Course syllabus

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Contact information

Lecturer and examiner

Co-lecturer and examiner for VSM167

Teaching assistants

    Calling name is underlined.

 

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.

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 and physics. The course content includes both deriving finite element equations as well as implementing these in some 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.

Schedule

TimeEdit

Also, see the detailed course schedule 

 

Literature

  • N Ottosen & H Petersson: Introduction to the Finite Element Method, Prentice Hall, New York, 1992. Some copies are available at STORE, otherwise, order it online. NOTE: This is a recommended book, but is not allowed during the exam and is therefore not considered mandatory.
  • CALFEM - A Finite Element Toolbox to MATLAB V3.4, Division of Structural Mechanics and the Department of Solid Mechanics, Lund University, 2004. (Available for download).

 

Organization 

  • The learning activities consist of about 35 h of lectures, 12 h of tutorials, and 20 h of consultations in computer rooms.
  • 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. Most weeks there is also a tutorial class focusing on problem-solving and numerical implementation.
  • Each week, there is a four hour computer lab during which the students work on the hand-in assignments with teachers available for consultation.
  • If possible, a guest speaker will be invited to describe and show how the finite element method is used in industry.

 

Changes made since the last occasion

  •  Ambition is to include more use of industrial FE-software as part of the projects. 

 

Examination

Projects

The main course work consists of one mandatory and two semi-mandatory projects. The computer projects involve FE-analysis using the Matlab toolbox CALFEM as well as the commercial FE-software COMSOL. The CALFEM package can be downloaded from the course homepage or from http://sourceforge.net/projects/calfem/Links to an external site.

The projects are to be solved in groups of maximum two students where both students should contribute equally in solving the tasks. Computer classes are scheduled where you will have access to teaching assistants for asking questions about the projects. Please note however, that we will not answer direct questions like “Is this correct?”. It is up to each individual group to make sure that the answers are reasonable.

A written report for each project must be submitted via Canvas by the deadline (see course schedule). Reports may be handwritten, but care should be taken so that reports are readable and understandable. In order to be able to submit the report, you need to sign-up for one of the project groups.

It is mandatory to fully complete the first project, which means that students will be required to update their reports until everything is correct. This must be done before the exam and will then render 4 credit points. For projects 2 and 3, a maximum of 4 credits can be obtained. Altogether, 12 credit points can thus be obtained towards the final grade, see below. These points will remain valid until the course is given next time. Please note that project reports will be graded without any possibility for modifications or improvements in a second stage.

Some remarks about academic honesty

Since the projects are a significant part of the examination, it is essential that each group solves the tasks separately. To clarify, it is OK to:

  • collaborate and discuss derivations asked for in different subtasks. HOWEVER, each group should in the end do the derivations themselves and hand in their own written solutions
  • discuss implementation strategies in MATLAB for different subtasks. HOWEVER, limit the discussions to pen and paper discussions otherwise it is far too easy to cross the line to plagiarism.

It is NOT OK to:

  • Directly copy other groups derivations and hand them in as your own. This applies also to codes from students from previous years.
  • Show or share any amount of MATLAB-code or COMSOL files between groups. 
  • Copy and hand-in code which is partly written by another group. This is true for code written by other students this year as well as from previous years.

Project reports will be analysed trough Urkund. Any suspected cheating (including things like copying MATLAB code or parts of someone else’s report) will be directly reported to the Disciplinary Committee, please refer to the collection of rules:

https://student.portal.chalmers.se/en/chalmersstudies/joint-rules-and-directives/Documents/20090920_Academic_Honesty.pdf

Please note that failure to comply with the rules may lead to warnings or in worst case time limited suspension form Chalmers.

Regarding the ambition level and quality of the project reports, imagine the current situation: The person correcting and grading the report is your manager at the consultancy company where you work. You have been given an FE assignment and when you hand in the results your manager should be able to send the report directly to the client without further modification.

Written exam

The final written exam takes place in a computer room in the morning on Thursday, 12 January at 8:30. During the exam, you will have access to MATLAB and CALFEM, which are tools that will be used throughout the course. Short programming tasks may therefore be part of the exam. The only additional aids that will be allowed are a Chalmers type approved pocket calculator and a formula sheet that will be appended to the exam. The exam will have questions/problems which altogether can give 18 credit points.

Course grade

The final course grade is given as a combination of the credit points obtained at the exam and from the projects according to the table below:

Grades are awarded as follows:

Required credit points

Chalmers grade

<15

U

15 (minimum 6 from the final exam)

3

20

4

25

5

Learning objectives

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 and, using test functions according to the Galerkin method,
  • Derive a global and local FE formulation from the weak formulation.
  • Explain how different types of boundary conditions affect the weak 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.
  • 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 to evaluate stiffness matrices and load vectors.
  • 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 must fulfill to guarantee convergence, and give physical interpretations of these conditions.
  • Construct computer codes that solve any of the treated problem types by a FE method, and use the code to solve given examples.
  • Use industrial FE software to solve problems covered in the course.

 

 

 

Course summary:

Date Details Due