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## Syllabus for

MHA021 - Finite element method (FEM) Finit elementmetod (FEM)

Owner: TKMAS
7,5 Credits
Grading: TH - Five, Four, Three, Fail
Education cycle: First-cycle
Major subject: Mechanical Engineering, Civil and Environmental Engineering, Shipping and Marine Technology
Department: 30 - MECHANICS AND MARITIME SCIENCES

Teaching language: Swedish
Application code: 55136
Open for exchange students: No
Block schedule: A+
Only students with the course round in the programme plan

 Module Credit distribution Examination dates Sp1 Sp2 Sp3 Sp4 Summer course No Sp 0197 Examination 7,5c Grading: TH 7,5c 13 Jan 2020 pm SB_MU 08 Apr 2020 pm DIST 26 Aug 2020 pm J

#### In programs

TKAUT AUTOMATION AND MECHATRONICS ENGINEERING, Year 3 (elective)
TKMAS MECHANICAL ENGINEERING, Year 3 (elective)

Peter Möller

#### Eligibility:

In order to be eligible for a first cycle course the applicant needs to fulfil the general and specific entry requirements of the programme(s) that has the course included in the study programme.

#### Course specific prerequisites

No formal requirements; however, MATLAB is extensively used, so the student is expected to have a basic knowledge in how to use this software.
It is also beneficial to have some basic knowledge in mechanics and strength of materials, e.g. to be familiar with concepts such as stress, strain, Hookes law, equilibrium and the like.
Some basic knowledge in mathematics and linear algebra will also be necessary; e.g. rules for integration, derivatives, Taylor series, ordinary and partial differential equations, algebra of matrices.

#### 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 variational problem that has the same solution as the original boundary value problem and, using test functions according to the Galerkin method,
• Derive a FE formulation from the variational formulation.
• 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.
• Derive integration weights and integration point coordinates for Gauss quadrature schemes, and show the accuracy of an N point scheme.
• 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.
• Construct a computer code 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.
• Prove Galerkin orthogonality and that the energy in the error equals the error in energy.
• Describe adaptivity with emphasis on a posteriori error estimation and method to modify the FE approximation so as to be able to reduce the discretization error.
• Describe types of error sources and give examples of the different types, when a physical problem is cast into a mathematical model whose solution subsequently is approximated.
• Describe direct solution of large sparse systems of equations and explain how the node numbering will affect the computational time as well as the required amount of storage.
• Describe iterative solution of large sparse systems of equations and relate such method to the minimization problem.
• Construct computer codes that solves any of the treated problem types by a FE method, and use the code to solve given examples.

#### Content

Finte element methods are used to approximate solutions to partial differential equations. Here we concentrate on some of the most common problem types in engineering mechanics, such as stationary field problems (e.g. heat conduction and the Prandtl stress function) and linear elasticity (including bending of beams). 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 variational problems (e.g. the principle of virtual work) 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, solotion of large spase systems of equations, convergens, error estimation, and adaptivity.

#### Organisation

The course embrace a series of lectures and computer exercises.
Approximately 20 lectures are given. These embrace a heoretical description of the finite element method and related numerical techniques. We also solve a few numerical examples in order to illustrate some of the material. At the computer exercises we use MATLAB and a toolbox (CALFEM) to construct our own FE-programs. Four of the exercises embrace assignments that should be solved and reported; three of the assignments should result in a brief written report, while the solution of one is to be demonstarted at a computer terminal. Four hours each week, teachers are available in the computer lab to assist in the work with the assignments.
Guest speakers are invited to describe and show how the finite element method is used in an industrial environment.

#### Literature

N Ottosen & H Petersson: "Introduction to the Finite Element Method", Prentice Hall, New York, 1992.
CALFEM - A Finite Element Toolbox to MATLAB V3.3, Division of Structural Mechanics and the Department of Solid Mechanics, Lund University, 1999. (Available as a pdf-document on the web).
P W Möller: "Error Estimation and Adaptivity in the Finite Element Method", Publication U73, Department of Applied Mechanics, Chalmers, 1998.

#### Examination including compulsory elements

Written examination and 4 computer assignments.