Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May, 2009
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: Classification of optimization problems; Finite variable optimization vs. calculus of variations Necessary conditions for the finite variable unconstrained optimization Jan. 8: Holiday: Happy Moharram!	--	Homework #1
2	Jan. 13: Necessary conditions for unconstrained minimum in finite variables. Jan. 15: KKT conditions, sufficient conditions, and the concept of search algorithms	--	Homework #2
3	Jan. 20: A bit of history of calculus of variations Formulation of calculus of variations problems Jan. 22: Function vs. functional; metric space and metric; vector space and norm Banach space, Cauchy sequence; relevance to optimization process; function spaces and why we need them for mechanics and structural optimization? Inner product spaces; inner product; Hilbert space; Sobolev and Lebesgue norms; continuous and linear functionals.	Notes 1	No homework this week
4	Jan. 27: First variation of a functional; Gateux variation; Frechet differential; epsilon-method for taking the first variation. Fundamental lemma of calculus of variations Euler-Lagrange equations Jan. 29: Extension of Euler-Lagrage equations to to multiple derivatives and multiple functions in the functional. Calculus of variations in functionals involving two independent variables.	Notes 2 Notes 3	Homework #3
5	Feb. 3: Global constraints in calculus of variations. Local (finite subsidiary) constrains in calculus of variations. Feb. 5: General variation of a functional; transversality conditions. Broken extremals; Wierstrass-Erdmann corner conditions	Notes 4	Homework #4
6	Feb. 10: Invariants of Euler-Lagrange equations Summary of calculus of variations Feb. 12: Optimal cross-section area profile for the stiffest bar under arbitrary loading	-	Homework #5
7	Feb. 17: Including the governing equations in the weak form Imposing upper and lower limits on the area of cross-section Feb. 19: Min-max type problems with stress constraints Deflection constraint at a point Min-max type stress constraint	Last four years' midterms are here.	No home work this week. Midterm exam on 24th February, 2009.
8	Feb. 24: Mid-term examination during class time. Feb. 26: Worst load determination A case of a single scalar unknown along with an unknown function.	See project page.	Homework #6
9	Mar. 3: Beam design problem and an inkling of the optimality criteria method. Mar. 5: Worst load determination Beam optimization problem using the optimality criteria method. Design for deflection problem for the beam. Truss optimization problem for the desired deflection and an inkling of topology optimization problem.	--	Homework #7
10	Mar. 10: No class. Work on homework #7. Mar. 12: No class. Work on homework #7.	--	No homework this week.
11	Mar. 17: Sensitivity analysis for discretized structural optimization problems; direct and adjoint methods. Mar. 19: Governing equation for the free vibration of a taut string; derivation using Hamilton's principle to get the eigenvalue problem.	--	No homework this week; start the course project.
12	Mar. 24: Minimum characterization of the eigenvalue problem; Rayleigh quotient. Mar. 26: Derivation of the Rayleigh quotient for the beam vibration and column buckling Frequency and mode shape optimization problems for bars and beams.	--	Homework #8
13	Mar. 31: Structural optimization of 2D and 3D elasticity problems for stiffness and flexibility. Simple Isotropic Material with Penalty (SIMP) approach to topology optimization. Apr. 2: Structural optimization of electro-thermal-compliant actuator problem. Extra class on April 3rd, 2009, 9 AM in the M2D2 lab. Demonstration of the topology optimization programmes.	Stress constraints paper	Homework #9
13	Apr. 7: Holiday; no class. Apr. 9: Structural optimization for transient (dynamic) problems. Shape optimization and sensitivity analysis.	Shape optimization paper	Homework #10
14	Apr. 14: Holiday; no class. Apr. 16: Numerical optimization methods for structural optimization; separable problems and dual methods; convex linearization, method of moving asymtotes, generalized convex approximation.	CONLIN paper MMA paper	Work on the course project.
Final exam and project	Apr. 21: Final exam: 3:00 PM to 4:30 PM on April 21st, 2009 in the class-room May 7: Project presentations in the class room at starting at 10 AM. 15 minutes for each presentation.	Final exams papers of 2005-2008	Submit the PPT file of your project along with any Matlab and other supplementary files. Please look at project page for details regarding what the PPT file should contain.