Numerische Simulation – Vom Modell zur Visualisierung, Ferienakademie 2005 Finite element discretisation finite difference and finite element discretisation 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 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation Arbour with torque transfer D=40mm T=250Nm 12

Christian Maier 4 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation Arbour with torque transfer calculation of the mechanical stress and safety drag torsional moment shering stress

Christian Maier 5 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation Arbour with torque transfer definition of safety determination of the material stress safety number

Christian Maier 6 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation Arbour with torque transfer not considered details: –dynamic influences –surface quality (roughness) –notch sensifity

Christian Maier 7 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation Complex problem How can this problem be solved analytically in about one hour? from:

Christian Maier 8 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 9 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The Dirichlet problem for the Poisson equation Example: Poisson equation Conditions: –Solve within a rectangle –Area is: Open Not empty contiguous

Christian Maier 10 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The Dirichlet problem for the Poisson equation given functions: searched function: conditions:

Christian Maier 11 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation 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

Christian Maier 12 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation 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

Christian Maier 13 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method characteristics: –two dimensions in space –Constant mesh size (h>0) –h: truncation parameter from: Numerik partieller DGL, Knabner/Angermann

Christian Maier 14 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method definition of the latice points: a = l*h, b = m*h; short writing all points in the rectangle:

Christian Maier 15 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method form the system of equations: part one –given

Christian Maier 16 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method form the system of equations: part two –given

Christian Maier 17 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method lattice points within the rectangle lattice points on the boundary of the rectangle

Christian Maier 18 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method linear equation for approximate value definitions:

Christian Maier 19 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method the 5 points star of the difference method from: Numerik partieller DGL, Knabner/Angermann

Christian Maier 20 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method dimension of the boundary equation form of the equation system dimensions

Christian Maier 21 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation 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

Christian Maier 22 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method form of 0 0

Christian Maier 23 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-difference-method form of identity matrix form of matrix 0 0

Christian Maier 24 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 25 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-element-method short example for the calculating operations –given equations: –testing function:

Christian Maier 26 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-element-method next step: integration over the area

Christian Maier 27 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-element-method next step: discretisation

Christian Maier 28 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-element-method next step: use of a base

Christian Maier 29 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation The finite-element-method next step: use of a base

Christian Maier 30 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 31 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation Lattice points from: Skriptum zur Vorlesung Leichtbau, Prof. Baier

Christian Maier 32 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation Lattice points from: Skriptum zur Vorlesung Leichtbau, Prof. Baier

Christian Maier 33 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation Lattice points

Christian Maier 34 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation table of contents introduction method of FDM method of FEM lattice points technical applications –bend guider –parts of a robot arm

Christian Maier 35 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application two examples: –bend guider analytical numerical –parts of a robot arm

Christian Maier 36 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application z x

Christian Maier 37 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application equation of motion

Christian Maier 38 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application solution for this problem

Christian Maier 39 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application Human robot

Christian Maier 40 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application arm

Christian Maier 41 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application elbow joint

Christian Maier 42 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application elbow joint

Christian Maier 43 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application

Christian Maier 44 Numerische Simulation - Vom Modell zur Visualisierung, Ferienakademie 2005 finite element discretisation the technical application