Course Syllabus

Mechanical Engineering 432
Optimization Methods In Science & Engineering

Prerequisites

Calculus to the level of ODEs, partial derivative and multiple integrals, some knowledge of PDEs helpful but not essential.

Who takes it?

Since optimization problems are encountered in every branch of science and engineering, students with diverse backgrounds would benefit from it. In particular students with research interests in any area of mechanical engineering, physics, applied mathematics, chemistry, chemical engineering and biomedical engineering are encouraged to register. The course is primarily for graduate students (at any level) but advanced undergraduates may also benefit from it if they have the prerequisites mentioned above.

What is it about?

We often marvel at the manner in which animals seem so perfectly adapted to their environment: the cheetah is perfectly build for running at high speeds in open areas, the shark has all the adaptations for swimming at high speeds under water, certain eels use electric fields instead of vision to navigate in muddy waters and so on. Mathematically speaking each of these animals is a "solution" to an "optimization problem" that nature has solved through the iterative method of mutation and natural selection! In the machines that we build, as in natures "machines" the need to maximize or minimize certain things (such as energy consumption, cost, aerodynamic resistance etc.) is the essence of the design process. In this application oriented course we will look at the various common types of "optimization problems" that arise in diverse areas of science and engineering and learn some of the methods that have been developed to solve them; particularly, the area of applied mathematics known as the Calculus of Variations.

Minisyllabus

*    Introduction
*    Extremizing functions of several variables (review)
*    Extremizing with constraints - Lagrange multipliers
*    Functionals and the Euler-Lagrange equations
*    Constrained optimization of functionals
*    Some classical problems in the calculus of variations
*    Applications: classical mechanics, geometrical optics, elasticity, fluid mechanics, vibrations and waves

Assessment/Evaluation

Based on project (in class oral presentation & written report)

Study Material

Printed Lecture Notes (Primary Source): download

Additionally, the following books are suggested as reference material.

1. Gelfand and Fomin "Calculus of Variations" Dover

2. Feynman "The Feynman Lectures on Physics" Vol 1 Ch 26 & Vol 2 Ch 19 Addison-Wesley Longman

[Free to read online: http://www.feynmanlectures.info/ ]

3. Myskis "Advanced Mathematics for Engineers" Ch 6, Mir Publishers, Moscow

4. Courant & Hilbert "Methods of Mathematical Physics Vol 1" Ch 4, John Wiley