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Advanced Finite Elements (MA5337)

Exact representation of curved geometries, treatment of shocks in the solution of a PDE or coupled systems of PDEs with different numerical requirements for each variable are just a few examples, where classical finite elements are not the best choice to solve PDEs numerically. In this course we will discuss a variety of advanced finite element techniques ranging from isogeometric analysis over possibly non-conforming or H(div), H(curl) elements to mixed formulations. We will also put a focus on non-linear PDEs and dynamical problems, i.e. PDEs involving a time derivative and how to couple techniques from ODE-numerics like time-integrators with the spatial discretization via finite elements. All these approaches are accompanied by applications e.g. from elasticity, fluid- or wave-mechanics also involving implementation. We may also cover parameter dependent PDEs for example in the context of reduced basis methods, variational inequalities as they arrise e.g. in contact mechanics or some simple examples of parameter estimation/inverse problems or shape optimization, where the finite element techniques are used to solve sub-problems of the (huge) optimization problem and hence are embedded into a broader applicational context.

Flow.jpg WavePool.jpg Elasticity.jpg

This lecture is a superset of the lecture Theory and Numerics of Finite Elements (MA4303), which means that after successful completion it can be accounted in place of (MA4303), e.g. for the "Numerical Mathematics" block of the M.Sc. Mathematics in Science and Engineering.

Lecture Prof. Dr. B. Wohlmuth
Tuesday 12:00-14:00 in MI 03.10.011
Office hours by arrangement (eMail)
Exercises M.Sc. Markus Muhr
Tuesday 14:00-16:00 in MI 03.10.011
Programming
Tutorials
M.Sc. Markus Muhr
Thursday 10:00-14:00 in MI 02.04.011
Supplementary
Exercise hour
M.Sc. Markus Muhr
Thursday 08:30-10:00 in MI 02.04.011
Requirements Required:
- Numerik gewöhnlicher Differentialgleichungen (MA2304) or a similar course. You should be familiar with numerical integration and time stepping schemes.
- Numerical Methods for Partial Differential Equations (MA 3303) or a similar course. You should be familiar with the concepts of finite elements.
- (MATLAB)-Programming skills (MA8003) Just basic MATLAB. For all the special toolboxes etc. we will give introductions.

Helpful but not necessary:
- Functional Analysis (MA3001)
- Nonlinear Optimization: Advanced (MA 3503) We will use the concept of Lagrange multipliers in some of our settings, hence some finite dimensional experience with them might be helpful.
- Basic knowledge in continuum mechanics (fluid mechanics and elasticity) might help, but also here we can give a short introduction to the respective equations.
- Further programming languages like C++/Python are a plus, but even if not, you can do "learning by doing". As long as you have any experience in programming (like MATLAB) you only have to adapt to a new syntax.
TUMOnline Link to TUMOnline entry

General Information

Announcements

Examination

Content

Lecture

Slides


Exercises

Every week we will have a 90 min. exercise session, where the current exercise sheet will be discussed. These exercises will mostly be theoretical ones dealing with the mathematical content of the lecture like definitions, theorems and proofs. A solution sheet will be uploaded after each exercise.


Programming Tutorials

Also we will have 90 min. programming tutorials, where coding examples of the methods from the lecture are implemented. Attendance is not mandatory, however the content of these programming exercises is definitely relevant for the final exam! Solution code-files will be uploaded after each tutorial.


Supplementary exercise hour

We will have a supplementary exercise hour of 45 min. where you can ask questions about either the content of the lecture, the implementation or any exercises.

Software

MATLAB

FEniCS

Since it is always quite a mess to get FEniCS to run on each computer/laptop etc. please prepare your installation already in advance of the first programming tutorial so that you can directly start and do not have to download and install large files with the slow university WLAN. The following points might be helpful for the installation but they are certainly not a complete guide on how to install FEniCS on each operating system or machine.

General hints

A collection of commands that might work

FEniCS Box

Test your installation

Paraview

Literature

DiscontinuousGalerkin.jpg HughesBook.jpg Brezzi.jpg Ern.jpg Quarteroni.jpg Kaltenbacher.jpg ReducedBasis.jpg CertifiedReducedBasis.jpg VariationalInequality.jpg
Daniele Antonio Di Pietro,
Alexandre Ern
xxxxxxxxxxxxxxxxx
J. Austin Cottrell; Thomas J.R.
Hughes; Yuri Bazilevs
xxxxxxxxxxxxx
D. Boffi, F. Brezzi, M. Fortin xxxxx A. Ern, J. L. Guermond xxxxxxxx A. Quarteroni, A. Valli xxxxxxxx M. Kaltenbacher Quarteroni, Manzoni,
Negri
xxxxxxxxxxxxxxxxxxxxxxx
Hesthaven, Rozza,
Stamm
R. Glowinski
Mathematical Aspects of
Discontinuous Galerkin Methods
Isogeometric Analysis:
Toward Integration of CAD and FEA
Mixed Finite Element Methods
and Applications
Theory and practice of finite
elements
Numerical Approximation of
Partial Differential Equations
Numerical Simulation of
Mechatronic Sensors and Actuators
Reduced Basis Methods for
Partial Differential Equations
Certified Reduced Basis Methods
for Parametrized PDEs
Numerical Methods for
Nonlinear Variational Problems
Springer, 2012 Wiley, 2009 Springer, 2013 Springer, 2004 Springer, 1994 Springer, 2007 Springer, 2015 Springer, 2015 Springer, 1984
(eBook version available) (Not the book, but a summary Pfeil) (eBook version available) (eBook version available) (eBook version available)   (eBook version available) (eBook version available) (eBook version available)

The available eBook versions can be downloaded over your TUM account at Springer link Pfeil. Log in via Shibboleth (not Athens !) by choosing the TUM as your institution and entering your TUMonline username and password. After that, navigate to the book and klick the blue "Download Book"-button and you should receive the complete book as a pdf file (for some books you also have to download the single chapters individually but the login process should be the same).


For questions, suggestions or complaints please write an eMail to Markus Muhr.

-- MarkusMuhr - 04 Jul 2017