Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May, 2011
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. 6: Finite variable optimization vs. calculus of variations; Chatterjee's problem; minimum-time paths by light rays in reflection and refraction; Necessary and sufficient conditions for unconstrained minimization.	--	Homework #1
2	Jan. 11: Necessary conditions for the finite variable constrained optimization problem. Jan. 13: Karish-Kuhn-Tucker (KKT) necessary conditions Sufficient conditions for the finite variable constrained optimization problem; bordered hessian	--	--
3	Jan. 18: Vector spaces and functionals Metric, norm, normed vector spaces, Banach spaces Jan. 20: Inner product spaces, Hilbert spaces Gateaux variation: formal and operationally useful definitions Directional derivative of multi-variable functions Frechet differential	--	Homework #2
4	Jan. 25: Thumbrule for taking variation of functions involving integrals Fundamental lemms of calculus of variations Euler-Lagrange equations and the boundary conditions when the integrand is of the form F(y,y'). Jan. 27: Extension of Euler-Lagrange equations to multiple derivatives, F(y,y',y", ...y(n)). Extension of E-L equations to multiple functions Extension of E-L equations to two independent variables; Green's theorem	--	Homework #3
5	Feb. 1: Divergence theorem and extension to three independent variables Complete generalization of E-L equations Feb. 3: Local vs. global constraints	--	--
6	Feb. 8: Dealing with global constraints in calculus of variations; theorem and proof; the role of Lagrange multiplier Feb. 10: Dealing with local (finite susidiary) constraints in calculus of variations; Lagrange multiplier function theorem	--	Homework #4
7	Feb. 15: General variation Variable end conditions Weierstraas-Erdmann corner conditions Broken extremals Feb. 17: Curvilinear coordinates and Euler-Lagrange equations Noether's theorem and first intergrals	--	--
8	Feb. 22: Parametric form of Euler-Lagrange equations Canonical form and first integrals when the integrand is of the form F(x,y,y') Differential equation to functional; self-adjointness Functionals for dissipative systems: two solitions. Feb. 24: Structural optimization of the stiffest bar for a given volume.		homework #5
9	Mar. 1: Design and adjoint equilibrium equations Stiffest structure problem with weak formulation Minimum volume for specified stiffness of a bar Side constraints on the design variables and active regions Alternate objective functions for stiffness Mar. 3: Constraints on displacements Optimality criteria method illustrated with the bar optimiation problem Sensitivity analysis with continuous systems Sensitivity analysis for discretized systems based on sensitivities from continuous analysis.	Mid-term exam papers from 2005 to 2010	--
10	Mar. 8: Solving structural optimization problems using the optimality criteria method and mathematical programming Mar. 10: Mid-term exam	--	--
11	Mar. 15: Sensitivity analysis: direct and adjoint methods; comparing the computations involved. Mar. 17: Design for desired deflection; Barnett's paradox; unit dummy load method	--	--
12	Mar. 22: Minimization of maximum stress in a beam Mar. 24: Beam and bar optimization problem: implementation details of the optimality criteria method using Matlab	--	homework #6
13	Mar. 29: No class. Mar. 31: Minimum characterization of eigenvalue problems: vibrating strings	--	--
14	Apr. 5: Minimum characterization of eigenvalue problems (Contd.) Apr. 7: Rayleigh quotient and calculus of variations; Rayleigh quotient for column-buckling	--	Finalize and submit the project proposal.
15	Apr. 12: Optimality criteria method for the multi-physics problems; illustration with electro-thermal-compliant actuators. Apr. 14: Topology optimization and its capabilities; illsutration of YinSyn	YinSyn , a topology optimization software for stiff structures and compliant mechanisms.	Note the dates of final exam and course-project presentations!
16	Apr. 21: Final examination in the ME lecture hall from 2:00 pm to 4:00 pm.	Final exam papers of 2005-2010	Open book and open notes! You can bring anything.
17	May 3: Course-project presentations from 8:30 am onwards in the ME lecture hall.	--	Present for not 15 minutes. You should not have more than 15 slides.