Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May, 2012
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. 3: Finite variable optimization vs. calculus of variations; Types of optimization problems; scope of the course. Jan. 5: No class due to the ISSS conference	Harold Kuhn's paper on Duality (Karush's work recognized)	Homework #1
2	Jan. 10: Necessary conditions for the finite variable constrained optimization problem. Jan. 12: Concept of local and global minima Necessary and sufficient conditions for unconstrained minimization	-	Homework #2
3	Jan. 17: Karush-Kuhn-Tucker (KKT) necessary conditions Sufficient conditions for the finite variable constrained optimization problem; bordered hessian Jan. 19: No class due to MAMM 2012 conference.	Prof. K. Suresh's noted on finite variable optimization	Homework #2 is due on 24th Jan., 2012.
4	Jan. 24: Mathematical preliminaries for calculus of variations; Notions of vectors space, scalar field, norm, functional, Banach space, Lebegue and Sobolve space and why we need them. Jan. 26: No class due to Republic Day holiday.	Mathematical preliminaries for variational methods	Homework #3
5	Jan. 31: Linear and continuous functionals; Gateaux variation; Frechet differential; Fundamental lemma of calculus of variations Feb. 2: Euler-Lagrange equations for the integrad of the simplest type, F(y,y'); variational derivative and its implication Feb. 4, extra class for two hours: Generalization of Euler-Lagrange equations to multiple derivatives and multiple functions; Generalization to two and three independent variables.	--	-
6	Feb. 7: Minimization of a functional with functional type (global) constraints A dozen problems in calculus of variations Feb. 9: Minimization of a functional with "local" constraints General variation (begun)	--	Homework #4
7	Feb. 14: General variation continued variable end conditions; Weierstrass-Erdmann corner conditions Broken extremals Feb. 16: No lecture due to travel.	--	Homework #5
8	Feb. 21: First integrals of Euler-Lagrange equations; Noether's integral Parametric form and change of variables for functionals and their Euler-Lagrange equations Canonical form of Euler-lagrange Equations Feb. 23: Analytical structural optimization begins Size optimization of the stiffest bar of given weight and axial load.	--	Homework #6
9	Feb. 28: Self-adjointness and weak variational form Stiffest bar for given weight and size limits Stiffest beam for given weight; reinforcing the procedure. Mar. 1: Strength constraints on bars and beams	Midterm papers of previous years	--
10	Mar. 6: Optimization for given deflection of bars and beams Mar. 8: Mid-term examination	FEA notes for bars, beams, etc.	Exam!
11	Mar. 13: Truss optimization Adjoint method (ananlytical) Mar. 15: Sensitivity analysis for discretized systems Adjoint vs. direct methods and their relation to the analytical (Lagrangian) method of computing sensittivities.	Beam FEM codes in Matlab	Homework #7 One-page project proposal
12	Mar. 20: Design for deflection problem; beams with discretized and continuum formulations. Mar. 22: Hands-on computer lab session in the CoNe lab.	Beam FEM codes in Matlab	--
13	Mar. 27: Stress constraints; SIMP parameterization for 2D continuum problems; glimpses of homogenization theory and its relevance to structural topology optimization Mar. 29: No class.	Beam FEM codes in Matlab	--
14	Apr. 3: Optimization for coupled electro-thermal-elastic problem; 2D continuum parameterization Apr. 5: Hands-on computer lab session in the CoNe lab at 3 pm to 6 pm	Beam optimization code in Matlab	--
15	Apr. 10: Calculus of variations and eigenvalue problems; taut string example Apr. 12: Minimum characterization of eigenvalues problems; Rayleight quotient maximizing buckling load of a column for given volume	Column (maximizing buckling load) optimization code in Matlab	--
Finals and project presentations	Apr. 25: Final examination in the CoNe laboratory at 9 am to 11 am. May 4: Project presentations; 10 minutes per group; in the CoNe laboratory starting at 9 am.	Final examinations papers from 2005 to 2011	--