Course syllabus for Finite element method (FEM)

Course syllabus adopted 2025-02-04 by Head of Programme (or corresponding).

Overview

  • Swedish nameFinit elementmetod (FEM)
  • CodeMHA021
  • Credits7.5 Credits
  • OwnerTKMAS
  • Education cycleFirst-cycle
  • Main field of studyMechanical Engineering, Civil and Environmental Engineering, Shipping and Marine Technology
  • DepartmentMECHANICS AND MARITIME SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

  • Teaching language English
  • Application code 55119
  • Open for exchange studentsYes

Credit distribution

0197 Examination 7.5 c
Grading: TH		 7.5 c

In programmes

Examiner

Eligibility

General entry requirements for bachelor's level (first cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Specific entry requirements

The same as for the programme that owns the course.
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

Course specific prerequisites

Basic programming skills in Python. It is also expected to have knowledge in mechanics and strength of materials, e.g. to be familiar with concepts such as stress, strain, Hooke's law, equilibrium and related concepts. Some knowledge in mathematics and linear algebra is also necessary; e.g. rules for integration, derivatives, Taylor series, ordinary and partial differential equations, matrix algebra.

Aim

Mathematical modelling of phenomena studied in science and engineering frequently lead to integral equations or boundary value problems. The finite element method (FEM) is a powerful tool to obtain approximate solutions to such equations and, thus, a foundation for computational engineering; the method has become a standard tool in analysis, design and simulation. Our primary aim is therefore to show how and why FEM works as well as to show how use the method to solve some of the most common problems in mechanical engineering and physics. The course content is such that the participants should be able to program their own FEM codes using some high level programming language. Furthermore, the course should give some insight into modern computational mechanics and show examples on how FEM is used in industrial applications. Finally, a firm basis for studies in advanced FEM (such as e.g. methods for transient and non linear problems) and related topics (e.g. advanced solid mechanics, continuum mechanics, structural mechanics/dynamics) should be provided.

Learning outcomes (after completion of the course the student should be able to)

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.

Content

Finte element methods are used to approximate solutions to partial differential equations. In the course we concentrate on some of the most common problem types in engineering mechanics, such as stationary field problems (e.g. heat conduction) and linear elasticity (torsion, plates, bending). The mathematical modelling of physical problems (i.e. the derivation of the governing differential equations and appropriate boundary conditions) are only briefly described. Instead we focus on how the respective boundary value problems may be formulated as weak forms or minimization problems (e.g. the the principle of minimum potential energy), and on how finite element methods approximate the solutions of these latter formulations. We also treat various numerical methods and techniques that are common in this context: numerical integration, mappings, substitution of variables, element approximations, solutions to systems of equations, and convergence. The course also contains learning activities where relevant problems are solved using industrial FE-software. Time dependent approximations and mass matrices. Eigenfrequency analysis.

Organisation

The course embrace a series of lectures and computer exercises. The lectures, approximately 15, cover the theory. We also solve a few numerical examples in order to illustrate some of the material. During the computer labs, Python is used to construct your own finite element programs. Supervision of the computer labs consists of two sessions each week. Guest lecturers are invited to talk about how FE computations are used in industry.

Literature

Recommended literature (not allowed on the exam): N Ottosen & H Petersson: "Introduction to the Finite Element Method", Prentice Hall, New York, 1992. 

Examination including compulsory elements

Hand-in assignments (3) and written examination. Grading: TH

The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers about disability study support.