Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Aug.-Dec. 2005
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	Aug. 4: Calculus of variations and its applications; a bit of history.	Galileo and BP, Galileo and BP2	Homework #1
2	Aug. 9: Mathematical preliminaries: function, operator, functional, metric space, metric, scalar field, vector space, norm, normed vector space, Banach space, Cauchy sequence, completeness Aug. 11: 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 Aug. 12: Examples of Gateaux variation of some functionals	Notes #1	Homework #2
3	Aug. 16: Some properties of Frechet differential, equivalence between Gateaux variation and Frechet differential, operationally useful form of Gateaux variation, fundamental lemma of calculus of variations and du Bois-Raymond's proof of the lemma, and lemmas involving higher derivatives with proofs. Aug. 18: Euler-Lagrange equations, example problems: minimum distance between two points in a plane and brachistochrone problem, principle of minimum potential energy and its application to the governing equilibrium equations for an axially deforming bar, significance of boundary conditions.	--	Homework #3
4	Aug. 23: Weak form of governing differential equations of elastic systems, principle of virtual work, boundary conditions for a beam, E-L equations for the case of an intergral constraint, the problem of enclosing maximum area by a planar curve of given length. Aug. 25: Variational derivative, statement of the E-L equations for the case of an integral constraint, constraint-qualification in finite-variable optimization and its equivalent in calculus of variations.	--	Homework #4
5	Aug. 30: multiple-function and multiple-derivative cases, proofs of E-L equations for the cases of an integral constraint and an algebraic constraint. Statement of the problem with general differential equation constraints. Sep. 1: General variation of functionals at the boundaries, transversality conditions, Weierstrass-Erdmann corner conditions and discontinuous extremals.	--	Homework #5
6	Sep. 6: Fundamental lemma of the two-variable case and its proof; E-L necessary conditions for the two independent variable case; the case of multiple independent variables and multiple (higher order) partial derivatives; Green's theorem in 3-D; example of a plate with uniform load Sep. 8: Size optimization of an axially loaded bar for maximum stiffness for given volume of material; mean compliance as a measure of stiffness; outline of the solution and its important features. Sep. 9: Examples	Lecture 11	No homework this week!
7	Sep. 12: Continuation of the stiffest bar problem with upper and lower bounds on the area of cross-section; emphasis on design equations, boundary conditions, adjoint variable, and physical meaning of the Lagrange (both scalar and function) multiplers; complementarity conditions, nature of the solution and all the equations that provide the complete solution; Switching the objective function and the volume constraint. Sep. 15: Solution of the stiffest bar problem using Maple; Inclusion of stress constrains; analyzing the nature of the solution by identifying various cases of active and inactive constraints; dividing the design domain into different intervals; analogy to solving finite-variable constrained optimization problems. Sep. 16: Beam-bar problem with a torsional-spring hinge at one end and guiding along a straight line at the other end; treating the boundary integrals in a general problem; transversality conditions for a functional involving second derivatives of the state variable.	--	Homework #6
8	Sep. 20: Worst distribution of a given load for an axially loaded bar; minimizing the maximum stress for a given volume of material of an axially loaded bar. Sep. 22: Clayperon's theorem and an alternate statement of the worst load problem; Stiffness requirement as a point-wise constraint and its implications. Sep. 23: Going over the questions submitted; Stiffest beam for a given volume; analogy to the bar problem;	--	Homework #7
9	Sep. 27: Design for a given deflection of beams; repetition of beam problems based on the formulations for the bars. Sep. 29: More on design for deflection of beams for flexibility and design; design of beams and bars for desired mode shapes; ill-posed (or ill-formulated) problems and ways to rectify them Sep. 30: Practice problems; finding the functional when the extremizing functions or the E-L equations are given.	--	Practice problems for mid-term
10	Oct. 4th: Optimal design of columns Oct. 6th: Elastic string in tension, equation of motion, minimum characterization of the eigenvalue problem with proof, Rayleigh quotient for the eigenvalue problem and buckling Oct. 7th: Practice problems and solutions of homeworks 6 and 7	--	No homework this week
11	Oct. 11th: Mid-term exam; open notes but closed book. Oct. 13th: Holiday; Happy Dasara! Oct. 14th: Numerical optimization with Matlab's optimization toolbox	Sample optimization files TrussFiles Test your intuition	Midterm exam (in class) Project proposal
12	Oct. 18th: Variable-thickness plates and sheets Oct. 20th: Finite variable otpimization; KKT-conditions and outline of their derivations and their significance Oct. 14th: More on KKT conditions; sufficiency conditions	Optimization theory	Homework #8