Finite Element Method Programming

Course Description

The finite element method (FEM) offers a powerful general framework for solving ordinary and partial differential equations. It is especially useful in solving engineering problems, for which domain is usually geometrically complex, and materials often have anisotropic and nonlinear properties, because in this case the application of analytical methods is generally not possible. Within this course, weak formulations for various differential equations are derived using variational methods that include the Rayleigh-Ritz method and the method of weighted residuals. The discretization of the obtained integral forms based on finite elements is introduced. The geometry of one-dimensional (linear), two-dimensional (triangular and quadrilateral) and three-dimensional (tetrahedral, hexahedral, and other) finite elements is analytically defined, and the transformation of the geometry of reference finite elements into real finite elements is considered. The shape functions and their derivatives in finite elements are constructed and the choice of the weighting functions is considered. The approximation of integral forms in finite elements, rounding error in finite elements and the procedure of numerical integration are analysed. Local and global matrices of the obtained systems of equations are formed and the properties of the obtained matrices such as symmetry and density are considered. Direct and iterative algorithms suitable for solving the obtained matrix equations are introduced. Nonlinear problems are considered and strategies for solving them are analysed, and the Newton-Raphson method and the incremental method are introduced. Students will be introduced to the programming techniques required for the computer implementation of the finite element method and will model various engineering problems using the finite element method in appropriate programming environments.

Learning Outcomes

  1. Explain approximation with finite elements
  2. Apply various types of elements in the approximation.
  3. Explain variational formulation of engineering problems
  4. Explain matrix formulation of the finite element method
  5. Apply various numerical procedures in the finite element method
  6. Apply programming techniques for the finite element method

Forms of Teaching

Lectures

participation in lectures

Seminars and workshops

project and team work

Independent assignments

project and team work

Laboratory

project and team work

Work with mentor

support in team work

Grading Method

Continuous Assessment Exam
Type Threshold Percent of Grade Threshold Percent of Grade
Seminar/Project 0 % 30 % 0 % 30 %
2. Mid Term Exam: Written 0 % 30 % 0 %
Final Exam: Written 0 % 30 %
Final Exam: Oral 10 %
Exam: Written 0 % 60 %
Exam: Oral 10 %

Week by Week Schedule

  1. Variational methods
  2. Variational Formulation of the Elliptic Linear Model Problem
  3. The Finite Elements in One-dimensional Space; Linear Finite Elements; Interpolation Operator and Interpolation Error
  4. The Finite Elements in Two-dimensional Space; Triangulation of the Domain; The Finite Element Space; Lagrange Finite Elements
  5. Discretization of weak formulation; Numerical integration
  6. Formation of local matrix in a finite element; Assembling global matrices over a problem domain
  7. Analysis of the obtained matrix equations: symmetry and density of matrices; Direct and iterative methods for solving matrix equations: numerical stability and convergence
  8. Midterm exam
  9. The Finite Elements in Three-dimensional Space
  10. Convergence of Conforming Methods
  11. Finite Element Method for Unsteady Problems: On the Weak formulation; Semi-Discretization by Finite Elements; Temporal Discretization; Error Control
  12. Finite Element Method for Unsteady Problems: On the Weak formulation; Semi-Discretization by Finite Elements; Temporal Discretization; Error Control
  13. Nonlinear Problems and Systems
  14. Nonlinear Problems and Systems
  15. Final exam

Study Programmes

University graduate
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Literature

Zijad Haznadar, Željko Štih (1997.), Elektromagnetizam 2, Školska knjiga
Gouri Dhatt, Gilbert Touzot, Emmanuel Lefrançois (2012.), Finite Element Method, , Inc,UK, 2012, Wiley-ISTE
Mats G. Larson, Fredrik Bengzon (2013.), The Finite Element Method: Theory, Implementation, and Applications, Spinger
Ian M. Smith, D. V. Griffiths (2004.), Programming the finite element method, Wiley

For students

General

ID 222724
  Summer semester
5 ECTS
L1 English Level
L2 e-Learning
30 Lectures
0 Seminar
0 Exercises
13 Laboratory exercises
0 Project laboratory

Grading System

86 Excellent
74 Very Good
62 Good
50 Acceptable