Schedule and Notes
| Week | Dates Topics | Notes | Assignments |
| 1 |
Jan. 6:
Panorama of optimization problems; scope of the course. Unconstrained minimization in finite number of variables: necessary and sufficient conditions Jan. 8: Constrained minimization with equality constraints: Lagrange multiplier concept Karush Kuhn-Tucker (KKT) conditions |
Lecture B Lecture C |
Homework #1 |
| 2 |
Jan. 13:
Constraint qualification Sufficient condition for a constrained minimum; Bordered Hessian Jan. 15: Happy Sankrnathi! No lecture. |
|
Homework #2 |
| 3 |
Jan. 20:
History of calculus of variations Formulating calculus of variations Jan. 22: Calculus of variations problems in geometry and mechanics |
Lecture F |
Homework #3 |
| 4 |
Jan. 27:
Vector spaces; metric spaces; function spaces; metric; norm; Banach and Hilbert spaces; Sobolev and Lebegue norms and spaces; their implications in calculus of variations and structural optimization Jan. 29: First variation; Gateux variation of a functional; examples |
Lecture H Lecture I |
Homework #4 |
| 5 |
Feb. 3rd:
Fundamental Lemma of calculus of variations Euler-Lagrange equations; boundary conditions Feb. 5th: Extension of E-L equations to multiple derivatives and multiple functions |
Lecture K |
Homework #5 |
| 6 |
Feb. 10th:
Beam equation using E-L equations. Derivation of the plate equation using the principle of minimum potential energy. Feb. 12th: Extension of E-L equations to two and three independent variables |
|
- |
| 7 |
Feb. 17th:
No class (Greetings on Maha Shivarathri!)
Feb. 19th: Global constraint and Euler-Lagrange equations. Local (differential or algebraic) equations and Euler-Lagrange equations. Feb. 21st; 8:30 to 10 am: variable end conditions; general (non-contemporaneus) variation Broken extrenals and Weierstrass-Erdmann corner conditions |
Lecture N |
- |
| 8 |
Feb. 24th: Integrals and invariants of Euler-Lagrange equations; Noether's theorem and its implications. Feb. 26th: Mid-term examination during class-time |
Midterm papers from 2007 to 2014 |
- |
| 9 |
Mar. 3rd: Stiffest bar for given volume; formulation and analytical solution. Mar. 5th: variants of the stiffest-lightest bar with governing equations in the weak form; Clayperon's theorem, Min-Max potential energy formulation |
Lecture S |
Homework #6 |
| 10 |
Mar. 10th: Optimality criteria method of solution to the stiffest-lightest bar Constraints on areas of cross-section; displacement, strain/stress Minimizing maximum stress in a bar subject to given axial load Mar. 12th: Stiffest beam for given volume and transverse load With the governing euqations in the weak form; displacement/strain/stress constraints. |
Lecture T Lecture U Lecture V |
Homework #7 |
| # |
April 23:
Final examination at 3 pm to 5 pm in MMCR. |
|
Solve final exam problems of past years for practice. |
| # |
May 4:
Project presentations at 2 pm to 5 pm in MMCR. |
|
12-minute presentation per student |
You can find the content-page of the previous years here.
2014
2013
2012
2011
2009
2007
2006
2005