Presentation is loading. Please wait.

Presentation is loading. Please wait.


Similar presentations

Presentation on theme: "CHAPTER 1: COMPUTATIONAL MODELLING"— Presentation transcript:

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

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

Mechanics for solids and structures Heat transfer Acoustics Fluid mechanics Others

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)


Similar presentations

Ads by Google