# ME205: Numerical Methods for Engineers

Unit 9: Additional Tools   This unit is not meant to be an in-depth study on additional tools for numerical methods in engineering.  Rather it is intended as an inspiration for further study. The topics chosen for a cursory introduction, Fourier transforms and finite element methods, find widespread use in the dynamics of mechanical systems. Hence, some acquaintance with the terminology and capabilities of the techniques and concepts may serve you well in future study and/or applications.*

*This unit covers two topics which may at first seem unrelated (Fourier transforms and finite element methods); through continued study you may find links between these areas and many of the other topics covered in this course.

This unit will take you approximately 14 hours to complete.

☐    Subunit 9.1: 9 hours

☐    Sub-subunit 9.1.1: 3 hours

☐    Sub-subunit  9.1.2: 6 hours

☐    Web Media: 3 hours

☐    Subunit 9.2:: 5 hours

☐    Web Media: 2 hours

Unit9 Learning Outcomes
Upon successful completion of this unit, the student will be able to:

• Define Fourier series and the Fourier transform.
• Find Fourier coefficients for a given data set or function and domain.
• Describe the finite element method for one-dimensional problems.

9.1 Fourier Series and the Fast Fourier Transform   9.1.1 The Fourier Series   - Lecture: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Fourier Series (Part 1)” and “Fourier Series (Part 2)” Links: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Fourier Series (Part 1)” (FLASH, MP4, or iTunes) and “Fourier Series (Part 2)” (FLASH, MP4, or iTunes)

Also available in:

Instructions: Please view these lectures before going on to the reading in this subunit.  Both videos are approximately 49 minutes.  You may also access the transcript for the lectures by clicking on the “transcript” tab on each webpage and then the “Download this transcript-PDF” link.

• Reading: MIT: Professor Gilbert Strang’s Computational Science and Engineering: Chapter 4.1: “Fourier Series for Periodic Functions” Link: MIT: Professor Gilbert Strang’s Computational Science and Engineering: Chapter 4.1:“Fourier Series for Periodic Functions” (PDF)

9.1.2 The Discrete Fourier Series   - Lecture: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Discrete Fourier Series”  Link: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Discrete Fourier Series” (FLASH, MP4, or iTunes)

Also available in:

Instructions: Please view the video lecture in its entirety (approximately 50 minutes). You may also access the transcript for the lectures by clicking on the “transcript” tab on each webpage and then the “Download this transcript-PDF” link.

• Reading: University of Maryland: Professor Ramani Duraiswami’s “Fast Fourier Transform” Link: University of Maryland: Professor Ramani Duraiswami’s “Fast Fourier Transform” (PDF)

Instructions: Scroll down the webpage linked here, and click on the hyperlink titled “Lecture 6” to download the PDF file.  Read this entire article (18 pages) and if interested, read the book chapter linked under “Lecture 6,” before proceeding to Professor Strang’s lectures on the Fourier Integral Transform and the Fast Fourier Transform in this subunit.

• Lecture: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Filters, Fourier Integral Transform,” “Fourier Integral Transform (Part 2),” and “Fast Fourier Transform, Convolution” Links: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Filters, Fourier Integral Transform,” (FLASH, MP4, or iTunes) “Fourier Integral Transform (Part 2),” (FLASH, MP4, or iTunes) and “Fast Fourier Transform, Convolution” (FLASH, MP4, or iTunes)

Also available in:
YouTube: Fourier Integral Transform (Part 2)

Instructions: Please view the first video lecture (“Filters, Fourier Integral Transform”) from 41:17 minutes to the end.  Then, view the entire “Fourier Integral Transform (Part 2)” video lecture (approximately 51 minutes).  Finally, view the “Fast Fourier Transform, Convolution” lecture up to 40:36 minutes.

• Reading: MIT Opencourseware: Professor Peter Shor’s Lecture Notes for Mathematics 18.310C: Principles of Applied Mathematics: “The Finite Fourier Transform” and “FFT” Links: MIT Opencourseware: Professor Peter Shor’s Lecture Notes for Mathematics 18.310C: Principles of Applied Mathematics: “The Finite Fourier Transform” (PDF) and “FFT” (PDF)

Instructions: Go to the websites linked here, and click on the hyperlink “L23” for the first set of lecture notes and on the hyperlink “L24-FFT” for the second set.  Please read through Professor Shor’s notes to ensure that you understand and can reproduce the development and application of the FFT.

9.2 Finite Element Methods (One-dimensional)   - Reading: California State University, Fullerton: Professor John Mathew’s “Galerkin’s Method” Link: California State University, Fullerton: Professor John Mathew’s “Galerkin’s Method” (HTML)

Instructions: Although these notes on Galerkin’s method are a bit advanced, it provides a good overview of the method.  Please read through these notes carefully, and re-read as necessary.

• Lecture: MIT Opencourseware: Professor Gilbert Strang’s Mathematics 18.085: “Finite Elements in 1D” Link: MIT Opencourseware : Professor Gilbert Strang’s Mathematics 18.085: “Finite Elements in 1D” (FLASH, MP4, or iTunes)

Also available in:

Instructions: Please view the video lecture in its entirety (about 54 minutes).  You may also access the transcript for the lectures by clicking on the “transcript” tab on each webpage and then the “Download this transcript-PDF” link.

• Reading: Cornell University: Professor Nicholas Zabaras’s “Introduction to the FEM for Elliptic Problems” Link: Cornell University: Professor Nicholas Zabaras’s “Introduction to the FEM for Elliptic Problems”(PDF)

Instructions: Go to the Cornell website linked here, and click on the “PDF” hyperlink after the title “Introduction to the FEM for elliptic problems.”  Read these notes (60 pages).