(Created 2009-08-11.)

THE FINITE ELEMENT METHOD | VSM040 |

**Aim**

The aim of the course is to give detailed theoretical and practical knowledge on the finite element method to be able to model and analyse general problems described from a physical context. The finite element method is utilised for solving common problems within the field of engineering, such as, heat flow, ground water flow, diffusion, 2- and 3-dimensional elasticity problems, beams and plates.

*Knowledge and understanding*

For a passing grade the student must

- be able to define linear static structural mechanics problems and field problems in 1d-3d and interpret their physical terms.
- from physical relations be able to formulate a mathematical model for the problem.
- be able to transfer a mathematical model, through the weak formulation, to a finite element formulation.
- be able to explain convergence, completeness and compatibility requirements for approximating functions.
- be able to describe the formulation of isoparametric elements and numerical integration.
- be able to define and utilise various types of boundary conditions and loadings.

*Skills and abilities*

For a passing grade the student must

- be able to create finite element models of real problems from a physical described context.
- be able to perform finite element analyses of differing types of engineering problems.

*Judgement and approach*

For a passing grade the student must

- be able to analyse and interpret results from a finite element simulation.
- be able to estimate the reliability of a finite element analysis.

**Contents**

The course consists of lectures, exercise sessions and three compulsory design assignments. In the first part of the course a detailed derivation of all the steps in the finite element formulation are given for a one-dimensional heat flow problem: direct approach, strong and weak formulations, approximating functions and weighted residual methods. More advanced problems are gradually added to this basic knowledge, such as, field problems and solid mechanics problems. Field problems that are studied: heat flow, groundwater flow and Saint Venant torsion. Solid mechanics problems that are studied: stresses and strains, 2 and 3D elasticity, beams and plates. At the end of the course the theory for isoparametric finite elements and numerical integration is introduced. The design assignments illustrate the procedure of transferring a design problem into a model suitable for finite element analysis.

**Literature**

Ottosen, N. & Petersson, H.: Introduction to the Finite Element Method. Prentice Hall, 1992. Olsson, K.-G and Heyden S.: Introduction to the finite element method, problems. Structural Mechanics, Lund 2001. CALFEM ver 3.4 - A finite element toolbox, KFS i Lund AB, 2004.