Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256 Variational Methods and Structural Optimization Jan.-May, 2013
Instructor: G. K. Ananthasuresh, Room 106, ME Building, suresh at mecheng.iisc.ernet.in
Course work
Assigned work
Books
Syllabus and notes
Project
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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. 10: Finite variable optimization vs. calculus of variations;
Types of optimization problems; scope of the course.
Prof. K. Suresh's notes on finite variable optimization
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2 Jan. 15: Explanation of what constitutes a "calculus of variations" problem;
Necessary and sufficient conditions for nconstrained minimization in finite number of variables.
Jan. 17: Constrained minimization in two variables with an equality constraint; the concept of Lagrange multiplier and Lagrangian
Unconstrained minimization
Homework 1
3 Jan. 22: Constrained minimization in n variables with equality and inequality constraints
Karush-Kuhn-Tucker (necessary) conditions
Jan. 24: Sufficient conditions for constrained minimization
Derivation of the sufficient condition for constrained minimization
Homework 2
4 Jan. 29: Review of continuous optimization theory.
Jan. 31: Sample problems of calculus of variations.
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5 Feb. 5: Mathematical preliminaries: vectors spaces, functional, Banach space, Hilbert space, Lebegue and Sobolev norms and spaces, etc.
Feb. 7: No class.
Mathematical preliminaries
Homework 3
6 Feb. 12: Euler-Lagrange-Ostrogradski euqations for a functional involving one derivative; boundary conditions;
principle of minimum potential energy; the weak form and the principle of virtual work. the principel of
Feb. 14: Euler-Lagrange-Ostrogradski equations for functional involving multiple derivatives and functions.
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7 Feb. 19: Functionals involving function of two and three independent variables
General (non-contemporaneous) variation and the associated boundary conditions.
Feb. 21: Trabsversality conditions; Weierstrass-Erdmann corner conditions for broken extremals.
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Homework #4
8 Feb. 26: Minimization of functionals with global and local constraints
Feb. 28: No class.
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Homework #5
9 Mar. 5: First integrals of Euler-Lagrange equations;
Change of coordinates and invariance of Euler-Lagrange-Ostrogradski equations; Noether's theorem.
Parametric form of Euler-Lagrange -Ostrogradsky equations
Mar. 7: Review for the mid-term examination
Mar. 9: Mid-term examination; open book; open notes; open Internet! 10 am to 12 noon in MMCR of the ME department.
Mid-term papers of past years 2005 to 2012
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10 Mar. 12: Stiffest bar for given volume
Design and adjoint equations
Mar. 14: Stiffest bar for given volume with weak form of the governing equation
Physical meaning of the Lagrange multipliers
Complete solution of the stiffest bar problem
Project proposal due as Homework #7; see Project page.
Homework #6
11 Mar. 19: Stiffest bar problem with upper and lower bound constraints on the area of cross-section(br> Strength considerations: two ways of posing the problem
Mar. 21: Optimality criteria method
General elastic structure for maximum stiffness for given volume
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Homework #7
12 Mar. 26: Truss, beam, and bar optimization for desired deflection
Mar. 28: Sensitivity analysis for the continous and discrete systems: comparison and physical interpretation
Mar. 30: Demonstration of the Matlab programs using the optimality criteria method
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13 Apr. 2: Sensitivity analysis: direct and adjoint methods Apr. 4: Matlab programs - a review
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14 Apr. 9: Electro-thermal-compliant atuator optimization
Apr. 11: Minimim characterizaton of eigenvalue problems
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15 Apr. 16: Minimim characterizaton of eigenvalue problems (contd); Rayleigh quotient; buckling problems as eigenvalue problems
Final examms from 2005 to 2012
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Exam and project Apr. 27: Final examination at 2:30 to 4:30
May 7: At 10 am
Open-book, open-notes, open Internet!
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You can find the content-page of the previous years here.
2012
2011
2009
2007
2006
2005