Numerische Simulation

Numerische Simulation – Vom Modell zur Visualisierung, Ferienakademie 2005

Finite element discretisation

finite difference andfinite difference andfinite element finite element discretisationdiscretisation

by

Christian Maier

Christian Maier 2

Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005

finite element discretisation

table of contents

• introduction

• method of FDM

• method of FEM

• lattice

• technical applications

Christian Maier 3



Arbour with torque transfer

D=40mm

T=250Nm

1 2

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• calculation of the mechanical stress and safety

• drag torsional moment

• shering stress

tW

Tmax

333

37,1256616

*40

16

*mm

mmdWt

2max 89,19mm

N

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• definition of safety

• determination of the material stress

• safety number

max tBS

22203350*58,0*

mm

N

mm

NRf mtB

20,10S

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• not considered details:

– dynamic influences

– surface quality (roughness)

– notch sensifity

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Complex problem

How can this problem be solved analytically in about one hour?

from: www.i-deus.hu/tablo/tablo7/fem.gif

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table of contents

• introduction

• method of FDM

• method of FEM

• lattice


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The Dirichlet problem for the Poisson equation

• Example: Poisson equation

• Conditions:

– Solve within a rectangle

– Area is:• Open• Not empty• contiguous

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The Dirichlet problem for the Poisson equation

• given functions:

• searched function:• conditions:

:

:

f

g

:u

within f

on g

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The finite-difference-method

• the idea:

– searching for an approximate function

• to get a solution

• for the marginal problem

• for a number of finite points

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• what is to do:

– derivative in first condition is replaced by quotients of differences

– second condition must be valid only on latice points

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• characteristics:– two dimensions in space– Constant mesh size (h>0)– h: truncation parameter

from: Numerik partieller DGL, Knabner/Angermann

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• definition of the latice points:

a = l*h, b = m*h;

• short writing

• all points in the rectangle:

N ml,

lmij xx

hhh :

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• form the system of equations: part one

– given

– given

xuhR

h

xuhxuxu

Cu

:2

xuhR

h

hxuxuxu

Cu

:2

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• form the system of equations: part two

– given

– given

xuRh

h

hxuhxuxu

Cu

02

3

2

:

xuRh

h

hxuxuhxuxu

Cu

22

4

2

:

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• lattice points within the rectangle

• lattice points on the boundary of the rectangle

jhihfh

hjihujhihuhjihuh

jhhiujhihujhhiu

,1,,21,

,1,2,1

2

2

jhihgjhihu ,,

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• linear equation for approximate value

• definitions:

jijijijijiji fuuuuuh ,1,,1,,11,2

41

limj

mjligu

mjli

ijij

,...,0,,0

or ,...,0,,0 if

1,...,1;1,...,1

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the 5 points star of the difference method

from: Numerik partieller DGL, Knabner/Angermann

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• dimension of the boundary equation

• form of the equation system

• dimensions

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• shape of the right side of the equation

if point is near the boundary and j is neighbour in 5 points star

in all other cases

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• form of hA

TI

ITI

ITI

IT

hAh 2

1

0

0

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• form of identity matrix

• form of matrix

1,1 llI

T

1,1

41

141

141

14

llT

0

0

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table of contents

• introduction

• method of FDM

• method of FEM

• lattice


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The finite-element-method

• short example for the calculating operations

– given equations:

– testing function:

on 0

inside

u

fu

0 ; Cvxv

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• next step: integration over the area :

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• next step: discretisation

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• next step: use of a base

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• next step: use of a base

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table of contents

• introduction

• method of FDM

• method of FEM

• lattice


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Lattice points

from: Skriptum zur Vorlesung Leichtbau, Prof. Baier

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Lattice points

from: Skriptum zur Vorlesung Leichtbau, Prof. Baier

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Lattice points

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table of contents

• introduction

• method of FDM

• method of FEM

• lattice points

• technical applications– bend guider– parts of a robot arm

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the technical application

• two examples:

– bend guider• analytical• numerical

– parts of a robot arm

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z

x

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• equation of motion

0*

2

2

4

4

t

w

xEI

xA

x

w

0*2

2

4

4

t

w

EI

A

x

w

const. , xEIxA

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• solution for this problem

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• Human robot

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• arm

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• elbow joint

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• elbow joint

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Numerische Simulation – Vom Modell zur Visualisierung, Ferienakademie 2005 Finite element discretisation finite difference and finite element discretisation.

Documents

christian maier slide

modell zur visualisierung

number of finite points

difference method

rectangle slide

given slide

knabnerangermann slide