## 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 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)

## Announcements

- No class on Thursday 8/18/2016. The first class will be on Tuesday 8/23/2016. One make up class will be held during the semester.
- Make up class Friday 9/9, usual class time: 11:10-12:25 am EST (10:10-11:25 am CST)
- 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 9/29/2016).
- UTSI accounts / Ansys commercial software:
- Installing Ansys and VPN – (contact the instructor first): Kanawful Massingille (kmassing@utsi.edu) and Terry Garner (tgarner@utsi.edu)
- Setting up Ansys and connecting to UTSI network. This
**link**summarizes the following steps.

- Coding Term project: Due 12/01/2016 for input files and 12/05/2016 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: FEM_517/CFEM_DONOT_MODIFY_IT_HERE 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

- HW2: link (Due 10/11/2016).
- HW3: link (Due 10/20/2016).
- Term project 1, use of commercial FEM software for a design problem: link (Due 10/25/2016).
- Make up class Thursday 10/20, 12:40-2:10 pm ET (11:40 am -1:10 pm CT)
- HW4: link (Due 11/08/2016).
- Coding Term project: Due 11/29/2016 for input files and 12/08/2016 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: FEM_517/CFEM_DONOT_MODIFY_IT_HERE 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: 12/01/2016; last problem is an extra credit problem with 50 points. It shows you
- How you can directly obtain quadrature points by solving for the roots of certain orthogonal polynomials.
- Extend the concept of Gauss quadrature to more general integrals (especially those that arise in probability theory). That is, quadrature points and weights can be defined for more general integrals involving kernel of integration (ρ). ) Bathe’s book (section 5.5.3 (equations 5.144-5.149) describes the process of obtaining Gauss points by finding roots of Legendre polynomials). In this assignment, you’ll use this to obtain Gauss points for n = 3 scheme. Also, In part (b), this concept is extended to more general integrals (with kernels (ρ)). Useful link: link

- HW6, Isoparametric in 2D, 3D: link Due: 12/08/2016.
- Final exam (take-home) link Due: 12/03/2016.

## Resources

- 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 (link)

## Class timeline

- 08/23/2016 Lecture: notes Topics: Introduction to topics covered throughout the course. Start of balance laws.
- 08/25/2016 Lecture: notes Topics: Balance laws for steady state (static) and dynamic problems.
- 08/30/2016 Lecture: notes Topics: Derivation of the strong form from a balance law; divergence and localization theorems; constitutive equations.
- 09/06/2016 Lecture: notes Topics: (Initial) Boundary Value Problem; Dirichlet (essential) and Neumann (natural) BCs.
- 09/08/2016 Lecture: notes Topics: Continuum weighted residual (WR) statement.
- 09/13/2016 Lecture: notes Topics: Continuum weak statements; examples from beam equations and elastostatics.
- 09/15/2016 Lecture: notes Topics: Energy method, part I.
- 09/16/2016 Lecture: notes Topics: Energy method, part II. 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.
- 09/20/2016 Lecture: notes Topics: Relation between energy method, virtual work, & strong form / Physical perspespective to FEM
- 09/22/2016 Lecture: notes Topics: Physical perspespective to FEM (part 2); Introduction to commecrial FEM software Ansys
- 09/27/2016 Lecture: notes Topics: Commecrial FEM software Ansys; Start of discretization.
- 09/29/2016 Lecture: notes Topics: Discretization part 2: Discretization of balance law, strong form, energy method, and least square method.
- 10/04/2016 Lecture: notes Topics: Discretization part 3: Weighted residual and weak forms for a numerical example.
- 10/11/2016 Lecture: notes Topics: Discretization part 4: Numerical example.
- 10/13/2016 Lecture: notes Topics: Discretization part 5: Numerical example.
- 10/18/2016 Lecture: notes Topics: Finite element formulation for bars. Global 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.

- Solid Mechanics weak formulation: link This part was covered earlier in the course and is for your reference.
- Strain-Stress relation: link Expression of stress & strain in 1-index array form (Voigt notation) and related by elasticity matrix.
- Solid Mechanics FEM formulation: link FEM formulation of stiffness matrix for 2D and 3D solid mechanics.
- 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.
- 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