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  11:10-12:25 am EST (10:10-11:25 am CST)
  • Location: UTK:  Dougherty 406                    UTSI: Main Academic Building E110 

Office Hours

By appointment (email or phone call)  

Office: B203, Main academic building, UTSI
Skype username: rpabedi


  • HW1: link (Due 9/26/2017).
  • UTSI accounts / Ansys commercial software:
  • Contact: UTSI IT helpdesk and contact information can be found here. (Installing Ansys and VPN – contact the instructor first).
    • Setting up Ansys and connecting to UTSI network:
    • VPN (link) must be installed to connect to UTSI network before launching Ansys (the executable can also be downloaded from here, saving you the first few steps in the previous file).
    • Download Ansys from here. Run the file Install.bat from unzipped file to install Ansys. Note: Ansys only installs on Windows and linux systems. For more information contact the instructor.
    • Instructions for installing Ansys (link).
  • HW2, Commerical code Truss example: link Due: 10/03/2017.
  • HW3, Discretization: link , Matlab code Due: 10/12/2017.
  • Commercial code term project, dental crown analysis: link Due: 10/19/2017.
  • Make-up class, Thursday 10/5, 11:10 am – 12:40 pm ET.
  • HW4: link (Due 10/31/2017).


  • 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 (Note bar area section should be entered under “sections” in new version of Ansys). There are many youtube demos as well, such as this video. In this project you need to select a group of elements to find min/max stresses. this video shows how this step is done.

Lecture Presentations (link)

Class timeline

  1. 08/24/2017 Lecture: notes          Topics: Introduction to topics covered throughout the course. Start of balance laws.
  2. 08/29/2017 Lecture: notes          Topics: Balance law to strong form.
  3. 08/29/2017 Lecture: notes          Topics: Constitutive equations; Boundary Value Problem; Dirichlet and Neumann BCs.
  4. 09/05/2017 Lecture: notes          Topics: Weighted Residual Statement.
  5. 09/07/2017 Lecture: notes          Topics: Weighted Residual Statement (various forms), weak form (introduction).
  6. 09/12/2017 Lecture: notes          Topics: Weak statements (beam and elastostatics). Engineering view of FEM (part 1).
  7. 09/14/2017 Lecture: notes          Topics: Engineering view of FEM (part 2); energy methods (part 1).
  8. 09/19/2017 Lecture: notes          Topics: Energy methods (part 2).
    Useful links for energy method (not necessary to apply energy approach in the derivation of weak statement) - link Functional optimization: How an equation for first variation of a functional (e.g. equations 93, 95 on slide 78) can be derived. You clearly do not need to read this document for this course and this is only provided as a related material for students that want to understand the logic behind the derivation of equations 93, 95. - link Exact calculation of total, first, and second variations for a simple example: In this document the total variation of the energy functional for the bar problem is directly calculated. The first and second variations are directly obtained and higher variations are zero for this simple functional. It is observed that the first variation is exactly the same as what we would have obtained by equation 96 on slide 78.
  9. 09/21/2017 Lecture: notes          Topics: Introduction to Ansys; Discretization (part 1).
  10. 09/26/2017 Lecture: notes          Topics: Numerical example for discretization:  subdomain, collocation, finite difference, least square, and Galerkin methods).
  11. 09/28/2017 Lecture: notes , Q&ASession for HW1, video        Topics: Numerical example for finite element and Ritz methods; comparison of solutions.
  12. 10/03/2017 Lecture: notes         Topics: Comparison of solutions, comments on applicability of weighted residual vs. weak form; relation of WRM to balance laws, PDEs, etc.
  13. 10/05/2017 Lecture: notes         Topics: Finite element formulation for bars. Global view, part 1.
  14. 10/10/2017 Lecture: notes         Topics: Finite element formulation for bars. Global view, part 2 (Dirichlet Boundary condition & concentrated forces).
  15. 10/12/2017 Lecture: notes         Topics: Finite element formulation for bars. Global view, part 3 (Numerical example).
  16. 10/17/2017 Lecture: notes         Topics: Finite element formulation for bars. Local view, part 1.

10/20/2016A Lecture: notes          Topics: Finite element formulation for bars. Global view, part 2.
10/20/2016B Lecture: notes          Topics: Finite element formulation for bars. Local view, part 1.
10/25/2016 Lecture: notes          Topics: Finite element formulation for bars. Local view, part 2.
10/27/2016 Lecture: notes          Topics: Finite element formulation for bars. Local view, part 3; 1D example / parent element concept.
11/01/2016A Lecture: notes          Topics: Finite element formulation for trusses.
11/01/2016B Lecture: notes          Topics: Finite element formulation for beams (part 1).
11/03/2016 Lecture: notes          Topics: Finite element formulation for beams (part 2).
11/08/2016 Lecture: notes          Topics: Finite element code development (part 1).
11/10/2016 Lecture: notes          Topics: Finite element code development (part 2).
11/17/2016 Lecture: notes          Topics: Finite element code development (part 3); Higher order elements (introduction).
11/22/2016A Lecture: notes          Topics: Higher order elements (introduction); quadrature (part 1).
11/22/2016B Lecture: notes          Topics: Quadrature (part 2).
11/29/2016 Lecture: notes          Topics: Isoparametric elements in 2D/3D (part 1).
12/01/2016 Lecture: notes          Topics: Isoparametric elements in 2D/3D (part 2).

Related material: 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