1. CHAPTER 1: COMPUTATIONAL MODELLING
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 Discrete system equations Equations solvers
Two major aspects when performing simulation:
Discrete system equations
Principles for discretization
Problem dependent
Equations solvers
Making use of computer architecture

20. Discrete system equations
Principle of virtual work or variational principle
Hamilton’s 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. For nonlinear problems, another iterative loop is needed
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)