The Finite Element Method for Problems in Physics

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About this course: 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 th…

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When you enroll for courses through Coursera you get to choose for a paid plan or for a free plan

  • Free plan: No certicification and/or audit only. You will have access to all course materials except graded items.
  • Paid plan: Commit to earning a Certificate—it's a trusted, shareable way to showcase your new skills.

About this course: 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.

Who is this class for: This class is aimed at the beginning graduate student, or the well-prepared undergraduate in engineering, mathematics or the physical sciences. A working knowledge of linear algebra (matrix-vector manipulations) is needed. Some exposure to partial differential equations would be very helpful. Experience with programming is a must. This could be Matlab or a language such as Fortran, C or Python.

Created by:  University of Michigan
  • Taught by:  Krishna Garikipati, Ph.D., Professor of Mechanical Engineering, College of Engineering - Professor of Mathematics, College of Literature, Science and the Arts

Level Intermediate Commitment 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. Language English Hardware Req It would be ideal if you have your own notebook/desktop computer to install the open-access code. How To Pass Pass all graded assignments to complete the course. User Ratings 4.5 stars Average User Rating 4.5See what learners said Coursework

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Syllabus


WEEK 1


1
This unit is an introduction to a simple one-dimensional problem that can be solved by the finite element method.


11 videos, 2 readings expand


  1. Reading: Help us learn more about you!
  2. Video: 01.01. Introduction. Linear elliptic partial differential equations - I
  3. Video: 01.02. Introduction. Linear elliptic partial differential equations - II
  4. Video: 01.03. Boundary conditions
  5. Video: 01.04. Constitutive relations
  6. Video: 01.05. Strong form of the partial differential equation. Analytic solution
  7. Video: 01.06. Weak form of the partial differential equation - I
  8. Video: 01.07. Weak form of the partial differential equation - II
  9. Video: 01.08. Equivalence between the strong and weak forms
  10. Video: 01.08ct.1. Intro to C++ (running your code, basic structure, number types, vectors)
  11. Video: 01.08ct.2. Intro to C++ (conditional statements, “for” loops, scope)
  12. Video: 01.08ct.3. Intro to C++ (pointers, iterators)
  13. Reading: "Paper and pencil" practice assignment on strong and weak forms

Graded: Unit 1 Quiz

WEEK 2


2
In this unit you will be introduced to the approximate, or finite-dimensional, weak form for the one-dimensional problem.


14 videos expand


  1. Video: 02.01. The Galerkin, or finite-dimensional weak form
  2. Video: 02.01q. Response to a question
  3. Video: 02.02. Basic Hilbert spaces - I
  4. Video: 02.03. Basic Hilbert spaces - II
  5. Video: 02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation
  6. Video: 02.04q. Response to a question
  7. Video: 02.05. Basis functions - I
  8. Video: 02.06. Basis functions - II
  9. Video: 02.07. The bi-unit domain - I
  10. Video: 02.08. The bi-unit domain - II
  11. Video: 02.09. The finite dimensional weak form as a sum over element subdomains - I
  12. Video: 02.10. The finite dimensional weak form as a sum over element subdomains - II
  13. Video: 02.10ct.1. Intro to C++ (functions)
  14. Video: 02.10ct.2. Intro to C++ (C++ classes)

Graded: Unit 2 Quiz

WEEK 3


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.


14 videos expand


  1. Video: 03.01. The matrix-vector weak form - I - I
  2. Video: 03.02. The matrix-vector weak form - I - II
  3. Video: 03.03. The matrix-vector weak form - II - I
  4. Video: 03.04. The matrix-vector weak form - II - II
  5. Video: 03.05. The matrix-vector weak form - III - I
  6. Video: 03.06. The matrix-vector weak form - III - II
  7. Video: 03.06ct.1. Dealii.org, running deal.II on a virtual machine with Oracle VirtualBox
  8. Video: 03.06ct.2. Intro to AWS, using AWS on Windows
  9. Video: 03.06ct.2c. In-Video Correction
  10. Video: 03.06ct.3. Using AWS on Linux and Mac OS
  11. Video: 03.07. The final finite element equations in matrix-vector form - I
  12. Video: 03.08. The final finite element equations in matrix-vector form - II
  13. Video: 03.08q. Response to a question
  14. Video: 03.08ct. Coding assignment 1 (main1.cc, overview of C++ class in FEM1.h)

Graded: Unit 3 Quiz
Graded: Coding Assignment 1

WEEK 4


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.


17 videos expand


  1. Video: 04.01. The pure Dirichlet problem - I
  2. Video: 04.02. The pure Dirichlet problem - II
  3. Video: 04.02c. In-Video Correction
  4. Video: 04.03. Higher polynomial order basis functions - I
  5. Video: 04.03c0. In-Video Correction
  6. Video: 04.03c1. In-Video Correction
  7. Video: 04.04. Higher polynomial order basis functions - I - II
  8. Video: 04.05. Higher polynomial order basis functions - II - I
  9. Video: 04.06. Higher polynomial order basis functions - III
  10. Video: 04.06ct. Coding assignment 1 (functions: class constructor to “basis_gradient”)
  11. Video: 04.07. The matrix-vector equations for quadratic basis functions - I - I
  12. Video: 04.08. The matrix-vector equations for quadratic basis functions - I - II
  13. Video: 04.09. The matrix-vector equations for quadratic basis functions - II - I
  14. Video: 04.10. The matrix-vector equations for quadratic basis functions - II - II
  15. Video: 04.11. Numerical integration -- Gaussian quadrature
  16. Video: 04.11ct.1. Coding assignment 1 (functions: “generate_mesh” to “setup_system”)
  17. Video: 04.11ct.2. Coding assignment 1 (functions: “assemble_system”)

Graded: Unit 4 Quiz

WEEK 5


5
This unit outlines the mathematical analysis of the finite element method.


12 videos expand


  1. Video: 05.01. Norms - I
  2. Video: 05.01c. In-Video Correction
  3. Video: 05.01ct.1. Coding assignment 1 (functions: “solve” to “l2norm_of_error”)
  4. Video: 05.01ct.2. Visualization tools
  5. Video: 05.02. Norms - II
  6. Video: 05.02. Response to a question
  7. Video: 05.03. Consistency of the finite element method
  8. Video: 05.04. The best approximation property
  9. Video: 05.05. The "Pythagorean Theorem"
  10. Video: 05.05q. Response to a question
  11. Video: 05.06. Sobolev estimates and convergence of the finite element method
  12. Video: 05.07. Finite element error estimates

Graded: Unit 5 Quiz

WEEK 6


6
This unit develops an alternate derivation of the weak form, which is applicable to certain physical problems.


4 videos expand


  1. Video: 06.01. Functionals. Free energy - I
  2. Video: 06.02. Functionals. Free energy - II
  3. Video: 06.03. Extremization of functionals
  4. Video: 06.04. Derivation of the weak form using a variational principle

Graded: Unit 6 Quiz

WEEK 7


7
In this unit, we develop the finite element method for three-dimensional scalar problems, such as the heat conduction or mass diffusion problems.


24 videos expand


  1. Video: 07.01. The strong form of steady state heat conduction and mass diffusion - I
  2. Video: 07.02. The strong form of steady state heat conduction and mass diffusion - II
  3. Video: 07.02q. Response to a question
  4. Video: 07.03. The strong form, continued
  5. Video: 07.03c. In-Video Correction
  6. Video: 07.04. The weak form
  7. Video: 07.05. The finite-dimensional weak form - I
  8. Video: 07.06. The finite-dimensional weak form - II
  9. Video: 07.07. Three-dimensional hexahedral finite elements
  10. Video: 07.08. Aside: Insight to the basis functions by considering the two-dimensional case
  11. Video: 07.08c In-Video Correction
  12. Video: 07.09. Field derivatives. The Jacobian - I
  13. Video: 07.10. Field derivatives. The Jacobian - II
  14. Video: 07.11. The integrals in terms of degrees of freedom
  15. Video: 07.12. The integrals in terms of degrees of freedom - continued
  16. Video: 07.13. The matrix-vector weak form - I
  17. Video: 07.14. The matrix-vector weak form II
  18. Video: 07.15.The matrix-vector weak form, continued - I
  19. Video: 07.15c. In-Video Correction
  20. Video: 07.16. The matrix-vector weak form, continued - II
  21. Video: 07.17. The matrix vector weak form, continued further - I
  22. Video: 07.17c. In-Video Correction
  23. Video: 07.18. The matrix-vector weak form, continued further - II
  24. Video: 07.18c. In-Video Correction

Graded: Unit 7 Quiz

WEEK 8


8
In this unit, you will complete some details of the three-dimensional formulation that depend on the choice of basis functions, as well as be introduced to the second coding assignment.


9 videos expand


  1. Video: 08.01. Lagrange basis functions in 1 through 3 dimensions - I
  2. Video: 08.01c. In-Video Correction
  3. Video: 08.02. Lagrange basis functions in 1 through 3 dimensions - II
  4. Video: 08.02ct. Coding assignment 2 (2D problem) - I
  5. Video: 08.03. Quadrature rules in 1 through 3 dimensions
  6. Video: 08.03ct.1. Coding assignment 2 (2D problem) - II
  7. Video: 08.03ct.2. Coding assignment 2 (3D problem)
  8. Video: 08.04. Triangular and tetrahedral elements - Linears - I
  9. Video: 08.05. Triangular and tetrahedral elements - Linears - II

Graded: Unit 8 Quiz
Graded: Coding Assignment 2

WEEK 9


9
In this unit, we take a detour to study the two-dimensional formulation for scalar problems, such as the steady state heat or diffusion equations.


6 videos expand


  1. Video: 09.01. The finite-dimensional weak form and basis functions - I
  2. Video: 09.02. The finite-dimensional weak form and basis functions - II
  3. Video: 09.03. The matrix-vector weak form
  4. Video: 09.03c. In-Video Correction
  5. Video: 09.04. The matrix-vector weak form - II
  6. Video: 09.04c. In-Video Correction

Graded: Unit 9 Quiz

WEEK 10


10
This unit introduces the problem of three-dimensional, linearized elasticity at steady state, and also develops the finite element method for this problem. Aspects of the code templates are also examined.


22 videos expand


  1. Video: 10.01. The strong form of linearized elasticity in three dimensions - I
  2. Video: 10.02. The strong form of linearized elasticity in three dimensions - II
  3. Video: 10.02c. In-Video Correction
  4. Video: 10.03. The strong form, continued
  5. Video: 10.04. The constitutive relations of linearized elasticity
  6. Video: 10.05. The weak form - I
  7. Video: 10.05q. Response to a question
  8. Video: 10.06. The weak form - II
  9. Video: 10.07. The finite-dimensional weak form - Basis functions - I
  10. Video: 10.08. The finite-dimensional weak form - Basis functions - II
  11. Video: 10.09. Element integrals - I
  12. Video: 10.09c. In-Video Correction
  13. Video: 10.10. Element integrals - II
  14. Video: 10.11. The matrix-vector weak form - I
  15. Video: 10.12. The matrix-vector weak form - II
  16. Video: 10.13. Assembly of the global matrix-vector equations - I
  17. Video: 10.14. Assembly of the global matrix-vector equations - II
  18. Video: 10.14c. In Video Correction
  19. Video: 10.14ct.1. Coding assignment 3 - I
  20. Video: 10.14ct.2. Coding assignment 3 - II
  21. Video: 10.15. Dirichlet boundary conditions - I
  22. Video: 10.16. Dirichlet boundary conditions - II

Graded: Unit 10 Quiz
Graded: Coding Assignment 3

WEEK 11


11
In this unit, we study the unsteady heat conduction, or mass diffusion, problem, as well as its finite element formulation.


27 videos expand


  1. Video: 11.01. The strong form
  2. Video: 11.01c In-Video Correction
  3. Video: 11.02. The weak form, and finite-dimensional weak form - I
  4. Video: 11.03. The weak form, and finite-dimensional weak form - II
  5. Video: 11.04. Basis functions, and the matrix-vector weak form - I
  6. Video: 11.04c In-Video Correction
  7. Video: 11.05. Basis functions, and the matrix-vector weak form - II
  8. Video: 11.05. Response to a question
  9. Video: 11.06. Dirichlet boundary conditions; the final matrix-vector equations
  10. Video: 11.07. Time discretization; the Euler family - I
  11. Video: 11.08. Time discretization; the Euler family - II
  12. Video: 11.09. The v-form and d-form
  13. Video: 11.09ct.1. Coding assignment 4 - I
  14. Video: 11.09ct.2. Coding assignment 4 - II
  15. Video: 11.10. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - I
  16. Video: 11.11. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - II
  17. Video: 11.11c. In-Video Correction
  18. Video: 11.12. Modal decomposition and modal equations - I
  19. Video: 11.13. Modal decomposition and modal equations - II
  20. Video: 11.14. Modal equations and stability of the time-exact single degree of freedom systems - I
  21. Video: 11.15. Modal equations and stability of the time-exact single degree of freedom systems - II
  22. Video: 11.15q. Response to a question
  23. Video: 11.16. Stability of the time-discrete single degree of freedom systems
  24. Video: 11.17. Behavior of higher-order modes; consistency - I
  25. Video: 11.18. Behavior of higher-order modes; consistency - II
  26. Video: 11.19. Convergence - I
  27. Video: 11.20. Convergence - II

Graded: Unit 11 Quiz
Graded: Coding Assignment 4

WEEK 12


12
In this unit we study the problem of elastodynamics, and its finite element formulation.


9 videos expand


  1. Video: 12.01. The strong and weak forms
  2. Video: 12.02. The finite-dimensional and matrix-vector weak forms - I
  3. Video: 12.03. The finite-dimensional and matrix-vector weak forms - II
  4. Video: 12.04. The time-discretized equations
  5. Video: 12.05. Stability - I
  6. Video: 12.06. Stability - II
  7. Video: 12.07. Behavior of higher-order modes
  8. Video: 12.08. Convergence
  9. Video: 12.08c. In-Video Correction

Graded: Unit 12 Quiz

WEEK 13


113
This is a wrap-up, with suggestions for future study.


1 video, 1 reading expand


  1. Video: Conclusion, and the Road Ahead
  2. Reading: Post-course Survey

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