# Introduction to Finite Element Methods

### Course Description

Here they are then, about 50 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.

It is hoped that these lectures on Finite Element Methods will complement the series on Continuum Physics to provide a point of departure from which the seasoned researcher or advanced graduate student can embark on work in (continuum) computational physics.

Prof. Krishna Garikipati explaining directional derivatives and the Jacobian.
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### Video Lectures & Study Materials

Visit the official course website for more study materials: http://open.umich.edu/education/engin/intro-finite-element-methods/2013

# Lecture Play Lecture
1 Introduction, Linear Elliptic Partial Differential Equations (Part 1) (14:47) Play Video
2 Introduction, Linear Elliptic Partial Differential Equations (Part 2) (13:02) Play Video
3 Boundary Conditions (22:19) Play Video
4 Constitutive relations (20:07) Play Video
5 Strong Form of the Partial Differential Equation, Analytic Solution (22:45) Play Video
6 Weak Form of the Partial Differential Equation (Part 1) (12:30) Play Video
7 Weak Form of the Partial Differential Equation (Part 2) (15:06) Play Video
8 Equivalence Between the Strong and Weak Forms (Part 1) (24:21) Play Video
9 The Galerkin, or finite dimensional weak form (23:15) Play Video
10 Response to a question (7:29) Play Video
11 Basic Hilbert Spaces (Part 1) (15:52) Play Video
12 Basic Hilbert Spaces (Part 2) (9:29) Play Video
13 FEM for the One Dimensional, Linear Elliptic PDE (22:54) Play Video
14 Response to a question (6:22) Play Video
15 Basis Functions (Part 1) (14:56) Play Video
16 Basis Functions (Part 2) (14:44) Play Video
17 The Bi-Unit Domain (Part 1) (11:45) Play Video
18 The Bi-Unit Domain (Part 2) (16:20) Play Video
19 Finite Dimensional Weak Form as a Sum Over Element Subdomains (Part 1) (16:09) Play Video
20 Finite Dimensional Weak Form as a Sum Over Element Subdomains (Part 2) (12:25) Play Video
21 The Matrix-Vector Weak Form - I (Part 1) (16:27) Play Video
22 The Matrix-Vector Weak Form - I (Part 2) (17:45) Play Video
23 The Matrix-Vector Weak Form - II (Part 1) (29:23) Play Video
24 The Matrix-Vector Weak Form - II (Part 2) (13:51) Play Video
25 The Matrix-Vector Weak Form - III (Part 1) (22:32) Play Video
26 The Matrix-Vector Weak Form - III (Part 2) (13:23) Play Video
27 The Final Finite Element Equations in Matrix-Vector form (Part 1) (21:03) Play Video
28 The Final Finite Element Equations in Matrix-Vector form (Part 2) (18:24) Play Video
29 Response to a question (4:36) Play Video
30 The Pure Dirichlet Problem (Part 1) (18:16) Play Video
31 The Pure Dirichlet Problem (Part 2) (17:42) Play Video
32 Correction to boardwork (1:01) Play Video
33 Higher Polynomial Order Basis Functions - I (22:56) Play Video
34 Correction to boardwork (:58) Play Video
35 Higher Polynomial Order Basis Functions - 1 (Part 2) (16:39) Play Video
36 Higher Polynomial Order Basis Functions - II (Part 1) (13:39) Play Video
37 Higher Polynomial Order Basis Functions - III (23:24) Play Video
38 The Matrix Vector Equations for Quadratic Basis Functions - I (Part 1) (21:20) Play Video
39 The Matrix Vector Equations for Quadratic Basis Functions - I (Part 2) (11:54) Play Video
40 The Matrix Vector Equations for Quadratic Basis Functions - II (Part 1) (19:10) Play Video
41 The Matrix Vector Equations for Quadratic Basis Functions - II (Part 2) (24:09) Play Video
42 Numerical Integration -- Gaussian Quadrature (13:58) Play Video
43 Norms (Part 1) (18:23) Play Video
44 Correction to boardwork (:57) Play Video
45 Norms (Part 2) (18:22) Play Video
46 Response to a question (5:46) Play Video
47 Consistency of the Finite Element Method (24:28) Play Video
48 The Best Approximation Property (21:33) Play Video
49 The "Pythagorean Theorem" (13:15) Play Video
50 Response to a question (3:32) Play Video
51 Sobolev Estimates and Convergence of the Finite Element Method (23:51) Play Video
52 Finite Element Error Estimates (22:08) Play Video
53 Functionals, Free Energy (Part 1) (17:39) Play Video
54 Functionals, Free Energy (Part 2) (13:21) Play Video
55 Extremization of Functionals (18:31) Play Video
56 Derivation of the Weak Form Using a Variational Principle (20:10) Play Video
57 The Strong Form of Steady State Heat Conduction and Mass Diffusion (Part 1) (18:25) Play Video
58 The Strong Form of Steady State Heat Conduction and Mass Diffusion (Part 2) (19:01) Play Video
59 Response to a question (1:28) Play Video
60 The Strong Form, continued (19:28) Play Video
61 Correction to boardwork (:43) Play Video
62 The Weak Form (24:34) Play Video
63 The Finite Dimensional Weak Form (Part 1) (12:36) Play Video
64 The Finite Dimensional Weak Form (Part 2) (15:57) Play Video
65 Three-Dimensional Hexahedral Finite Elements (21:31) Play Video
66 Aside: Insight to the Basis Functions by Considering the Two-Dimensional Case (16:44) Play Video
67 Field Derivatives: The Jacobian (Part 1) (12:39) Play Video
68 Field Derivatives: The Jacobian (Part 2) (14:21) Play Video
69 The Integrals in Terms of Degrees of Freedom (16:26) Play Video
70 The Integrals in Terms of Degrees of Freedom - Continued (20:56) Play Video
71 The Matrix-Vector Weak Form (Part 1) (17:20) Play Video
72 The Matrix-Vector Weak Form (Part 2) (11:20) Play Video
73 The Matrix-Vector Weak Form, continued (Part 1) (17:22) Play Video
74 Correction to boardwork (1:01) Play Video
75 The Matrix Vector Weak Form, continued (Part 2) (16:09) Play Video
76 The Matrix-Vector Weak Form, continued further (Part 1) (17:41) Play Video
77 Correction to boardwork (:48) Play Video
78 The Matrix-Vector Weak Form, continued further (Part 2) (17:19) Play Video
79 Correction to boardwork (:48) Play Video
80 Lagrange Basis Functions in 1 Through 3 Dimensions (Part 1) (18:59) Play Video
81 Lagrange Basis Functions in 1 through 3 dimensions (Part 2) (12:37) Play Video
82 Quadrature Rules in 1 Through 3 Dimensions (17:04) Play Video
83 Triangular and Tetrahedral Elements-Linears (Part 1) (10:26) Play Video
84 Triangular and Tetrahedral Elements Linears (Part 2) (16:30) Play Video
85 The Finite Dimensional Weak Form and Basis Functions (Part 1) (20:40) Play Video
86 The Finite Dimensional Weak Form and Basis Functions (Part 2) (19:13) Play Video
87 The Matrix Vector Weak Form (19:07) Play Video
88 The Matrix Vector Weak Form (Part 2) (9:43) Play Video
89 Correction to boardwork (1:53) Play Video
90 The Strong Form of Linearized Elasticity in Three Dimensions (Part 1) (9:59) Play Video
91 The Strong Form of Linearized Elasticity in Three Dimensions (Part 2) (15:45) Play Video
92 The Strong Form, continued (23:55) Play Video
93 The Constitutive Relations of Linearized Elasticity (21:10) Play Video
94 The Weak Form (Part 1) (17:38) Play Video
95 Response to a Question (7:56) Play Video
96 The Weak Form (Part 2) (20:24) Play Video
97 The Finite-Dimensional Weak Form-Basis Functions (Part 1) (18:24) Play Video
98 The Finite-Dimensional Weak Form-- Basis functions (Part 2) (10:01) Play Video
99 Element Integrals (Part 1) (20:46) Play Video
100 Correction to boardwork (:54) Play Video
101 Element Integrals (Part 2) (6:46) Play Video
102 The Matrix-Vector Weak Form (Part 1) (19:01) Play Video
103 The Matrix Vector-Weak Form (Part 2) (12:12) Play Video
104 Assembly of the Global Matrix-Vector Equations (Part 1) (20:41) Play Video
105 Assembly of the Global Matrix-Vector Equations II (9:17) Play Video
106 Correction to boardwork (2:54) Play Video
107 Dirichlet Boundary Conditions (Part 1) (21:24) Play Video
108 Dirichlet Boundary Conditions (Part 2) (14:00) Play Video
109 The Strong Form (16:30) Play Video
110 Correction to boardwork (:44) Play Video
111 The Weak Form, and Finite Dimensional Weak Form (Part 1) (18:45) Play Video
112 The Weak Form, and Finite Dimensional Weak Form (Part 2) (10:16) Play Video
113 Basis Functions, and the Matrix-Vector Weak Form (Part 1) (19:53) Play Video
114 Correction to Boardwork (:45) Play Video
115 Basis Functions, and the Matrix-Vector Weak Form (Part 2) (12:04) Play Video
116 Response to a question (:52) Play Video
117 Dirichlet Boundary Conditions; The Final Matrix Vector Equations (16:58) Play Video
118 Time Discretization; The Euler Family (Part 1) (22:38) Play Video
119 Time Discretization; The Euler Family (Part 2) (9:56) Play Video
120 The V-Form and D-Form (20:55) Play Video
121 Integration Algorithms for First-Order, Parabolic, Equations-Modal Decomposition (Part 1) (17:25) Play Video
122 Integration Algorithms for First-Order, Parabolic, Equations-Modal Decomposition (Part 2) (12:56) Play Video
123 Modal Decomposition and Modal Equations (Part 1) (16:01) Play Video
124 Modal Decomposition and Modal Equations (Part 2) (16:02) Play Video
125 Modal Equations and Stability of the Time Exact Single Degree of Freedom Systems (Part 1) (10:50) Play Video
126 Modal Equations and Stability of the Time-Exact Single Degree of Freedom Systems (Part 2) (17:39) Play Video
127 Stability of the Time-Discrete Single Degree of Freedom Systems (23:26) Play Video
128 Behavior of Higher-Order Modes; Consistency (Part 1) (18:58) Play Video
129 Behavior of Higher-Order Modes; consistency (Part 2) (19:52) Play Video
130 Convergence (Part 1) (20:50) Play Video
131 Convergence (Part 2) (16:39) Play Video
132 The Strong and Weak Forms (16:38) Play Video
133 The Finite-Dimensional and Matrix-Vector Weak Forms (Part 1) (10:38) Play Video
134 The Finite-Dimensional and Matri-Vector Weak Forms (Part 2) (16:01) Play Video
135 The Time-Discretized Equations (23:16) Play Video
136 Stability (Part 1) (12:58) Play Video
137 Stability (Part 2) (14:36) Play Video
138 Behavior of High-Order Modes (19:33) Play Video
139 Convergence (20:55) Play Video
140 Correction to boardwork (2:45) Play Video