F inite Element Method

1Finite Element Method by G. R. Liu and S. S. Quek

FFinite Element Methodinite Element Method

CHAPTER 1:

COMPUTATIONAL MODELLING

for readers of all backgroundsfor readers of all backgrounds

G. R. Liu and S. S. Quek

2Finite Element Method by G. R. Liu and S. S. Quek

CONTENTSCONTENTS

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

3Finite Element Method by G. R. Liu and S. S. Quek

INTRODUCTIONINTRODUCTION

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

4Finite Element Method by G. R. Liu and S. S. Quek

Conceptual design

Modelling Physical, mathematical , computational , and

operational, economical

Simulation Experimental, analytical, and computational

Analysis Photography, visual -tape, and

computer graphics, visual reality

Design

Prototyping

Testing

Fabrication

Vir

tual

pro

toty

ping

5Finite Element Method by G. R. Liu and S. S. Quek

PHYSICAL PROBLEMS IN PHYSICAL PROBLEMS IN ENGINEERINGENGINEERING

Mechanics for solids and structures Heat transfer Acoustics Fluid mechanicsOthers

6Finite Element Method by G. R. Liu and S. S. Quek

COMPUTATIONAL COMPUTATIONAL MODELLING USING FEMMODELLING USING FEM

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

conditions

7Finite Element Method by G. R. Liu and S. S. Quek

Modelling of geometryModelling 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.

8Finite Element Method by G. R. Liu and S. S. Quek

Modelling of geometryModelling 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

9Finite Element Method by G. R. Liu and S. S. Quek

Modelling of geometryModelling of geometry

Eventually represented by discretized elements

Note that curved lines/surfaces may not be well represented if elements with linear edges are used.

10Finite Element Method by G. R. Liu and S. S. Quek

Meshing (Discretization)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)

11Finite Element Method by G. R. Liu and S. S. Quek

A complex function is represented by A complex function is represented by piecewise linear functionspiecewise linear functions

x

F ( x )

nodes elements

Unknown function of field variable

Unknown discrete values of field variable at nodes

12Finite Element Method by G. R. Liu and S. S. Quek

Meshing (Discretization)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.

13Finite Element Method by G. R. Liu and S. S. Quek

Mesh for the design of scaled model of aircraft for dynamic Mesh for the design of scaled model of aircraft for dynamic analysisanalysis

14Finite Element Method by G. R. Liu and S. S. Quek

MMesh for a boom showing the stress distribution (Picture used by esh for a boom showing the stress distribution (Picture used by

courtesy of EDS PLM Solutions)courtesy of EDS PLM Solutions)

15Finite Element Method by G. R. Liu and S. S. Quek

Mesh of a hinge jointMesh of a hinge joint

16Finite Element Method by G. R. Liu and S. S. Quek

Axisymmetric meshAxisymmetric mesh of part of a dental implant of part of a dental implant (The CeraOne(The CeraOne abutment system, Nobel Biocare) abutment system, Nobel Biocare)

17Finite Element Method by G. R. Liu and S. S. Quek

Property of Property of mmaterial or aterial or mmediaedia

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

18Finite Element Method by G. R. Liu and S. S. Quek

Boundary, Boundary, iinitial and nitial and lloading oading cconditionsonditions

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)

19Finite Element Method by G. R. Liu and S. S. Quek

SIMULATIONSIMULATION

Two major aspects when performing simulation:

– Discrete system equationsPrinciples for discretizationProblem dependent

– Equations solversProblem dependentMaking use of computer architecture

20Finite Element Method by G. R. Liu and S. S. Quek

Discrete Discrete ssystem ystem eequationsquations

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

21Finite Element Method by G. R. Liu and S. S. Quek

Discrete Discrete ssystem ystem eequationsquations

Taylor series– For traditional FDM

Control of conservation laws– For Finite Volume Method (FVM)

22Finite Element Method by G. R. Liu and S. S. Quek

Equations solversEquations 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

23Finite Element Method by G. R. Liu and S. S. Quek

Equations solversEquations 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)

24Finite Element Method by G. R. Liu and S. S. Quek

VISUALIZATIONVISUALIZATION

Vast volume of digital dataMethods to interpret, analyse and for

presentationUse post-processors 3D object representation

– Wire-frames– Collection of elements– Collection of nodes

25Finite Element Method by G. R. Liu and S. S. Quek

VISUALIZATIONVISUALIZATION

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

26Finite Element Method by G. R. Liu and S. S. Quek

Air flow in a virtually designed Air flow in a virtually designed buildingbuilding

(Image courtesy of Institute of (Image courtesy of Institute of High Performance Computing)High Performance Computing)

27Finite Element Method by G. R. Liu and S. S. Quek

Air flow in a virtually designed Air flow in a virtually designed buildingbuilding

(Image courtesy of Institute of (Image courtesy of Institute of High Performance Computing)High Performance Computing)