Schedule and Notes
| Week | Dates Topics | Notes | Assignments |
| 1 |
Jan. 5:
Panorama of optimization problems; scope of the course. Unconstrained minimization in finite number of variables: necessary and sufficient conditions Jan. 7: Constrained minimization with equality constraints: Lagrange multiplier concept |
Lecture B Lecture C |
Homework #1 |
| 2 |
Jan. 12:
Constraint qualification Jan. 14: Sufficient conditions for a constrained minimum; Bordered Hessian History of calculus of variations
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Lecture E
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| 3 |
Jan. 19:
Calculus of variations in geometry and mechanics
Jan. 21: No class |
|
Homework #2 |
| 4 |
Jan. 26:
No class; Happy Republic Day!
Jan. 28: Functional, metric spaces, vectors spaces, norm, inner product, Banach space, Hilbert space, Lebesgue and Sobolev norms; Significance of all of these in calculus of variations and structural optimization |
Lecture H |
Homework #3 |
| 4 |
Feb. 2:
Gateux variation and Frechet differential; related concepts. Fundamental lemma of calculus of variations Euler-Lagrange equations Feb. 4: Euler's way of deriving Euler-Lagrange's equations; variational derivative Euler-Lagrange equation for a bar under axial loading; strain energy, work potential and the principle of minimum potential energy; principle of virtual work and the weak form |
Lecture J |
- |
| 5 |
Feb. 9:
Boundary conditions in calculus of variations
Euler-Lagrange equations and their generalization to
many derivatives and many functions Integrals and invariants of Euler-Lagrange equations Change of variables in Euler-Lagrange equations Feb. 11: Noether's theorem (guest lecture by Prof. Ramsharan Rangarajan) |
Lecture H Noether theorem |
Homework #4 |
| 6 |
Feb. 16:
Functionals involving two or three independent variables. Functionals with spatial and temporal variables Feb. 18: Global (functional) constraints in calculus of variations |
Lecture J |
Homework #5 |
| 7 |
Feb. 23:
Local (pointwise or continuous) constraints in calculus of variations
Feb. 25: General variation Transversality conditions, broken extremals, and Weierstrass-Erdmann corner conditions Obtaining the functional from the differential equation: self-adjointness, two methods for dissipative systems (integrating factor and parallel generative system) |
Lecture L |
Homework #6 |
| 8 |
Mar. 1:
Stiffest bar for given volume; design and adjoint equations; uniformly stressed design
Mar. 3: Midterm examination |
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- |
| 9 |
Mar. 8:
Discussion of the midterm problems Optimality criteria method Mar. 10: Variants of the stiffest bar problem: switching the objective function and resource constraint, using the weak form of the governing equation; Clayperon's theorem; Stiffest beam for given volume |
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- |
| 10 |
Mar. 15:
Upper and lower bounds on the design variables in the bar problem Optimality criteria method for the bar Stress constraints Mar. 17: Optimization for flexibility |
|
Homework #7 |
| 11 |
Mar. 22:
Sensitivity analysis: continuous vs. discretized Direct method vs. adjoint method Mar. 24: Optimization for natural frequency Eigenvalue equation for bars using Hamilton's principle |
|
Homework #8 |
| 12 |
Mar. 29:
Minimum characterization of eigenvalue problems Rayleigh quotient Mar. 31: Maximizing the frequency of a bar for given volume |
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- |
| 13 |
Apr. 5:
Buckling phenomenon from the energy and stability viewpoint Derivation of the eigenvalue problem for column buckling and Rayleigh quotient Most stable column for given volume of material Apr. 7: Demonstration of bar, beam, and column optimization codes in Matlab using the optimality criteria method Clarification on handling the inner loop and upper and lower bounds on area of cross-section variables. |
Matlab code for stiffest beam optimization Matlab code for most stable column optimization |
Homework #9 |
| 14 |
Apr. 12:
Structural optimization in a multiphysics problem--electro-thermal-elastic actuator optimization Transient problem--sensitivity analysis |
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- |
| 15 |
Apr. 20:
Final Examination, 2 pm in the classroom
May 2: Project presentations starting at 8:30 am; 15 min for each person. |
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