# Introduction to Finite Element Methods Term:
2013
Published:
December 11, 2013
Revised:
September 2, 2014

## Welcome to Finite Element Methods

The idea for an online version of Finite Element Methods first came a little more than a year ago. Articles about Massively Open Online Classes (MOOCs) had been rocking the academic world (at least gently), and it seemed that your writer had scarcely experimented with teaching methods. Particularly compelling was the fact that there already had been some successes reported with computer programming classes in the online format, especially as MOOCs. Finite Element Methods, with the centrality that computer programming has to the teaching of this topic, seemed an obvious candidate for experimentation in the online format. From there to the video lectures that you are about to view took nearly a year. I first had to take a detour through another subject, Continuum Physics, for which video lectures also are available, and whose recording in this format served as a trial run for the present series of lectures on Finite Element Methods.

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.

There are a number of people that I need to thank: Shiva Rudraraju and Greg Teichert for their work on the coding framework, Tim O'Brien for organizing the recordings, Walter Lin and Alex Hancook for their camera work and post-production editing, and Scott Mahler for making the studios available.

Krishna Garikipati
Ann Arbor, December 2013 ## Krishna Garikipati

Dr. Garikipati's work draws from nonlinear mechanics, materials physics, applied mathematics and numerical methods. He's particularly interested in problems of mathematical biology, biophysics and the materials physics. Current research interests include: (1) mathematical and physical modelling of tumor growth, (2) cell mechanics (3) chemo-mechanically driven phenomena in materials, such as phase transformations and stress-influenced mass transport. more... Term:
2013
Published:
December 11, 2013
Revised:
September 2, 2014

### Assignments

03.08ct. Coding Assignment 01

Krishna Garikipati

03.08ct. Coding Assignment 01 Template

Krishna Garikipati

08.02ct. Coding Assignment 02

Krishna Garikipati

08.02ct. Coding Assignment 02 Template

Krishna Garikipati

10.14ct. 1. Coding Assignment 03

Krishna Garikipati

10.14ct. 1. Coding Assignment 03 Template

Krishna Garikipati

11.09ct. 1. Coding Assignment 04

Krishna Garikipati

11.09ct. 1. Coding Assignment 04 Template

Krishna Garikipati

### Lectures

01.01. Introduction. Linear elliptic partial differential equations - I (14:46)

Krishna Garikipati

01.02. Introduction. Linear elliptic partial differential equations - II (13:01)

Krishna Garikipati

01.03. Boundary conditions (22:18)

Krishna Garikipati

01.04. Constitutive relations (20:06)

Krishna Garikipati

01.05. Strong form of the partial differential equation. Analytic solution (22:44)

Krishna Garikipati

01.06. Weak form of the partial differential equation - I (12:29)

Krishna Garikipati

01.07. Weak form of the partial differential equation - II (15:05)

Krishna Garikipati

01.08. Equivalence between the strong and weak forms - 1 (25:10)

Krishna Garikipati

01.08ct. 1. Intro to C++ (Running Your Code, Basic Structure, Number Types, Vectors) (21:09)

Gregory Teichert

01.08ct. 2. Intro to C++ (Conditional Statements, "for" Loops, Scope) (19:27)

Gregory Teichert

01.08ct. 3. Intro to C++ (Pointers, Iterators) (14:01)

Gregory Teichert

02.01. The Galerkin, or finite-dimensional weak form (23:14)

Krishna Garikipati

02.01.Response to a question (7:28)

Krishna Garikipati

02.02. Basic Hilbert spaces - I (15:51)

Krishna Garikipati

02.03. Basic Hilbert spaces - II (9:28)

Krishna Garikipati

02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation (22:53)

Krishna Garikipati

02.04.Response to a question (6:21)

Krishna Garikipati

02.05. Basis functions - I (14:55)

Krishna Garikipati

02.06. Basis functions - II (14:43)

Krishna Garikipati

02.07. The bi-unit domain - I (11:43)

Krishna Garikipati

02.08. The bi-unit domain - II (16:19)

Krishna Garikipati

02.09. The finite dimensional weak form as a sum over element subdomains - I (16:08)

Krishna Garikipati

02.10. The finite dimensional weak form as a sum over element subdomains - II (12:24)

Krishna Garikipati

02.10ct. 1. Intro to C++ (Functions) (13:27)

Gregory Teichert

02.10ct. 2. Intro to C++ (C++ Classes) (16:43)

Gregory Teichert

03.01. The matrix-vector weak form - I - I (16:26)

Krishna Garikipati

03.02. The matrix-vector weak form - I - II (17:44)

Krishna Garikipati

03.03. The matrix-vector weak form - II - I (15:37)

Krishna Garikipati

03.04. The matrix-vector weak form - II - II (13:50)

Krishna Garikipati

03.05. The matrix-vector weak form - III - I (22:31)

Krishna Garikipati

03.06. The matrix-vector weak form - III - II (13:22)

Krishna Garikipati

03.06ct. 1. Dealii.org, Running Deal.II on a Virtual Machine with Oracle Virtualbox (12:59)

Gregory Teichert

03.06ct. 2. Intro to AWS; Using AWS on Windows (24:43)

Gregory Teichert

03.06ct. 2c. Correction (3:31)

Gregory Teichert

03.06ct. 3. Using AWS on Linux and Mac OS (7:42)

Gregory Teichert

03.07. The final finite element equations in matrix-vector form - I (21:02)

Krishna Garikipati

03.08. The final finite element equations in matrix-vector form - II (18:23)

Krishna Garikipati

03.08.Response to a question (4:35)

Krishna Garikipati

03.08ct. Coding Assignment 1 (main1.cc, Overview of C++ Class in FEM1.h) (19:34)

Gregory Teichert

04.01. The pure Dirichlet problem - I (18:14)

Krishna Garikipati

04.02. The pure Dirichlet problem - II (17:41)

Krishna Garikipati

04.03. Higher polynomial order basis functions - I (22:55)

Krishna Garikipati

04.04. Higher polynomial order basis functions - I - II (16:38)

Krishna Garikipati

04.05. Higher polynomial order basis functions - II - I (13:38)

Krishna Garikipati

04.06. Higher polynomial order basis functions - III (23:23)

Krishna Garikipati

04.06ct. Coding Assignment 1 (Functions: Class Constructor to "basis_gradient") (14:40)

Gregory Teichert

04.07. The matrix-vector equations for quadratic basis functions - I - I (21:19)

Krishna Garikipati

04.08. The matrix-vector equations for quadratic basis functions - I - II (11:53)

Krishna Garikipati

04.09. The matrix-vector equations for quadratic basis functions - II - I (19:09)

Krishna Garikipati

04.10. The matrix-vector equations for quadratic basis functions - II - II (24:08)

Krishna Garikipati

04.11. Numerical integration -- Gaussian quadrature (13:57)

Krishna Garikipati

04.11ct. 1. Coding Assignment 1 (Functions: "generate_mesh" to "setup_system") (14:21)

Gregory Teichert

04.11ct.2. Coding Assignment 1 (Functions: "assemble_system") (26:58)

Gregory Teichert

05.01. Norms - I (18:22)

Krishna Garikipati

05.01ct. 1. Coding Assignment 1 (Functions: "solve" to "I2norm_of_error") (10:57)

Gregory Teichert

05.01ct. 2. Visualization Tools (7:17)

Gregory Teichert

05.02. Norms - II (18:21)

Krishna Garikipati

05.02. Response to a question (5:45)

Krishna Garikipati

05.03. Consistency of the finite element method (24:27)

Krishna Garikipati

05.04. The best approximation property (21:32)

Krishna Garikipati

05.05. Response to a question (3:31)

Krishna Garikipati

05.05. The Pythagorean Theorem (13:14)

Krishna Garikipati

05.06. Sobolev estimates and convergence of the finite element method (23:50)

Krishna Garikipati

05.07. Finite element error estimates (22:07)

Krishna Garikipati

06.01. Functionals. Free energy - I (17:38)

Krishna Garikipati

06.02. Functionals. Free energy - II (13:20)

Krishna Garikipati

06.03. Extremization of functionals (18:30)

Krishna Garikipati

06.04. Derivation of the weak form using a variational principle (20:09)

Krishna Garikipati

07.01. The strong form of steady state heat conduction and mass diffusion - I (18:24)

Krishna Garikipati

07.02. Response to a question (1:27)

Krishna Garikipati

07.02. The strong form of steady state heat conduction and mass diffusion - II (19:00)

Krishna Garikipati

07.03. The strong form, continued (19:27)

Krishna Garikipati

07.04. The weak form (24:33)

Krishna Garikipati

07.05. The finite-dimensional weak form - I (12:35)

Krishna Garikipati

07.06. The finite-dimensional weak form - II (15:56)

Krishna Garikipati

07.07. Three-dimensional hexahedral finite elements (21:30)

Krishna Garikipati

07.08. Aside: Insight to the basis functions by considering the two-dimensional case (16:43)

Krishna Garikipati

07.09. Field derivatives. The Jacobian - I (12:38)

Krishna Garikipati

07.10. Field derivatives. The Jacobian - II (14:20)

Krishna Garikipati

07.11. The integrals in terms of degrees of freedom (16:25)

Krishna Garikipati

07.12. The integrals in terms of degrees of freedom - continued (20:55)

Krishna Garikipati

07.13. The matrix-vector weak form - I (17:19)

Krishna Garikipati

07.14. The matrix-vector weak form II (11:20)

Krishna Garikipati

07.15.The matrix-vector weak form, continued - I (17:21)

Krishna Garikipati

07.16. The matrix-vector weak form, continued - II (16:08)

Krishna Garikipati

07.17. The matrix vector weak form, continued further - I (17:40)

Krishna Garikipati

07.18. The matrix-vector weak form, continued further - II (17:18)

Krishna Garikipati

08.01. Lagrange basis functions in 1 through 3 dimensions - I (18:58)

Krishna Garikipati

08.02. Lagrange basis functions in 1 through 3 dimensions - II (12:36)

Krishna Garikipati

08.02ct. Coding Assignment 2 (2D Problem) - I

Gregory Teichert

08.03. Quadrature rules in 1 through 3 dimensions (17:03)

Krishna Garikipati

08.03ct. 1. Coding Assignment 2 (2D Problem) - II (13:50)

Gregory Teichert

08.03ct. 2. Coding Assignment 2 (3D Problem)

Gregory Teichert

08.04. Triangular and tetrahedral elements - Linears - I (10:25)

Krishna Garikipati

08.05. Triangular and tetrahedral elements - Linears - II (16:29)

Krishna Garikipati

09.01. The finite-dimensional weak form and basis functions - I (20:39)

Krishna Garikipati

09.02. The finite-dimensional weak form and basis functions - II (19:12)

Krishna Garikipati

09.03. The matrix-vector weak form (19:06)

Krishna Garikipati

09.04. The matrix-vector weak form - II (9:42)

Krishna Garikipati

10.01. The strong form of linearized elasticity in three dimensions - I (09:58)

Krishna Garikipati

10.02. The strong form of linearized elasticity in three dimensions - II (15:44)

Krishna Garikipati

10.03. The strong form, continued (23:54)

Krishna Garikipati

10.04. The constitutive relations of linearized elasticity (21:09)

Krishna Garikipati

10.05. Response to a question (07:55)

Krishna Garikipati

10.05. The weak form - I (17:37)

Krishna Garikipati

10.06. The weak form - II (20:23)

Krishna Garikipati

10.07. The finite-dimensional weak form - Basis functions - I (18:23)

Krishna Garikipati

10.08. The finite-dimensional weak form - Basis functions - II (10:00)

Krishna Garikipati

10.09. Element integrals - I (20:45)

Krishna Garikipati

10.10. Element integrals - II (6:45)

Krishna Garikipati

10.11. The matrix-vector weak form - I (19:00)

Krishna Garikipati

10.12. The matrix-vector weak form - II (12:11)

Krishna Garikipati

10.13. Assembly of the global matrix-vector equations - I (20:40)

Krishna Garikipati

10.14. Assembly of the global matrix-vector equations - II (9:16)

Krishna Garikipati

10.14ct. 1. Coding Assignment 3 - I (10:19)

Gregory Teichert

10.14ct. 2. Coding Assignment 3 - II (19:55)

Gregory Teichert

10.15. Dirichlet boundary conditions - I (21:23)

Krishna Garikipati

10.16. Dirichlet boundary conditions - II (13:59)

Krishna Garikipati

11.01. The strong form (16:29)

Krishna Garikipati

11.02. The weak form, and finite-dimensional weak form - I (18:44)

Krishna Garikipati

11.03. The weak form, and finite-dimensional weak form - II (10:15)

Krishna Garikipati

11.04. Basis functions, and the matrix-vector weak form - I (19:52)

Krishna Garikipati

11.05. Basis functions, and the matrix-vector weak form - II (12:03)

Krishna Garikipati

11.05. Response to a question (00:51)

Krishna Garikipati

11.06. Dirichlet boundary conditions; the final matrix-vector equations (16:57)

Krishna Garikipati

11.07. Time discretization; the Euler family - I (22:37)

Krishna Garikipati

11.08. Time discretization; the Euler family - II (9:55)

Krishna Garikipati

11.09. The v-form and d-form (20:54)

Krishna Garikipati

11.09ct. 1. Coding Assignment 4 - I (11:10)

Gregory Teichert

11.09ct. 2. Coding Assignment 4 - II (13:53)

Gregory Teichert

11.10. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - I (17:24)

Krishna Garikipati

11.11. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - II (12:55)

Krishna Garikipati

11.12. Modal decomposition and modal equations - I (16:00)

Krishna Garikipati

11.13. Modal decomposition and modal equations - II (16:01)

Krishna Garikipati

11.14. Modal equations and stability of the time-exact single degree of freedom systems - I (10:49)

Krishna Garikipati

11.15. Modal equations and stability of the time-exact single degree of freedom systems - II (17:38)

Krishna Garikipati

11.16. Stability of the time-discrete single degree of freedom systems (23:25)

Krishna Garikipati

11.17. Behavior of higher-order modes; consistency - I (18:57)

Krishna Garikipati

11.18. Behavior of higher-order modes; consistency - II (19:51)

Krishna Garikipati

11.19. Convergence - I (20:49)

Krishna Garikipati

11.20. Convergence - II (16:38)

Krishna Garikipati

12.01. The strong and weak forms (16:37)

Krishna Garikipati

12.02. The finite-dimensional and matrix-vector weak forms - I (10:37)

Krishna Garikipati

12.03. The finite-dimensional and matrix-vector weak forms - II (16:00)

Krishna Garikipati

12.04. The time-discretized equations (23:15)

Krishna Garikipati

12.05. Stability - I (12:57)

Krishna Garikipati

12.06. Stability - II (14:35)

Krishna Garikipati

12.07. Behavior of higher-order modes (19:32)

Krishna Garikipati

12.08. Convergence (20:54)

Krishna Garikipati Term:
2013
Published:
December 11, 2013
Revised:
September 2, 2014

### Unit 01: Linear and elliptic partial differential equations in one dimension. Elasticity; heat conduction; and mass diffusion.

01.01. Introduction. Linear elliptic partial differential equations - I (14:46)

Krishna Garikipati

01.02. Introduction. Linear elliptic partial differential equations - II (13:01)

Krishna Garikipati

01.03. Boundary conditions (22:18)

Krishna Garikipati

01.04. Constitutive relations (20:06)

Krishna Garikipati

01.05. Strong form of the partial differential equation. Analytic solution (22:44)

Krishna Garikipati

01.06. Weak form of the partial differential equation - I (12:29)

Krishna Garikipati

01.07. Weak form of the partial differential equation - II (15:05)

Krishna Garikipati

01.08. Equivalence between the strong and weak forms - 1 (25:10)

Krishna Garikipati

01.08ct. 1. Intro to C++ (Running Your Code, Basic Structure, Number Types, Vectors) (21:09)

Gregory Teichert

01.08ct. 2. Intro to C++ (Conditional Statements, "for" Loops, Scope) (19:27)

Gregory Teichert

01.08ct. 3. Intro to C++ (Pointers, Iterators) (14:01)

Gregory Teichert

### Unit 02: Approximation. The finite-dimensional weak form.

02.01. The Galerkin, or finite-dimensional weak form (23:14)

Krishna Garikipati

02.01.Response to a question (7:28)

Krishna Garikipati

02.02. Basic Hilbert spaces - I (15:51)

Krishna Garikipati

02.03. Basic Hilbert spaces - II (9:28)

Krishna Garikipati

02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation (22:53)

Krishna Garikipati

02.04.Response to a question (6:21)

Krishna Garikipati

02.05. Basis functions - I (14:55)

Krishna Garikipati

02.06. Basis functions - II (14:43)

Krishna Garikipati

02.07. The bi-unit domain - I (11:43)

Krishna Garikipati

02.08. The bi-unit domain - II (16:19)

Krishna Garikipati

02.09. The finite dimensional weak form as a sum over element subdomains - I (16:08)

Krishna Garikipati

02.10. The finite dimensional weak form as a sum over element subdomains - II (12:24)

Krishna Garikipati

02.10ct. 1. Intro to C++ (Functions) (13:27)

Gregory Teichert

02.10ct. 2. Intro to C++ (C++ Classes) (16:43)

Gregory Teichert

### Unit 03: Linear algebra; the matrix-vector form.

03.01. The matrix-vector weak form - I - I (16:26)

Krishna Garikipati

03.02. The matrix-vector weak form - I - II (17:44)

Krishna Garikipati

03.03. The matrix-vector weak form - II - I (15:37)

Krishna Garikipati

03.04. The matrix-vector weak form - II - II (13:50)

Krishna Garikipati

03.05. The matrix-vector weak form - III - I (22:31)

Krishna Garikipati

03.06. The matrix-vector weak form - III - II (13:22)

Krishna Garikipati

03.06ct. 1. Dealii.org, Running Deal.II on a Virtual Machine with Oracle Virtualbox (12:59)

Gregory Teichert

03.06ct. 2. Intro to AWS; Using AWS on Windows (24:43)

Gregory Teichert

03.06ct. 2c. Correction (3:31)

Gregory Teichert

03.06ct. 3. Using AWS on Linux and Mac OS (7:42)

Gregory Teichert

03.07. The final finite element equations in matrix-vector form - I (21:02)

Krishna Garikipati

03.08. The final finite element equations in matrix-vector form - II (18:23)

Krishna Garikipati

03.08.Response to a question (4:35)

Krishna Garikipati

03.08ct. Coding Assignment 01

Krishna Garikipati

03.08ct. Coding Assignment 01 Template

Krishna Garikipati

03.08ct. Coding Assignment 1 (main1.cc, Overview of C++ Class in FEM1.h) (19:34)

Gregory Teichert

### Unit 04: More on boundary conditions; basis functions; numerics.

04.01. The pure Dirichlet problem - I (18:14)

Krishna Garikipati

04.02. The pure Dirichlet problem - II (17:41)

Krishna Garikipati

04.03. Higher polynomial order basis functions - I (22:55)

Krishna Garikipati

04.04. Higher polynomial order basis functions - I - II (16:38)

Krishna Garikipati

04.05. Higher polynomial order basis functions - II - I (13:38)

Krishna Garikipati

04.06. Higher polynomial order basis functions - III (23:23)

Krishna Garikipati

04.06ct. Coding Assignment 1 (Functions: Class Constructor to "basis_gradient") (14:40)

Gregory Teichert

04.07. The matrix-vector equations for quadratic basis functions - I - I (21:19)

Krishna Garikipati

04.08. The matrix-vector equations for quadratic basis functions - I - II (11:53)

Krishna Garikipati

04.09. The matrix-vector equations for quadratic basis functions - II - I (19:09)

Krishna Garikipati

04.10. The matrix-vector equations for quadratic basis functions - II - II (24:08)

Krishna Garikipati

04.11. Numerical integration -- Gaussian quadrature (13:57)

Krishna Garikipati

04.11ct. 1. Coding Assignment 1 (Functions: "generate_mesh" to "setup_system") (14:21)

Gregory Teichert

04.11ct.2. Coding Assignment 1 (Functions: "assemble_system") (26:58)

Gregory Teichert

### Unit 05: Analysis of the finite element method.

05.01. Norms - I (18:22)

Krishna Garikipati

05.01ct. 1. Coding Assignment 1 (Functions: "solve" to "I2norm_of_error") (10:57)

Gregory Teichert

05.01ct. 2. Visualization Tools (7:17)

Gregory Teichert

05.02. Norms - II (18:21)

Krishna Garikipati

05.02. Response to a question (5:45)

Krishna Garikipati

05.03. Consistency of the finite element method (24:27)

Krishna Garikipati

05.04. The best approximation property (21:32)

Krishna Garikipati

05.05. Response to a question (3:31)

Krishna Garikipati

05.05. The Pythagorean Theorem (13:14)

Krishna Garikipati

05.06. Sobolev estimates and convergence of the finite element method (23:50)

Krishna Garikipati

05.07. Finite element error estimates (22:07)

Krishna Garikipati

### Unit 06: Variational principles.

06.01. Functionals. Free energy - I (17:38)

Krishna Garikipati

06.02. Functionals. Free energy - II (13:20)

Krishna Garikipati

06.03. Extremization of functionals (18:30)

Krishna Garikipati

06.04. Derivation of the weak form using a variational principle (20:09)

Krishna Garikipati

### Unit 07: Linear and elliptic partial differential equations for a scalar variable in three dimensions. Heat conduction and mass diffusion at steady state.

07.01. The strong form of steady state heat conduction and mass diffusion - I (18:24)

Krishna Garikipati

07.02. Response to a question (1:27)

Krishna Garikipati

07.02. The strong form of steady state heat conduction and mass diffusion - II (19:00)

Krishna Garikipati

07.03. The strong form, continued (19:27)

Krishna Garikipati

07.04. The weak form (24:33)

Krishna Garikipati

07.05. The finite-dimensional weak form - I (12:35)

Krishna Garikipati

07.06. The finite-dimensional weak form - II (15:56)

Krishna Garikipati

07.07. Three-dimensional hexahedral finite elements (21:30)

Krishna Garikipati

07.08. Aside: Insight to the basis functions by considering the two-dimensional case (16:43)

Krishna Garikipati

07.09. Field derivatives. The Jacobian - I (12:38)

Krishna Garikipati

07.10. Field derivatives. The Jacobian - II (14:20)

Krishna Garikipati

07.11. The integrals in terms of degrees of freedom (16:25)

Krishna Garikipati

07.12. The integrals in terms of degrees of freedom - continued (20:55)

Krishna Garikipati

07.13. The matrix-vector weak form - I (17:19)

Krishna Garikipati

07.14. The matrix-vector weak form II (11:20)

Krishna Garikipati

07.15.The matrix-vector weak form, continued - I (17:21)

Krishna Garikipati

07.16. The matrix-vector weak form, continued - II (16:08)

Krishna Garikipati

07.17. The matrix vector weak form, continued further - I (17:40)

Krishna Garikipati

07.18. The matrix-vector weak form, continued further - II (17:18)

Krishna Garikipati

### Unit 08: Lagrange basis functions and numerical quadrature in 1 through 3 dimensions

08.01. Lagrange basis functions in 1 through 3 dimensions - I (18:58)

Krishna Garikipati

08.02. Lagrange basis functions in 1 through 3 dimensions - II (12:36)

Krishna Garikipati

08.02ct. Coding Assignment 02

Krishna Garikipati

08.02ct. Coding Assignment 02 Template

Krishna Garikipati

08.02ct. Coding Assignment 2 (2D Problem) - I

Gregory Teichert

08.03. Quadrature rules in 1 through 3 dimensions (17:03)

Krishna Garikipati

08.03ct. 1. Coding Assignment 2 (2D Problem) - II (13:50)

Gregory Teichert

08.03ct. 2. Coding Assignment 2 (3D Problem)

Gregory Teichert

08.04. Triangular and tetrahedral elements - Linears - I (10:25)

Krishna Garikipati

08.05. Triangular and tetrahedral elements - Linears - II (16:29)

Krishna Garikipati

### Unit 09: Linear; elliptic; partial differential equations for a scalar variable in two dimensions

09.01. The finite-dimensional weak form and basis functions - I (20:39)

Krishna Garikipati

09.02. The finite-dimensional weak form and basis functions - II (19:12)

Krishna Garikipati

09.03. The matrix-vector weak form (19:06)

Krishna Garikipati

09.04. The matrix-vector weak form - II (9:42)

Krishna Garikipati

### Unit 10: Linear and elliptic partial differential equations for vector unknowns in three dimensions (Linearized elasticity)

10.01. The strong form of linearized elasticity in three dimensions - I (09:58)

Krishna Garikipati

10.02. The strong form of linearized elasticity in three dimensions - II (15:44)

Krishna Garikipati

10.03. The strong form, continued (23:54)

Krishna Garikipati

10.04. The constitutive relations of linearized elasticity (21:09)

Krishna Garikipati

10.05. Response to a question (07:55)

Krishna Garikipati

10.05. The weak form - I (17:37)

Krishna Garikipati

10.06. The weak form - II (20:23)

Krishna Garikipati

10.07. The finite-dimensional weak form - Basis functions - I (18:23)

Krishna Garikipati

10.08. The finite-dimensional weak form - Basis functions - II (10:00)

Krishna Garikipati

10.09. Element integrals - I (20:45)

Krishna Garikipati

10.10. Element integrals - II (6:45)

Krishna Garikipati

10.11. The matrix-vector weak form - I (19:00)

Krishna Garikipati

10.12. The matrix-vector weak form - II (12:11)

Krishna Garikipati

10.13. Assembly of the global matrix-vector equations - I (20:40)

Krishna Garikipati

10.14. Assembly of the global matrix-vector equations - II (9:16)

Krishna Garikipati

10.14ct. 1. Coding Assignment 03

Krishna Garikipati

10.14ct. 1. Coding Assignment 03 Template

Krishna Garikipati

10.14ct. 1. Coding Assignment 3 - I (10:19)

Gregory Teichert

10.14ct. 2. Coding Assignment 3 - II (19:55)

Gregory Teichert

10.15. Dirichlet boundary conditions - I (21:23)

Krishna Garikipati

10.16. Dirichlet boundary conditions - II (13:59)

Krishna Garikipati

### Unit 11: Linear and parabolic partial differential equations for a scalar unknown in three dimensions (Unsteady heat conduction and mass diffusion)

11.01. The strong form (16:29)

Krishna Garikipati

11.02. The weak form, and finite-dimensional weak form - I (18:44)

Krishna Garikipati

11.03. The weak form, and finite-dimensional weak form - II (10:15)

Krishna Garikipati

11.04. Basis functions, and the matrix-vector weak form - I (19:52)

Krishna Garikipati

11.05. Basis functions, and the matrix-vector weak form - II (12:03)

Krishna Garikipati

11.05. Response to a question (00:51)

Krishna Garikipati

11.06. Dirichlet boundary conditions; the final matrix-vector equations (16:57)

Krishna Garikipati

11.07. Time discretization; the Euler family - I (22:37)

Krishna Garikipati

11.08. Time discretization; the Euler family - II (9:55)

Krishna Garikipati

11.09. The v-form and d-form (20:54)

Krishna Garikipati

11.09ct. 1. Coding Assignment 04

Krishna Garikipati

11.09ct. 1. Coding Assignment 04 Template

Krishna Garikipati

11.09ct. 1. Coding Assignment 4 - I (11:10)

Gregory Teichert

11.09ct. 2. Coding Assignment 4 - II (13:53)

Gregory Teichert

11.10. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - I (17:24)

Krishna Garikipati

11.11. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - II (12:55)

Krishna Garikipati

11.12. Modal decomposition and modal equations - I (16:00)

Krishna Garikipati

11.13. Modal decomposition and modal equations - II (16:01)

Krishna Garikipati

11.14. Modal equations and stability of the time-exact single degree of freedom systems - I (10:49)

Krishna Garikipati

11.15. Modal equations and stability of the time-exact single degree of freedom systems - II (17:38)

Krishna Garikipati

11.16. Stability of the time-discrete single degree of freedom systems (23:25)

Krishna Garikipati

11.17. Behavior of higher-order modes; consistency - I (18:57)

Krishna Garikipati

11.18. Behavior of higher-order modes; consistency - II (19:51)

Krishna Garikipati

11.19. Convergence - I (20:49)

Krishna Garikipati

11.20. Convergence - II (16:38)

Krishna Garikipati

### Unit 12: Linear and hyperbolic partial differential equations for a vector unknown in three dimensions (Linear elastodynamics)

12.01. The strong and weak forms (16:37)

Krishna Garikipati

12.02. The finite-dimensional and matrix-vector weak forms - I (10:37)

Krishna Garikipati

12.03. The finite-dimensional and matrix-vector weak forms - II (16:00)

Krishna Garikipati

12.04. The time-discretized equations (23:15)

Krishna Garikipati

12.05. Stability - I (12:57)

Krishna Garikipati

12.06. Stability - II (14:35)

Krishna Garikipati

12.07. Behavior of higher-order modes (19:32)

Krishna Garikipati

12.08. Convergence (20:54)

Krishna Garikipati