Module 1 : Numerical Methods for PDEs : Course Introduction, Lecture 1

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1 Module 1 : Numerical Methods for PDEs : Course Introduction, Lecture 1 David J. Willis September 7, 2016 David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 1 / 38

2 References and Acknowledgements The following materials were used in the preparation of this lecture: Course notes and slides from MIT Lecture 1 2 L.N. Treffethen and D.Bau III, Numerical Linear Algebra, SIAM. 3 G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press. The author of these slides wishes to thank these sources for helping with the current lecture. David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 2 / 38

3 Table of contents 1 Announcements Enrollment 2 Syllabus and Course Administration Class Format Grading Scheme Grading Scheme Grading Scheme Textbook Schedule 3 Getting Matlab 4 Introducing Partial Differential Equations 5 Programming, Matlab and Some Linear Algebra David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 3 / 38

4 Announcements Enrollment Course (Over-)Enrollment At present there are 19 seats in the class: There is 1 permission number not being employed. 7+ people are on the wait list Exploring the possibility of increased enrollment David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 4 / 38

5 Syllabus and Course Administration Introductions Website: faculty.uml.edu/dwillis/mech5200/ David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 5 / 38

6 Syllabus and Course Administration Introductions Numerical Methods for PDEs : Graduate & Seniors Lecture: Mon/Weds: 5:00pm-6:15pm* Office hours: Monday. 3:00-5:00pm, Tuesday 1pm-2pm. Office: EB-322 (Perry Hall 322) david willis@uml.edu David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 6 / 38

7 Syllabus and Course Administration What is the purpose of this course? This course is a combination of: David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 7 / 38

8 Syllabus and Course Administration What is the purpose of this course? Specifically, you will learn: [FDM] Finite Difference Methods: Theory, Discretization, and Numerical Solutions. [FVM] Finite Volume Methods: Theory, Discretization, and Numerical Solutions. [FEM] Finite Element Methods: Theory, Discretization, and Numerical Solutions. [BEM] Boundary Integral Methods: Theory, Discretization, and Numerical Solutions. And supporting this will be: [MAT] Numerical methods for solving linear and nonlinear systems. [GEO] Discrete Geometry Representation and Mesh Generation David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 8 / 38

9 Syllabus and Course Administration Class Format Blended and Flipped Classroom My challenges Theoretical math in evening is tough to do well! Material is heavily applied and learned best this way. Computer coding takes practice and interaction. My proposed solution: Semi-blended, semi-flipped classroom (modern, woooaah!): minute videos a week on core concepts (watch before class). Meet in class to do blended instruction and coding (matlab). Sometimes code/classwork is due for credit. Go home and do the homework projects. Thoughts? David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 1 9 / 38

10 Syllabus and Course Administration Class Format Videos Video 1-3 uploaded to YouTube sign-up sheet (will send links weekly) David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 110 / 38

11 Syllabus and Course Administration Grading Scheme Grades and Deliverables 60% Homework individually performed homework problems with computer programming (6-7 total, with 3-4 being larger mini-project assignments). 30% Computer Programming Project An individual FDM, FVM or FEM project in a field of your choice. 10% In-class exercises in-class exercises and assignments. David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 111 / 38

12 Syllabus and Course Administration Grading Scheme Grades and Deliverables Homework will cover main concepts discussed in online videos and in class (theory, coding and application). Homework policy: Study and workgroups are permitted; however, what you write and hand-in must be individual. You may not code/implement homework problems together outside of class. Any computer code that is similar to other students will result in a grade of zero. All homeworks & projects will require Matlab or another programming language (matlab/python/c/c++/fortran etc.) Homework will be due on paper, with OneDrive used for electronic code. Since the homework is the source of the majority of your grade, a zero-tolerance policy will be in effect in accordance with the UML academic integrity policy: Integrity.aspx David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 112 / 38

13 Syllabus and Course Administration Grading Scheme Grades and Deliverables Class exercises will be hands-on, pair programming efforts. We will mix groups weekly. Aim is to work together to get a deeper understanding. We ll see this in weeks to come. David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 113 / 38

14 Syllabus and Course Administration Grading Scheme Frequently Asked Questions Questions? Do I need to type my answers for questions that are not implemented on a computer? Do I need to type my project proposal and final report? How do I hand in the portions of the homework that are implemented on a computer? Do I need to hand in color printouts of matlab (or other coding language) results? Can I ask for additional computer coding help? Do I need to know matlab? What if I know no programming languages? How is this course diffferent than John White s? MIT s? Will the lecture videos be available for students to watch? David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 114 / 38

15 Syllabus and Course Administration Textbook Online Materials and Textbook(s) MIT Opencourseware : , , ) Suggestions for texts will appear as references in the lecture notes buy if you want, when you want. Lecture videos to be posted. David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 115 / 38

16 Syllabus and Course Administration Schedule Approximate Schedule Introduction, matlab & some linear algebra today Finite difference methods for elliptical, parabolic and hyperbolic equations (4-5 weeks) Finite volume methods for conservation laws (2-3 weeks) Finite element methods (4-5 weeks) Boundary element methods (whatever time is left at the end) David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 116 / 38

17 Getting Matlab Class Plan & Announcements Virtual labs Login to a virtual machine to use matlab. UMass Lowell Educational Matlab License: David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 117 / 38

18 Getting Matlab Questions? Any questions...? David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 118 / 38

19 Getting Matlab Table of contents 1 Announcements Enrollment 2 Syllabus and Course Administration Class Format Grading Scheme Grading Scheme Grading Scheme Textbook Schedule 3 Getting Matlab 4 Introducing Partial Differential Equations 5 Programming, Matlab and Some Linear Algebra David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 119 / 38

20 Introducing Partial Differential Equations Example PDEs PDE: A differential equation involving an unknown function(s) of several variables and their partial derivatives w.r.t those variables. Partial differential equations are convenient models of physical system behavior e.g., Indicates how u changes with respect to x, y and/or t? They are found everywhere in engineering and mathematics Can you give and example of a PDE? David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 120 / 38

21 Introducing Partial Differential Equations PDEs w/ Smooth Solutions The Poisson Equation (Elliptic): Laplace s Equation (Elliptic): ( κ 2 2 ) u u = κ x u y 2 = f (1) 2 u = 2 u x u y 2 = 0 (2) Electrostatics and capacitance extraction, Potential flow, Torsion in a bar of constant cross section, Gravity driven viscous flow in a channel, Membrane deflection, Ground water flow, Stationary heat transfer... What does the equation mean? David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 121 / 38

22 Introducing Partial Differential Equations Laplace Operator David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 122 / 38

23 Introducing Partial Differential Equations PDEs w/ Smooth Solutions Example: ArbTwoD FD.m Figure : Torsion in a bar solved using Finite Differences (HW assignment?) David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 123 / 38

24 Introducing Partial Differential Equations PDEs w/ Smooth Solutions Let s make the equation a bit more complicated and see what happens. Unsteady heat conduction equation (Parabolic): u = κ 2 u + f (3) t f = u ( 2 ) t κ u x u y 2 (4) (5) We ve simply added a u t term. What does this mean? How have things changed? David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 124 / 38

m Figure : Model problem for unsteady temperature problem David J.

25 Introducing Partial Differential Equations Unsteady heat conduction equation Example: OneD FD Temp EX.m Example: TempImplicit.m Figure : Model problem for unsteady temperature problem David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 125 / 38

26 Introducing Partial Differential Equations PDEs w/ Non-smooth Solutions The 1st Order Wave Equation (Hyperbolic) one direction of information propagation: u + c u = 0 (6) t What does this equation mean? David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 126 / 38

27 Introducing Partial Differential Equations First Order Wave Equation Example: HyperWave EX.m Figure : Model for the wave equation (showing time and space) David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 127 / 38

28 Introducing Partial Differential Equations Combining ideas together yields the convection diffusion equation (transport phenomena): u t + U u = κ 2 u + f (7) Where U, κ > 0, f may be constant or functions in space. U is typically a field velocity, f is a source term and κ is a physical parameter related to the physics of the problem. David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 128 / 38

29 Introducing Partial Differential Equations PDEs : The Convection-Diffusion Equation: DISCUSSION David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 129 / 38

30 Introducing Partial Differential Equations PDEs : The Convection-Diffusion Equation u t + U u = κ 2 u + f (8) u = T (temperature) this is the Heat Transfer Equation u = P (pollutant concentration) Coastal/groundwater engineering u = p (price of an option) Financial Engineering u = u (momentum per unit mass or velocity) Navier Stokes Equation Obviously, the convection-diffusion equation is an important PDE. David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 130 / 38

31 Introducing Partial Differential Equations Additional Important PDEs : The Second Order Wave Equation Some other equations of interest The wave equation: 2 u t 2 c2 2 u x 2 = 0 (9) This equation describes the propagation of a wave through the domain being considered (bi-directional). David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 131 / 38

32 Introducing Partial Differential Equations PDEs : The KdV Equation The KdV (Korteweg - de Vries) equation: u t + 3 u u 6u x 3 x = 0 (10) Weakly non-linear, shallow water equation, optics signal propagation, etc. Example: Solitons InClass.m David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 132 / 38

33 Introducing Partial Differential Equations PDEs : The Hamilton-Jacobi equation and Eikonal equation The H-J equation: The Eikonal equation: u t + F (x, y, z) u(x, y) = 0 (11) F (x, y, z) T (x, y) = 1 (12) What does this equation mean? Use heavily in level sets and path planning. Used at UML for machine tool path planning (CNC) David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 133 / 38

34 Introducing Partial Differential Equations PDEs : The Hamilton-Jacobi equation and Eikonal equation Figure : Solution to the Eikonal Equation for Path Planning David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 134 / 38

35 Introducing Partial Differential Equations Table of contents 1 Announcements Enrollment 2 Syllabus and Course Administration Class Format Grading Scheme Grading Scheme Grading Scheme Textbook Schedule 3 Getting Matlab 4 Introducing Partial Differential Equations 5 Programming, Matlab and Some Linear Algebra David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 135 / 38

36 Programming, Matlab and Some Linear Algebra Compiled vs. Interpreted Coding Languages Computer programming languages are classified as: Interpreted languages (eg. Matlab, Python, Octave, etc.) Code is saved as written Intructions are sequentially compiled as they are reached Slow run-time Fast to prototype and code (less errors) Usually used for prototyping Compiled languages (eg: C/C++/Fortran) Code is converted to machine instructions and saved as an executable Fast run-time just need to run the instructions Slower to prototype and code (more errors) Usually used for production and released codes David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 136 / 38

37 Programming, Matlab and Some Linear Algebra Interpreted Coding Languages Matlab is an interpreted language designed for vector and matrix algebra. Thus, there are ways to make matlab interpreted code more efficient by making use of built in compiled functions. Many of the matrix-matrix, matrix-vector, and vector-vector operations are compiled. Writing vectorized code is much more efficient than loop strucutres example Try the following: Generate a matrix, A, and multiply it by a vector, b. Use for loops in one case and use the operator in the other case We can see that matlabs built-in operations are much better. Try to keep for, while, etc. loops to a minimum. David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 137 / 38

38 Programming, Matlab and Some Linear Algebra Interpreted Coding Languages Octave For those who are interested in a cheaper version of Matlab, octave & FreeMat are very similar to matlab (identical but slower for most things): Also, python is not a significant departure from Matlab and is rapidly gaining/gained popularity. David J. Willis Module 1 : Numerical Methods for PDEs : CourseSeptember Introduction, 7, 2016 Lecture 138 / 38

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