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The Finite Element Method for Problems in Physics

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Course Summary

This course offers a comprehensive introduction to the Finite Element Method, a popular numerical technique used for solving engineering problems.

Key Learning Points

Gain an understanding of the fundamental concepts of the Finite Element Method
Learn how to apply the Finite Element Method to solve engineering problems
Develop skills in using Finite Element software packages

Learning Outcomes

Understand the basic principles of the Finite Element Method
Apply the Finite Element Method to solve engineering problems
Develop proficiency in using Finite Element software packages

Prerequisites or good to have knowledge before taking this course

Basic knowledge of calculus and linear algebra
Familiarity with engineering concepts and principles

Course Difficulty Level

Intermediate

Course Format

Online
Self-paced

Similar Courses

Applied Finite Element Analysis
Finite Element Analysis for Solids and Structures

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Description

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.

Outline

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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Summary of User Reviews

Learn the Finite Element Method on Coursera. This course has received positive reviews from its users. Many users have found the course to be comprehensive and informative.

Key Aspect Users Liked About This Course

The course is comprehensive and informative.

Pros from User Reviews

The course is well-structured and easy to follow.
The instructor is knowledgeable and engaging.
The course provides a good balance of theory and practical applications.
The course offers valuable insights into real-world engineering problems.
The course materials are of high quality and easy to access.

Cons from User Reviews

The course may be too technical for beginners.
The course requires a significant amount of time and effort to complete.
The course may be too focused on theory and not enough on practical applications.
The course may require additional resources to fully understand the material.
The course may not be suitable for those without a strong background in mathematics or engineering.

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