Finite Element Method - Elsevier · 2013-09-03 · The Finite Element Method by G. R. Liu and S. S....

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

The Finite Element Method A Practical Course

G. R. Liu and S. S. Quek

Chapter 1: Computational modeling

An overview


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


C onceptual 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

Virt

ual p

roto

typi

ng

Design process for an advanced engineering system


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


PHYSICAL PROBLEMS IN ENGINEERING

√ Mechanics for solids and structures Heat transfer Acoustics Fluid mechanics Others


COMPUTATIONAL MODELLING USING FEM

Four major aspects:

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

conditions


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.



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



Eventually represented by discretized elements

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


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)


A complex function is represented by piecewise linear functions

x

F ( x )

nodes elements

Unknown function of field variable

Unknown discrete values of field variable at nodes


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.


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


Mesh for a boom showing the stress distribution (Picture used by

courtesy of EDS PLM Solutions)


Mesh of a hinge joint


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


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


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)


SIMULATION

Two major aspects when performing simulation: – Discrete system equations

Principles for discretization Problem dependent

– Equations solvers Problem dependent Making use of computer architecture


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


Discrete system equations

Taylor series – For traditional FDM

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


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


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)


VISUALIZATION

Vast volume of digital data Methods to interpret, analyze and for

presentation Use post-processors 3D object representation

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


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


Air flow in a virtually designed building

(Image courtesy of Institute of High Performance Computing)

Finite Element Method - Elsevier · 2013-09-03 · The Finite Element Method by G. R. Liu and S. S....

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