Finite Element Method

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

• CHAPTER 1
• COMPUTATIONAL MODELLING

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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

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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

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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

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PHYSICAL PROBLEMS IN ENGINEERING

• Mechanics for solids and structures
• Heat transfer
• Acoustics
• Fluid mechanics
• Others

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COMPUTATIONAL MODELLING USING FEM

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

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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.

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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

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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.

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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)

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A complex function is represented by piecewise
linear functions
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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.

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Mesh for the design of scaled model of aircraft
for dynamic analysis
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Mesh for a boom showing the stress distribution
(Picture used by courtesy of EDS PLM Solutions)
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Mesh of a hinge joint
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Axisymmetric mesh of part of a dental implant
(The CeraOne? abutment system, Nobel Biocare)
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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

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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)

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SIMULATION

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

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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

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Discrete system equations

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

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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

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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)

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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

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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

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Air flow in a virtually designed building(Image
courtesy of Institute of High Performance
Computing)
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Air flow in a virtually designed building (Image
courtesy of Institute of High Performance
Computing)