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University of Michigan

The Finite Element Method for Problems in Physics

This course is an introduction to the finite element method as applicable to a range of problems in physics and engineering sciences. The treatment is mathematical, but only for the purpose of clarifying the formulation. The emphasis is on coding up the formulations in a modern, open-source environment that can be expanded to other applications, subsequently. The course includes about 45 hours of lectures covering the material I normally teach in an introductory graduate class at University of Michigan. The treatment is mathematical, which is natural for a topic whose roots lie deep in functional analysis and variational calculus. It is not formal, however, because the main goal of these lectures is to turn the viewer into a competent developer of finite element code. We do spend time in rudimentary functional analysis, and variational calculus, but this is only to highlight the mathematical basis for the methods, which in turn explains why they work so well. Much of the success of the Finite Element Method as a computational framework lies in the rigor of its mathematical foundation, and this needs to be appreciated, even if only in the elementary manner presented here. A background in PDEs and, more importantly, linear algebra, is assumed, although the viewer will find that we develop all the relevant ideas that are needed. The development itself focuses on the classical forms of partial differential equations (PDEs): elliptic, parabolic and hyperbolic. At each stage, however, we make numerous connections to the physical phenomena represented by the PDEs. For clarity we begin with elliptic PDEs in one dimension (linearized elasticity, steady state heat conduction and mass diffusion). We then move on to three dimensional elliptic PDEs in scalar unknowns (heat conduction and mass diffusion), before ending the treatment of elliptic PDEs with three dimensional problems in vector unknowns (linearized elasticity). Parabolic PDEs in three dimensions come next (unsteady heat conduction and mass diffusion), and the lectures end with hyperbolic PDEs in three dimensions (linear elastodynamics). Interspersed among the lectures are responses to questions that arose from a small group of graduate students and post-doctoral scholars who followed the lectures live. At suitable points in the lectures, we interrupt the mathematical development to lay out the code framework, which is entirely open source, and C++ based. Books: There are many books on finite element methods. This class does not have a required textbook. However, we do recommend the following books for more detailed and broader treatments than can be provided in any form of class: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, T.J.R. Hughes, Dover Publications, 2000. The Finite Element Method: Its Basis and Fundamentals, O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu, Butterworth-Heinemann, 2005. A First Course in Finite Elements, J. Fish and T. Belytschko, Wiley, 2007. Resources: You can download the deal.ii library at dealii.org. The lectures include coding tutorials where we list other resources that you can use if you are unable to install deal.ii on your own computer. You will need cmake to run deal.ii. It is available at cmake.org.

状态:Mathematical Modeling
状态:Mathematical Software
中级课程小时

精选评论

RD

5.0评论日期:Sep 4, 2020

Well worth the time if you wish to understand the mathematical origin of the FEM methods used in solving various physical situations such as heat/mass transfer and solid mechanics

S

5.0评论日期:Nov 22, 2020

Excellent course, subject matter is presented very methodically, and the instructor's command over the subject taught is outstanding.

AI

5.0评论日期:Mar 8, 2017

A rigorous and organized introduction to the subject with the additional benefit of learning through implementation.

SM

4.0评论日期:Nov 14, 2019

A good primer of the theoretical fundamentals of the Finite Element Methods. The coding assignments were good too but could have benefited more with support from the mentors via the forums.

GD

4.0评论日期:May 20, 2019

Well structured course. It builds up from the basics of finite elements to more complex problems.

VV

4.0评论日期:Dec 29, 2018

The course was was great. However, illustrative examples solving real engineering problems could be introduced in lecture.

IK

5.0评论日期:Jul 20, 2019

The course is great and the tutors are very helpful. I just have a suggestion that there should be more coding assignment like one for every week.Thank you

DV

5.0评论日期:Mar 23, 2021

Excelente para personas con conocimiento básico o nulo de elementos finitos. Super recomendado :)

MP

5.0评论日期:Aug 19, 2023

Hello, thank you for your efforts in the course. It was very useful and beautiful. I hope you will always be healthy and happy. Mohammad Pakzad from Iran.

AM

5.0评论日期:Feb 13, 2021

Perfect Course ! I recommend it to everyone who is interested in mathematics and physics.

SS

5.0评论日期:Mar 12, 2017

It is very well structured and Dr Krishna Garikipati helps me understand the course in very simple manner. I would like to thank coursera community for making this course available.

MA

5.0评论日期:Sep 7, 2020

This course is really didactic course. If you are interested in FEM, please take this course. Thank you Professor Krishna Garikipat.Sincerely....

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