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 12:55 – 2:10 pm ET (11:55 am – 1:10 pm CT)
  • Location: UTK:  Dougherty 406 and students at UTSI: Main Academic Building E110 will be remotely connected).

Office Hours

By appointment, zoom: http://tennessee.zoom.us/j/4441721751

Announcements

  • HW1, Commercial code Truss example: link Due: 9/1/2025
    • Please refer to this link on how a 3-element truss is solved: link.
    • I didn’t have a problem entering forces using Ansys GUI. If you have a problem doing so, use the command line option (Acknowledgment: Matthew Carter):
      • F, node number, label(ie.FY for y dir.), force value → for example F,1,FY,-1.0 for the 3 node truss solved in the class.
  • Ansys:
    • Installation: (free academic version, link). While this is a limited version, it is sufficient for your project and is recommended due to the ease of installation.
    • Make sure in ADPL launcher you use “Shared Memory” under High Performance Computing Setup. 
    • Link to command lines (Acknowledgment: Matthew Carter). 
  • Commercial code term project, link Due: 9/15/2025.
  • HW2, Weighted residual and weak statement: link Due: 09/29/2025
  • HW2: link (Due 10/09/2025).
  • HW3, Discretization: Link , Matlab code figures Due: 10/22/2025.
  • HW4: link Due: 11/19/2025; hint for the last problem: link.
  • Coding Term project: (Due 12/12/2025 by 8 am; No late submission)
    • Truss example in course notes (TrussExt.txt, TrussTest.txt, TrussTestOutput.txt, TrussTestOutputVerbose.txt) and a sample L-shaped frame problem with fixed boundaries at the ends and a moment of value 1.5 applied at L-connection (FrameSmall.txt, FrameSmallOutput.txt, FrameSmallOutputVerbose.txt)
    • Input files for the term project (TrussExt.txt, FrameExt.txt). Make sure your executable runs the files with the correct format, otherwise I cannot check your code.  You can download all these files from here.
    • The output files TrussExtOutput_Incomplete.txt, FrameExtOutput_Incomplete.txt include the solution for the first few nodes for your reference.
    • 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.
    • Note: You do not need to have access to the website address I have cited as the source of the problem. All needed information is already included in the project description.
    • A sample C++ implementation with a few functions was shared with you in the beginning of the course along with come references on C++. The incomplete CFEM code can be downloaded from here. A pre-recorded lecture on coding the FEM solver in C++ can be found here: mp4.
  • HW5: link Due: 12/01/2025 (no late submission). Matlab files: link. You don’t need to submit the problems in green font, but if you do, you’ll receive 10% extra credit on those parts of the grade.
  • Final exam: in-person: Announced on canvas.

Resources

  • Resources for C++: link Read README file, refer to RelevantC++Concepts.docx (skip PhyElement, … discussion near the end as that’s related to my code), read .. RelevantSections.docx.
  • Ansys: There are many online resources for Ansys. In addition, by typing help, N, where N is an element or topic number in the 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)

  • FYI: Material that are more advanced and you can easily skip. Some, like elastodynamic concepts are relevant and are not covered in FEM implementation in this course, as all FEM examples are for steady-state (no time involved).
  • FYI,rel: I suggest that you review them, as they help in having a better understanding of the course. If it seems difficult, at least try to get he big picture from these slides.

Class timeline

Link to all notes.

  1. 08/18/2025 Lecture: notes,video   Topics: Introduction to topics covered throughout the course. Engineering perspective to formulate FEM for bar problems.
  2. 08/20/2025 Lecture: notes,video  Topic: Introduction to Ansys and bar / truss elements. Simple truss example. 2D example (part 1).
  3. 08/25/2025 Lecture: notes,video  Topic: Ansys: a 2D example. Balance laws (part 2); Balance laws (part 1).
  4. 08/27/2025 Lecture: notes,video  Topic: Balance laws (part 2); Closing the system of equations (constitutive equations, kinematic compatibility).
  5. 09/03/2025 Lecture: notes,video  Topic: Boundary conditions; Beam problem; Boundary conditions.
  6. 09/08/2025 Lecture: notes,video  Topic: Weighted Residual Statement and Weak Statement for elastistatics  (part 1).
  7. 09/10/2025 Lecture: notes,video  Topics: Elastostatics (part 2: different forms of weighted residual method); Beam problem; Discrete solution (part 1).
  8. 09/15/2025 Lecture: notes,video  Topics: Energy method (part 1).
  9. 09/17/2025 Lecture: notes,video  Topics: Energy method (part 2).
  10. 09/22/2025 Lecture: notes,video  Topics: Discretization: Numerical examples (part 1), subdomain method.
  11. 09/24/2025 Lecture: notes,video  Topics: Discretization: Collocation and finite difference methods (WRM and weak statement) for 1E bar (part 2)
  12. 09/29/2025 Lecture: notes,video  Topics: Discretization: finite Difference, Galerkin method (WRM and weak statement) for 1E bar, Ritz method  and Least Square Method (part 2)
  13. 10/01/2025 Lecture: notes,video  Topics: Discretization: Finite Element Method, Least Square Method (part 3). Comparison of WRS methods.
  14. 10/08/2025 Lecture: notes,video  Topics:Discretization: Galerkin (spectral and FEM), comparing different methods; Discretization: Error analysis, form of stiffness matrix (sparsity, symmetry), Spectral method versus; function space (FYI); Force vectors from natural BC
  15. 10/13/2025 Lecture: notes,video  Topics: FEM global perspective (nodes, elements) (part 1): Force vector for essential BC; general expression for stiffness matrix.
  16. 10/15/2025 Lecture: notes,video  Topics: Force vectors from natural BC; FEM global perspective (nodes, elements); A bar example; Element (local) FEM approach (part 1).
  17. 10/20/2025 Lecture: notes,video  Topics: dof and nodes; comparison with the global approach., Element FEM approach (part 2).
  18. 10/22/2025 Lecture: notes,video  Topics: Element FEM approach (part 3).
  19. 10/27/2025 Lecture: notes,video  Topics: FEM and direct calculation of element stiffness matrix. Truss formulation; Truss element example (part 1).
  20. 10/29/2025 Lecture: notes,video  Topics: Truss element example (part 2); Beam elements (part 1): continuity requirement; stiffness matrix.
  21. 11/03/2025 Lecture: notes,video  Topics: Beam (part 1). B and K formulas; Beam numerical example.
  22. 11/05/2025 Lecture: notes,video  Topics: Beam  (part 2).
  23. 11/10/2025 Lecture: notes,video  Topics: Beam (part 3): y,  theta, M, and V calculation; Frame elements; Finite Element implementation (part 1).
  24. 11/12/2025 Lecture: notes (Video was not recorded, but you can refer to notes and videos of FEM 2024, end of session 25: 11/18/2024 -> steps 8, 9, 10; and beginning of session 26: 11/20/2024 -> steps 11, 12, 13) Topics: Frame elements; Finite Element implementation (part 2).
  25. 11/17/2025 Lecture: notes,video  Topics: Finite Element implementation (part 3); Higher order elements: Motivation. 1D elements; Quadrature (part 1).
  26. 11/19/2025 Lecture: notes,video  Topics: Quadrature (part 2): Newton-Cotes & Gauss quadrature.
  27. 11/24/2025 Lecture: notes,video  Topics: Quadrature (part 3): Full and reduced integration order; zero modes and rank of the stiffness matrix.
  28. 12/01/2025 Lecture: notes,video  Topics: 2D elements: Heat conduction isoparametric formulation for p = 1 quadrilateral element; Quadrature. 

From previous classes:

  1. 12/04/2023 Lecture: notes,video Topics: Quadrature (part 3), Rank of stiffness and relation to reduced order integration (already covered above); 2D and 3D elements: Coordinate transformation between parent and actual element coordinates.
  2. 12/06/2022 Lecture: notes,video Topics: 2D and 3D elements: Higher order elements, h- and p-adaptivity, isoparametric and subparametric elements, connecting elements.

Notes on higher order elements; hints on HW6 and final exam problems (from 2020): notes , video

12/01/Related material: Documents under item 1 are related to energy methods. Subsequent items: Apart from document 2 which was discussed earlier in the course, only documents 3 and 4 are relevant to elastostatic formulation discussed in the class. Documents 5 and 6 are for your information.

  1. 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.
  2. Derivation of Gauss quadrature points and weights     link               (optional): Also relation to Legendre polynomials.
  3. Solid Mechanics weak formulation:      link              This part was covered earlier in the course and is for your reference.
  4. Strain-Stress relation:                           link              Expression of stress & strain  in 1-index array form (Voigt notation) and related by elasticity matrix.
  5. Solid Mechanics FEM formulation:       link               FEM formulation of stiffness matrix for 2D and 3D solid mechanics.
  6. 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.
  7. 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

  • Jacob, Fish, and Belytschko Ted. A first course in finite elements. Wiley, 2007. link
  • 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