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. It is for this reason this course is a prerequisite for the masters program Computational Mechanics.

 

3. Literature and software

Main literature is in the form of lecture notes but there is one reference that is good if you plan on continuing with studies in FEM:

  • 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. 

Softwares used during the course are:

  • Python and some standard packages, together with VS code as and IDE.
  • For Finite element analyssis we will use ANSYS which has a free student version that can be downloaded from https://ansys.synopsys.com/academic/students note that it is only available for PC.
  • Ansys is also available through Chalmers Apps where you should use version 2024. There are bugs with the later versions on Chalmers Apps causing many problems with crashes and corrupt files - please do not use version 25 or 26! If you download and install it locally on your own computer, later versions are fine.  

 

4. Organization 

4.1 Activities

  • The learning activities consist of lectures (~35 h),  tutorials (~12 h), and consultations (~20  h) in computer rooms. Separate bookings for oral presentations for the project assignments.
  • 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, numerical implementation and .
  • Computer labs
    • Each week, there are 2x 2 hour computer labs, during which you can work on the project assignments with access to teaching assistants for asking questions about the projects.
  • Guest speakers from industry will talk about how finite element simulations are used.
  • CANVAS is used for:
    • Distributing all course material (lecture notes, project pm, etc.)
    • submitting assignments,
    • receiving feedback on assignments,
    • posting questions to the discussion board.

Schedule: The general schedule is found through TimeEdit

4.2 Changes made since the last occasion

  • Formally this is a new course: MHA021 -> MHA022 but in practice much is the same. Some of the changes are:
    • The hand-in assignments is a separate course module and will be examined through report and oral examination
    • There will be more hands-on practice doing Finite Element Analysis
    • Some updated learning outcomes 

 

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 are:

Please contact one of your representatives if you would like to convey information to the examiner (me Jim) or for any reason would prefer to stay anonymous. For more information on what it means to be a student representative see the link Being a Student representative

If you are not listed above but would like to volunteer as a student representative you are more than welcome! Please contact me at jim.brouzoulis@chalmers.se and I will add you (must be done the first study week).

 

5. Examination

The course has two modules:

  • Hand-in projects - 3.0 credits (grade: pass/fail)
  • Written exam - 4.5 credits (grade: fail/3/4/5)

Below are details and how the course grade is determined.

5.1 Projects - 3 credits

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

  • The projects are to be solved in groups of 1-2 students where both students should contribute equally in solving the tasks.
  • Written report
    • 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. 
    • There are report guidelines that must be followed.
    • A report that has been marked as passed makes the group elligable for oral presentation of the project.
  • Oral presentation
    • There will be specific bookings for the presentations. 
    • Each project is examined orally (~20 min/group) with two teaching assistants present which will ask questions.
    • Each group has two opportunities to present each project (1 + 1 extra if you fail the first attempt)
      • Note: If the report has been handed in after the deadline there will be only one opportunity to present.
    • Groups which do not pass the oral presentation will have to redo the assignments next year.  

 

5.2 Written exam - 4.5 credits

The final written exam takes place in a computer room on Thursday, January 14 at 8:30. During the exam, you will have access to Python (but no internet and no AI-support) 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)
    • Reference book in Python programming

 

5.3 Course grade

The final course grade is determined by the written exam (max 30) as shown below.

Grade distribution table:

Required total points

Grade

0 - 11.5

U

12 - 17.5

3

18 - 23.5

4

24-30

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 fulfill in order to prove 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.
    • Have knowledge about structural elements like bars, shafts and beams, as well as compute sectional forces and stresses.
  • 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