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: Tuesdays and Thursdays 2:50-4:05 pm ET (1:50-3:05 pm CT)
  • Location: Online (If not, I’ll be teaching at UTK:  Dougherty 406 and students at UTSI: Main Academic Building E110 will be remotely connected).

Office Hours

By appointment (email or phone call)  [one_third last=”no”]Office: B203, Main academic building, UTSI[/one_third] [one_third last=”no”][/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, Commercial code Truss example: link Due: 9/06/2021
    • In 2021 version, the GUI does not work for entering forces at FE nodes. Instead, one should 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.
      • Link to the command line for entering point forces. 
  • 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: 9/16/2021.
  • HW2: link (Due 9/28/2021).
  • HW3, Discretization: Link , Matlab code Due: 10/14/2021.
  • HW4: link Due: 11/16/2021.
  • HW5: link Due: 11/29/2021. Matlab files: link.
  • Coding Term project: (Due 12/13/2021 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. 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 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.
  • HW6: link (Due 12/09/2021).
  • Final exam (take-home) link Due: 12/05/2021.

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. 08/19/2021 Lecture: notes,video    Topics: Introduction to topics covered throughout the course. Start of balance laws.
  2. 08/24/2021 Lecture: notes,video    Topic: Introduction to Ansys and bar / truss elements. Simple truss example.
  3. 08/26/2021 Lecture: notesAnsys,notesTheory,video   Topic: Ansys: a 2D example. Derivation of PDEs (divergence theorem).
  4. 08/31/2021 Lecture: notes,video    Topics: Closing the system of equations (constitutive equations, kinematic compatibility, etc.); Boundary conditions (beginning).
  5. 09/02/2021 Lecture: notes,video   Topics: Boundary conditions Beam equation.
  6. 09/07/2021 Lecture: notes,video   Topics: Boundary conditions, Weighted residual method (part 1: different forms of weighted residual method).
  7. 09/14/2021 Lecture: notes,video   Topics:  Derivation of weak form.
  8. 09/16/2021 Lecture: notes,video   Topics:  Weak form for the beam problem; discretization of the solution; energy methods (part 1).
  9. 09/17/2021 Lecture: notes,video   Topics:  Weak form for the beam problem; discretization of the solution; energy methods (part 1).
  10. 09/21/2021 Lecture: notes,video   Topics: Energy methods (part 2); Discretization: Forming a solution that strongly satisfies essential BCs (part 1)
  11. 09/23/2021 Lecture: notes,video   Topics: Discretization: Least square and WRS relation; Subdomain method for 1E bar (part 2)
  12. 09/28/2021 Lecture: notes,video   Topics: Discretization: Collocation method; Finite Difference; Galerkin method (WRM and weak statement) (part 3)
  13. 10/05/2021 Lecture: notes,video   Topics: Discretization: Galerkin (spectral and FEM), Ritz method, least square method, comparing different methods (part 4)
  14. 10/07/2021 Lecture: notes,video   Topics: Discretization: Error analysis, form of stiffness matrix (sparsity, symmetry), Spectral method versus FEM, function space.
  15. 10/12/2021 Lecture: notes,video   Topics: Force vectors from natural BC; FEM global perspective (nodes, elements)
  16. 10/14/2021 Lecture: notes,video   Topics: Force vectors essential BC, General expression for stiffness matrix.
  17. 10/19/2021 Lecture: notes,video   Topics: Force vectors from natural BC; FEM global perspective (nodes, elements); A bar example.
  18. 10/21/2021 Lecture: notes,video   Topics: Element (local) FEM approach, dof and nodes; comparison with the global approach.
  19. 10/26/2021 Lecture: notes,video   Topics: Element (local) FEM approach, dof and nodes; comparison with the global approach.
  20. 10/28/2021 Lecture: notes,video   Topics: FEM and direct calculation of element stiffness matrix; Trusses (part 1).
  21. 11/02/2021 Lecture: notes,video   Topics: Truss element example; comparison of assembly of all dofs vs. only free dofs; Beam elements (part 1): continuity requirement.
  22. 11/04/2021 Lecture: notes,video   Topics: Beam elements (part 2): Stiffness matrix and different types of forces.
  23. 11/09/2021 Lecture: notesA,notesB, video   Topics: Beam elements (part 2): Stiffness matrix and different types of forces. Equivalent versus element reaction forces; Frame Elements; Finite Element implementation (part 1).
  24. 11/11/2021 Lecture: notes, video   Topics: Finite Element implementation (part 2).
  25. 11/16/2021 Lecture: notes, video   Topics: Finite Element implementation (part 3);Higher order elements: Motivation. 1D elements: part 1.
  26. 11/18/2021 Lecture: notes, video Topics: Higher order elements: Motivation. 1D elements: part 2. Quadrature (part 1).
  27. 11/19/2021 Lecture: notes, video Topics: Quadrature: Newton Cotes and Gauss Quadrature.
  28. 11/23/2021 Lecture: notes, video Topics: Full integration, reduced order integration; reduced order integration and the rank of stiffness. 
  29. 11/30/2021 Lecture: notes, video Topics: 2D and 3D elements: Coordinate transformation between parent and actual element coordinates; Topics: 2D and 3D elements: Integration.

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.

Office hours timeline

  1. 09/30/2021 Lecture: notes,video   Topics: HW2.

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