Class Information

Course Description

Modern computational theory applied to conservation principles across the engineering sciences. Weak forms, extremization, boundary conditions, discrete implementation via finite element, finite difference, finite volume methods. Asymptotic error estimates, accuracy, convergence, stability. Linear problem applications in 1, 2 and 3 dimensions, extensions to non-linearity, non-smooth data, unsteady, spectral analysis techniques, coupled equation systems. Computer projects in heat transfer, structural mechanics, mechanical vibrations, fluid mechanics, heat/mass transport.

Syllabus

Class Information

• Hours: Mondays and Wednesdays  12:40-1:55 pm EST (11:40am-12:55 pm CST)
• Location: UTK:  Doherty 406                    UTSI: Main Academic Building E110

Office Hours

Wednesday 2:40-4:40 pm EST [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 02/02/2015
• HW2: link. Due 02/23/2015
• HW3: link; link to Matlab file. Due 03/15/2015
• Midterm survey
• HW4: link. Due 04/06/2015
• Midterm exam (Take-home): Due 03/26/2015 (9 am local time).
• Term project: Due 04/24/2015 for input files and 05/04/2014 for the entire project. A sample C++ implementation with a few functions implemented can be found in the shared Dropbox folder. If you have not already accessed the folder contact the instructor.
• Optional class for FEM C++ programming 4/17/2015 12:40-1:55 pm EST.
• HW5: link; link to Matlab files: LoadGPs.m, ingegrandSolid1DExample.m, computeK1DSolid.m. Maximum score 350 out of 380 available points (above 350 will be extra credit score); Due 04/20/2015
• HW6: link. Due 05/06/2015
• Final exam (Take-home): Due 05/08/2015 (9 am local time).
• Courses offerings for the  next semesters:
  • Continuum Mechanics (Fall 2015)
  • Fracture Mechanics (Fall 2015)
  • Dynamic FEMs (Spring or Fall 2016): Classification of dynamic problems into parabolic and hyperbolic groups; Solution schemes such as FEM and Finite Volume (FV); Time marching schemes; Spacetime methods; Stability and error analysis.
  • Advanced Finite Element Methods: Error and stability analysis; A deeper look at mathematical foundation of FEMs; Very useful for students whose research involves FE formulation.
  • Discontinuous Galerkin Finite Element Methods

Lecture Presentations

Course notes from the last semester can be downloaded as one file from here.

Class timeline

1. 01/12/2015 Lecture: notes          Video: mp4          Topics: Introduction to topics covered throughout the course. Start of balance laws.
2. 01/14/2015 Lecture: notes          Video: mp4          Topics: Static and dynamic balance laws (pages 1-7)
3. 01/16/2015 Lecture: notes          Video: mp4          Topics: General form of static and dynamic balance laws (pages 8-14)
4. 01/21/2015 Lecture: notes          Video: mp4          Topics: Balance law to strong form (pages 15-23)
5. 01/26/2015 Lecture: notes          Video: part1, part2          Topics: Boundary value problem: PDEs, Constitutive equations, Boundary condition (pages 24-31)
6. 01/28/2015 Lecture: notes          Video: part1, part2          Topics: Weighted Residual statement (pages 36-44)
7. 02/02/2015 Lecture: notes          Video: mp4          Topics: Weighted residual to Weak form (pages 37-57)
8. 02/04/2015 Lecture: notes          Video: mp4          Topics: Weak form and comparison of various representations, Least square method (pages 58-68; 48-49)
9. 02/09/2015 Lecture: notes          Video: mp4          Topics: Energy method (pages 68-77)
10. 02/11/2015 Lecture: notes          Video: mp4          Topics: Energy method (pages 78-92)
11. 02/18/2015 Lecture: notes          Video: mp4          Topics: Discretization (pages 93-96)
12. 02/23/2015 Lecture: notes          Video: mp4          Topics: Discretization (pages 97-105)
13. 02/25/2015 Lecture: none          Video: none          Topics: Discretization (pages 106-120)
14. 03/02/2015 Lecture: notes          Video: mp4          Topics: Discretization and numerical examples (pages 121-152)
15. 03/04/2015 Lecture: notes          Video: mp4          Topics: Numerical examples for weighted residual discretization methods (pages 153-179)
16. 03/06/2015 Lecture: notes          Video: mp4          Topics: Numerical examples for weighted residual discretization methods / function spaces (pages 180-224)
17. 03/09/2015 Lecture: notes          Video: mp4          Topics: General form of (stiffness) matrix K and force vector F for FEM solution (pages 225-240)
18. 03/11/2015 Lecture: notes          Video: mp4          Topics: Global and local view of FEM assembly  (pages 241-270)
19. 03/13/2015 Lecture: notes          Video: mp4          Topics: Local view (partially) of FEM / truss elements (coordinate transformation) (pages 271-320)
20. 03/23/2015 Lecture: notes          Video: mp4          Topics: Local coordinate system; comparison of FEM and direct approaches for calculating stiffness  matrix (pages 301-311)  truss elements example (pages 321-323)
21. 03/25/2015 Lecture: notes          Video: mp4          Topics: Truss elements example and comparison of two approaches for FEM solution (pages 324-335); FEM implementation: FEM objects (pages 389-402)
22. 03/27/2015 Lecture: notes          Video: mp4          Topics: Overview of the implementation of the term project in C++.
23. 03/30/2015 Lecture: notes          Video: mp4          Topics: Beam and frame elements (pages 336-388)
24. 04/01/2015 Lecture: notes          Video: mp4          Topics: FEM implementations, Steps 1-10 (pages 403-410)
25. 04/06/2015 Lecture: notes          Video: mp4          Topics: FEM implementations  (pages 411-457) and start of higher order elements.
26. 04/08/2015 Lecture: notes          Video: mp4          Topics: higher order elements and numerical quadrature: Newton-Cotes and Gauss quadrature  (1D). Bathe section 5.5 (sections 5.5.1-5.5.4)
27. 04/13/2015 Lecture: notes          Video: mp4          Topics: 2D and 3D problems: Thermal problem, FEM formulation, local conductivity matrix and heat source vectors, element shape functions (beginning). Bathe section 7.2 (up to 7.2.2), Hughes (H2.3 – 2.6)
28. 04/15/2015 Lecture: notes          Video: mp4          Topics: 2D and 3D problems: Thermal problem, shape functions and parent to global coordinate transformation.
29. 04/17/2015 Lecture: notes          Video: mp4          Topics: Term project implementation, Q/A session.
30. 04/20/2015 Lecture: notes          Video: mp4          Topics: 2D and 3D problems: Higher order elements, sub-,super-, and iso-parametric formulations, static condensation, quadrature. Bathe B5.3.1. quad element; Hughes H3.6 Higher order elements; Lagrange polynomials;  3.7 Elements with variable number of nodes.
31. 04/22/2015 Lecture: notes          Video: mp4          Topics: 2D and 3D elastostatics: Rectangular and cubic elements. Bathe B.6.3.3-5 (1D to 3D elastostatics); Hughes H2.8, H2.9.
32. 05/01/2015 Lecture: notes          Video: mp4          Topics: Term project implementation in C++.

• Simplex elements (triangle, tetrahedron, etc.) notes

Selected Bibliography

•  K. J. Bathe; Finite Element Procedures. Cambridge, MA: Klaus-Jurgen Bathe, 2007. ISBN: 9780979004902 (B). link
• T. J. R. Hughes; The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, 2000. ISBN: 978-0486411811 (H). link
• R.D. Cook, D.S. Malkus, M.E. Plesha, R.J. Witt, Concepts and Applications of Finite Element Analysis, Wiley, 4th Edition, 2001.ISBN: 0471356050 (C). link
• o O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu; The Finite Element Method: Its Basis and Fundamentals, Butterworth-Heinemann; 7th edition, 2013. ISBN: 1856176339 (Z). link