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
About the Creators
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...
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03.08ct. Coding Assignment 01 |
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08.02ct. Coding Assignment 02 |
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08.02ct. Coding Assignment 02 Template |
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10.14ct. 1. Coding Assignment 03 |
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11.09ct. 1. Coding Assignment 04 |
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11.09ct. 1. Coding Assignment 04 Template |
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01.01. Introduction. Linear elliptic partial differential equations - I (14:46) |
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01.02. Introduction. Linear elliptic partial differential equations - II (13:01) |
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01.03. Boundary conditions (22:18) |
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01.04. Constitutive relations (20:06) |
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01.05. Strong form of the partial differential equation. Analytic solution (22:44) |
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01.06. Weak form of the partial differential equation - I (12:29) |
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01.07. Weak form of the partial differential equation - II (15:05) |
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01.08. Equivalence between the strong and weak forms - 1 (25:10) |
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01.08ct. 1. Intro to C++ (Running Your Code, Basic Structure, Number Types, Vectors) (21:09) |
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01.08ct. 2. Intro to C++ (Conditional Statements, "for" Loops, Scope) (19:27) |
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01.08ct. 3. Intro to C++ (Pointers, Iterators) (14:01) |
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02.01. The Galerkin, or finite-dimensional weak form (23:14) |
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02.01.Response to a question (7:28) |
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02.02. Basic Hilbert spaces - I (15:51) |
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02.03. Basic Hilbert spaces - II (9:28) |
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02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation (22:53) |
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02.04.Response to a question (6:21) |
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02.05. Basis functions - I (14:55) |
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02.06. Basis functions - II (14:43) |
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02.07. The bi-unit domain - I (11:43) |
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02.08. The bi-unit domain - II (16:19) |
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02.09. The finite dimensional weak form as a sum over element subdomains - I (16:08) |
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02.10. The finite dimensional weak form as a sum over element subdomains - II (12:24) |
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02.10ct. 1. Intro to C++ (Functions) (13:27) |
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02.10ct. 2. Intro to C++ (C++ Classes) (16:43) |
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03.01. The matrix-vector weak form - I - I (16:26) |
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03.02. The matrix-vector weak form - I - II (17:44) |
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03.03. The matrix-vector weak form - II - I (15:37) |
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03.04. The matrix-vector weak form - II - II (13:50) |
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03.05. The matrix-vector weak form - III - I (22:31) |
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03.06. The matrix-vector weak form - III - II (13:22) |
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03.06ct. 1. Dealii.org, Running Deal.II on a Virtual Machine with Oracle Virtualbox (12:59) |
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03.06ct. 2. Intro to AWS; Using AWS on Windows (24:43) |
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03.06ct. 2c. Correction (3:31) |
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03.06ct. 3. Using AWS on Linux and Mac OS (7:42) |
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03.07. The final finite element equations in matrix-vector form - I (21:02) |
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03.08. The final finite element equations in matrix-vector form - II (18:23) |
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03.08.Response to a question (4:35) |
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03.08ct. Coding Assignment 1 (main1.cc, Overview of C++ Class in FEM1.h) (19:34) |
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04.01. The pure Dirichlet problem - I (18:14) |
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04.02. The pure Dirichlet problem - II (17:41) |
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04.03. Higher polynomial order basis functions - I (22:55) |
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04.04. Higher polynomial order basis functions - I - II (16:38) |
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04.05. Higher polynomial order basis functions - II - I (13:38) |
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04.06. Higher polynomial order basis functions - III (23:23) |
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04.06ct. Coding Assignment 1 (Functions: Class Constructor to "basis_gradient") (14:40) |
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04.07. The matrix-vector equations for quadratic basis functions - I - I (21:19) |
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04.08. The matrix-vector equations for quadratic basis functions - I - II (11:53) |
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04.09. The matrix-vector equations for quadratic basis functions - II - I (19:09) |
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04.10. The matrix-vector equations for quadratic basis functions - II - II (24:08) |
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04.11. Numerical integration -- Gaussian quadrature (13:57) |
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04.11ct. 1. Coding Assignment 1 (Functions: "generate_mesh" to "setup_system") (14:21) |
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04.11ct.2. Coding Assignment 1 (Functions: "assemble_system") (26:58) |
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05.01. Norms - I (18:22) |
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05.01ct. 1. Coding Assignment 1 (Functions: "solve" to "I2norm_of_error") (10:57) |
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05.01ct. 2. Visualization Tools (7:17) |
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05.02. Norms - II (18:21) |
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05.02. Response to a question (5:45) |
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05.03. Consistency of the finite element method (24:27) |
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05.04. The best approximation property (21:32) |
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05.05. Response to a question (3:31) |
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05.05. The Pythagorean Theorem (13:14) |
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05.06. Sobolev estimates and convergence of the finite element method (23:50) |
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05.07. Finite element error estimates (22:07) |
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06.01. Functionals. Free energy - I (17:38) |
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06.02. Functionals. Free energy - II (13:20) |
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06.03. Extremization of functionals (18:30) |
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06.04. Derivation of the weak form using a variational principle (20:09) |
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07.01. The strong form of steady state heat conduction and mass diffusion - I (18:24) |
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07.02. Response to a question (1:27) |
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07.02. The strong form of steady state heat conduction and mass diffusion - II (19:00) |
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07.03. The strong form, continued (19:27) |
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07.04. The weak form (24:33) |
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07.05. The finite-dimensional weak form - I (12:35) |
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07.06. The finite-dimensional weak form - II (15:56) |
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07.07. Three-dimensional hexahedral finite elements (21:30) |
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07.08. Aside: Insight to the basis functions by considering the two-dimensional case (16:43) |
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07.09. Field derivatives. The Jacobian - I (12:38) |
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07.10. Field derivatives. The Jacobian - II (14:20) |
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07.11. The integrals in terms of degrees of freedom (16:25) |
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07.12. The integrals in terms of degrees of freedom - continued (20:55) |
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07.13. The matrix-vector weak form - I (17:19) |
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07.14. The matrix-vector weak form II (11:20) |
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07.15.The matrix-vector weak form, continued - I (17:21) |
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07.16. The matrix-vector weak form, continued - II (16:08) |
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07.17. The matrix vector weak form, continued further - I (17:40) |
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07.18. The matrix-vector weak form, continued further - II (17:18) |
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08.01. Lagrange basis functions in 1 through 3 dimensions - I (18:58) |
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08.02. Lagrange basis functions in 1 through 3 dimensions - II (12:36) |
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08.02ct. Coding Assignment 2 (2D Problem) - I |
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08.03. Quadrature rules in 1 through 3 dimensions (17:03) |
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08.03ct. 1. Coding Assignment 2 (2D Problem) - II (13:50) |
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08.03ct. 2. Coding Assignment 2 (3D Problem) |
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08.04. Triangular and tetrahedral elements - Linears - I (10:25) |
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08.05. Triangular and tetrahedral elements - Linears - II (16:29) |
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09.01. The finite-dimensional weak form and basis functions - I (20:39) |
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09.02. The finite-dimensional weak form and basis functions - II (19:12) |
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09.03. The matrix-vector weak form (19:06) |
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09.04. The matrix-vector weak form - II (9:42) |
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10.01. The strong form of linearized elasticity in three dimensions - I (09:58) |
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10.02. The strong form of linearized elasticity in three dimensions - II (15:44) |
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10.03. The strong form, continued (23:54) |
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10.04. The constitutive relations of linearized elasticity (21:09) |
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10.05. Response to a question (07:55) |
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10.05. The weak form - I (17:37) |
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10.06. The weak form - II (20:23) |
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10.07. The finite-dimensional weak form - Basis functions - I (18:23) |
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10.08. The finite-dimensional weak form - Basis functions - II (10:00) |
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10.09. Element integrals - I (20:45) |
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10.10. Element integrals - II (6:45) |
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10.11. The matrix-vector weak form - I (19:00) |
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10.12. The matrix-vector weak form - II (12:11) |
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10.13. Assembly of the global matrix-vector equations - I (20:40) |
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10.14. Assembly of the global matrix-vector equations - II (9:16) |
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10.14ct. 1. Coding Assignment 3 - I (10:19) |
Gregory Teichert
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10.14ct. 2. Coding Assignment 3 - II (19:55) |
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10.15. Dirichlet boundary conditions - I (21:23) |
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10.16. Dirichlet boundary conditions - II (13:59) |
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11.01. The strong form (16:29) |
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11.02. The weak form, and finite-dimensional weak form - I (18:44) |
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11.03. The weak form, and finite-dimensional weak form - II (10:15) |
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11.04. Basis functions, and the matrix-vector weak form - I (19:52) |
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11.05. Basis functions, and the matrix-vector weak form - II (12:03) |
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11.05. Response to a question (00:51) |
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11.06. Dirichlet boundary conditions; the final matrix-vector equations (16:57) |
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11.07. Time discretization; the Euler family - I (22:37) |
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11.08. Time discretization; the Euler family - II (9:55) |
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11.09. The v-form and d-form (20:54) |
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11.09ct. 1. Coding Assignment 4 - I (11:10) |
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11.09ct. 2. Coding Assignment 4 - II (13:53) |
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11.10. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - I (17:24) |
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11.11. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - II (12:55) |
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11.12. Modal decomposition and modal equations - I (16:00) |
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11.13. Modal decomposition and modal equations - II (16:01) |
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11.14. Modal equations and stability of the time-exact single degree of freedom systems - I (10:49) |
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11.15. Modal equations and stability of the time-exact single degree of freedom systems - II (17:38) |
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11.16. Stability of the time-discrete single degree of freedom systems (23:25) |
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11.17. Behavior of higher-order modes; consistency - I (18:57) |
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11.18. Behavior of higher-order modes; consistency - II (19:51) |
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11.19. Convergence - I (20:49) |
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11.20. Convergence - II (16:38) |
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12.01. The strong and weak forms (16:37) |
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12.02. The finite-dimensional and matrix-vector weak forms - I (10:37) |
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12.03. The finite-dimensional and matrix-vector weak forms - II (16:00) |
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12.04. The time-discretized equations (23:15) |
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12.05. Stability - I (12:57) |
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12.06. Stability - II (14:35) |
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12.07. Behavior of higher-order modes (19:32) |
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12.08. Convergence (20:54) |
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Jump to:
| Document Title | Creator | Downloads | License |
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01.01. Introduction. Linear elliptic partial differential equations - I (14:46) |
Krishna Garikipati
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01.02. Introduction. Linear elliptic partial differential equations - II (13:01) |
Krishna Garikipati
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01.03. Boundary conditions (22:18) |
Krishna Garikipati
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01.04. Constitutive relations (20:06) |
Krishna Garikipati
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01.05. Strong form of the partial differential equation. Analytic solution (22:44) |
Krishna Garikipati
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01.06. Weak form of the partial differential equation - I (12:29) |
Krishna Garikipati
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01.07. Weak form of the partial differential equation - II (15:05) |
Krishna Garikipati
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01.08. Equivalence between the strong and weak forms - 1 (25:10) |
Krishna Garikipati
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01.08ct. 1. Intro to C++ (Running Your Code, Basic Structure, Number Types, Vectors) (21:09) |
Gregory Teichert
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01.08ct. 2. Intro to C++ (Conditional Statements, "for" Loops, Scope) (19:27) |
Gregory Teichert
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01.08ct. 3. Intro to C++ (Pointers, Iterators) (14:01) |
Gregory Teichert
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| Document Title | Creator | Downloads | License |
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02.01. The Galerkin, or finite-dimensional weak form (23:14) |
Krishna Garikipati
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02.01.Response to a question (7:28) |
Krishna Garikipati
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02.02. Basic Hilbert spaces - I (15:51) |
Krishna Garikipati
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02.03. Basic Hilbert spaces - II (9:28) |
Krishna Garikipati
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02.04. The finite element method for the one-dimensional, linear, elliptic partial differential equation (22:53) |
Krishna Garikipati
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02.04.Response to a question (6:21) |
Krishna Garikipati
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02.05. Basis functions - I (14:55) |
Krishna Garikipati
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02.06. Basis functions - II (14:43) |
Krishna Garikipati
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02.07. The bi-unit domain - I (11:43) |
Krishna Garikipati
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02.08. The bi-unit domain - II (16:19) |
Krishna Garikipati
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02.09. The finite dimensional weak form as a sum over element subdomains - I (16:08) |
Krishna Garikipati
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02.10. The finite dimensional weak form as a sum over element subdomains - II (12:24) |
Krishna Garikipati
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02.10ct. 1. Intro to C++ (Functions) (13:27) |
Gregory Teichert
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02.10ct. 2. Intro to C++ (C++ Classes) (16:43) |
Gregory Teichert
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03.01. The matrix-vector weak form - I - I (16:26) |
Krishna Garikipati
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03.02. The matrix-vector weak form - I - II (17:44) |
Krishna Garikipati
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03.03. The matrix-vector weak form - II - I (15:37) |
Krishna Garikipati
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03.04. The matrix-vector weak form - II - II (13:50) |
Krishna Garikipati
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03.05. The matrix-vector weak form - III - I (22:31) |
Krishna Garikipati
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03.06. The matrix-vector weak form - III - II (13:22) |
Krishna Garikipati
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03.06ct. 1. Dealii.org, Running Deal.II on a Virtual Machine with Oracle Virtualbox (12:59) |
Gregory Teichert
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03.06ct. 2. Intro to AWS; Using AWS on Windows (24:43) |
Gregory Teichert
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03.06ct. 2c. Correction (3:31) |
Gregory Teichert
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03.06ct. 3. Using AWS on Linux and Mac OS (7:42) |
Gregory Teichert
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03.07. The final finite element equations in matrix-vector form - I (21:02) |
Krishna Garikipati
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03.08. The final finite element equations in matrix-vector form - II (18:23) |
Krishna Garikipati
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03.08.Response to a question (4:35) |
Krishna Garikipati
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03.08ct. Coding Assignment 01 |
Krishna Garikipati
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03.08ct. Coding Assignment 01 Template |
Krishna Garikipati
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03.08ct. Coding Assignment 1 (main1.cc, Overview of C++ Class in FEM1.h) (19:34) |
Gregory Teichert
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04.01. The pure Dirichlet problem - I (18:14) |
Krishna Garikipati
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04.02. The pure Dirichlet problem - II (17:41) |
Krishna Garikipati
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04.03. Higher polynomial order basis functions - I (22:55) |
Krishna Garikipati
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04.04. Higher polynomial order basis functions - I - II (16:38) |
Krishna Garikipati
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04.05. Higher polynomial order basis functions - II - I (13:38) |
Krishna Garikipati
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04.06. Higher polynomial order basis functions - III (23:23) |
Krishna Garikipati
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04.06ct. Coding Assignment 1 (Functions: Class Constructor to "basis_gradient") (14:40) |
Gregory Teichert
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04.07. The matrix-vector equations for quadratic basis functions - I - I (21:19) |
Krishna Garikipati
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04.08. The matrix-vector equations for quadratic basis functions - I - II (11:53) |
Krishna Garikipati
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04.09. The matrix-vector equations for quadratic basis functions - II - I (19:09) |
Krishna Garikipati
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04.10. The matrix-vector equations for quadratic basis functions - II - II (24:08) |
Krishna Garikipati
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04.11. Numerical integration -- Gaussian quadrature (13:57) |
Krishna Garikipati
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04.11ct. 1. Coding Assignment 1 (Functions: "generate_mesh" to "setup_system") (14:21) |
Gregory Teichert
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04.11ct.2. Coding Assignment 1 (Functions: "assemble_system") (26:58) |
Gregory Teichert
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05.01. Norms - I (18:22) |
Krishna Garikipati
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05.01ct. 1. Coding Assignment 1 (Functions: "solve" to "I2norm_of_error") (10:57) |
Gregory Teichert
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05.01ct. 2. Visualization Tools (7:17) |
Gregory Teichert
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05.02. Norms - II (18:21) |
Krishna Garikipati
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05.02. Response to a question (5:45) |
Krishna Garikipati
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05.03. Consistency of the finite element method (24:27) |
Krishna Garikipati
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05.04. The best approximation property (21:32) |
Krishna Garikipati
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05.05. Response to a question (3:31) |
Krishna Garikipati
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05.05. The Pythagorean Theorem (13:14) |
Krishna Garikipati
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05.06. Sobolev estimates and convergence of the finite element method (23:50) |
Krishna Garikipati
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05.07. Finite element error estimates (22:07) |
Krishna Garikipati
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06.01. Functionals. Free energy - I (17:38) |
Krishna Garikipati
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06.02. Functionals. Free energy - II (13:20) |
Krishna Garikipati
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06.03. Extremization of functionals (18:30) |
Krishna Garikipati
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06.04. Derivation of the weak form using a variational principle (20:09) |
Krishna Garikipati
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07.01. The strong form of steady state heat conduction and mass diffusion - I (18:24) |
Krishna Garikipati
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07.02. Response to a question (1:27) |
Krishna Garikipati
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07.02. The strong form of steady state heat conduction and mass diffusion - II (19:00) |
Krishna Garikipati
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07.03. The strong form, continued (19:27) |
Krishna Garikipati
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07.04. The weak form (24:33) |
Krishna Garikipati
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07.05. The finite-dimensional weak form - I (12:35) |
Krishna Garikipati
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07.06. The finite-dimensional weak form - II (15:56) |
Krishna Garikipati
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07.07. Three-dimensional hexahedral finite elements (21:30) |
Krishna Garikipati
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07.08. Aside: Insight to the basis functions by considering the two-dimensional case (16:43) |
Krishna Garikipati
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07.09. Field derivatives. The Jacobian - I (12:38) |
Krishna Garikipati
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07.10. Field derivatives. The Jacobian - II (14:20) |
Krishna Garikipati
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07.11. The integrals in terms of degrees of freedom (16:25) |
Krishna Garikipati
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07.12. The integrals in terms of degrees of freedom - continued (20:55) |
Krishna Garikipati
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07.13. The matrix-vector weak form - I (17:19) |
Krishna Garikipati
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07.14. The matrix-vector weak form II (11:20) |
Krishna Garikipati
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07.15.The matrix-vector weak form, continued - I (17:21) |
Krishna Garikipati
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07.16. The matrix-vector weak form, continued - II (16:08) |
Krishna Garikipati
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07.17. The matrix vector weak form, continued further - I (17:40) |
Krishna Garikipati
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07.18. The matrix-vector weak form, continued further - II (17:18) |
Krishna Garikipati
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08.01. Lagrange basis functions in 1 through 3 dimensions - I (18:58) |
Krishna Garikipati
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08.02. Lagrange basis functions in 1 through 3 dimensions - II (12:36) |
Krishna Garikipati
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08.02ct. Coding Assignment 02 |
Krishna Garikipati
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08.02ct. Coding Assignment 02 Template |
Krishna Garikipati
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08.02ct. Coding Assignment 2 (2D Problem) - I |
Gregory Teichert
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08.03. Quadrature rules in 1 through 3 dimensions (17:03) |
Krishna Garikipati
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08.03ct. 1. Coding Assignment 2 (2D Problem) - II (13:50) |
Gregory Teichert
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08.03ct. 2. Coding Assignment 2 (3D Problem) |
Gregory Teichert
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08.04. Triangular and tetrahedral elements - Linears - I (10:25) |
Krishna Garikipati
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08.05. Triangular and tetrahedral elements - Linears - II (16:29) |
Krishna Garikipati
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09.01. The finite-dimensional weak form and basis functions - I (20:39) |
Krishna Garikipati
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09.02. The finite-dimensional weak form and basis functions - II (19:12) |
Krishna Garikipati
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09.03. The matrix-vector weak form (19:06) |
Krishna Garikipati
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09.04. The matrix-vector weak form - II (9:42) |
Krishna Garikipati
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10.01. The strong form of linearized elasticity in three dimensions - I (09:58) |
Krishna Garikipati
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10.02. The strong form of linearized elasticity in three dimensions - II (15:44) |
Krishna Garikipati
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10.03. The strong form, continued (23:54) |
Krishna Garikipati
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10.04. The constitutive relations of linearized elasticity (21:09) |
Krishna Garikipati
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10.05. Response to a question (07:55) |
Krishna Garikipati
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10.05. The weak form - I (17:37) |
Krishna Garikipati
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10.06. The weak form - II (20:23) |
Krishna Garikipati
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10.07. The finite-dimensional weak form - Basis functions - I (18:23) |
Krishna Garikipati
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10.08. The finite-dimensional weak form - Basis functions - II (10:00) |
Krishna Garikipati
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10.09. Element integrals - I (20:45) |
Krishna Garikipati
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10.10. Element integrals - II (6:45) |
Krishna Garikipati
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10.11. The matrix-vector weak form - I (19:00) |
Krishna Garikipati
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10.12. The matrix-vector weak form - II (12:11) |
Krishna Garikipati
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10.13. Assembly of the global matrix-vector equations - I (20:40) |
Krishna Garikipati
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10.14. Assembly of the global matrix-vector equations - II (9:16) |
Krishna Garikipati
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10.14ct. 1. Coding Assignment 03 |
Krishna Garikipati
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10.14ct. 1. Coding Assignment 03 Template |
Krishna Garikipati
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10.14ct. 1. Coding Assignment 3 - I (10:19) |
Gregory Teichert
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10.14ct. 2. Coding Assignment 3 - II (19:55) |
Gregory Teichert
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10.15. Dirichlet boundary conditions - I (21:23) |
Krishna Garikipati
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10.16. Dirichlet boundary conditions - II (13:59) |
Krishna Garikipati
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11.01. The strong form (16:29) |
Krishna Garikipati
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11.02. The weak form, and finite-dimensional weak form - I (18:44) |
Krishna Garikipati
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11.03. The weak form, and finite-dimensional weak form - II (10:15) |
Krishna Garikipati
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11.04. Basis functions, and the matrix-vector weak form - I (19:52) |
Krishna Garikipati
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11.05. Basis functions, and the matrix-vector weak form - II (12:03) |
Krishna Garikipati
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11.05. Response to a question (00:51) |
Krishna Garikipati
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11.06. Dirichlet boundary conditions; the final matrix-vector equations (16:57) |
Krishna Garikipati
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11.07. Time discretization; the Euler family - I (22:37) |
Krishna Garikipati
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11.08. Time discretization; the Euler family - II (9:55) |
Krishna Garikipati
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11.09. The v-form and d-form (20:54) |
Krishna Garikipati
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11.09ct. 1. Coding Assignment 04 |
Krishna Garikipati
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11.09ct. 1. Coding Assignment 04 Template |
Krishna Garikipati
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11.09ct. 1. Coding Assignment 4 - I (11:10) |
Gregory Teichert
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11.09ct. 2. Coding Assignment 4 - II (13:53) |
Gregory Teichert
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11.10. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - I (17:24) |
Krishna Garikipati
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11.11. Analysis of the integration algorithms for first order, parabolic equations; modal decomposition - II (12:55) |
Krishna Garikipati
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11.12. Modal decomposition and modal equations - I (16:00) |
Krishna Garikipati
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11.13. Modal decomposition and modal equations - II (16:01) |
Krishna Garikipati
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11.14. Modal equations and stability of the time-exact single degree of freedom systems - I (10:49) |
Krishna Garikipati
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11.15. Modal equations and stability of the time-exact single degree of freedom systems - II (17:38) |
Krishna Garikipati
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11.16. Stability of the time-discrete single degree of freedom systems (23:25) |
Krishna Garikipati
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11.17. Behavior of higher-order modes; consistency - I (18:57) |
Krishna Garikipati
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11.18. Behavior of higher-order modes; consistency - II (19:51) |
Krishna Garikipati
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11.19. Convergence - I (20:49) |
Krishna Garikipati
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11.20. Convergence - II (16:38) |
Krishna Garikipati
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| Document Title | Creator | Downloads | License |
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12.01. The strong and weak forms (16:37) |
Krishna Garikipati
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12.02. The finite-dimensional and matrix-vector weak forms - I (10:37) |
Krishna Garikipati
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12.03. The finite-dimensional and matrix-vector weak forms - II (16:00) |
Krishna Garikipati
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12.04. The time-discretized equations (23:15) |
Krishna Garikipati
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12.05. Stability - I (12:57) |
Krishna Garikipati
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12.06. Stability - II (14:35) |
Krishna Garikipati
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12.07. Behavior of higher-order modes (19:32) |
Krishna Garikipati
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12.08. Convergence (20:54) |
Krishna Garikipati
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