1 Finite Element Method by G. R. Liu and S. S. Quek F F inite Element inite Element Method Method CHAPTER 1: COMPUTATIONAL MODELLING for readers of all backgrounds for readers of all backgrounds G. R. Liu and S. S. Quek
Mar 27, 2015
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)
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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
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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)
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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
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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)