Class Information
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 12:40-1:55 pm ET (11:40am-10:55 pm CT)
- Location: UTK: Doherty 406 UTSI: Main Academic Building E110
Office Hours
Tuesday 2:15-4:15 pm and Wednesday 2:15-3:15 pm ET
[one_third last=”no”]Office: F245, Main academic building, UTSI[/one_third]
[one_third last=”no”]Phone: (931) 393-7334[/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
- Midterm exam: 02/27/2014 – sections 1.1 to 1.6 – open notes.
- Final exam: 05/01/2014 from 11-1 CST (12-2 EST): All the material after (and including) solid bar elements – open notes.
- Some suggested problems for mainly for 2D/3D isoparametric elements can be found here.
Lecture Presentations
All presentations can be downloaded as one file from here.
- Overview
- Section 1: continuum and discrete statements (lectures 1:10) Note: All the proofs in section 1 are only for students’ information.
- 1.1 Balance Laws
- 1.2 Strong Form and Boundary Value Problem
- 1.3 Weak Form and Weighted Residual Statement
- 1.4 Energy Methods
- 1.5 Discretization: Weighted Residual Method, Ritz Energy Method
- 1.6 Numerical Example
- 1.7 Appendix: Function spaces
- Section 2: 1D structural Finite Elements (Bar, Truss, Beam)
- 2.1 Global (weighted residual perspective) using bar element
- 2.2 Local perspective: nodes, elements; local to global assembly; local coordinate system
- 2.3 Trusses: Local coordinate rotation; Assembly of free vs (free + prescribed) dof stiffness
- 2.4 Beams and Frames; FE for higher order Differential equations; [latex]C^1[/latex] elements
- Section 3: Computer Implementation of FEM
Lecture Handwritten notes
- 01/23/2014 (Weighted Residual Statement for Euler Bernoulli beam).
- 01/28/2014 (Summary: balance law approach to weak statement; sections of lecture on energy methods).
- 01/30/2014 Energy Methods.
- 02/04/2014 Discrete systems: How to obtain finite number of equations from continuum statements.
- 02/06/2014 Discrete weighted residual method.
- 02/11/2014 Discrete weighted residual, 1D bar example.
- 02/18/2014 Discrete weighted residual, 1D bar example continued.
- 02/19/2014 Discrete weighted residual, 1D bar example continued. Spectral metods.
- 02/20/2014 Function spaces. Start of solid bar finite element.
- 02/25/2014 Finite Element for solid bar element (global view).
- 02/26/2014 Finite Element for solid bar element, assembly and local stiffness matrix.
- 03/04/2014 Finite Element for solid bar element (local view).
- 03/11/2014 Physical interpretation of local stiffness matrix and assembly routine.
- 03/13/2014 Stiffness matrix for all degrees of freedoms (dofs) vs. that for only free dofs; coordinate rotation for truss elements.
- 03/14/2014 Assembly and solution of trusses.
- 03/28/2014 Stages of FEM solution: Preprocessor, solver, and postprocessor.
- 04/03/2014 FEM implementation in Matlab.
- 04/08/2014 Stationary heat equation FEM formulation.
- 04/10/2014 Stationary heat equation FEM formulation (continued).
- 04/15/2014 Element conductivity matrix and heat source vectors (e.g. force vectors).
- 04/17/2014 Parent coordinate system for 2D rectangular elements.
- 04/22/2014 Calculation of stiffness matrix in the parent coordinate system; introduction to quadrature.
- 04/24/2014 Gauss quadrature and higher order finite elements.
- 04/25/2014 Higher order rectangular shape functions; Sub-, iso-, and super-parametric elements.
- Material that were not fully elaborated during class hours:
- Derivation of Gauss quadrature points and weights (optional): Also relation to Legendre polynomials.
- Quadrature scheme and order: Taught in class 04/25/2014; Included for final exam.
- Definition of full and reduced integration order.
- Stiffness/Conductivity matrix rank and zero eigenvalues: Relation to reduced order integration.
- Choosing appropriate integration order.
- Simplex elements (triangles and tetrahedron):
- Why Simplex elements may be preferred over quad/cube elements?
- Natural coordinates and their use for geometry/solution representation (shape functions).
- Jacobian matrix for simplex elements.
- Formation of higher order simplicial elements.
- Quadrature for simplicial elements.
- 2D and 3D elastostatic elements:
- Section 1: Review of balance law, strong form, weighted residual, and weak form for elastostatics.
- Section 2: Vector representation of strain and stress in 2D and 3D; 6 x 6 material stiffness matrix (3D).
- Section 3: FEM formulation for 2D and 3D solid mechanics. Contrast to thermal problem:
- Instead of scalar temperature field T, we solve the vector field of displacment.
- Differential operation L that maps displacement to strain.
Homework Assignments, Term Project, and Final Exam
- HW01: Balance Laws (Due: 01/30/2014)
- HW02: Discretization, Weighted Residual Method (Due: 02/20/2014)
- HW03: Bar and Beam Elements (Due: 04/10/2014)
- Term Project: FEM implementation in Matlab (Due: 04/24/2014 and 05/08/2014)
- Final Exam
Reading Assignments
Students are encouraged to read the material provided before the given classes. Letters B and H stand for:
K. J. Bathe; Finite Element Procedures. Cambridge, MA: Klaus-Jurgen Bathe, 2007. ISBN: 9780979004902 (B).
T. J. R. Hughes; The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, 2000. ISBN: 978-0486411811 (H)
When references are provided for both books, one reference would be enough.
- Lecture 4 (1/21/2014): Strong and Weak statements: H1.1 to H1.4. Optional: Function spaces: H1.10 up to equation 1.10.7 and H Appendix 1.1
- Lecture 5 (1/23/2014): Energy variational formulation and Weighted Residual (WR) method: B3.3.2 to B.3.3.4. Optional: Self-adjoint operators in regard to WR method: B3.3.3.
- Lecture 6 (1/28/2014): Energy methods continued (B 3.3.2 to B.3.3.4); Discrete weighted residual method (WRM): H1.5 or B3.3.3.
- Lecture 7 (1/30/2014): Finite Element 1D Example, bar elements: H1.6-H1.9, H1.11; Finite Difference (FD) in 1D: B3.3.5.
- Lecture 9 (2/6/2014): Course notes section 1.6.
- Lecture 10 (2/11/2014 & 2/13/2014) Course notes section 1.6 and 1.7.
- Lecture 16-17 (02/26/2014 – 03/04/2014) Local element view and load vectors: H1.15, B4.2.4.
- Lecture 19 (3/13/2014) Fixed and free dofs: B4.2.2.
- Lecture 20 (3/14/2014) Coordinate rotation for trusses: B4.2.1 (pages 182 to 187).
- Lecture 23-24 (4/1/2014-4/3/2014): Heat conduction: H2.3-2.6; B7.2 (up to 7.2.2).
- Lectures 25-28 (4/8/2014 – 4/17/2014) 2D isoparametric elements:
- H3.2 Bilinear quadrilateral elements, H3.3 isoparametric elements, H3.6 Higher order elements: Lagrange polynomials; 3.7 Elements with variable number of nodes.
- B5.1 intro, B5.2 1D element, B5.3.1. quad element, B5.3.2 tri element (only triangular elements by area coordinates).
- Lecture 29 (4/22/2014): Numerical integration (quadrature): H3.8; B5.5: Introduction, B5.5.3 Gauss quadrature 1D, B5.5.4 Gauss Quadrature higher dimensions, B5.5.5 quadrature order, B5.5.6 Reduced integration order.
- 04/26/2014: Elastostatics: H2.7, H2.8, H2.9.