MAE4700/5700: Finite Element Analysis for Mechanical and Aerospace Design

Cornell University, Fall 2009

Lectures: PHL 403, Tuesdays and Thursdays, 10:10-11:25 am

Recitations (ANSYS instruction): Rhodes 471, Fr 1:25-2:15 pm

Professor Nicholas Zabaras

To see what others think of us and the class, visit RateMyProfessor.com

Reading assignments

  • Lecture 1: Introduction to the course pdf (also read the following MATLAB tutorial.) In preparation for your first recitation (9/4/09) read this handout.
  • Lecture 2: Direct approach and basic finite element program structure pdf
  • Lecture 3: Two- and three-dimensional truss structures pdf
  • Lecture 4: Finite element formulation of beam problems pdf
  • Lecture 5: Strong and weak forms for one-dimensional boundary value problems pdf
  • Lecture 6: One-dimensional shape functions, numerical integration, Gauss quadrature pdf
  • Lecture 7: Finite element approach to one-dimensional boundary value problems pdf
  • Lecture 8: Error analysis in finite element computations pdf
  • Lecture 9: Review of the MatLab program structure for 1D boundary value problems pdf
  • Lecture 10: Strong and weak forms of 2D boundary value problems for scalar fields pdf
  • Lecture 11: Finite element discretization in 2D, triangular and rectangular elements pdf
  • Lecture 12: Finite element approximations for 2D boundary value problems pdf
  • Lecture 13: Element calculations, the master element and transformation equations, finite element calculations in the master element, quadrature rules pdf
  • Lecture 14: An overview of finite elements for 2D calculations pdf
  • Lecture 15: Two-dimensional linear elastic problems, strong and weak formulations pdf
  • Lecture 16: Finite element discretization for two-dimensional linear elasticity problems pdf
  • Lecture 17: Finite element analysis of three-dimensional elasticity problems, solid elements, patch test, stress recovery pdf
  • Lecture 18: Introduction to computational design using finite element analysis pdf
  • Lecture 19: Transient & Dynamics problems: pdf
  • Lecture 20: Introduction to finite element methods for incompressible flows (draft) pdf

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Homework

This class teaches the fundamentals of FEM with hands-on experience in numerical implementation and applications. All homework assignments (~12) involve significant computer modeling using MATLAB versions 7.0 or higher. Each HW will include theoretical (hand calculations) problem(s), MatLab programming of FEM and ANSYS programming. The MatLab programs will be used to reenforce the fundamentals whereas the ANSYS programs will be used to verify your MatLab calculations and to build your ANSYS training that maybe useful for your final project. The weekly recitations will introduce and discuss the ANSYS homework exercises and thus your attendance to recitations is important.

Group HW (3 students maximum) is strongly encouraged. Homework done as a group will be graded as a group. Submit only one solution per group clearly listing the names of the group's members. Groups may change without need for any justification at any time during the semester.

All HWs need to be typed and submitted electronically to this Email address by the designated time. Copying software or results from another group is a strict violation of the class honor code.
  • Homework 1- FEM analysis of 2D truss structures (due Monday September 7th, midnight): HW. This HW requires the following software for plane truss structures.

  • Homework 2- FEM analysis of plane (2D) and space (3D) truss structures (due Monday September 14th, midnight): HW. This HW requires the following software for plane and space truss structures.

  • Homework 3- Truss structures with inclined supports, beam analysis and plane frames (due Monday September 21st, midnight): HW. This HW requires the following software for plane truss structures and beam analysis.

  • Homework 4- FEM solution of one-dimensional boundary value problems (1DBVP) (due Monday September 28th, midnight): HW. This HW requires the following software for 1DBVP.

  • Homework 5- Solution of 1D boundary value problems: Error analysis, higher-order elements and adaptivity (due Monday October 5th, midnight): HW. For the 1st problem, use the 1DBVP software. The second problem requires the development of an adaptive FEM version of the 1DBVP. For this, you can start modifying this code.

  • Homework 6- Solution of 2D boundary value problems: Weak formulations, element calculations, Gauss integration and applying boundary conditions (due Monday October 12th, midnight): HW. This HW requires the 2DBVP software.
    • Also you will need to download this information and software for transfering mesh information from Ansys to the MatLab toolbox.
  • Homework 7- FEM solution of 2DBVP: Heat transfer, torsion problems, confined flow around a wall (due Monday October 19th, midnight): HW. This HW requires the MatLab libraries from HW 6.
  • Homework 8- FEM solution of plane stress and plane strain elasticity problems (due Monday October 26th, midnight): HW. This HW requires the 2DStressAnalysis software. Also you will need to use this software for transfering mesh information from Ansys to the MatLab toolbox.

  • Homework 9- Two-dimensional stress analysis problems with singularities, axisymmetric stress analysis (due Monday November 2nd, midnight): HW. This HW requires that you modify and extend the 2DStressAnalysis software.

  • Homework 10 - Introduction to computational design using finite element analysis (due Monday November 9th, midnight): HW.

  • Homework 11 - Finite elements for transient analysis and dynamical systems (due Monday November 16nd, midnight): HW.

  • Homework 12 - Finite elements in fluid mechanics (due Monday November 23th, midnight): HW. This HW requires extension of the 1DBVP software, the 2DBVP libraries and the attached implementation of a stabilized projection FEM method for incompressible flows.


  • Homework 1 solutions pdf
  • Homework 2 solutions pdf
  • Homework 3 solutions pdf
  • Homework 4 solutions pdf
  • Homework 5 solutions pdf
  • Homework 6 solutions pdf
  • Homework 7 solutions pdf
  • Homework 8 solutions pdf
  • Homework 9 solutions pdf
  • Homework 10 solutions pdf
  • Homework 11 solutions pdf
  • Homework 12 solutions pdf

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Exams

  • Prelim 1 solutions pdf
  • Prelim 2 solutions pdf

Policy for make-up exams: If there is a documented conflict with another exam, we will provide a make-up exam 2-3 hours before our regular exam. We strongly discourage you from requesting a make-up exam at another time and day. This way all students take the same exam and issues of varying degree of difficulty between exams do not arise. We will work with each of you case-by-case if there is an unexpected `emergency'. We are very sorry if the days of the exams are not convenient to your overall (travel) plans. Failure to take any of the exams will result in an automatic loss of grade.

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Student Presentations to the 2009 Swanson Program Advisory Committee Meeting

As part of the 2009 meeting of the Swanson program advisory board on October 26, 2009, a small number of MAE4700 undergraduate students introduced their work. Their presentations are given below:

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Basic course info

Credit hours: 3-4. 3 credits as MAE 4700 for seniors, 4 credits as MAE 570 for graduate students. Each HW set may contain an additional computational problem for those registered under MAE 5700. MAE seniors taking MAE 4700 can enroll for an extra one credit course (MAE 429) to fulfill the MAE design requirement. These students will have to address in a substantial manner design issues in their final project.

Lectures: Tues./Thurs. 10:10 -- 11:25 am, PHL 403 (Prof. N. Zabaras)

Recitation (ANSYS instruction): Fridays 1:25 -- 2:15 pm, 471 Rhodes Hall (Dr. R. Bhaskaran)

Professor: Nicholas Zabaras, 101 Frank H. T. Rhodes Hall.

Teaching Assistant: Xiang Ma

ANSYS instructor: Dr. Rajesh Bhaskaran

HW or exam re-grading: For regrades, please submit your HW or prelim within two days after its return to Xiang Ma. No regrades will be considered after that time. Be sure to attach to your HW/prelim a clear statement explaining what you need to be regraded and why. It is advisable to discuss your concern with the TA during office hrs before officially requesting a regrade.

Office hours (to be held in the Swanson Lab, Rhodes Hall 163):

You are encouraged to attend office hours. The office hours can be of much higher benefit if you already have spent some time working the HW problems and you come prepared to ask the right questions. Note that the office hours are not for us to locate bugs in your computer programs!
  • Mondays: 5:00-6:00 pm (Prof. N. Zabaras).
  • Tuesdays: 8:00-9:00 pm (Xiang Ma).
  • Wednesdays: 4:00-5:00 pm (Dr. Rajesh Bhaskaran, only for ANSYS consultations).
  • Wednesdays: 5:00-6:00 pm (Prof. N. Zabaras)
  • Thursdays: 8:00-9:00 pm (Xiang Ma).
  • Fridays: 2:30-3:30 pm (Dr. Rajesh Bhaskaran, only for ANSYS consultations)

Exam and final project presentation schedule:

The exams are with closed books & notes. You need to be able to demonstrate your understanding of the fundamentals of the FEM as well as your understanding of implementation aspects. Electronic and wireless communications are serious violations of the course policy and prohibited.
  • Exam 1: 10/27/2009, 7:30-9:30 pm, Hollister Hall B14.
  • Exam 2: 11/24/2009, 7:30-9:30 pm, Hollister Hall B14.
  • Final project presentations: Sat. December 5th, 8:00 am- 2:00 pm, Rhodes 178.

Required textbook:

Due to the nature of this course, there is no required textbook. Class notes will be provided as appropriate (but never as a substitute of the lectures). If you plan to study FEM further and you would like to have an introductory textbook, any of the books listed below will serve you well.

Optional references:

The first two books are used as introductory textbooks in many other universities and are rather easy to follow. The books by Oden et al. and Fish & Belytschko cover introductory material at about the same level as our course. The book by Bathe is one of the classics in FEM but outdated in many subjects. The books by Zienkiewicz et al. provide panoramic coverage of FEM including recent topics but are not appropriate as introductory textbooks. The book by Hughes (~$20) provides a very detailed introduction to linear FEM analysis with many implementation aspects included. However, it requires some reasonable mathematical background to follow. You can visit the links below and read reviews by other students.

Final project: All students are required to complete a project (an abstract for your project is due October 15th). Individual or group projects are accepted. To make sure that you work on the project on a continuous basis, drafts of your final report are due on October 31 (draft I) and November 15th (draft II). Your topic cannot be re-defined in a substantive manner as you move from the abstract to the two drafts and to the final report. The final project will involve an oral presentation as well as a final report (based on your drafts I and II). The presentations will take place on Saturday December 5th. Your final report (6 pages + Appendices providing technical details, drawings, etc. + programs that you wrote with the required information (how to run them, input files, etc.)) and your final power point presentation are due by December 3nd, midnight.

For those registered for MAE 4290 design credit, the final project needs to emphasize design applications. Based on guidelines of the Sibley School for ABET accreditation of courses that provide design credit, the following form needs to be submitted as part of your final report documenting the design aspects of your project. While ANSYS is expected to be the main software used for design applications other options can be considered as well. Other non-design oriented projects may emphasize the (i) development of MATLAB (or other programming language) FEM modules for applications substantially beyond those discussed in class, or (ii) using ANSYS for challenging engineering applications. Please consult the following document on guidelines for projects.

Grading: A-F: Homework 50%, two prelims 30%, project 20%.

Prerequisites: Generally senior or graduate student in Engineering or permission of instructor. The course with review and make extensive use of elementary principles from solid mechanics, heat transfer and fluid mechanics. Familiarity with calculus, differential equations and linear algebra is essential. The course will use MATLAB programming for all homework assignments and project. Access to MATLAB 7.0 or higher is required.

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Course description

Introduction to linear finite element static and dynamic analysis for discrete and distributed mechanical and aerospace structures. Prediction of load, deflection, stress, strain, temperature and flow distributions. Major emphasis on underlying physics, numerical methods and implementation. All homework assignments involve realistic numerical implementation of engineering problems using MATLAB based FEM software tools. To allow the whole class to be exposed to industrial computational design tools and for providing an opportunity to those interested to perform large scale FEM projects, ANSYS will also be introduced during recitations.

Course objectives:

  • Understand the mathematical and physical principles underlying the Finite Element Method (FEM) as applied to solid mechanics, thermal analysis and fluid mechanics.
  • Be able to create his/her own FEM computer programs, for simple problems, on MATLAB.
  • Be able to analyze more complex problems (in solid mechanics or thermal analysis) using commercial FEM software such as ANSYS
  • Demonstrate the ability to design a component using FEM analysis.
  • Make clear and effective technical presentations, both in terms of form as well as content, of his/her work.
  • Write clear technical reports describing his/her work.
  • Understand the importance of analysis and design, using the FEM, in the broader context of engineering practice.

Intended audience: Seniors, M.Eng and M.S./Ph.D. students.

Syllabus

  1. Introduction to the finite element method.
  2. Structural analysis: Truss 2D and 3D FEM analysis.
  3. Bending of beams using a minimum potential energy approach, analysis of frames (combining truss and beam analysis)
  4. FEM solution of 1D boundary value problems: weak forms, interpolation, numerical integration, boundary conditions, element calculations, assembly, solution, error analysis, postprocessing.
  5. Finite element analysis for 2D scalar field problems: element calculations, isoparametric transformation, quadrature rules, assembly, etc.
  6. Applications to steady-state heat conduction, torsion, irrotational flow, etc.
  7. 2D solid mechanics problems, linear elasticity, plane stress and plane strain.
  8. 3D stress analysis
  9. Transient heat conduction.
  10. Introduction to elasto-dynamics - natural frequencies, modal analysis, transient response.
  11. Introduction to optimization and design, sensitivity analysis, integration of FEM with optimization, applications in the design of solids and structures.
  12. Finite element modeling of incompressible flows.

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