Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May, 2007
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 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. 2: Classes have not begun yet. Jan. 4: Classification of optimization problems; Calculus of variations and its applications; a bit of history.	Galileo and BP, Galileo and BP2	Homework #1
2	Jan. 9: A quick review of the theory of finite-variable optimization: KKT conditions, sufficient conditions, and the concept of search algorithms. Jan. 11: Formulation of a few classical problems of calculus of variations. Motivation for a rigirous understanding of a functional and function spaces.	--	Homework #2
3	Jan. 16: Function, functional, metric spaces, vector spaces, normed vector spaces, Banach spaces Jan. 18: Inner product space, Hilbert space, Lebegue and Sobolov norms; function spaces that we deal with.	Notes #1 Notes #2	Homework #3
4	Jan. 23: Gateaux variation and Frechet differential Jan. 25: Fundamental lemmas of calculus of variations; Euler-Lagrange equations.	Notes #3	Homework #4
5	Jan. 30: Gandhi Jayanthi--Holiday. Feb. 1st: Six generalizations of the E-L equations. Example problems of E-L equations; Clarifications on Bordered Hessian.	No notes yet	Homework #5
6	Feb. 6: No class. Feb. 8: Glimpses of topology optimization by Girish Krishnan.	--	No homework this week.
7	Feb. 13: Functionals consisting of two independent variables; variational derivative; E-L equation for a functional with a global constraint: theorem and proof. Feb. 15: E-L equation for a functional with a local constraint: theorem and proof; variable end conditions; transversality conditions.	Notes #4	Homework #6
8	Feb. 20: Size optimization of an axially loaded bar for maximum stiffness; Clayperon's theorem; variation of PE as the statement of the principle of virtual work. Feb. 22: No class.	Notes #5	--
9	Feb. 26: Size optimization of a bar contd.; with weak form of governing equations; upper and lower bounds on the area of cross-section; duality in constrainted optimization. Mar. 1: Stress constraints in bar optimization; worst load scenario for a given problem. Mar. 3: Mid-term exam; covers all of calculus of variations; open notes and open book; 9:30 - 10:30 AM	Practice problems Mid-term 2005 Mid-term 2006	Homework #7; written down in the class on 26 February, 2007.
10	Mar. 6: Design for given deflection Minimum characterization of eigenvalue problems Mar. 8: Optimization of bars with frequency, mode shape and buckling constraints	--	Project proposal
11	Mar. 13: Truss optimization and sensitivity analysis by direct method Mar. 15: Adjoint method of sensitivity analyis Truss topology optimization	--	homework #8
12	Mar. 20: Optimality criteria method; Sigmund's 99 line code; Material interpolation using SIMP and other techniques; basis from homogenization theory, implementation issues in structural topology optimization: checkerboard problems, filters, permimeter constraints, etc. Mar. 22: Design for desired deflection of compliant mechanismms. Different formulations.	--	--
13	Mar. 27: Shape optimization of compliant mechanisms. Mar. 29: Multi-physics optimization: electro-thermal-ealstic optimization example in continuous form and sensitivity analysis. Mar. 30: Optimization for transient response and its adjoint sensitivity analysis.	--	--
14	Apr. 3: Reciprocal linearization, convex linearization, method of moving asymptotes (MMA). Apr. 5: MMA (contd.), generalized convex approximation, some implementation issues.	Handouts of Fleury and Braibant's 1986 paper & Svanaberg's 1987 paper.	--
15	Apr. 10: More on sensitivity analysis; for nonlinear problems. Apr. 12: Project problem statements by the students.	Link to Dr. Moller's PhD thesis	--
16	Apr. 17: Review Apr. 19: Final exam: Open book and open notes. May 1 at 9 AM: Project presentations Each group or individual will have 15 minutes to present. If there are two people in a group, both should present equal parts. Prepare a nice and clear Power Point presentation. Bring your presentation on a CD or a pen drive.	Final exam 2005 Final exam 2006	Due on May 1st: Project report (Power point file is fine but make sure that it has all the details. If it does not have all the details, please submit a report separately.