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Finite DifferencesC++C Sharp (C#) (Programming Language)Matrices

1

## 1

This unit is an introduction to a simple one-dimensional problem that can be solved by the finite element method....
11 個視頻 （總計 200 分鐘）, 2 個閱讀材料, 1 個測驗
11 個視頻
01.02. Introduction. Linear elliptic partial differential equations - II 13分鐘
01.03. Boundary conditions 22分鐘
01.04. Constitutive relations 20分鐘
01.05. Strong form of the partial differential equation. Analytic solution 22分鐘
01.06. Weak form of the partial differential equation - I 12分鐘
01.07. Weak form of the partial differential equation - II 15分鐘
01.08. Equivalence between the strong and weak forms 24分鐘
01.08ct.1. Intro to C++ (running your code, basic structure, number types, vectors) 21分鐘
01.08ct.2. Intro to C++ (conditional statements, “for” loops, scope) 19分鐘
01.08ct.3. Intro to C++ (pointers, iterators) 14分鐘
2 個閱讀材料
Help us learn more about you!10分鐘
"Paper and pencil" practice assignment on strong and weak forms
1 個練習
Unit 1 Quiz8分鐘
2

## 2

In this unit you will be introduced to the approximate, or finite-dimensional, weak form for the one-dimensional problem....
14 個視頻 （總計 202 分鐘）, 1 個測驗
14 個視頻
02.01q. Response to a question 7分鐘
02.02. Basic Hilbert spaces - I 15分鐘
02.03. Basic Hilbert spaces - II 9分鐘
02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation 22分鐘
02.04q. Response to a question 6分鐘
02.05. Basis functions - I 14分鐘
02.06. Basis functions - II 14分鐘
02.07. The bi-unit domain - I 11分鐘
02.08. The bi-unit domain - II 16分鐘
02.09. The finite dimensional weak form as a sum over element subdomains - I 16分鐘
02.10. The finite dimensional weak form as a sum over element subdomains - II 12分鐘
02.10ct.1. Intro to C++ (functions) 13分鐘
02.10ct.2. Intro to C++ (C++ classes) 16分鐘
1 個練習
Unit 2 Quiz6分鐘
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 個視頻 （總計 213 分鐘）, 2 個測驗
14 個視頻
03.02. The matrix-vector weak form - I - II 17分鐘
03.03. The matrix-vector weak form - II - I 15分鐘
03.04. The matrix-vector weak form - II - II 13分鐘
03.05. The matrix-vector weak form - III - I 22分鐘
03.06. The matrix-vector weak form - III - II 13分鐘
03.06ct.1. Dealii.org, running deal.II on a virtual machine with Oracle VirtualBox12分鐘
03.06ct.2. Intro to AWS, using AWS on Windows24分鐘
03.06ct.2c. In-Video Correction3分鐘
03.06ct.3. Using AWS on Linux and Mac OS7分鐘
03.07. The final finite element equations in matrix-vector form - I 22分鐘
03.08. The final finite element equations in matrix-vector form - II 18分鐘
03.08q. Response to a question 4分鐘
03.08ct. Coding assignment 1 (main1.cc, overview of C++ class in FEM1.h) 19分鐘
1 個練習
Unit 3 Quiz6分鐘
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 個視頻 （總計 262 分鐘）, 1 個測驗
17 個視頻
04.02. The pure Dirichlet problem - II 17分鐘
04.02c. In-Video Correction 1分鐘
04.03. Higher polynomial order basis functions - I 23分鐘
04.03c0. In-Video Correction 57
04.03c1. In-Video Correction 34
04.04. Higher polynomial order basis functions - I - II 16分鐘
04.05. Higher polynomial order basis functions - II - I 13分鐘
04.06. Higher polynomial order basis functions - III 23分鐘
04.06ct. Coding assignment 1 (functions: class constructor to “basis_gradient”) 14分鐘
04.07. The matrix-vector equations for quadratic basis functions - I - I 21分鐘
04.08. The matrix-vector equations for quadratic basis functions - I - II 11分鐘
04.09. The matrix-vector equations for quadratic basis functions - II - I 19分鐘
04.10. The matrix-vector equations for quadratic basis functions - II - II 24分鐘
04.11. Numerical integration -- Gaussian quadrature 13分鐘
04.11ct.1. Coding assignment 1 (functions: “generate_mesh” to “setup_system”) 14分鐘
04.11ct.2. Coding assignment 1 (functions: “assemble_system”) 26分鐘
1 個練習
Unit 4 Quiz8分鐘
5

## 5

This unit outlines the mathematical analysis of the finite element method....
12 個視頻 （總計 170 分鐘）, 1 個測驗
12 個視頻
05.01c. In-Video Correction 56
05.01ct.1. Coding assignment 1 (functions: “solve” to “l2norm_of_error”) 10分鐘
05.01ct.2. Visualization tools7分鐘
05.02. Norms - II 18分鐘
05.02. Response to a question 5分鐘
05.03. Consistency of the finite element method 24分鐘
05.04. The best approximation property 21分鐘
05.05. The "Pythagorean Theorem" 13分鐘
05.05q. Response to a question 3分鐘
05.06. Sobolev estimates and convergence of the finite element method 23分鐘
05.07. Finite element error estimates 22分鐘
1 個練習
Unit 5 Quiz8分鐘
6

## 6

This unit develops an alternate derivation of the weak form, which is applicable to certain physical problems....
4 個視頻 （總計 70 分鐘）, 1 個測驗
4 個視頻
06.02. Functionals. Free energy - II 13分鐘
06.03. Extremization of functionals 18分鐘
06.04. Derivation of the weak form using a variational principle 20分鐘
1 個練習
Unit 6 Quiz4分鐘
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 個視頻 （總計 322 分鐘）, 1 個測驗
24 個視頻
07.02. The strong form of steady state heat conduction and mass diffusion - II 19分鐘
07.02q. Response to a question 1分鐘
07.03. The strong form, continued 19分鐘
07.03c. In-Video Correction 42
07.04. The weak form 24分鐘
07.05. The finite-dimensional weak form - I 12分鐘
07.06. The finite-dimensional weak form - II 15分鐘
07.07. Three-dimensional hexahedral finite elements 21分鐘
07.08. Aside: Insight to the basis functions by considering the two-dimensional case 17分鐘
07.08c In-Video Correction 44
07.09. Field derivatives. The Jacobian - I 12分鐘
07.10. Field derivatives. The Jacobian - II 14分鐘
07.11. The integrals in terms of degrees of freedom 16分鐘
07.12. The integrals in terms of degrees of freedom - continued 20分鐘
07.13. The matrix-vector weak form - I 17分鐘
07.14. The matrix-vector weak form II 11分鐘
07.15.The matrix-vector weak form, continued - I 17分鐘
07.15c. In-Video Correction 1分鐘
07.16. The matrix-vector weak form, continued - II 16分鐘
07.17. The matrix vector weak form, continued further - I 17分鐘
07.17c. In-Video Correction 47
07.18. The matrix-vector weak form, continued further - II 20分鐘
07.18c. In-Video Correction 3分鐘
1 個練習
Unit 7 Quiz10分鐘
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 個視頻 （總計 108 分鐘）, 2 個測驗
9 個視頻
08.01c. In-Video Correction 1分鐘
08.02. Lagrange basis functions in 1 through 3 dimensions - II 12分鐘
08.02ct. Coding assignment 2 (2D problem) - I 13分鐘
08.03. Quadrature rules in 1 through 3 dimensions 17分鐘
08.03ct.1. Coding assignment 2 (2D problem) - II 13分鐘
08.03ct.2. Coding assignment 2 (3D problem) 6分鐘
08.04. Triangular and tetrahedral elements - Linears - I 6分鐘
08.05. Triangular and tetrahedral elements - Linears - II 16分鐘
1 個練習
Unit 8 Quiz6分鐘
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 個視頻 （總計 73 分鐘）, 1 個測驗
6 個視頻
09.02. The finite-dimensional weak form and basis functions - II 19分鐘
09.03. The matrix-vector weak form 19分鐘
09.03c. In-Video Correction 38
09.04. The matrix-vector weak form - II 11分鐘
09.04c. In-Video Correction 1分鐘
1 個練習
Unit 9 Quiz4分鐘
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 個視頻 （總計 306 分鐘）, 2 個測驗
22 個視頻
10.02. The strong form of linearized elasticity in three dimensions - II 17分鐘
10.02c. In-Video Correction 1分鐘
10.03. The strong form, continued 23分鐘
10.04. The constitutive relations of linearized elasticity 21分鐘
10.05. The weak form - I 17分鐘
10.05q. Response to a question 7分鐘
10.06. The weak form - II 20分鐘
10.07. The finite-dimensional weak form - Basis functions - I 18分鐘
10.08. The finite-dimensional weak form - Basis functions - II 9分鐘
10.09. Element integrals - I 20分鐘
10.09c. In-Video Correction 53
10.10. Element integrals - II 6分鐘
10.11. The matrix-vector weak form - I 19分鐘
10.12. The matrix-vector weak form - II 12分鐘
10.13. Assembly of the global matrix-vector equations - I 20分鐘
10.14. Assembly of the global matrix-vector equations - II 9分鐘
10.14c. In Video Correction 2分鐘
10.14ct.1. Coding assignment 3 - I 10分鐘
10.14ct.2. Coding assignment 3 - II 19分鐘
10.15. Dirichlet boundary conditions - I 21分鐘
10.16. Dirichlet boundary conditions - II 13分鐘
1 個練習
Unit 10 Quiz8分鐘
11

## 11

In this unit, we study the unsteady heat conduction, or mass diffusion, problem, as well as its finite element formulation....
27 個視頻 （總計 378 分鐘）, 2 個測驗
27 個視頻
11.01c In-Video Correction 43
11.02. The weak form, and finite-dimensional weak form - I 18分鐘
11.03. The weak form, and finite-dimensional weak form - II 10分鐘
11.04. Basis functions, and the matrix-vector weak form - I 19分鐘
11.04c In-Video Correction 44
11.05. Basis functions, and the matrix-vector weak form - II 12分鐘
11.05. Response to a question 51
11.06. Dirichlet boundary conditions; the final matrix-vector equations 16分鐘
11.07. Time discretization; the Euler family - I 22分鐘
11.08. Time discretization; the Euler family - II 9分鐘
11.09. The v-form and d-form 20分鐘
11.09ct.1. Coding assignment 4 - I 11分鐘
11.09ct.2. Coding assignment 4 - II 13分鐘
11.10. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - I 17分鐘
11.11. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - II 14分鐘
11.11c. In-Video Correction 1分鐘
11.12. Modal decomposition and modal equations - I 16分鐘
11.13. Modal decomposition and modal equations - II 16分鐘
11.14. Modal equations and stability of the time-exact single degree of freedom systems - I 10分鐘
11.15. Modal equations and stability of the time-exact single degree of freedom systems - II 17分鐘
11.15q. Response to a question 10分鐘
11.16. Stability of the time-discrete single degree of freedom systems 23分鐘
11.17. Behavior of higher-order modes; consistency - I 18分鐘
11.18. Behavior of higher-order modes; consistency - II 19分鐘
11.19. Convergence - I 20分鐘
11.20. Convergence - II 16分鐘
1 個練習
Unit 11 Quiz8分鐘
12

## 12

In this unit we study the problem of elastodynamics, and its finite element formulation....
9 個視頻 （總計 141 分鐘）, 1 個測驗
9 個視頻
12.02. The finite-dimensional and matrix-vector weak forms - I 10分鐘
12.03. The finite-dimensional and matrix-vector weak forms - II 16分鐘
12.04. The time-discretized equations 23分鐘
12.05. Stability - I12分鐘
12.06. Stability - II 14分鐘
12.07. Behavior of higher-order modes 19分鐘
12.08. Convergence 24分鐘
12.08c. In-Video Correction 3分鐘
1 個練習
Unit 12 Quiz4分鐘
13

## 113

This is a wrap-up, with suggestions for future study....
1 個視頻 （總計 9 分鐘）, 1 個閱讀材料
1 個視頻
1 個閱讀材料
Post-course Survey10分鐘
4.7
58 個審閱

## 67%

### 熱門審閱

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.

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

## 講師 ### Krishna Garikipati, Ph.D.

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

## 關於 密歇根大学

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....

## 常見問題

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• 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.