Variational methods and structural optimization

Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May, 2017
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at mecheng.iisc.ernet.in
Lectures: Tu, Th: 08:30 AM - 09:55 AM; Venue: ME Lecture Hall

Schedule and Notes

Week Dates Topics Notes Assignments

1 Jan. 3: Panorama of optimization problems; scope of the course.
Unconstrained minimization in finite number of variables: necessary and sufficient conditions
Jan. 5: Constrained minimization with equality constraints: Lagrange multiplier concept
Lecture 1
Lecture 2
Lecture 3 Homework #1

2 Jan. 10: KKT conditions for constrained minimization.
Jan. 12: Sufficient conditions for constrained minimization; Bordered Hessian Lecture 3
KKTdemo.m (Example file for familiarity with KKT conditons)
obj.m (Objective function file)
g1.m (Nonlinear constraints file)
Lecture 4 Homework #2

3 Jan. 17: Genesis of calculus of variations
Calculus of variations problems in geometry and mechanics
Jan. 19: Calculus of variations problems in geometry and mechanics (contd.)
Formulating calculus of variations problems. Lecture 5
Lecture 6 -

4 Jan. 24: Mathematical preliminaries to calculus of variations: vector spaces and their properties; function spaces
Jan. 26: No class; Happy Republic Day! Lecture 7
Lecture 8
-

5 Jan. 31: Mathematical preliminaries to calculus of variations (contd.): Gateaux variation, Frechet differential,
Feb. 2: Euler-Lagrange equations; How did Lagrange derive them? How did Euler derive them?
Variational derivative; fundamental lemma of calculus of variations Lecture 9
Lecture 10 Homework #3

6 Feb. 7: Extension of Euler-Lagrange equations to multiple derivatives and multiple functions
Feb. 9: Euler-Lagrnage equations when there are two and three independent variables of the unknown functon. Lecture 11
Lecture 12 Homework #4

7 Feb. 14: Global (functional type) constraints in variational calculus
Feb. 16: Local (point-wise or function type) constraints in variational calculus Lecture 13
Lecture 14 -

8 Feb. 21: Variable end conditions in calculus of variations; Weierstrass-Erdmann corner conditions; broken extremails.
Feb. 23: First integrals of Euler-Lagrange equations; change of variables; parametric form; transformation with a parameter and Noether's theorm. Lecture 15
Lecture 16 -

9 Feb. 28: "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 systems.
Mar. 2: Practice problems in calculus of variations 1 Lecture 17
Lecture 18: Some problems in calculus of variations
Homework #5

10 Mar 7: Practice problems in calculus of variations 2
Mar 9: Midterm examination during the class-time: 8:30 am to 10:30 am. Midterm papers from 2005 to 2016
Questions posed by you for Homework #3 Midterm examination

11 Mar 14: Optimization of cross-section area of an axially loaded bar; multiple formulations involving volume, strain energy, potential energy, displacement, and stress.
Mar 16: Optimality criteria method implemented for an axially loaded bar. Lecture 18: Solutions to midterm 2017
Lecture 19a: More than a dozen problems pertaining to optimization of a bar
Lecture 19b: Solutions to Problems 1 and 8
Download Matlab files of bar optimization problems solved using the optimality criteria method Homework #6

12 Mar 21: Optimization of cross-section area of a beam in multiple settings.
Mar 23: Optimality criteria method implemented for a beam. Lecture 20a: Many problems pertaining to optimization of a bar
Download Matlab files of beam optimization problem solved using the optimality criteria method Homework #7

13 Mar. 28: Optimization of a truss and its implementation in Matlab.
Mar. 30: Sensitivity analysis and optimality criterion; adjoint method - -

14 Apr. 4: Free vibration problem as a calculus of variations problem
Apr. 6: Minimization characterization of Sturm-Liouville problems
Strongest column: optimization for buckling load. Strongest column Matalb code
Minimum characterization of structural optimization problems -

15 Apr. 11: Optimization for transient problems.
Apr. 13: Structural optimization in multi-physics problems Transient loading problem and their sensitivity analysis
Electro-thermal-elastic structure optimization -

16 Apr. 19: Final examination at 3 pm in the ME MMCR on April 19, 2017.
May 4: Project presentations on May 4, 2017, starting at 9 am. Each project gets 15 min. Past final examination papers Project presentation; pdf file of the PPT file to be submitted soon after the presentation.

You can find the content-page of the previous years here.
2016
2015
2014
2013
2012
2011
2009
2007
2006
2005

Week	Dates Topics	Notes	Assignments
1	Jan. 3: Panorama of optimization problems; scope of the course. Unconstrained minimization in finite number of variables: necessary and sufficient conditions Jan. 5: Constrained minimization with equality constraints: Lagrange multiplier concept	Lecture 1 Lecture 2 Lecture 3	Homework #1
2	Jan. 10: KKT conditions for constrained minimization. Jan. 12: Sufficient conditions for constrained minimization; Bordered Hessian	Lecture 3 KKTdemo.m (Example file for familiarity with KKT conditons) obj.m (Objective function file) g1.m (Nonlinear constraints file) Lecture 4	Homework #2
3	Jan. 17: Genesis of calculus of variations Calculus of variations problems in geometry and mechanics Jan. 19: Calculus of variations problems in geometry and mechanics (contd.) Formulating calculus of variations problems.	Lecture 5 Lecture 6	-
4	Jan. 24: Mathematical preliminaries to calculus of variations: vector spaces and their properties; function spaces Jan. 26: No class; Happy Republic Day!	Lecture 7 Lecture 8	-
5	Jan. 31: Mathematical preliminaries to calculus of variations (contd.): Gateaux variation, Frechet differential, Feb. 2: Euler-Lagrange equations; How did Lagrange derive them? How did Euler derive them? Variational derivative; fundamental lemma of calculus of variations	Lecture 9 Lecture 10	Homework #3
6	Feb. 7: Extension of Euler-Lagrange equations to multiple derivatives and multiple functions Feb. 9: Euler-Lagrnage equations when there are two and three independent variables of the unknown functon.	Lecture 11 Lecture 12	Homework #4
7	Feb. 14: Global (functional type) constraints in variational calculus Feb. 16: Local (point-wise or function type) constraints in variational calculus	Lecture 13 Lecture 14	-
8	Feb. 21: Variable end conditions in calculus of variations; Weierstrass-Erdmann corner conditions; broken extremails. Feb. 23: First integrals of Euler-Lagrange equations; change of variables; parametric form; transformation with a parameter and Noether's theorm.	Lecture 15 Lecture 16	-
9	Feb. 28: "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 systems. Mar. 2: Practice problems in calculus of variations 1	Lecture 17 Lecture 18: Some problems in calculus of variations	Homework #5
10	Mar 7: Practice problems in calculus of variations 2 Mar 9: Midterm examination during the class-time: 8:30 am to 10:30 am.	Midterm papers from 2005 to 2016 Questions posed by you for Homework #3	Midterm examination
11	Mar 14: Optimization of cross-section area of an axially loaded bar; multiple formulations involving volume, strain energy, potential energy, displacement, and stress. Mar 16: Optimality criteria method implemented for an axially loaded bar.	Lecture 18: Solutions to midterm 2017 Lecture 19a: More than a dozen problems pertaining to optimization of a bar Lecture 19b: Solutions to Problems 1 and 8 Download Matlab files of bar optimization problems solved using the optimality criteria method	Homework #6
12	Mar 21: Optimization of cross-section area of a beam in multiple settings. Mar 23: Optimality criteria method implemented for a beam.	Lecture 20a: Many problems pertaining to optimization of a bar Download Matlab files of beam optimization problem solved using the optimality criteria method	Homework #7
13	Mar. 28: Optimization of a truss and its implementation in Matlab. Mar. 30: Sensitivity analysis and optimality criterion; adjoint method	-	-
14	Apr. 4: Free vibration problem as a calculus of variations problem Apr. 6: Minimization characterization of Sturm-Liouville problems Strongest column: optimization for buckling load.	Strongest column Matalb code Minimum characterization of structural optimization problems	-
15	Apr. 11: Optimization for transient problems. Apr. 13: Structural optimization in multi-physics problems	Transient loading problem and their sensitivity analysis Electro-thermal-elastic structure optimization	-
16	Apr. 19: Final examination at 3 pm in the ME MMCR on April 19, 2017. May 4: Project presentations on May 4, 2017, starting at 9 am. Each project gets 15 min.	Past final examination papers	Project presentation; pdf file of the PPT file to be submitted soon after the presentation.