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 11:20 am -12:40 pm ET (10:20-11:40 am CT)
  • Location: UTK:  Dougherty 406 and students at UTSI: Main Academic Building E110 will be remotely connected).

Office Hours

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

Announcements

  • HW1, Commercial code Truss example: link Due: 9/11/2023
    • 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: 9/25/2023.
  • HW2: link (Due 10/11/2023).
  • HW3, Discretization: Link , Matlab code figures Due: 10/23/2023.
  • HW4: link Due: 11/15/2023.
  • HW5: link Due: 12/06/2023. Matlab files: link. You don’t need to submit the problems in green font, but if you do, you’ll get 50% of the grade of those parts as extra credit.
  • Coding Term project: (Due 12/17/2023 by 3 pm; 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 desciption.
    • 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.
  • Final exam (take-home) link Due: 12/14/2023.

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 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. 08/23/2023 Lecture: notes,video   Topics: Introduction to topics covered throughout the course. Engineering perspective to formulate FEM for bar problems.
  2. 08/28/2023 Lecture: notes,video  Topic: Introduction to Ansys and bar / truss elements. Simple truss example.
  3. 08/28/2023 Lecture: notes,video Topic: Ansys: a 2D example. Balance laws (part 1).
  4. 09/06/2023 Lecture: notes,video Topic: Balance laws (part 2); Closing the system of equations (constitutive equations, kinematic compatibility, etc., part 1);
  5. 09/11/2023 Lecture: notes,video Topics: Closing the system of equations (constitutive equations, kinematic compatibility, etc.); Boundary conditions.
  6. 09/13/2023 Lecture: notes,video Topics: Boundary conditions, Weighted residual method and weak statement for the bar problem; Elastostatics (part 1: different forms of weighted residual method).
  7. 09/18/2023 Lecture: notes,video Topics: Elastostatics (part 1: different forms of weighted residual method); Beam problem; Discrete solution (part 1).
  8. 09/20/2023 Lecture: notes,video Topics: Discrete solution (part 2); Weak statement; Energy method (part 1).
  9. 09/25/2023 Lecture: notes,video Topics: Energy method (part 2).
  10. 09/27/2023 Lecture: notes,video Topics: Energy methods (part 3); Discretization: Forming a solution that strongly satisfies essential BCs quation.
  11. 10/02/2023 Lecture: notes,video Topics: Discretization: Numerical examples (part 1).
  12. 10/04/2023 Lecture: notes,video Topics: Discretization: Subdomain, collocation, and finite Difference, Galerkin method (WRM and weak statement) for 1E bar (part 2)
  13. 10/11/2023 Lecture: notes,video Topics: Discretization: Ritz Method, Finite Element Method, Least Square Method (part 3).
  14. 10/16/2023 Lecture: notes,video Topics: Discretization: Galerkin (spectral and FEM), comparing different methods (part 4); Discretization: Error analysis, form of stiffness matrix (sparsity, symmetry), Spectral method versus FEM (par 1).
  15. 10/18/2023 Lecture: notes,video Topics: Spectral method versus FEM (par 2); Function space (FYI); Force vectors from natural BC; FEM global perspective (nodes, elements) (part 1).
  16. 10/23/2023 Lecture: notes,video Topics: Force vector for essential BC; general expression for stiffness matrix; a numerical example.
  17. 10/25/2023 Lecture: notes,video Topics: Force vectors from natural BC; FEM global perspective (nodes, elements); A bar example; Element (local) FEM approach
  18. 10/25/2023 Lecture: notes,video Topics:Topics: dof and nodes; comparison with the global approach., Element FEM approach (part 2).
  19. 11/01/2023 Lecture: notes,video Topics: Element FEM approach (part 3). FEM and irect calculation of element stiffness matrix.
  20. 11/06/2023 Lecture: notes,video Topics: Truss formulation; Truss element example.
  21. 11/06/2023 Lecture: notes,video Topics: Beam elements (part 1): continuity requirement; stiffness matrix and source term force.
  22. 11/13/2023 Lecture: notes,video Topics: Beam numerical example, Equivalent versus element reaction forces.
  23. 11/15/2023 Lecture: notes,video Topics: Frame elements; Finite Element implementation (part 1).
  24. 11/20/2023 Lecture: notes,video Topics: Frame elements; Finite Element implementation (part 2).
  25. 11/27/2023 Lecture: notes,video Topics: Higher order elements: Motivation. 1D elements; Quadrature (part 1).
  26. 11/29/2023 Lecture: notes,video Topics: Quadrature (part 2), full integration order. 
  27. 12/04/2023 Lecture: notes,video Topics: Quadrature (part 3), Rank of stiffness and relation to reduced order integration; 2D and 3D elements: Coordinate transformation between parent and actual element coordinates.
  28. 12/06/2023 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