Experiments for accurate material property

18
Boundary, initial and loading conditions
  • Very important for accurate simulation of
    engineering systems
  • Usually involves the input of conditions with the
    aid of a graphical interface using preprocessors
  • Can be applied to geometrical identities (points,
    lines/curves, surfaces, and solids) and mesh
    identities (elements or grids)

19
SIMULATION
  • Two major aspects when performing simulation
  • Discrete system equations
  • Principles for discretization
  • Problem dependent
  • Equations solvers
  • Problem dependent
  • Making use of computer architecture

20
Discrete system equations
  • Principle of virtual work or variational
    principle
  • Hamiltons principle
  • Minimum potential energy principle
  • For traditional Finite Element Method (FEM)
  • Weighted residual method
  • PDEs are satisfied in a weighted integral sense
  • Leads to FEM, Finite Difference Method (FDM) and
    Finite Volume Method (FVM) formulations
  • Choice of test (weight) functions
  • Choice of trial functions

21
Discrete system equations
  • Taylor series
  • For traditional FDM
  • Control of conservation laws
  • For Finite Volume Method (FVM)

22
Equations solvers
  • Direct methods (for small systems, up to 2D)
  • Gauss elimination
  • LU decomposition
  • Iterative methods (for large systems, 3D onwards)
  • Gauss Jacobi method
  • Gauss Seidel method
  • SOR (Successive Over-Relaxation) method
  • Generalized conjugate residual methods
  • Line relaxation method

23
Equations solvers
  • For nonlinear problems, another iterative loop is
    needed
  • For time-dependent problems, time stepping is
    also additionally required
  • Implicit approach (accurate but much more
    computationally expensive)
  • Explicit approach (simple, but less accurate)

24
VISUALIZATION
  • Vast volume of digital data
  • Methods to interpret, analyse and for
    presentation
  • Use post-processors
  • 3D object representation
  • Wire-frames
  • Collection of elements
  • Collection of nodes

25
VISUALIZATION
  • Objects rotate, translate, and zoom in/out
  • Results contours, fringes, wire-frames and
    deformations
  • Results iso-surfaces, vector fields of
    variable(s)
  • Outputs in the forms of table, text files, xy
    plots are also routinely available
  • Visual reality
  • A goggle, inversion desk, and immersion room

26
Air flow in a virtually designed building(Image
courtesy of Institute of High Performance
Computing)
27
Air flow in a virtually designed building (Image
courtesy of Institute of High Performance
Computing)
About PowerShow.com