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finite difference and finite element discretisation

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finite difference and finite element discretisation by Christian Maier – PowerPoint PPT presentation

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Title: 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
  • latice points within the rectangle
  • latice 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

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
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