Course Description

This course is intended to serve as a sequel to an introductory finite element or computational mechanics courses. It is designed to deepen student’s understanding of the characteristics of elliptic, parabolic, and hyperbolic partial differential equations (PDE) and get familiar with solution techniques for dynamic problems.

Course Objectives

  1. Provide sufficient mathematical background to read the current literature and understand new developments in the field
  2. Familiarize the students with various numerical schemes for continuum dynamics
  3. Relate theory to practical applications in computational science and engineering.
  4. Develop the student’s capabilities for technical communication and independent research in computational science and engineering.

Syllabus

Class Information

  • Hours: Mondays and Wednesdays  2:10-3:25 pm EST (1:10 -2:25 pm CST)
  • Location: UTK:  Doherty 406                    UTSI: Main Academic Building E110

Office Hours

By appointment (email or phone call)  [one_third last=”no”]Office: B203, Main academic building, UTSI[/one_third] [one_third last=”no”]Phone: (931) 393-7334[/one_third] [one_third last=”yes”]Skype username: rpabedi[social_links colorscheme=”” linktarget=”_self” rss=”” facebook=”” twitter=”” dribbble=”” google=”” linkedin=”” blogger=”” tumblr=”” reddit=”” yahoo=”” deviantart=”” vimeo=”” youtube=”” pinterest=”” digg=”” flickr=”” forrst=”” myspace=”” skype=”skype:rpabedi?add”][/one_third]

Announcements

  • HW1: link. Due 02/12/2020
  • HW2: link. Due 03/09/2020                link to supplementary plots
  • HW3: link. Due 03/21/2020               link to supplementary plots (they can also be found in the dropbox folder)
  • HW4: link. Due 04/11/2020
  • HW5: link. Due 04/29/2020
  • HW6: link. Due 05/09/2020               Note: The return of this assignment is optional. It will help you to better understand the concepts of dispersion and dissipation from the last chapter (chapter 7) and well-posedness, dynamic stability, and robustness from chapter 7.
    If you return this assignment half of your total grade (that is a maximum of 100 points) will be added to your total points as extra credit points.
  • Final presentation: 5/6 11am – 1 pm EST (10 am – 12 pm CST) in the usual classrooms.

Presentation file

I am changing the format of the presentation file. Until the transition is complete, please use part1 for previous sections and part2 for new sections. part1     part2

Class timeline

  1. 01/13/2020 Lecture: notes          Topics: Discussion of material to be covered in the course and Classification of PDEs: 1.a and 1.b
  2. 01/15/2020 Lecture: notes          Topics: Classification of PDEs: 1.c  (i. characteristics for 1st order PDEs, classification of 2nd order PDEs).
  3. 01/22/2020 Lecture: notes          Topics: 1.c.iii D’Alembert solution of the wave equation; 1.c.iv systems of PDEs (part 1l).
  4. 01/29/2020 Lecture: notes          Topics: Systems of first order PDEs 1.c.iii.
  5. 02/03/2020 Lecture: notes          Topics: Quasi-linear systems, shocks and expansion waves (brief discussion) 1.c.iv; Riemann solution and 2 methods for linear systems 1.c.v.
  6. 02/05/2020 Lecture: notes          Topics: Riemann solution and 2 methods for linear systems 1.c.v. (part 2); Transmission & Reflection coefficients.
  7. 02/10/2020 Lecture: notes          Topics: 2.a Finite Difference method (part 1).
  8. 02/12/2020 Lecture: notes          Topics: 2.a Finite Difference method (part 2).
  9. 02/17/2020 Lecture: notes          Topics: 2.a Finite Difference method (part 3); 2.b Finite Volume method (part 1).
  10. 02/19/2020 Lecture: notes          Topics:  2.b Finite Volume method (part 2).
  11. 02/24/2020 Lecture: notes          Topics:  2.b Finite Volume method (part 3); 2.c Finite Element formulation for dynamic problems (part 1).
  12. 02/26/2020 Lecture: notes          Topics:  2.c Finite Element formulation for dynamic problems (part 2).
  13. 03/02/2020 Lecture: notes          Topics: 3.a Exact temporal integration; 3.b time marching schemes; 3.c Modal superposition (part 1).
  14. 03/04/2020 Lecture: notes          Topics: 3.c Modal superposition (part 2).
  15. 03/09/2020 Lecture: notes         Topics: 3.c Modal superposition (part 3).
  16. 03/11/2020 Lecture: notes          Topics: 3.c Modal superposition (part 4).
  17. 03/23/2020 Lecture: notes          Topics: 4.a Time marching schemes (part 1): Linear Multi-step methods.
  18. 03/25/2020 Lecture: notes  video        Topics: 4.a Time marching schemes (part 2): Multivariate single-step methods.
  19. 03/30/2020 Lecture: notes  video        Topics: 4.a Time marching schemes (part 3): Runge-Kutta methods.
  20. 04/01/2020 Lecture: notes  video        Topics: Runge-Kutta methods; 5. Analysis of MDOF/SDOF for generalized alpha method (part 1).
  21. 04/06/2020 Lecture: notes  video        Topics: 5. Stability, consistency, and convergence analysis of MDOF/SDOF for generalized alpha method (part 2).
  22. 04/08/2020 Lecture: notes  video          Topics: 5. Stability analysis of Linear Multi-step (LMS) methods (e.g. Central difference, Houbolt) and multivariate single step methods (Mewmark, θ-Wilson).
  23. 04/13/2020 Lecture: notes  video          Topics: 5. Absolute stability, practical consideration in using numerical methods (part 1).
  24. 04/15/2020 Lecture: notes  video          Topics: 5. practical consideration (part 2).
  25. 04/20/2020 Lecture: notes  video          Topic 6: FD analysis consistency; stability (part 1).
  26. 04/22/2020 Lecture: notes  video          Topic 6: FD stability, Introduction to von Neumann analysis (part 2).
  27. 04/27/2020 Lecture: notes  video          Topic 6: FD stability, von Neumann analysis: FTBS, Lax-Friedrichs, and leapfrog schemes (part 3).
  28. 04/29/2020 Lecture: notes  video          Topic 6: Dispersion analysis: Dispersion relation, phase velocity, and physical dispersion and dissipation; Numerical dispersion and dissipation errors with examples from FD methods.

Selected Bibliography

  • [Strikwerda, 2004] Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations. SIAM.
  • [Hughes, 2012] Hughes, T. J. (2012). The finite element method: linear static and dynamic finite element analysis. Courier Corporation.
  • [Bathe, 2006] Bathe, K.-J. (2006). Finite element procedures. Klaus-Jurgen Bathe.
  • [Farlow, 2012] Farlow, S. J. (2012). Partial differential equations for scientists and engineers. Courier Corporation.
  • [LeVeque, 2002] LeVeque, R. L. (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.
  • [Chapra and Canale, 2010] Chapra, S. C. and Canale, R. P. (2010). Numerical methods for engineers, volume 2. McGraw-Hill. 6th edition.