MECH 309 - Numerical Methods in Mechanical Engineering

Numerical techniques for problems commonly encountered in Mechanical Engineering including systems of linear equations, eigenvalue problems, least-squares estimation, optimization, ordinary differential equations, and interpolation. The emphasis is on understanding the underlying numerical methods and their application to engineering problems.

Instructor: Prof. James Richard Forbes

Course Overview

MECH 309 introduces the theory and application of numerical methods, also known as scientific computing. Students learn how to reformulate engineering problems into mathematical forms that can be solved computationally and how to assess the accuracy and reliability of numerical solutions.

Topics include:

  • Error analysis and conditioning
  • Systems of linear equations
  • Eigenvalue problems
  • Eigen and Singular value decompositions
  • Interpolation and splines
  • Linear and nonlinear least squares
  • Optimization
  • Ordinary differential equations (IVPs and BVPs)
  • Partial differential equations

Prerequisite and Corequisite Courses

Students are expected to be comfortable with:

  • Calculus
  • Differential equations
  • Linear algebra
  • Basic Python programming
Prerequisite Courses
  • MATH 263
  • MATH 271
  • COMP 208

Python Resources

The course makes extensive use of Python together with NumPy and SciPy.

Useful resources:

Learning Outcomes

By the end of the course, students will be able to:

  • Reformulate engineering problems as numerical problems suitable for computation.
  • Solve systems of linear equations and eigenvalue problems numerically.
  • Formulate and solve linear and nonlinear least-squares problems.
  • Apply optimization techniques to engineering problems.
  • Numerically solve initial value and boundary value problems.
  • Assess the accuracy, conditioning, and reliability of numerical solutions.
  • Apply modern scientific-computing tools to engineering analysis and design.

Textbooks

There is no required textbook. However, lectures are based on the following references.

  • M. T. Heath, Scientific Computing: An Introductory Survey, 2nd ed. New York, NY: McGraw-Hill, 2002.
  • S. Boyd and L. Vandenberghe, Introduction to Applied Linear Algebra. Cambridge, UK: Cambridge University Press, 2018. Available here.
  • J. Solomon, Numerical Algorithms: Methods for Computer Vision, Machine Learning and Graphics. Boca Raton, FL: CRC Press, 2015.
  • T. Sauer, Numerical Analysis, 3rd ed. Boston, MA: Pearson, 2018.

Additional Resources

  • P. E. Gill, W. Murray, and M. H. Wright, Numerical Linear Algebra and Optimization. Philadelphia, PA: SIAM, 2021.
  • P. R. Turner, T. Arildsen, and K. Kavanagh, Applied Scientific Computing with Python. Cham, Switzerland: Springer, 2018.
  • J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. New York, NY: Springer, 2006.
  • C. D. Meyer, Matrix Analysis and Applied Linear Algebra. Philadelphia, PA: SIAM, 2000.
  • S. Toledo, Location Estimation from the Ground Up. Philadelphia, PA: SIAM, 2020.
  • G. Strang and K. Borre, Linear Algebra, Geodesy, and GPS. Wellesley, MA: Wellesley-Cambridge Press, 1997.

Schedule

Week Date Topic Materials
1 Sep/Jan Errors and Conditioning

Binary numbers, floating point arithmetic, error analysis, and conditioning.
2 Sep/Jan Linear Algebra Review

$Ax = b$ and the four fundamental subspaces.
3 Sep/Jan Linear Systems of Equations

Conditioning, Gaussian elimination, LU decomposition, and Cholesky factorization, interpolation posed as a $Ax = b$ problem.
4 Sep/Jan Iterative Methods

Gradient descent, steepest descent, and iterative solution methods.
5 Oct/Feb QR Factorization

Orthogonal factorization methods and applications.
6 Oct/Feb Eigenvalue Problems

Eigenvalues, eigenvectors, power iteration, and inverse power iteration.
7 Oct/Feb Singular Value Decomposition

SVD computation and engineering applications.
8 Oct/Feb Linear Least Squares

Problem formulation, normal equations, and weighted least squares.
9 Nov/Mar Nonlinear Least Squares

Gauss-Newton and Levenberg-Marquardt algorithms.
10 Nov/Mar Optimization

Unconstrained and constrained optimization methods.
11 Nov/Mar Ordinary Differential Equations - Initial Value Problems

Euler and Runge-Kutta methods for initial value problems.
12 Nov/Mar Ordinary Differential Equations - Boundary Value Problems

Shooting methods and finite difference methods.
13 Dec/Apr Partial Differential Equations

Finite difference methods.