F inite 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 computergraphics, visualreality 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 • 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. 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)