Below find a list of the topics to be discussed in class. The references refer to the lecture notes to be handed out. During the semester the order of presentation may change, the changes should show on this page.

Topics to be treated:

• Introduction : 22.10.2001 - 12.11.2001
  • Introduction, general information
  • Minimization of functions, §1
  • system of bars, element stiffness matrix, global stiffness matrix, §2.1
  • stretching of a bar of variable cross section, §2.2
• Sample problems in two variables
  • Derivation of the heat equation, §4.1
  • General second order linear partial differential equations, §6.3, Table 6.1
• Using the Matlab PDE Toolbox
  • Introduction to Matlab, the demos and §9.1
  • Solve a heat conduction problem, §9.2
  • Solving a three dimensional problem, using polar coordinates, §9.3
• Using the FEM Octave
• Calculus of Variations in one Variable
  • Shortest connection between two points, §3.1.2
  • Fundamental Lemma, §3.1.1
  • Euler Lagrange Equation, §3.1.3, §3.1.4
  • Quadratic functionals, §3.1.5, connection to §6.3
  • Radial symmetric heat problem, §4.1.4 and §9.3
• From minimisation to differential equations
  • Extrema of functions, §1
  • Electrostatic problem as a minimisation problem §6.1
  • Simplified version of a quadratic functional, §6.3
  • Minimal surface problem, §6.4
  • General minimisation problem, §6.2
• Finite Elements in two Variables
  • General description of the algorithms, §7.1
  • From finite elements to finite differences §7.2
  • Coding in Mathematica with the help of EasyMesh, §7.3
  • Coding in Octave
• Calculus of Variations in two Variables
  • Euler Lagrange Equation, §6.2
  • Quadratic functionals and linear differential equations §6.3
• Applications of Calculus of Variations (tentative)
  • First integrals, §3.1.6
  • Deflection of a string, §3.3
  • Hamilton's Principle with examples, §3.3
  • Deflection of a laser beam by a heat source, §3.5
• Finite Elements in one variable
  • Horizontal bar problem, §2
  • General quadratic functionals, §4.3
  • First order elements, §4.4
  • Second order elements, Gauss integration, §4.5
  • Code in Mathematica, §4.6
  • Examples, §4.7
  • Convergence issues, §5
• Transversal vibrations of a bar, §4.8
• Matrix Computations, §10
• Plane Linear Elasticity, §8
• Further topics

Homework assignments

1. Coding
   one of the following programming problems has to be solved to pass the class
   • Write code comparable to §7.3. Use your favourite programming language, e.g. Octave, Matlab, C, Pascal, Java, ...
     The code has to produce correct results and has to be well documented.
   • Modify FEMoctave to take advantage of the sparse matrix package by A. Adler
   • Export results from FEMoctave to be read by Mathematica and then create a graphical presentation
   • Extend FEMoctave to use more of Gnuplot. Visualize vector fields and flow lines
   • Use the Visualization Toolkit VTK to provide better visualization. The book on VTK is in the HTA-Biel library 004.08.03 SCHRO 98-803 (Bitte nicht ausleihen)
   • Optimize the Mathematica code in §7.3 for speed
   Assistance will be given by the instructor
2. Sample problems
   • mandatory to pass the class
     • s simple fluid flow problem, to be solve with the Matlab toolbox
     • specified during the semester and posted here
   • facultative
     • specified during the semester and posted here

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October, 2001 by Andreas.Stahel@hta-bi.bfh.ch
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