Schedule and Notes
Panorama of optimization problems; scope of the course.||
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. 11: Necessary conditions for constrained minimization (N variables).
Sufficient conditions for constrained minimization; Bordered Hessian
Genesis of calculus of variations
Calculus of variations problems in geometry and mechanics
Jan. 18: 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 5, Lecture 6
Mathematical preliminaries to calculus of variations: vector spaces and their properties; function spaces
Jan. 25: Mathematical preliminaries to calculus of variations (contd.): Gateaux variation
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?
Feb. 1: Extension of Euler-Lagrange equations to multiple derivatives; beam problem
Extension of Euler-Lagrange equations to multiple derivatives and multiple functions
Euler-Lagrnage equations when there are two independent variables of the unknown function.
Feb. 8: Euler-Lagrange equations when there are three independent variables of the unknown function.
Happy jagaran! (Holiday for Mahasivarathri)
Feb. 15: Global (functional type) constraints in variational calculus
Local (point-wise or function type) constraints in variational calculus
Variable end conditions in calculus of variations; Weierstrass-Erdmann corner conditions; broken extremails.
Feb. 22: First integrals of Euler-Lagrange equations; change of variables; parametric form; transformation with a parameter and Noether's theorm.
"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.
Mar. 1: Practice problems in calculus of variations
Lecture 18: Some problems in calculus of variations
Midterm examination during the class-time: 8:30 AM to 10:30 AM.
Mar 8: Glimpses of structural optimization
Lecture 18: Solutions to midterm 2017
Optimization of cross-section area of an axially loaded bar; multiple formulations involving volume,
strain energy, potential energy, displacement, and stress.
Mar 15: Optimality criteria method implemented for an axially loaded bar.
Lecture 19b: Solutions to Problems 1 and 8
Download Matlab files of bar optimization problems solved using the optimality criteria method
Optimization of cross-section area of a beam in multiple settings.
Mar 22: Optimality criteria method implemented for a beam.
Download Matlab files of beam optimization problem solved using the optimality criteria method
Optimization of a truss and its implementation in Matlab.
Sensitivity analysis and optimality criterion; adjoint method
Mar. 29: Holiday for Mahaveer Jayanthi
Truss FEA files
Free vibration problem as a calculus of variations problem
Apr. 5: Minimization characterization of Sturm-Liouville problems
Strongest column: optimization for buckling load.
Minimum characterization of structural optimization problems
Optimization for transient problems.
Apr. 12: Structural optimization in multi-physics problems
A short discussion on Electro-thermal-elastic structure optimization
Final examination at 2:00 PM to 3:30 PM in the ME MMCR on April 18, 2018. It will be followed by two project presentations.
April 30: Project presentations on April 30, 2018, starting at 8:30 AM in ME MMCR. Each project gets 15 min (including Q&A time of 2 min).
||Project presentation; pdf file of the PPT file to be submitted soon after the presentation by email.|
You can find the content-page of the previous years here.