Slide 2a.1 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Lecture 2a Mathematical Preliminaries for Optimal Design Essential basics of calculus of variations and constrained minimization

Slide 2a.2 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Contents Minimum-time problems –Fermat’s problem and Snell’s law –Brachistochrone problem Constrained minimization –Lagrangian and conventions –Karush-Kuhn-Tucker necessary conditions –Sufficient conditions Calculus of variations –Functional and its variation –Fundamental lemma –Euler-Lagrange equations –Extensions to other situations –Constrained variational calculus problems

Slide 2a.3 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Fermat’s light-ray problem (Feynman’s “life-guard on the beach” problem) What is the minimum-time path from A to B? Can be solved as a constrained minimization problem Leads to Snell’s law of refraction. Speed of light = c2 Speed of light = c1 Lifeguard’s swimming speed = c2 Lifeguard’s running speed = c1 A B A B

Slide 2a.4 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Brachistochrone (minimum time) problem x Y=f(x) The bead slides along a wire under the action of gravity. g A B What shape of the wire (i.e., what f(x)) will lead to the minimum descent time for the bead? Posed as a challenge by Johann Bernoulli. Solved by Leibnitz, Newton, L’Hospital, and Jacob Bernoulli… Functional

Slide 2a.5 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Unconstrained minimization Necessary condition: Sufficient condition:is positive definite Gradient Hessian i.e., If is a solution… ( equations)

Slide 2a.6 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Equality constrained minimization Necessary condition: Sufficient condition: If is a solution… Define a Lagrangian, ( equations) satisfying ( equations) and Lagrange multiplier(s)

Slide 2a.7 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh General constrained minimization Define a Lagrangian, If is a solution… An inequality constraint can be active (= sign) or inactive (<sign). Karush-Kuhn-Tucker (KKT) necessary conditions Complementarity conditions A Beautiful Mind

Slide 2a.8 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Solution to Fermat/Feynman’s minimum-time problem speed = c2 speed = c1 A B & Snell’s law

Slide 2a.9 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Return to brachistochrone problem What is different now? The unknown is a function. The objective is a function of the unknown function and its derivative(s). First variation Operationally useful definition:

Slide 2a.10 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Fundamental lemma of calculus of variations for any then If

Slide 2a.11 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Now, it is only integration by parts… Necessary condition for a minimum: Consider Boundary conditions Euler-Lagrange necessary conditions By the fundamental lemma

Slide 2a.12 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Extensions Second variations; sufficient conditions –Refer to any standard text, e.g., Gelfand and Fomin. To multiple derivatives of –Simply integrate by parts as many times as necessary and collect the boundary terms carefully. To multiple unknown functions, i.e., –Straightforward; write the same set of equations for each. To multiple independent variables, i.e., –Need to use the divergence theorem instead of integrating by parts.

Slide 2a.13 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Constrained variational calculus problems Integral (global) constraint Differential (local) constraint What is the Lagrangian now? Single scalar variable Scalar valued function

Slide 2a.14 Stiff Structures, Compliant Mechanisms, and MEMS: A short course offered at IISc, Bangalore, India. Aug.-Sep., G. K. Ananthasuresh Main points KKT necessary conditions for constrained minimization Euler-Lagrange necessary conditions for a functional The KKT conditions can be used for variational calculus problems as well.