Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May, 2013
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at mecheng.iisc.ernet.in

Syllabus, schedule, and notes

Topics to be covered in this course--not necessarily in this order.

Motivating examples of calculus of variations Mathematical preliminaries: normed vectors spaces, functionals (continuous and linear), directional derivative, concept of variation, Gateaux variation, Frechet differential, etc. Fundamental lemmas of calculus of variation Euler-Lagrange (E-L) equations Applications of E-L equation Extensions of E-L equation to multiple derivatives, independent variables, multiple state variables Isoperimetric problems--global and local (finite subsidiary) constraints Applications of optimizing functionals subject to constraints Applications in mechanics: strong and weak forms of governing equations Variable end conditions--transversality conditions

Size optimization of a bar for maximum stiffness Self-adjointness and optimization with weak variational form Optimization with side constraints (variable bounds) Worst load scenario for an axially loaded stiffest bar Min-max type problem with stress constraints Beam problems for stiffmess and strength Optimization of a beam for given deflection Variational formulations for the eigenvalue problems: strings, bars, beams, and other elastic strucrures. Optimum design of a column Variable-thickness optimization of plates

Sufficient conditions for E-L optimum Applications of sufficient conditions Finite dimensional optimization--A summary and highlights Numerical optimization techniques Using the optimization tool-box in Matlab

Truss topology optimization Sensitivity analysis Frame topology optimization Compliant mechanism design using topology optimization of trusses and frames Topology optimization using continuum elements Optimality criteria method Shape optimization of structures Applications to multi-physics problems Material interpolation techniques for topology optimization

Week	Dates Topics Recitations	Notes	Assignments
1	Jan. 10: Finite variable optimization vs. calculus of variations; Types of optimization problems; scope of the course.	Prof. K. Suresh's notes on finite variable optimization	-
2	Jan. 15: Explanation of what constitutes a "calculus of variations" problem; Necessary and sufficient conditions for nconstrained minimization in finite number of variables. Jan. 17: Constrained minimization in two variables with an equality constraint; the concept of Lagrange multiplier and Lagrangian	Unconstrained minimization	Homework 1
3	Jan. 22: Constrained minimization in n variables with equality and inequality constraints Karush-Kuhn-Tucker (necessary) conditions Jan. 24: Sufficient conditions for constrained minimization	Derivation of the sufficient condition for constrained minimization	Homework 2
4	Jan. 29: Review of continuous optimization theory. Jan. 31: Sample problems of calculus of variations.	-	-
5	Feb. 5: Mathematical preliminaries: vectors spaces, functional, Banach space, Hilbert space, Lebegue and Sobolev norms and spaces, etc. Feb. 7: No class.	Mathematical preliminaries	Homework 3
6	Feb. 12: Euler-Lagrange-Ostrogradski euqations for a functional involving one derivative; boundary conditions; principle of minimum potential energy; the weak form and the principle of virtual work. the principel of Feb. 14: Euler-Lagrange-Ostrogradski equations for functional involving multiple derivatives and functions.	-	-
7	Feb. 19: Functionals involving function of two and three independent variables General (non-contemporaneous) variation and the associated boundary conditions. Feb. 21: Trabsversality conditions; Weierstrass-Erdmann corner conditions for broken extremals.	-	Homework #4
8	Feb. 26: Minimization of functionals with global and local constraints Feb. 28: No class.	-	Homework #5
9	Mar. 5: First integrals of Euler-Lagrange equations; Change of coordinates and invariance of Euler-Lagrange-Ostrogradski equations; Noether's theorem. Parametric form of Euler-Lagrange -Ostrogradsky equations Mar. 7: Review for the mid-term examination Mar. 9: Mid-term examination; open book; open notes; open Internet! 10 am to 12 noon in MMCR of the ME department.	Mid-term papers of past years 2005 to 2012	-
10	Mar. 12: Stiffest bar for given volume Design and adjoint equations Mar. 14: Stiffest bar for given volume with weak form of the governing equation Physical meaning of the Lagrange multipliers Complete solution of the stiffest bar problem	Project proposal due as Homework #7; see Project page.	Homework #6
11	Mar. 19: Stiffest bar problem with upper and lower bound constraints on the area of cross-section(br> Strength considerations: two ways of posing the problem Mar. 21: Optimality criteria method General elastic structure for maximum stiffness for given volume	-	Homework #7
12	Mar. 26: Truss, beam, and bar optimization for desired deflection Mar. 28: Sensitivity analysis for the continous and discrete systems: comparison and physical interpretation Mar. 30: Demonstration of the Matlab programs using the optimality criteria method	-	-
13	Apr. 2: Sensitivity analysis: direct and adjoint methods Apr. 4: Matlab programs - a review	-	-
14	Apr. 9: Electro-thermal-compliant atuator optimization Apr. 11: Minimim characterizaton of eigenvalue problems	-	-
15	Apr. 16: Minimim characterizaton of eigenvalue problems (contd); Rayleigh quotient; buckling problems as eigenvalue problems	Final examms from 2005 to 2012	-
Exam and project	Apr. 27: Final examination at 2:30 to 4:30 May 7: At 10 am	Open-book, open-notes, open Internet!	-

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