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

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

Christian Maier 4




• 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

Christian Maier 5




• definition of safety

• determination of the material stress

• safety number

max tBS

22203350*58,0*

mm

N

mm

NRf mtB

20,10S

Christian Maier 6




• not considered details:

– dynamic influences

– surface quality (roughness)

– notch sensifity

Christian Maier 7



Complex problem

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

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

Christian Maier 8



table of contents

• introduction

• method of FDM

• method of FEM

• lattice


Christian Maier 9



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

Christian Maier 24



table of contents

• introduction

• method of FDM

• method of FEM

• lattice


Christian Maier 25



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


Christian Maier 31



Lattice points

from: Skriptum zur Vorlesung Leichtbau, Prof. Baier

Christian Maier 32



Lattice points

from: Skriptum zur Vorlesung Leichtbau, Prof. Baier

Christian Maier 33



Lattice points

Christian Maier 34



table of contents

• introduction

• method of FDM

• method of FEM

• lattice points

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

Christian Maier 35



the technical application

• two examples:

– bend guider• analytical• numerical

– parts of a robot arm

Christian Maier 36




z

x

Christian Maier 37




• 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

Christian Maier 38




• solution for this problem

Christian Maier 39




• Human robot

Christian Maier 40




• arm

Christian Maier 41




• elbow joint

Christian Maier 42




• elbow joint

Christian Maier 43




Christian Maier 44




Documents

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