ME 565 - Mechanical Engineering Analysis

This course will provide an in-depth overview of powerful mathematical techniques for the analysis of engineering systems. In addition to developing core analytical capabilities, students will gain proficiency with various computational approaches used to solve these problems. Applications will be emphasized, including fluid mechanics, elasticity and vibrations, weather and climate systems, epidemiology, space mission design, and applications in control.

Computing

In this course, we will develop many powerful analytic tools. Equally important is the ability to implement these tools on a computer. The instructor and TAs use Python or Matlab, and all examples in class will be in these two languages.

MATLAB: Student Edition (recommended if you do not have access)

ICL: Matlab laboratory & login access (Communications Bldg B022 and B027)

(MATLAB Intro) (Matlab_Introduction.m, ode_test.m)

(Python Intro) (Python_Introduction.ipynb)

Syllabus

Part 1 - Linear Algebra and Vector Calculus

(Lecture 00) Linear algebra in 2D and 3D: inner product, norm of a vector, and cross product (notes)

(Lecture 01) Div, Grad, and Curl (notes)

(Lecture 02) Gauss's Divergence Theorem (notes)

(Lecture 03) Directional derivative, continuity equation, and examples of vector fields (notes)

(Lecture 04) Stokes' theorem and conservative vector fields (notes)

(Lecture 05) Potential flow and Laplace's equation (notes)

(Lecture 06) Potential flow, stream functions, and examples (notes)

(Extra Lecture) Example of ODE for particle trajectories in a time-varying vector field: The double gyre (doublegyreVEC.m, integrateDG.m, rk4singlestep.m).

Part 2 - Partial Differential Equations and Transform Methods (Laplace and Fourier)

(Lecture 07) Canonical Linear PDEs: Wave equation, Heat equation, and Laplace's equation (notes)

(Lecture 08) Heat Eqaution: derivation and equilibrium solution in 1D (i.e., Laplace's equation) (notes)

(Lecture 09) Heat Equation in 2D and 3D. 2D Laplace Equation (on rectangle) (notes)

(Lecture 10) Analytic Solution to Laplace's Equation in 2D (on rectangle) (notes)

(Lecture 11) Numerical Solution to Laplace's Equation in Matlab. Intro to Fourier Series. (notes, L11_Laplace.m, L11_AnalyticLaplacian.m, L11_HardAnalyticLaplace.m)

(Lecture 12) Fourier Series (notes, L12_Fourier.m)

(Lecture 13) Infinite Dimensional Function Spaces and Fourier Series (notes)

(Lecture 14) Fourier Transforms (notes)

(Lecture 15) Properties of Fourier Transforms and Examples (notes)

(Lecture 16) Discrete Fourier Transforms (DFT) (notes)

(Lecture 17) Fast Fourier Transforms (FFT) and Audio (notes, EX1_FFT.m, EX2_FFT.m)

(Lecture 18) FFT and Image Compression (notes, compress.m)

(Lecture 19) Fourier Transform to Solve PDEs: 1D Heat Equation on Infinite Domain (notes)

(Lecture 20) Numerical Solutions to PDEs Using FFT (notes, HeatConvolution.m, HeatFFT.m, HeatBoth.m, SpectralDerivative.m, benchSpectralDerivative.m, smoothFFTDeriv.m)

(Lecture 21) The Laplace Transform (notes)

(Lecture 22) Laplace Transform and ODEs (notes)

(Lecture 23) Laplace Transform and ODEs with Forcing (step, impulse, and frequency response from transfer functions) (notes)

(Lecture 24) Convolution integrals, impulse and step responses (notes)

(Lecture 25) Laplace transform solutions to PDEs (notes)

(Lecture 26) Solving PDEs in Matlab using FFT (notes)

(Lectures 27-29) Singular value decomposition (SVD) and Data Science (notes)

(Lecture 29) SVD and facial recognition (eigenfaces) (EIGENFACE.zip)