Об этом курсе
4.6
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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....
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Approx. 23 hours to complete

Предполагаемая нагрузка: You should expect to watch about 3 hours of video lectures a week. Apart from the lectures, expect to put in between 3 and 5 hours a week....
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Приобретаемые навыки

Finite DifferencesC++C Sharp (C#) (Programming Language)Matrices
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Только онлайн-курсы

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Calendar

Гибкие сроки

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Intermediate Level

Промежуточный уровень

Clock

Approx. 23 hours to complete

Предполагаемая нагрузка: You should expect to watch about 3 hours of video lectures a week. Apart from the lectures, expect to put in between 3 and 5 hours a week....
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English

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Программа курса: что вы изучите

Week
1
Clock
6 ч. на завершение

1

This unit is an introduction to a simple one-dimensional problem that can be solved by the finite element method....
Reading
11 видео (всего 200 мин.), 2 материалов для самостоятельного изучения, 1 тест
Video11 видео
01.01. Introduction. Linear elliptic partial differential equations - I 14мин
01.02. Introduction. Linear elliptic partial differential equations - II 13мин
01.03. Boundary conditions 22мин
01.04. Constitutive relations 20мин
01.05. Strong form of the partial differential equation. Analytic solution 22мин
01.06. Weak form of the partial differential equation - I 12мин
01.07. Weak form of the partial differential equation - II 15мин
01.08. Equivalence between the strong and weak forms 24мин
01.08ct.1. Intro to C++ (running your code, basic structure, number types, vectors) 21мин
01.08ct.2. Intro to C++ (conditional statements, “for” loops, scope) 19мин
01.08ct.3. Intro to C++ (pointers, iterators) 14мин
Reading2 материала для самостоятельного изучения
Help us learn more about you!10мин
"Paper and pencil" practice assignment on strong and weak formsмин
Quiz1 практическое упражнение
Unit 1 Quiz8мин
Week
2
Clock
3 ч. на завершение

2

In this unit you will be introduced to the approximate, or finite-dimensional, weak form for the one-dimensional problem....
Reading
14 видео (всего 202 мин.), 1 тест
Video14 видео
02.01. The Galerkin, or finite-dimensional weak form 23мин
02.01q. Response to a question 7мин
02.02. Basic Hilbert spaces - I 15мин
02.03. Basic Hilbert spaces - II 9мин
02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation 22мин
02.04q. Response to a question 6мин
02.05. Basis functions - I 14мин
02.06. Basis functions - II 14мин
02.07. The bi-unit domain - I 11мин
02.08. The bi-unit domain - II 16мин
02.09. The finite dimensional weak form as a sum over element subdomains - I 16мин
02.10. The finite dimensional weak form as a sum over element subdomains - II 12мин
02.10ct.1. Intro to C++ (functions) 13мин
02.10ct.2. Intro to C++ (C++ classes) 16мин
Quiz1 практическое упражнение
Unit 2 Quiz6мин
Week
3
Clock
7 ч. на завершение

3

In this unit, you will write the finite-dimensional weak form in a matrix-vector form. You also will be introduced to coding in the deal.ii framework....
Reading
14 видео (всего 213 мин.), 2 тестов
Video14 видео
03.01. The matrix-vector weak form - I - I 16мин
03.02. The matrix-vector weak form - I - II 17мин
03.03. The matrix-vector weak form - II - I 15мин
03.04. The matrix-vector weak form - II - II 13мин
03.05. The matrix-vector weak form - III - I 22мин
03.06. The matrix-vector weak form - III - II 13мин
03.06ct.1. Dealii.org, running deal.II on a virtual machine with Oracle VirtualBox12мин
03.06ct.2. Intro to AWS, using AWS on Windows24мин
03.06ct.2c. In-Video Correction3мин
03.06ct.3. Using AWS on Linux and Mac OS7мин
03.07. The final finite element equations in matrix-vector form - I 22мин
03.08. The final finite element equations in matrix-vector form - II 18мин
03.08q. Response to a question 4мин
03.08ct. Coding assignment 1 (main1.cc, overview of C++ class in FEM1.h) 19мин
Quiz1 практическое упражнение
Unit 3 Quiz6мин
Week
4
Clock
5 ч. на завершение

4

This unit develops further details on boundary conditions, higher-order basis functions, and numerical quadrature. You also will learn about the templates for the first coding assignment....
Reading
17 видео (всего 262 мин.), 1 тест
Video17 видео
04.01. The pure Dirichlet problem - I 18мин
04.02. The pure Dirichlet problem - II 17мин
04.02c. In-Video Correction 1мин
04.03. Higher polynomial order basis functions - I 23мин
04.03c0. In-Video Correction мин
04.03c1. In-Video Correction мин
04.04. Higher polynomial order basis functions - I - II 16мин
04.05. Higher polynomial order basis functions - II - I 13мин
04.06. Higher polynomial order basis functions - III 23мин
04.06ct. Coding assignment 1 (functions: class constructor to “basis_gradient”) 14мин
04.07. The matrix-vector equations for quadratic basis functions - I - I 21мин
04.08. The matrix-vector equations for quadratic basis functions - I - II 11мин
04.09. The matrix-vector equations for quadratic basis functions - II - I 19мин
04.10. The matrix-vector equations for quadratic basis functions - II - II 24мин
04.11. Numerical integration -- Gaussian quadrature 13мин
04.11ct.1. Coding assignment 1 (functions: “generate_mesh” to “setup_system”) 14мин
04.11ct.2. Coding assignment 1 (functions: “assemble_system”) 26мин
Quiz1 практическое упражнение
Unit 4 Quiz8мин
4.6

Лучшие рецензии

автор: SS•Mar 13th 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.

автор: YW•Jun 21st 2018

Great class! I truly hope that there are further materials on shell elements, non-linear analysis (geometric nonlinearity, plasticity and hyperelasticity).

Преподаватель

Krishna Garikipati, Ph.D.

Professor of Mechanical Engineering, College of Engineering - Professor of Mathematics, College of Literature, Science and the Arts

О University of Michigan

The mission of the University of Michigan is to serve the people of Michigan and the world through preeminence in creating, communicating, preserving and applying knowledge, art, and academic values, and in developing leaders and citizens who will challenge the present and enrich the future....

Часто задаваемые вопросы

  • Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.

  • When you purchase a Certificate you get access to all course materials, including graded assignments. Upon completing the course, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.

  • You will need computing resources sufficient to install the code and run it. Depending on the type of installation this could be between a 13MB download of a tarred and gzipped file, to 45MB for a serial MacOSX binary and 192MB for a parallel MacOSX binary. Additionally, you will need a specific visualization program that we recommend. Altogether, if you have 1GB you should be fine. Alternately, you could download a Virtual Machine Interface.

  • You will be able to write code that simulates some of the most beautiful problems in physics, and visualize that physics.

  • You will need to know about matrices and vectors. Having seen partial differential equations will be very helpful. The code is in C++, but you don't need to know C++ at the outset. We will point you to resources that will teach you enough C++ for this class. However, you will need to have done some programming (Matlab, Fortran, C, Python, C++ should all do).

  • Apart from the lectures, expect to put in between 5 and 10 hours a week.

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