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, 2018
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at iisc.ac.in
Lectures: Tu, Th: 08:30 AM - 09:55 AM; Venue: ME MMCR

Schedule and Notes

Week Dates Topics Notes Assignments

1 Jan. 4: Panorama of optimization problems; scope of the course.
Lecture 1
Lecture 2 Homework 1

2 Jan. 9: 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). Lecture 3 Homework 2

3 Jan. 16: 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. KKTdemo.m (Example file for familiarity with KKT conditons)
obj.m (Objective function file)
g1.m (Nonlinear constraints file)
Lecture 4
Lecture 5, Lecture 6 -

4 Jan. 23: Mathematical preliminaries to calculus of variations: vector spaces and their properties; function spaces
Jan. 25: Mathematical preliminaries to calculus of variations (contd.): Gateaux variation Lecture 7
Lecture 8
Homework 3

5 Jan. 30: 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
Feb. 1: Extension of Euler-Lagrange equations to multiple derivatives; beam problem Lecture 9
Lecture 10
Lecture 11 Homework #4

6 Feb. 6: 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. Lecture 12 -

7 Feb. 13: Happy jagaran! (Holiday for Mahasivarathri)
Feb. 15: Global (functional type) constraints in variational calculus
Local (point-wise or function type) constraints in variational calculus Lecture 13
Lecture 14 Homework 5

8 Feb. 20: 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. Lecture 15
Lecture 16 Homework 6

9 Feb. 27: "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 17
Lecture 18: Some problems in calculus of variations
-

10 Mar 6: Midterm examination during the class-time: 8:30 AM to 10:30 AM.
Mar 8: Glimpses of structural optimization Midterm papers from 2005 to 2017
Lecture 18: Solutions to midterm 2017 Midterm examination

11 Mar 13: 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 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 #7

12 Mar 20: Optimization of cross-section area of a beam in multiple settings.
Mar 22: 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 -

13 Mar. 27: Optimization of a truss and its implementation in Matlab.
Sensitivity analysis and optimality criterion; adjoint method
Mar. 29: Holiday for Mahaveer Jayanthi FEA notes for bars, beams, and trusses
Truss FEA files Homework 8

14 Apr. 3: Free vibration problem as a calculus of variations problem
Apr. 5: Minimization characterization of Sturm-Liouville problems
Strongest column: optimization for buckling load. Strongest column Matalb code
Minimum characterization of structural optimization problems -

15 Apr. 10: Optimization for transient problems.
Apr. 12: Structural optimization in multi-physics problems Transient loading problem and their sensitivity analysis
A short discussion on Electro-thermal-elastic structure optimization -

16 Apr. 18: 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). Past final examination papers 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.
2017
2016
2015
2014
2013
2012
2011
2009
2007
2006
2005

Week	Dates Topics	Notes	Assignments
1	Jan. 4: Panorama of optimization problems; scope of the course.	Lecture 1 Lecture 2	Homework 1
2	Jan. 9: 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).	Lecture 3	Homework 2
3	Jan. 16: 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.	KKTdemo.m (Example file for familiarity with KKT conditons) obj.m (Objective function file) g1.m (Nonlinear constraints file) Lecture 4 Lecture 5, Lecture 6	-
4	Jan. 23: Mathematical preliminaries to calculus of variations: vector spaces and their properties; function spaces Jan. 25: Mathematical preliminaries to calculus of variations (contd.): Gateaux variation	Lecture 7 Lecture 8	Homework 3
5	Jan. 30: 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 Feb. 1: Extension of Euler-Lagrange equations to multiple derivatives; beam problem	Lecture 9 Lecture 10 Lecture 11	Homework #4
6	Feb. 6: 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.	Lecture 12	-
7	Feb. 13: Happy jagaran! (Holiday for Mahasivarathri) Feb. 15: Global (functional type) constraints in variational calculus Local (point-wise or function type) constraints in variational calculus	Lecture 13 Lecture 14	Homework 5
8	Feb. 20: 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.	Lecture 15 Lecture 16	Homework 6
9	Feb. 27: "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 17 Lecture 18: Some problems in calculus of variations	-
10	Mar 6: Midterm examination during the class-time: 8:30 AM to 10:30 AM. Mar 8: Glimpses of structural optimization	Midterm papers from 2005 to 2017 Lecture 18: Solutions to midterm 2017	Midterm examination
11	Mar 13: 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 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 #7
12	Mar 20: Optimization of cross-section area of a beam in multiple settings. Mar 22: 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	-
13	Mar. 27: Optimization of a truss and its implementation in Matlab. Sensitivity analysis and optimality criterion; adjoint method Mar. 29: Holiday for Mahaveer Jayanthi	FEA notes for bars, beams, and trusses Truss FEA files	Homework 8
14	Apr. 3: Free vibration problem as a calculus of variations problem Apr. 5: Minimization characterization of Sturm-Liouville problems Strongest column: optimization for buckling load.	Strongest column Matalb code Minimum characterization of structural optimization problems	-
15	Apr. 10: Optimization for transient problems. Apr. 12: Structural optimization in multi-physics problems	Transient loading problem and their sensitivity analysis A short discussion on Electro-thermal-elastic structure optimization	-
16	Apr. 18: 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).	Past final examination papers	Project presentation; pdf file of the PPT file to be submitted soon after the presentation by email.