The Finite Element Method for Problems in Physics
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The Finite Element Method for Problems in Physics at Coursera Overview
Duration | 61 hours |
Total fee | Free |
Mode of learning | Online |
Difficulty level | Intermediate |
Official Website | Explore Free Course  |
Credential | Certificate |
- Overview
- Highlights
- Course Details
- Curriculum
The Finite Element Method for Problems in Physics at Coursera Highlights
- Shareable Certificate Earn a Certificate upon completion
- 100% online Start instantly and learn at your own schedule.
- Flexible deadlines Reset deadlines in accordance to your schedule.
- Intermediate Level
- Approx. 61 hours to complete
- English Subtitles: Arabic, French, Portuguese (European), Italian, Vietnamese, German, Russian, English, Spanish
The Finite Element Method for Problems in Physics at Coursera Course details
- 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.
The Finite Element Method for Problems in Physics at Coursera Curriculum
1
01.01. Introduction. Linear elliptic partial differential equations - I
01.02. Introduction. Linear elliptic partial differential equations - II
01.03. Boundary conditions
01.04. Constitutive relations
01.05. Strong form of the partial differential equation. Analytic solution
01.06. Weak form of the partial differential equation - I
01.07. Weak form of the partial differential equation - II
01.08. Equivalence between the strong and weak forms
01.08ct.1. Intro to C++ (running your code, basic structure, number types, vectors)
01.08ct.2. Intro to C++ (conditional statements, ?for? loops, scope)
01.08ct.3. Intro to C++ (pointers, iterators)
Help us learn more about you!
"Paper and pencil" practice assignment on strong and weak forms
Unit 1 Quiz
2
02.01. The Galerkin, or finite-dimensional weak form
02.01q. Response to a question
02.02. Basic Hilbert spaces - I
02.03. Basic Hilbert spaces - II
02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation
02.04q. Response to a question
02.05. Basis functions - I
02.06. Basis functions - II
02.07. The bi-unit domain - I
02.08. The bi-unit domain - II
02.09. The finite dimensional weak form as a sum over element subdomains - I
02.10. The finite dimensional weak form as a sum over element subdomains - II
02.10ct.1. Intro to C++ (functions)
02.10ct.2. Intro to C++ (C++ classes)
Unit 2 Quiz
3
03.01. The matrix-vector weak form - I - I
03.02. The matrix-vector weak form - I - II
03.03. The matrix-vector weak form - II - I
03.04. The matrix-vector weak form - II - II
03.05. The matrix-vector weak form - III - I
03.06. The matrix-vector weak form - III - II
03.06ct.1. Dealii.org, running deal.II on a virtual machine with Oracle VirtualBox
03.06ct.2. Intro to AWS, using AWS on Windows
03.06ct.2c. In-Video Correction
03.06ct.3. Using AWS on Linux and Mac OS
03.07. The final finite element equations in matrix-vector form - I
03.08. The final finite element equations in matrix-vector form - II
03.08q. Response to a question
03.08ct. Coding assignment 1 (main1.cc, overview of C++ class in FEM1.h)
Unit 3 Quiz
4
04.01. The pure Dirichlet problem - I
04.02. The pure Dirichlet problem - II
04.02c. In-Video Correction
04.03. Higher polynomial order basis functions - I
04.03c0. In-Video Correction
04.03c1. In-Video Correction
04.04. Higher polynomial order basis functions - I - II
04.05. Higher polynomial order basis functions - II - I
04.06. Higher polynomial order basis functions - III
04.06ct. Coding assignment 1 (functions: class constructor to ?basis_gradient?)
04.07. The matrix-vector equations for quadratic basis functions - I - I
04.08. The matrix-vector equations for quadratic basis functions - I - II
04.09. The matrix-vector equations for quadratic basis functions - II - I
04.10. The matrix-vector equations for quadratic basis functions - II - II
04.11. Numerical integration -- Gaussian quadrature
04.11ct.1. Coding assignment 1 (functions: ?generate_mesh? to ?setup_system?)
04.11ct.2. Coding assignment 1 (functions: ?assemble_system?)
Unit 4 Quiz
5
05.01. Norms - I
05.01c. In-Video Correction
05.01ct.1. Coding assignment 1 (functions: ?solve? to ?l2norm_of_error?)
05.01ct.2. Visualization tools
05.02. Norms - II
05.02. Response to a question
05.03. Consistency of the finite element method
05.04. The best approximation property
05.05. The "Pythagorean Theorem"
05.05q. Response to a question
05.06. Sobolev estimates and convergence of the finite element method
05.07. Finite element error estimates
Unit 5 Quiz
6
06.01. Functionals. Free energy - I
06.02. Functionals. Free energy - II
06.03. Extremization of functionals
06.04. Derivation of the weak form using a variational principle
Unit 6 Quiz
7
07.01. The strong form of steady state heat conduction and mass diffusion - I
07.02. The strong form of steady state heat conduction and mass diffusion - II
07.02q. Response to a question
07.03. The strong form, continued
07.03c. In-Video Correction
07.04. The weak form
07.05. The finite-dimensional weak form - I
07.06. The finite-dimensional weak form - II
07.07. Three-dimensional hexahedral finite elements
07.08. Aside: Insight to the basis functions by considering the two-dimensional case
07.08c In-Video Correction
07.09. Field derivatives. The Jacobian - I
07.10. Field derivatives. The Jacobian - II
07.11. The integrals in terms of degrees of freedom
07.12. The integrals in terms of degrees of freedom - continued
07.13. The matrix-vector weak form - I
07.14. The matrix-vector weak form II
07.15.The matrix-vector weak form, continued - I
07.15c. In-Video Correction
07.16. The matrix-vector weak form, continued - II
07.17. The matrix vector weak form, continued further - I
07.17c. In-Video Correction
07.18. The matrix-vector weak form, continued further - II
07.18c. In-Video Correction
Unit 7 Quiz
8
08.01. Lagrange basis functions in 1 through 3 dimensions - I
08.01c. In-Video Correction
08.02. Lagrange basis functions in 1 through 3 dimensions - II
08.02ct. Coding assignment 2 (2D problem) - I
08.03. Quadrature rules in 1 through 3 dimensions
08.03ct.1. Coding assignment 2 (2D problem) - II
08.03ct.2. Coding assignment 2 (3D problem)
08.04. Triangular and tetrahedral elements - Linears - I
08.05. Triangular and tetrahedral elements - Linears - II
Unit 8 Quiz
9
09.01. The finite-dimensional weak form and basis functions - I
09.02. The finite-dimensional weak form and basis functions - II
09.03. The matrix-vector weak form
09.03c. In-Video Correction
09.04. The matrix-vector weak form - II
09.04c. In-Video Correction
Unit 9 Quiz
10
10.01. The strong form of linearized elasticity in three dimensions - I
10.02. The strong form of linearized elasticity in three dimensions - II
10.02c. In-Video Correction
10.03. The strong form, continued
10.04. The constitutive relations of linearized elasticity
10.05. The weak form - I
10.05q. Response to a question
10.06. The weak form - II
10.07. The finite-dimensional weak form - Basis functions - I
10.08. The finite-dimensional weak form - Basis functions - II
10.09. Element integrals - I
10.09c. In-Video Correction
10.10. Element integrals - II
10.11. The matrix-vector weak form - I
10.12. The matrix-vector weak form - II
10.13. Assembly of the global matrix-vector equations - I
10.14. Assembly of the global matrix-vector equations - II
10.14c. In Video Correction
10.14ct.1. Coding assignment 3 - I
10.14ct.2. Coding assignment 3 - II
10.15. Dirichlet boundary conditions - I
10.16. Dirichlet boundary conditions - II
Unit 10 Quiz
11
11.01. The strong form
11.01c In-Video Correction
11.02. The weak form, and finite-dimensional weak form - I
11.03. The weak form, and finite-dimensional weak form - II
11.04. Basis functions, and the matrix-vector weak form - I
11.04c In-Video Correction
11.05. Basis functions, and the matrix-vector weak form - II
11.05. Response to a question
11.06. Dirichlet boundary conditions; the final matrix-vector equations
11.07. Time discretization; the Euler family - I
11.08. Time discretization; the Euler family - II
11.09. The v-form and d-form
11.09ct.1. Coding assignment 4 - I
11.09ct.2. Coding assignment 4 - II
11.10. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - I
11.11. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - II
11.11c. In-Video Correction
11.12. Modal decomposition and modal equations - I
11.13. Modal decomposition and modal equations - II
11.14. Modal equations and stability of the time-exact single degree of freedom systems - I
11.15. Modal equations and stability of the time-exact single degree of freedom systems - II
11.15q. Response to a question
11.16. Stability of the time-discrete single degree of freedom systems
11.17. Behavior of higher-order modes; consistency - I
11.18. Behavior of higher-order modes; consistency - II
11.19. Convergence - I
11.20. Convergence - II
Unit 11 Quiz
12
12.01. The strong and weak forms
12.02. The finite-dimensional and matrix-vector weak forms - I
12.03. The finite-dimensional and matrix-vector weak forms - II
12.04. The time-discretized equations
12.05. Stability - I
12.06. Stability - II
12.07. Behavior of higher-order modes
12.08. Convergence
12.08c. In-Video Correction
Unit 12 Quiz
113
Conclusion, and the Road Ahead
Post-course Survey
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