finite difference and finite element discretisation

1
finite difference andfinite element
discretisation

• by
• Christian Maier

2
table of contents

• introduction
• method of FDM
• method of FEM
• lattice
• technical applications

3
Arbour with torque transfer
D40mm
1
2
T250Nm
4
Arbour with torque transfer

• calculation of the mechanical stress and safety
• drag torsional moment
• shering stress

5
Arbour with torque transfer

• definition of safety
• determination of the material stress
• safety number

6
Arbour with torque transfer

• not considered details
• dynamic influences
• surface quality (roughness)
• notch sensifity

7
Complex problem
from www.i-deus.hu/tablo/tablo7/fem.gif
How can this problem be solved analytically in
about one hour?
8
table of contents

• introduction
• method of FDM
• method of FEM
• lattice
• technical applications

9
The Dirichlet problem for the Poisson equation

• Example Poisson equation
• Conditions
• Solve within a rectangle
• Area is
• Open
• Not empty
• contiguous

10
The Dirichlet problem for the Poisson equation

• given functions
• searched function
• conditions

11
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

12
The finite-difference-method

• what is to do
• derivative in first condition is replaced by
  quotients of differences
• second condition must be valid only on latice
  points

13
The finite-difference-method
from Numerik partieller DGL, Knabner/Angermann

• characteristics
• two dimensions in space
• Constant mesh size (hgt0)
• h truncation parameter

14
The finite-difference-method

• definition of the latice points
• a lh, b mh
• short writing
• all points in the rectangle

15
The finite-difference-method

• form the system of equations part one
• given
• given

16
The finite-difference-method

• form the system of equations part two
• given
• given

17
The finite-difference-method

• lattice points within the rectangle
• lattice points on the boundary of the rectangle

18
The finite-difference-method

• linear equation for approximate value
• definitions

19
The finite-difference-method
from Numerik partieller DGL, Knabner/Angermann

• the 5 points star of the difference method

20
The finite-difference-method

• dimension of the boundary equation
• form of the equation system
• dimensions

21
The finite-difference-method

• 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
22
The finite-difference-method

• form of

0
0
23
The finite-difference-method

• form of identity matrix
• form of matrix

0
0
24
table of contents

• introduction
• method of FDM
• method of FEM
• lattice
• technical applications

25
The finite-element-method

• short example for the calculating operations
• given equations
• testing function

26
The finite-element-method

• next step integration over the area

27
The finite-element-method

• next step discretisation

28
The finite-element-method

• next step use of a base

29
The finite-element-method

• next step use of a base

30
table of contents

• introduction
• method of FDM
• method of FEM
• lattice
• technical applications

31
Lattice points
from Skriptum zur Vorlesung Leichtbau, Prof.
Baier
32
Lattice points
from Skriptum zur Vorlesung Leichtbau, Prof.
Baier
33
Lattice points
34
table of contents

• introduction
• method of FDM
• method of FEM
• lattice points
• technical applications
• bend guider
• parts of a robot arm

35
the technical application

• two examples
• bend guider
• analytical
• numerical
• parts of a robot arm

36
the technical application
x
z
37
the technical application

• equation of motion

38
the technical application

• solution for this problem

39
the technical application

• Human robot

40
the technical application

• arm

41
the technical application

• elbow joint

42
the technical application

• elbow joint

43
the technical application
44
the technical application