Continuum Physics by Krishna Garikipati (U of Michigan)

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source: openmichigan    2014年5月30日
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The idea for these Lectures on Continuum Physics grew out of a short series of talks on materials physics at University of Michigan, in the summer of 2013. Those talks were aimed at advanced graduate students, post-doctoral scholars, and faculty colleagues. From this group the suggestion emerged that a somewhat complete set of lectures on continuum aspects of materials physics would be useful. The lectures that you are about to dive into were recorded over a six-week period at the University. Given their origin, they are meant to be early steps on a path of research in continuum physics for the entrant to this area, and I daresay a second opinion for the more seasoned exponent of the science. The potential use of this series as an enabler of more widespread research in continuum physics is as compelling a motivation for me to record and offer it, as is its potential as an open online class.
This first edition of the lectures appears as a collection of around 130 segments (I confess, I have estimated, but not counted) of between 12 and 30 minutes each. The recommended single dose of online instruction is around 15 minutes. This is a recommendation that I have flouted with impunity, hiding behind the need to tell a detailed and coherent story in each segment. Still, I have been convinced to split a number of the originally longer segments. This is the explanation for the proliferation of Parts I, II and sometimes even III, with the same title. Sprinkled among the lecture segments are responses to questions that arose from a small audience of students and post-doctoral scholars who followed the recordings live. There also are assignments and tests.
The roughly 130 segments have been organized into 13 units, each of which may be a chapter in a book. The first 10 units are standard fare from the continuum mechanics courses I have taught at University of Michigan over the last 14 years. As is my preference, I have placed equal emphasis on solids and fluids, insisting that one cannot fully appreciate the mechanical state of one of these forms of matter without an equal appreciation of the other. At my pace of classroom teaching, this stretch of the subject would take me in the neighborhood of 25 lectures of 80 minutes each. At the end of the tenth of these units, I have attempted, perhaps clumsily, to draw a line by offering a roadmap of what the viewer could hope to do with what she would have learned up to that point. It is there that I acknowledge the modern masters of continuum mechanics by listing the books that, to paraphrase Abraham Lincoln, will enlighten the reader far above my poor power to add or detract.
At this point the proceedings also depart from the script of continuum mechanics, and become qualified for the mantle of Continuum Physics. The next three units are on thermomechanics, variational principles and mass transport--subjects that I have learned from working in these areas, and have been unable to incorporate in regular classes for a sheer want of time. In the months and years to come, new editions of these Lectures on Continuum Physics will feature an enhancement of breadth and depth of these three topics, as well as topics in addition to them.
Finally, a word on the treatment of the subject: it is mathematical. I know of no other way to do continuum physics. While being rigorous (I hope) it is, however, neither abstract nor formal. In every segment I have taken pains to make connections with the physics of the subject. Props, simple but instructive, have been used throughout. A deformable plastic bottle, water and food color have been used--effectively, I trust. The makers of Lego, I believe, will find reason to be pleased. Finally, the time-honored continuum potato has been supplanted by an icon of American life: the continuum football.
Krishna Garikipati
Ann Arbor, December 2013

01.01. Introduction 18:20
01.01. Response to a question 1:34
01.02. Vectors I 14:56
01.02. Response to a question 2:04
01.03. Vectors II 27:25
01.04. Vectors III 24:41
02.01. Tensors I 15:31
02.02. Tensors II 11:59
02.02. Response to a question 17:18
02.03. Tensors III 21:16
02.04. Tensor properties I 14:04
02.05. Tensor properties I 16:07
02.06. Tensor properties II 15:43
02.07. Tensor properties II 13:55
02.08. Tensor properties III 1:02
02.09. Vector and tensor fields 9:22
02.10. Vector and tensor fields 16:54
03.01. Configurations 14:29
03.02. Configurations 14:31
03.03. Motion 18:32
03.03. Response to a question 2:56
03.03. Response to a follow up question 2:59
03.04. The Lagrangian description of motion 14:32
03.05. The Lagrangian description of motion 15:35
03.06. The Eulerian description of motion 14:10
03.07. The Eulerian description of motion 14:07
03.08. The material time derivative 14:11
03.09. The material time derivative 16:52
03.09. Response to a question 1:21
04.01. The deformation gradient: mapping of curves 23:41
04.02. The deformation gradient: mapping of surfaces and volumes 18:08
04.03. The deformation gradient: mapping of surfaces and volumes 14:26
04.04. The deformation gradient: a first order approximation of the deformation 22:51
04.05. Stretch and strain tensors 21:00
04.06. Stretch and strain tensors 10:14
04.06. Response to a question 4:43
04.07. The polar decomposition I 22:03
04.08. The polar decomposition I 11:31
04.09. The polar decomposition II 14:21
04.10. The polar decomposition II 16:12
04.10. Response to a question 3:00
04.11. Velocity gradients, and rates of deformation 15:36
04.12. Velocity gradients, and rates of deformation 15:46
05.01. Balance of mass I 19:35
05.02. Balance of mass I 8:33
05.03. Balance of mass II 27:08
05.04. Balance of mass II 14:51
05.05. Reynolds' transport theorem I 19:07
05.06. Reynolds' transport theorem I 10:25
05.07. Reynolds' transport theorem II 20:36
05.08. Reynolds' transport theorem III 23:33
05.08. Response to a question 8:47
05.09. Linear and angular momentum I 24:38
05.10. Linear and angular momentum II 18:18
05.11. The moment of inertia tensor 16:41
05.12. The moment of inertia tensor 27:50
05.13. The rate of change of angular momentum 18:37
05.14. The balance of linear and angular momentum for deformable, continuum bodies 26:31
05.15. The balance of linear and angular momentum for deformable, continuum bodies 18:38
05.16. The Cauchy stress tensor 26:17
05.17. Stress-- An Introduction 19:55
06.01. Balance of energy 23:03
06.01. Response to a question 11:13
06.01. Response to a follow up question 6:58
06.02. Additional measures of stress 25:05
06.03. Additional measures of stress 15:54
06.03. Response to a question 6:50
06.03. Response to a follow up question 12:34
06.04. Work conjugate forms 25:57
06.05. Balance of linear momentum in the reference configuration 29:06
07.01. Equations and unknowns--constitutive relations 15:29
07.01. Response to a question 5:35
07.02. Constitutitve equations 27:03
07.03. Elastic solids and fluids--hyperelastic solids 29:16
07.03. Response to a question 5:54
08.01. Objectivity--change of observer 17:44
08.02. Objectivity--change of observer 14:47
08.03. Objective tensors, and objective constitutive relations 20:10
08.04. Objective tensors, and objective constitutive relations 11:06
08.05. Objectivity of hyperelastic strain energy density functions 29:18
08.06. Examples of hyperelastic strain energy density functions 23:21
08.07. Examples of hyperelastic strain energy density functions 10:08
08.07. Response to a question 4:36
08.08. The elasticity tensor in the reference configuration 14:27
08.09. Elasticity tensor in the current configuration--objective rates 25:00
08.10. Elasticity tensor in the current configuration--objective rates 11:40
08.11. Objectivity of constitutive relations for viscous fluids 19:06
08.12. Models of viscous fluids 28:15
08.12. Response to a question 3:50
08.13. Summary of initial and boundary value problems of continuum mechanics 25:21
08.14. An initial and boundary value problem of fluid mechanics--the Navier Stokes equations 18:23
08.15. An initial and boundary value problem of fluid mechanics--the Navier Stokes equation 14:58
08.16. An initial and boundary value problem of fluid mechanics II 20:38
08.17. Material symmetry 1--Isotropy 28:26
08.17. Response to a question 3:18
08.18. Material symmetry 2--Isotropy 11:04
08.19. Material symmetry 2--Isotropy 23:16
08.20. Material symmetry 3--Isotropy 23:54
09.01. A boundary value problem in nonlinear elasticity I 17:16
09.02. A boundary value problem in nonlinear elasticity I 13:25
09.02. Response to a question 7:15
09.03. A boundary value problem in nonlinear elasticity II--The inverse method 17:43
09.03. Response to another question 12:02
10.01. Linearized elasticity I 12:19
10.02. Linearized elasticity I 18:36
10.03. Linearized elasticity II 16:03
10.04. Linearized elasticity II 16:24
10.04. Response to a question 3:44
10.05. Classical continuum mechanics: Books, and the road ahead 22:26
11.01. The first law of thermodynamics the balance of energy 15:57
11.02. The first law of thermodynamics the balance of energy 12:30
11.03. The first law of thermodynamics the balance of energy 16:41
11.04. The second law of thermodynamics the entropy inequality 14:48
11.05. Legendre transforms the Helmholtz potential 20:23
11.06. The Clausius Planck inequality 17:24
11.07. The Clausius Duhem inequality 22:57
11.07. Response to a question 5:33
11.08. The heat transport equation 19:11
11.09. Thermoelasticity 25:23
11.10. The heat flux vector in the reference configuration 22:53
12.01. The free energy functional 12:37
12.02. The free energy functional 19:59
12.03. Extremization of the free energy functional variational derivatives 26:44
12.04. Euler Lagrange equations corresponding to the free energy functional 27:36
12.05. The weak form and strong form of nonlinear elasticity 23:42
12.06. The weak form and strong form of nonlinear elasticity 20:17
13.01. The setting for mass transport 19:20
13.02. The setting for mass transport 11:38
13.03. Aside A unified treatment of boundary conditions 20:25
13.04. The chemical potential 20:19
13.05. The chemical potential 14:52
13.06. Phase separation non convex free energy 16:05
13.07. Phase separation non convex free energy 17:31
13.08. The role of interfacial free energy 27:07
13.09. The Cahn Hilliard formulation 23:27
13.10. The Cahn Hilliard formulation 18:22

Introduction to Finite Element Methods by Krishna Garikipati (U of Michigan)

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source: openmichigan    2014年5月30日
View course on Open.Michigan: http://open.umich.edu/find/open-educa...
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Help us caption & translate this video! http://amara.org/v/PcPx/

01.01. Introduction, Linear Elliptic Partial Differential Equations (Part 1) 14:47
01.02. Introduction, Linear Elliptic Partial Differential Equations (Part 2) 13:02
01.03. Boundary Conditions 22:19
01.04. Constitutive relations 20:07
01.05. Strong Form of the Partial Differential Equation, Analytic Solution 22:45
01.06. Weak Form of the Partial Differential Equation (Part 1) 12:30
01.07. Weak Form of the Partial Differential Equation (Part 2) 15:06
01.08. Equivalence Between the Strong and Weak Forms (Part 1) 24:21
01.08ct. 1. Intro to C++ (Running Your Code, Basic Structure, Number Types, Vectors) 21:10
01.08ct. 2. Intro to C++ (Conditional Statements, "for" Loops, Scope) 19:28
01.08ct. 3. Intro to C++ (Pointers, Iterators) 14:02
02.01. The Galerkin, or finite dimensional weak form 23:15
02.01. Response to a question 7:29
02.02. Basic Hilbert Spaces (Part 1) 15:52
02.03. Basic Hilbert Spaces (Part 2) 9:29
02.04. FEM for the One Dimensional, Linear Elliptic PDE 22:54
02.04. Response to a question 6:22
02.05. Basis Functions (Part 1) 14:56
02.06. Basis Functions (Part 2) 14:44
02.07. The Bi-Unit Domain (Part 1) 11:45
02.08. The Bi-Unit Domain (Part 2) 16:20
02.09. Finite Dimensional Weak Form as a Sum Over Element Subdomains (Part 1) 16:09
02.10. Finite Dimensional Weak Form as a Sum Over Element Subdomains (Part 2) 12:25
02.10ct. 1. Intro to C++ (Functions) 13:28
02.10ct. 2. Intro to C++ (C++ Classes) 16:44
03.01. The Matrix-Vector Weak Form - I (Part 1) 16:27
03.02. The Matrix-Vector Weak Form - I (Part 2) 17:45
03.03. The Matrix-Vector Weak Form - II (Part 1) 15:38
03.04. The Matrix-Vector Weak Form - II (Part 2) 13:51
03.05. The Matrix-Vector Weak Form - III (Part 1) 22:32
03.06. The Matrix-Vector Weak Form - III (Part 2) 13:23
03.06ct1 Dealii.org, Running Deal.II on a Virtual Machine with Oracle Virtualbox 13:00
03.06ct. 2. Intro to AWS; Using AWS on Windows 24:44
03.06ct2. Correction 3:32
03.06ct. 3. Using AWS on Linux and Mac OS 7:43
03.07. The Final Finite Element Equations in Matrix-Vector form (Part 1) 22:05
03.08. The Final Finite Element Equations in Matrix-Vector form (Part 2) 18:24
03.08. Response to a question 4:36
03.08ct Coding Assignment 1 (main1.cc, Overview of C++ Class in FEM1.h) 19:35
04.01. The Pure Dirichlet Problem (Part 1) 18:16
04.02. The Pure Dirichlet Problem (Part 2) 17:42
04.02. Correction to boardwork 1:01
04.03. Higher Polynomial Order Basis Functions - I 22:56
04.03. Correction to boardwork 0:58
04.04. Higher Polynomial Order Basis Functions - 1 (Part 2) 16:39
04.05. Higher Polynomial Order Basis Functions - II (Part 1) 13:39
04.06. Higher Polynomial Order Basis Functions - III 23:24
04.06ct. Coding Assignment 1 (Functions: Class Constructor to "basis_gradient") 14:41
04.07. The Matrix Vector Equations for Quadratic Basis Functions - I (Part 1) 21:20
04.08. The Matrix Vector Equations for Quadratic Basis Functions - I (Part 2) 11:54
04.09. The Matrix Vector Equations for Quadratic Basis Functions - II (Part 1) 19:10
04.10. The Matrix Vector Equations for Quadratic Basis Functions - II (Part 2) 24:09
04.11. Numerical Integration -- Gaussian Quadrature 13:58
04.11ct. 1. Coding Assignment 1 (Functions: "generate_mesh" to "setup_system") 14:22
04.11ct.2. Coding Assignment 1 (Functions: "assemble_system") 26:59
05.01. Norms (Part 1) 18:23
05.01. Correction to boardwork 0:57
05.01ct. 1. Coding Assignment 1 (Functions: "solve" to "I2norm_of_error") 10:58
05.01ct.2. Visualization Tools 7:18
05.02. Norms (Part 2) 18:22
05.02. Response to a question 5:46
05.03. Consistency of the Finite Element Method 24:28
05.04. The Best Approximation Property 21:33
05.05. The "Pythagorean Theorem" 13:15
05.05. Response to a question 3:32
05.06. Sobolev Estimates and Convergence of the Finite Element Method 23:51
05.07. Finite Element Error Estimates 22:08
06.01. Functionals, Free Energy (Part 1) 17:39
06.02. Functionals, Free Energy (Part 2) 13:21
06.03. Extremization of Functionals 18:31
06.04. Derivation of the Weak Form Using a Variational Principle 20:10
07.01. The Strong Form of Steady State Heat Conduction and Mass Diffusion (Part 1) 18:25
07.02. The Strong Form of Steady State Heat Conduction and Mass Diffusion (Part 2) 19:01
07.02. Response to a question 1:28
07.03. The Strong Form, continued 19:28
07.03. Correction to boardwork 0:43
07.04. The Weak Form 24:34
07.05. The Finite Dimensional Weak Form (Part 1) 12:36
07.06. The Finite Dimensional Weak Form (Part 2) 15:57
07.07. Three-Dimensional Hexahedral Finite Elements 21:31
07.08. Aside: Insight to the Basis Functions by Considering the Two-Dimensional Case 16:44
07.09. Field Derivatives: The Jacobian (Part 1) 12:39
07.10. Field Derivatives: The Jacobian (Part 2) 14:21
07.11. The Integrals in Terms of Degrees of Freedom 16:26
07.12. The Integrals in Terms of Degrees of Freedom - Continued 20:56
07.13. The Matrix-Vector Weak Form (Part 1) 17:20
07.14. The Matrix-Vector Weak Form (Part 2) 11:20
07.15. The Matrix-Vector Weak Form, continued (Part 1) 17:22
07.15. Correction to boardwork 1:01
07.16. The Matrix Vector Weak Form, continued (Part 2) 16:09
07.17. The Matrix-Vector Weak Form, continued further (Part 1) 17:41
07.17. Correction to boardwork 0:48
07.18. The Matrix-Vector Weak Form, continued further (Part 2) 17:19
08.01. Lagrange Basis Functions in 1 Through 3 Dimensions (Part 1) 18:59
08.02. Lagrange Basis Functions in 1 through 3 dimensions (Part 2) 12:37
08.02ct. Coding Assignment 2 (2D Problem) - I 13:34
08.03. Quadrature Rules in 1 Through 3 Dimensions 17:04
08.03ct. 1. Coding Assignment 2 (2D Problem) - II 13:51
08.03ct. 2. Coding Assignment 2 (3D Problem) 6:53
08.04. Triangular and Tetrahedral Elements-Linears (Part 1) 6:52
08.05. Triangular and Tetrahedral Elements Linears (Part 2) 16:30
09.01. The Finite Dimensional Weak Form and Basis Functions (Part 1) 20:40
09.02. The Finite Dimensional Weak Form and Basis Functions (Part 2) 19:13
09.03. The Matrix Vector Weak Form 19:07
09.04. The Matrix Vector Weak Form (Part 2) 9:43
09.04. Correction to boardwork 1:53
10.01. The Strong Form of Linearized Elasticity in Three Dimensions (Part 1) 9:59
10.02. The Strong Form of Linearized Elasticity in Three Dimensions (Part 2) 15:45
10.03 The Strong Form, continued 23:55
10.04. The Constitutive Relations of Linearized Elasticity 21:10
10.05. The Weak Form (Part 1) 17:38
10.05. Response to a Question 7:56
10.06. The Weak Form (Part 2) 20:24
10.07. The Finite-Dimensional Weak Form-Basis Functions (Part 1) 18:24
10.08. The Finite-Dimensional Weak Form-- Basis functions (Part 2) 10:01
10.09. Element Integrals (Part 1) 20:46
10.09. Correction to boardwork 0:54
10.10 Element Integrals (Part 2) 6:46
10.11. The Matrix-Vector Weak Form (Part 1) 19:01
10.12. The Matrix Vector-Weak Form (Part 2) 12:12
10.13. Assembly of the Global Matrix-Vector Equations (Part 1) 20:41
10.14 Assembly of the Global Matrix-Vector Equations II 9:17
10.14. Correction to boardwork 2:54
10.14ct. 1. Coding Assignment 3 - I 10:20
10.14ct. 2. Coding Assignment 3 - II 19:56
10.15 Dirichlet Boundary Conditions (Part 1) 21:24
10.16 Dirichlet Boundary Conditions (Part 2) 14:00
11.01 The Strong Form 16:30
11.01. Correction to boardwork 0:44
11.02 The Weak Form, and Finite Dimensional Weak Form (Part 1) 18:45
11.03 The Weak Form, and Finite Dimensional Weak Form (Part 2) 10:16
11.04. Basis Functions, and the Matrix-Vector Weak Form (Part 1) 19:53
11.04. Correction to Boardwork 0:45
11.05. Basis Functions, and the Matrix-Vector Weak Form (Part 2) 12:04
11.05. Response to a question 0:52
11.06. Dirichlet Boundary Conditions; The Final Matrix Vector Equations 16:58
11.07. Time Discretization; The Euler Family (Part 1) 22:38
11.08. Time Discretization; The Euler Family (Part 2) 9:56
11.09. The V-Form and D-Form 20:55
11.09ct. 1. Coding Assignment 4 - I 11:11
11.09ct. 2. Coding Assignment 4 - II 13:54
11.10. Integration Algorithms for First-Order, Parabolic, Equations-Modal Decomposition (Part 1) 17:25
11.11. Integration Algorithms for First-Order, Parabolic, Equations-Modal Decomposition (Part 2) 12:56
11.12. Modal Decomposition and Modal Equations (Part 1) 16:01
11.13. Modal Decomposition and Modal Equations (Part 2) 16:02
11.14. Modal Equations and Stability of the Time Exact Single Degree of Freedom Systems (Part 1) 10:50
11.15. Modal Equations and Stability of the Time-Exact Single Degree of Freedom Systems (Part 2) 17:39
11.16. Stability of the Time-Discrete Single Degree of Freedom Systems 23:26
11.17. Behavior of Higher-Order Modes; Consistency (Part 1) 18:58
11.18. Behavior of Higher-Order Modes; consistency (Part 2) 19:52
11.19. Convergence (Part 1) 20:50
11.20. Convergence (Part 2) 16:39
12.01. The Strong and Weak Forms 16:38
12.02. The Finite-Dimensional and Matrix-Vector Weak Forms (Part 1) 10:38
12.03. The Finite-Dimensional and Matri-Vector Weak Forms (Part 2) 16:01
12.04. The Time-Discretized Equations 23:16
12.05. Stability (Part 1) 12:58
12.06. Stability (Part 2) 14:36
12.07. Behavior of High-Order Modes 19:33
12.08. Convergence 20:55
13.01. Conclusion, and the Road Ahead 9:26