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.


Class Information

  • Hours: Tuesdays and Thursdays  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)  

Office: B203, Main academic building, UTSI
Phone: (931) 393-7334
Skype username: rpabedi


  • Relevant course offered by the instructor in Spring 2016: Computer Methods in Dynamics of Continua.
  • HW0: link (NOT required to return this HW assignment, it is just for your information and review on material related to balance laws)
  • HW1: link Due 02/25/2016.
  • UTSI accounts / Ansys commercial software:
    • Contact: Kanawful Massingille (Installing Ansys and VPN – contact the instructor first).
    • Setting up Ansys and connecting to UTSI network. This link summarizes the following steps.
      • VPN (link) must be installed to connect to UTSI network before launching Ansys
      • Ansys can be downloaded from here (Look for Anys 15.0). Run the file Install.bat from unzipped file to install Ansys. Note: Ansys only installed on Windows and linux systems. For more information contact the instructor.
  • HW2, Commerical code Truss example: link Due: 03/01/2016.
  • HW3, Discretization: link Due: 03/08/2016.
  • Commercial code term project, dental crown analysis: link Due: 03/24/2016.
  • Midterm: link Due: 03/31/2016 (late submission of the midterm by 04/05 is accepted) .
  • HW4, 1D elements: link Due: 04/05/2016.
  • Coding Term project: Due 04/26/2015 for input files and 05/05/2014 for the entire project. You can do the project in groups of two if you have no programming background or are using computer programming languages such as C++, Fortran, rather than programs such as Matlab, Mathematica, Maple, etc.. You need to confirm your group members in case you do not want to do the project individually.
    • 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. The name of the folder in dropbox is:
      You can also download the zipped file from here
    • A pre-recorded lecture on coding the FEM solver in C++ can be found here: mp4
  • HW5, Quadrature: link Due: 04/26/2016.
  • HW6, Isoparametric in 2D, 3D: link Due: 05/08/2016.
  • Final exam (take-home) link Due: 05/03/2016.


  • Ansys: There are many online resources for Ansys. In addition by typing help, N where N is an element or topic number in Ansys command line you can get help on the given topic.
    • For bar elements this demo from Rice University is very detailed and useful. There are many youtube demos as well, such as this video.

Lecture Presentations

Lecture notes can be downloaded from here.

The entire set of handwritten notes can be downloaded from here (onenote file).

Class timeline

  1. 01/14/2016 Lecture: notes          Topics: Introduction to topics covered throughout the course. Start of balance laws.
  2. 01/19/2016 Lecture: notes          Topics: Balance laws, divergence theorem and the derivation of strong forms from balance laws.
  3. 01/21/2016 Lecture: notes          Topics: Derivation of strong forms from balance laws; constitutive equations; essential and natural boundary conditions.
  4. 01/26/2016 Lecture: notes          Topics: Essential and boundary conditions; Derivation of weighted residual statement from the strong form and BCs.
  5. 01/28/2016 Lecture: notes          Topics: Derivation of weak statement from the weighted residual form.
  6. 02/02/2016 Lecture: notes          Topics: Engineering perspective on finite element formulation; Using Ansys for solving truss problems.
  7. 02/04/2016 Lecture: notes          Topics: Using Ansys for solving truss problems, a plate problem; Energy methods (part 1).
  8. 02/09/2016 Lecture: notes          Topics: Energy methods (part 2).
  9. 02/11/2016 Lecture: notes          Topics: Discretization (part 1): Subdomain and collocation methods.
  10. 02/16/2016 Lecture: notes          Topics: Discretization (part 2); Start of numerical examples.
  11. 02/18/2016 Lecture: notes          Topics: Discretization (part 3).
  12. 02/23/2016 Lecture: notes          Topics: Discretization (part 4).
  13. 02/25/2016 Lecture: notes          Topics: Discretization (part 5): Spectral vs. Finite element methods; FEM function spaces; Section 2. Bar Element (part 1).
  14. 03/01/2016 Lecture: notes          Topics: FEM function spaces; Section 2. Bar Element (part 2).
  15. 03/03/2016 Lecture: notes          Topics: FEM function spaces; Section 2. Bar Element (part 3).
  16. 03/08/2016 Lecture: notes          Topics: FEM function spaces; Section 2. Bar Element (part 4): Local (element-level) perspective.
  17. 03/10/2016 Lecture: notes          Topics: FEM function spaces; Section 2. Bar Element (part 5): Local (element-level) perspective. Truss Element.
  18. 03/22/2016 Lecture: notes          Topics: Truss Elements.
  19. 03/24/2016 Lecture: notes          Topics: Beam elements (part 1).
  20. 03/29/2016 Lecture: notes          Topics: Beam elements (part 2); Frame elements.
  21. 03/31/2016 Lecture: notes          Topics: FEM implementation (part 1).
  22. 04/05/2016 Lecture: notes          Topics: FEM implementation (part 2).
  23. 04/07/2016 Lecture: notes          Topics: FEM implementation (part 3).
  24. 04/12/2016 Lecture: notes          Topics: Isoparametric elements, introduction by using a 1D example.
  25. 04/14/2016 Lecture: notes          Topics: Quadrature: Newton-Cotes and Gauss quadrature methods.
  26. 04/19/2016 Lecture: notes          Topics: Isoparametric formulation in 1D: skewedness of coordinate transformation in 1D, full and reduced order quadrature.
  27. 04/21/2016 Lecture: notes          Topics: Isoparametric formulation in 2D: Thermal problem formulation.
  28. 04/22/2016 Lecture: notes          Topics: Isoparametric formulation in 2D: Thermal problem formulation, parent element coordinate.
  29. 04/22/2016 Lecture: notes          Topics: FEM coding project Q & A.
  30. 04/26/2016 Lecture: notes          Topics: Isoparametric formulation: Higher order elements and transition elements.
  31. 04/28/2016 Lecture: notes          Topics: Isoparametric formulation: Solid Mechanics; A very brief review of formulation in dynamic setting and simplicial elements. The following documents were discussed during the class. Apart from document 1 which was discussed earlier in the course, only documents 2 and 3 are relevant to elastostatic formulation discussed in the class. Documents 4 and 5 are for your information.
    1. Solid Mechanics weak formulation:      link              This part was covered earlier in the course and is for your reference.
    2. Strain-Stress relation:                           link              Expression of stress & strain  in 1-index array form (Voigt notation) and related by elasticity matrix.
    3. Solid Mechanics FEM formulation:       link               FEM formulation of stiffness matrix for 2D and 3D solid mechanics.
    4. Elastodynamics:                                    link                This document is an overview of the previous 3 files in less detail but includes intertia (Mä) and damping terms (Cå). It also has an example of the assembly of M and C. This instructor’s computer methods in dynamics of continua discusses dynamic problems in much more detail.
    5. Simplicial elements:                             link                This document discusses simplicial natural coordinates, how FE shape functions are formed for simplicial elements (triangle and tetrahedron), and the quadrature points for simplicial elements. You can skip the parts about proofs of some concept in the document.

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