Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization May-Jun., 2006
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

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 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	May 1: Calculus of variations and its applications; a bit of history and a few example problems and how we can formulate them. May 3: Mathematical preliminaries: function, operator, functional, metric space, metric, scalar field, vector space, norm, normed vector space, Banach space, Cauchy sequence, completeness May 5: Some familiar problems of calculus of variations.	Galileo and BP, Galileo and BP2 Notes #1	Homework #1
2	May. 8 and 10: Mathametical preliminaries: Inner product, inner product space, Hilbert space, function spaces, energy space, Lebesgue space, Sobolev space, linear, continuous functional, Gateaux variation, necessary condition for the extremum of a functional, Frechet differential May 12: Examples of Gateaux variation of some functionals	Notes #2	No new homework this week.
3	May 15: Review of last week's content Fundamental lemmas of calculus of variations May 17: Euler-Lagrange equations and boundary conditions, I generalization to several derivatives May 19: II generalization of E-L equations to multiple functions and multiple derivatives; III generalization to two independent variables and their boundary conditions.	Notes #3	Homework #2
4	May 23: IV generalization: Global constraint in minimizing a functional; V generaization: local constraints (algebraic and differential equations); motivation for the general variation of a functional May 24: VI generalization: Gthe general variation of a functional; transversality conditions May 26: Stiffest axially-loaded bar for a given volume of material	TBP	Homework #3
5	May 29: Active constraints and how to identify; complementarity conditions; mention of KKT conditions Complete solution of the stiffest bar for given volume. Swicthing of the objective and resource constraint in the above problem. With upper and lower bounds on the cross-section area Posing the giverning equations in weak form. May 31: Including stress constraints in the bar problem. Min-max stress problem. Worst load scenario for the bar. Clayperon's theorem and min-max formulation of the stiffest bar problem. Jun. 2: Repetition of the bar problems for the case of beams. Review for exam 1	Notes #5	Homework #4
6	Jun. 5: Exam 1 Jun. 7: Design for given/desired deflection of bars Unit virtual load method Optimality criterion method--the first glimpse Jun. 9: Beam optimization for various old cases that were considered for bars Mode shape optimization--difference between the inverse problem approach and optimization Optical design of columns	--	Homework #5
7	Jun. 12: Demonstration of the "fmincon" routine in Matlab's optimization toolbox with a two-variable example Demonstration of the truss finite element program including the sensitivity analysis. Jun. 14: Dynamic behavior-based optimization objectives: natural frequency, normal modes, and transient response. Giverning equation of a string under tension using Hamilton's principle Rayleigh quotient; Minimum characterization of the eigenvalue problems: theorem and its proof for finding the kth eigenvalue. Optimal design of a column to maximize the buckling load. Jun. 16: Desired mode shape problem; difference and similarity between an "inverse problem" and "structural optimization problem". Mode shape optimization--difference between the inverse problem approach and optimization	--	Homework #6
8	Jun. 19: Continuum structure parameterization using fictititous density parameterization; 3D problems and optimality criteria method; demonstration and discussion of the 99-line code of Professor Sigmund. Jun. 21: Sensitivity analysis and subtle difference between the direct and adjoint methods Compliant mechanisms synthesis: objectives and constraints. Jun. 23: Discussion and demonstration of PennSyn-a program for stiff-structure and compliant mechanism topology synthesis. Plate and sheet optimization based on thickness.	--	Homework #7
9	Jun. 26: Topology optimization involving multiple energy domains; electro-thermal-complant actuator; problem statement, sensitivity analysis and optimality criteria method Jun. 28: More applications of topology optimization problem; multi-components, modeling convection, contact problems, protein design, etc. Jun. 30: Exam 2 (open-book and open-notes) 10 AM - 12 noon 2005Final	Problems formulated by the class of which one will be modified and given in the exam. Arjun.pdf * aseem.pdf * Joey.pdf * Krishna.pdf * manish.pdf * Navendu.pdf * Parag.pdf * Phani.pdf * rajesh.pdf * Rupesh.pdf * Shijo.pdf * Vivek.pdf *	Project report due on Jul. 3