Title: finite difference and finite element discretisation
1finite difference andfinite element
discretisation
2table of contents
- introduction
- method of FDM
- method of FEM
- lattice
- technical applications
3Arbour with torque transfer
D40mm
1
2
T250Nm
4Arbour with torque transfer
- calculation of the mechanical stress and safety
- drag torsional moment
- shering stress
5Arbour with torque transfer
- definition of safety
- determination of the material stress
- safety number
6Arbour with torque transfer
- not considered details
- dynamic influences
- surface quality (roughness)
- notch sensifity
7Complex problem
from www.i-deus.hu/tablo/tablo7/fem.gif
How can this problem be solved analytically in
about one hour?
8table of contents
- introduction
- method of FDM
- method of FEM
- lattice
- technical applications
9The Dirichlet problem for the Poisson equation
- Example Poisson equation
- Conditions
- Solve within a rectangle
- Area is
- Open
- Not empty
- contiguous
10The Dirichlet problem for the Poisson equation
- given functions
- searched function
- conditions
11The finite-difference-method
- the idea
- searching for an approximate function
- to get a solution
- for the marginal problem
- for a number of finite points
12The 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
13The finite-difference-method
from Numerik partieller DGL, Knabner/Angermann
- characteristics
- two dimensions in space
- Constant mesh size (hgt0)
- h truncation parameter
14The finite-difference-method
- definition of the latice points
- a lh, b mh
- short writing
- all points in the rectangle
15The finite-difference-method
- form the system of equations part one
- given
- given
16The finite-difference-method
- form the system of equations part two
- given
- given
17The finite-difference-method
- latice points within the rectangle
- latice points on the boundary of the rectangle
18The finite-difference-method
- linear equation for approximate value
- definitions
19The finite-difference-method
from Numerik partieller DGL, Knabner/Angermann
- the 5 points star of the difference method
20The finite-difference-method
- dimension of the boundary equation
- form of the equation system
- dimensions
21The finite-difference-method
- shape of the right side of the equation
22The finite-difference-method
0
0
23The finite-difference-method
- form of identity matrix
- form of matrix
0
0
24table of contents
- introduction
- method of FDM
- method of FEM
- lattice
- technical applications
25The finite-element-method
- short example for the calculating operations
- given equations
- testing function
26The finite-element-method
- next step integration over the area
27The finite-element-method
28The finite-element-method
29The finite-element-method
30table of contents
- introduction
- method of FDM
- method of FEM
- lattice
- technical applications
31Lattice points
from Skriptum zur Vorlesung Leichtbau, Prof.
Baier
32Lattice points
from Skriptum zur Vorlesung Leichtbau, Prof.
Baier
33Lattice points
34table of contents
- introduction
- method of FDM
- method of FEM
- lattice points
- technical applications
- bend guider
- parts of a robot arm
35the technical application
- two examples
- bend guider
- analytical
- numerical
- parts of a robot arm
36the technical application
x
z
37the technical application
38the technical application
- solution for this problem
39the technical application
40the technical application
41the technical application
42the technical application
43the technical application
44the technical application