Course syllabus

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For general course information see Chalmers homepage: Course syllabus for Finite element method (FEM) 

1. Contact information

Lecturer and examiner

Teaching assistants

    Calling name is underlined.

 

2. Aim

Mathematical modeling of phenomena studied in science and engineering frequently leads to partial differential equations or boundary value problems. The finite element method (FEM) is a powerful tool to obtain approximate solutions to such equations and has become a standard tool in analysis, design, and simulation for many engineers.

The course aims to show the theory behind FEM and how the method is used to solve some of the most common problems in mechanical engineering. The course content includes both deriving finite element equations as well as implementing these in a programming language. Furthermore, the course also gives some insight into modern computational mechanics and shows examples of how FEM is used in industrial applications.

Finally, the course gives a firm basis for continued studies in advanced FEM (for example methods for transient and nonlinear problems) and related topics such as advanced solid mechanics, continuum mechanics, structural mechanics/dynamics, etc.

 

3. Literature

Main literature is in the form of lecture notes but there is one reference:

  • N Ottosen & H Petersson: Introduction to the Finite Element Method, Prentice Hall, New York, 1992. Some copies are available at STORE, otherwise, one can order it online.
    • NOTE: This is only a recommended book, but is not allowed during the exam and is not considered mandatory.

 

4. Organization 

  • The learning activities consist of lectures (~35 h),  tutorials (~12 h), and consultations (~20  h) in computer rooms.
  • Lectures:
    • The lectures cover the theory of the finite element method and related numerical techniques. During the lectures, numerical examples are also solved to illustrate some of the course material.
  • Tutorials:
    • Most weeks there is also a tutorial class focusing on problem-solving and numerical implementation.
  • Computer labs
    • Each week, there is a four hour computer lab during which the students work on the hand-in assignments with access to teaching assistants for asking questions about the projects.
  • Guest speakers have been invited to explain and show how the finite element method is used in industry.
  • CANVAS is used for:
    • Used for distributing all course material (lecture notes, project pm, etc.), submitting assignments, receiving feedback on assignments, posting of scanned exams, posing questions to the discussion board 

4.1 Schedule

The general schedule is found through TimeEdit

4.2 Changes made since the last occasion

  • Course lectures have been given jointly with VSM167 for a number of years. This year the courses are given independently and all lectures are given by Jim.
  • Introduction of sub-tasks of the hand-ins using AI.
  • Modification of exam structure: 
    • 5 smaller problems instead of 3 large ones.
  • Change in points system for hand-ins and exam.
  • Removed the Python package CALFEM. 
  • Change in order of course content: there is less focus on heat transfer and it comes in the end of the course.
  • Added an introduction to structural dynamics. 

4.3 Quality work

As an examiner, I always strive to improve the course and a natural part of this is receiving feedback from the class. Here it is important to have course representatives which can act as an neutral link between us teachers and you students. We will meet during a mid-course evaluation meeting and also after the course. 

This years representatives and contact information can be found on the page Student representatives.  

 

5. Examination

The course has mandatory hand-in projects as well as a final written exam. Below are details and how the course grade is determined.

5.1 Projects

The main course work (how you will spend your time) consists of three mandatory projects. The projects involve deriving equations and implementing these in Python as well as using the industrial FE-software ANSYS. 

  • A maximum of 5 points can be obtained per assignment. Altogether, 15 points can thus be obtained towards the final grade (see 5.3 below).
  • The projects are to be solved in groups of 1-2 students where both students should contribute equally in solving the tasks.
  • A written report is submitted for each project via Canvas by each deadline.
  • Each submission must contain a full solution to the project otherwise it will be returned without being graded. 
  • Make sure you follow the guidelines for the report, otherwise this may lead to deduction in points.
  • For each project, one resubmission of the report is allowed without a point penalty. Each consecutive submission will lead to a point penalty.
  • All resubmissions must be done within 5 working days after receiving feedback and no reports will be graded after Jan 15, 2026.
  • The project points remain valid until the course is given next time (exam + 2 re-exams). 

 

5.2 Written exam

The final written exam takes place in a computer room in on Thursday, 15 January at 8:30. During the exam, you will have access to Python (but no internet) to carry out computations. 

  • The exam can maximum give 30 points (5 tasks x 6 points).
  • The exam consists of theory and short programming tasks.
  • Allowed aids:
    • Chalmers type approved pocket calculator
    • Course formula sheet (also available digitally on the computers)
    • Mathematical handbook (e.g. Beta)

 

5.3 Course grade

The final course grade is determined by combination of the points obtained from the written exam (max 30) and from the projects (max 15) according to the table below:

Grades are awarded as follows:

Required total points

Grade

<25

U

25

3

31

4

37

5

5.4 Academic honesty and use of AI

See the dedicated page for details about what is ok and not: Academic honesty and AI

Additional information on recommended use of AI will be posted during the course.

 

6. Learning objectives

Note: a more comprehensive list, suitable for studying, will be made available during the course.

The course treats primarily linear stationary problems with applications on field equations (such as e.g. heat conduction, torsion of prismatic members, deflection of membranes, porous media flow, etc.), theory of elasticity, and beam bending. For each of these problems and subsequent to the course, the student should be able to:

  • Derive a weak form that has the same solution as the original boundary value problem.
  • Derive a FE formulation from the weak formulation, using test functions according to the Galerkin method,
  • Explain how different types of boundary conditions affect the variational formulation as well as the FE formulation, and show how the different types of conditions are approximated.
  • Show how the FE approximation is constructed when a problem involves one or more unknown functions and show how to obtain a sufficient number of equations to solve for the unknown variables in the approximation.
  • Derive expressions for element stiffness matrices and element load vectors and explain how these are assembled to structure stuffiness and structure loads.
  • Use numerical integration (Gauss quadrature schemes) to compute stiffness matrices and load vectors. 
  • Describe benefits and drawbacks with so-called reduced integration.
  • Conclude the suitable number of integration points for a given element type.
  • Derive expressions for element stiffness matrices using isoparametric mapping, and describe restrictions on element geometries in such a context.
  • Implement, in a given programming language, a function that uses numerical integration to obtain the element stiffness matrix of an isoparametric element.
  • Describe the conditions a FE approximation have to fulfil in order to be certain to obtain convergence, and give physical interpretations of these conditions; be able to distinguish between sufficient conditions on one hand, and necessary conditions on the other.
  • Describe convergence and rate of convergence and how the rate is affected by the type of element approximation and the presence of singularities in the exact solution.
  • Describe situations that give rise to singularities and find out how to best construct a FE approximation in these cases.
  • Formulate a minimization problem that has the same solution as a given boundary value problem, and show that the minimization problem has a unique solution.
  • Prove that FEM minimizes the potential energy (or corresponding quantity) and prove that a conform FE approximation yields higher energy that the exact solution.
  • List different sources of errors, and give examples, when a physical problem is described ny a mathematical model and then approximated.
  • Construct computer codes that solves any of the treated problem types using FEM, and use the code to solve given examples.
  • Describe the how an industrial FE-software is structured.
  • Use an industrial FE-software to solve problems covered in the course.
  • Introduce time dependent FE approximations and in particular apply it the equations of motion to derive the equations of elastodynamics.
  • Derive, compute and solve the equations governing eigenfrequency analysis.

 

7. Prerequisites

  • Basic programming skills in Python.
  • Knowledge in mechanics and strength of materials:
    • be familiar with concepts such as stress, strain, Hooke's law, equilibrium and related concepts.
  • Knowledge in mathematics and linear algebra:
    • rules for integration, derivatives, Taylor series, ordinary and partial differential equations, matrix algebra.

 

Course summary:

Course Summary
Date Details Due