## 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

- 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).
- Coding Term project: (Due 12/07/2017) 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.
- HW5: link (Due 11/28/2017).
- HW6: link (Due 12/12/2017).
- Final exam (take-home) link Due: 12/05/2017.

## 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 (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

- 08/24/2017 Lecture: notes Topics: Introduction to topics covered throughout the course. Start of balance laws.
- 08/29/2017 Lecture: notes Topics: Balance law to strong form.
- 08/29/2017 Lecture: notes Topics: Constitutive equations; Boundary Value Problem; Dirichlet and Neumann BCs.
- 09/05/2017 Lecture: notes Topics: Weighted Residual Statement.
- 09/07/2017 Lecture: notes Topics: Weighted Residual Statement (various forms), weak form (introduction).
- 09/12/2017 Lecture: notes Topics: Weak statements (beam and elastostatics). Engineering view of FEM (part 1).
- 09/14/2017 Lecture: notes Topics: Engineering view of FEM (part 2); energy methods (part 1).
- 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. - 09/21/2017 Lecture: notes Topics: Introduction to Ansys; Discretization (part 1).
- 09/26/2017 Lecture: notes Topics: Numerical example for discretization: subdomain, collocation, finite difference, least square, and Galerkin methods).
- 09/28/2017 Lecture: notes , Q&ASession for HW1, video Topics: Numerical example for finite element and Ritz methods; comparison of solutions.
- 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*. - 10/05/2017 Lecture: notes Topics: Finite element formulation for bars. Global view, part 1.
- 10/10/2017 Lecture: notes Topics: Finite element formulation for bars. Global view, part 2 (Dirichlet Boundary condition & concentrated forces).
- 10/12/2017 Lecture: notes Topics: Finite element formulation for bars. Global view, part 3 (Numerical example).
- 10/17/2017 Lecture: notes Topics: Finite element formulation for bars. Local view, part 1.
- 10/19/2017 Lecture: notes Topics: Finite element formulation for bars. Local view, part 2, Trusses, part 1.
- 10/24/2017 Lecture: notes Topics: Finite element formulation for bars. Local view, part 2, Trusses, part 2 ( f + p versus f approaches).
- 10/26/2017 Lecture: notes Topics: Finite element formulation for beams (part 1).
- 10/31/2017 Lecture: notes Topics: Finite element formulation for beams (part 2); finite element code development (part 1).
- 11/02/2017 Lecture: notes Topics: Finite element code development (part 2).
- 11/07/2017 Lecture: notes Topics: Finite element code development (part 3).
- 11/14/2017 Lecture: notes Topics: Higher order elements, example 2nd order bar element.
- 11/16/2017 Lecture: notes Topics: Newton-Cotes and Gauss quadrature rules.
- 11/21/2017 Lecture: notes Topics: Isoparametric elements in 2D/3D (part 1).
- 11/28/2017 Lecture: notes Topics: Isoparametric elements in 2D/3D (part 2, parent element coordinate).
- 11/30/2017 Lecture: notes Topics: Isoparametric elements in 2D/3D (part 3, integration, zero modes, higher order elements).
- 12/05/2017 Lecture: notes Topics: Isoparametric elements in 2D/3D (part 4, connecting elements, simplicial elements).

12/01/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