Schedule and Notes
| Week | Dates Topics | Notes | Assignments |
| 1 |
Jan. 3:
Panorama of optimization problems; scope of the course. |
Lecture 2 |
Homework 1 |
| 2 |
Jan. 8:
Unconstrained minimization in finite number of variables: necessary and sufficient conditions
Constrained minimization with equality constraints: Lagrange multiplier concept Necessary conditions for constrained minimization (two variables). Jan. 10: Necessary conditions for constrained minimization (N variables). |
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| 3 |
Jan. 15:
Sufficient conditions for constrained minimization; Bordered Hessian
Genesis of calculus of variations Calculus of variations problems in geometry and mechanics Jan. 17: Calculus of variations problems in geometry and mechanics (contd.) Formulating calculus of variations problems. |
obj.m (Objective function file) g1.m (Nonlinear constraints file) Lecture 4 Lecture 5, Lecture 6 |
Homework 2 |
| 4 |
Jan. 22:
Mathematical preliminaries to calculus of variations: vector spaces and their properties; function spaces
Jan. 24: Mathematical preliminaries to calculus of variations (contd.): Gateaux variation |
Lecture 8 |
- |
| 5 |
Jan. 29:
Frechet differential, Frechet derivative Euler-Lagrange equations; How did Lagrange derive them? fundamental lemma of calculus of variations Euler-Lagrange equations; How did Lagrange derive them? How did Euler derive them? Variational derivative Jan. 31: Extension of Euler-Lagrange equations to multiple derivatives; beam problem Extension of Euler-Lagrange equations to multiple functions |
Lecture 10 Lecture 11 |
Homework #3 |
| 6 |
Feb. 5:
Euler-Lagrnage equations when there are two independent variables of the unknown function. Euler-Lagrange equations when there are three independent variables of the unknown function. Feb. 7: Discussion on Gateaux variation, Frechet differential, Frechet derivative, and variational derivative; why and what we should know about Banach, Hilbert, and Sobolev spaces" Q & A on what is covered until now. |
Lecture 13 |
Homework #4 |
| 7 |
Feb. 12:
Global (functional type) constraints in variational calculus Local (point-wise or function type) constraints in variational calculus Feb. 14: Variable end conditions in calculus of variations; Weierstrass-Erdmann corner conditions; broken extremals. |
Lecture 15 |
Homework #5 |
| 8 |
Feb. 19:
First integrals of Euler-Lagrange equations; change of variables; parametric form; transformation with a parameter and Noether's theorm.
Feb. 21: "Inverse" Euler-Lagrange equations problem: going from the differential equation to the functional to be optimized: three methods, (i) for self-adjoint operators, (ii) integrating factor method for dissipative systems, and (iii) parallel generative system for dissipative cases. |
Lecture 17 |
- |
| 9 |
Feb. 26:
Practice problems in calculus of variations
Review of lecture material until now; Q&A Feb. 28: Midterm examination; 1 h duration during the class-time; open-books, open-notes,open-Internet; interacting with any live person is not permitted. |
Midterm examination papers from 2005 to 2018 |
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| 10 |
Mar. 5:
Bar optimization: stiffest bar for a given volume
Mar. 7: Bar optimization: Interpretation of boundary conditions; lower and upper bounds on the design variable; other objective function; |
Lecture 19b Bar optimization files |
Homework #6 |
| 11 |
Mar. 12:
Optimality criteria method; Matlab code for bar optimization Beam optimization: stiffness for given volume; Mar. 14: Bar and beam optimization: constraints on displacement, stress, etc. |
Beam optimization files |
- |
| 12 |
Mar. 19:
Mutual strain energy and design for deflection of a beam problem for indeterminate beams
Mar. 21: Class cancelled due to NCMDAO (optimization conference) |
Beam optimization files |
- |
| 13 |
Mar. 26:
Optimal design for deflection of beams for statically determinate beams 2D continuum; design for deflection SIMP method for topology optimization Mar. 28: 2D continuum and 3D continuum stiffness problems Hierarchy in structural optimization: topolgy, shape, and size |
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| 14 |
Apr. 2:
Demonstration of software YinSyn, TopOpt, etc. Truss optimization Sensitivity analysis: direct and ajoing methods Apr. 4: Sensitvity analysis (contd.) Stress constraints; transient loading; and multi-physics formulations |
TopOpt.m: 99-line Matlab code of O. Sigmund 99-line code paper of O. Sigmund Truss FEA Matlab files Transient loading sensitivity analysis Electro-thermal-elastic structure problem |
Homework #7 |
| 15 |
Apr. 9:
Eigenvalue problems and their minimum characterization
Apr. 11: Raleigh quotient for eigenvalue problems Optimization of a column for maximum buckling load |
Link to NPTEl lecture on stability analysis and buckling using calculus of variations Link to NPTEL lecture on strongest most stable column |
Homework #8 |
| 16 |
Apr. 22:
Final examination 2 PM to 5 PM in ME MMCR
May 6 : Project presentations from 8:30 AM to 12 noon in ME MMCR |
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