Material for second exam

I. Approximations and surrogates

· Local approximatios

· Surrogates

· Local-Global approximation.

Sample problems:

· Construct the convex (conservative) approximation to f=x1/x2 at the point (1,1). Explain why it is called conservative?

· What are the assumptions that are embodied in Kriging surrogates?

· Explain without equations what is the local-global approximation including the ingredients used in it.

II. Sensitivity analysis

· Finite differences and logarithmic derivatives.

· Alternative methods for calculating derivatives.

· Sensitivity of static response.

· Eigenproblem sensitivity.

Sample problems:

· For the function f=x^7.3, when we change x by 1%, by how much would f change?

· We solve the differential equation y’²-4y=0, y(1)=p for p=1 and obtain the solution y=x². What differential equation and boundary condition does the derivative of y with respect to p satisfy?

· When is the semi-analytical method for calculating static sensitivity most preferable compared to finite difference calculation?

· A simply supported beam with square cross section has uniform properties across its span. Use the equation for the sensitivity of the frequency to indicate how you would change the area distribution across the span (sticking to a square cross section) in order to increase the frequency without increasing the mass.

I. RBDO

· Basic concepts (CDF, PDF, standard deviation, variance, coefficient of variation)

· Monte Carlo simulation and its accuracy.

· FORM.

· Uncertainty control.

Sample problems

· Write the expressions for the PDF and CDF of the random variable that you get from Matlab from the function rand.

· Your response y is given in terms of the random variable x as y=7+10x, where x varies uniformly in [0,1]. The failure criterion is y>16, and you simulate it by setting ys=7+10*rand(1,100) and counting the number of failures. If you repeat the Monte Carlo simulation many times, what will be the standard deviation of your estimate of the probability of failure?

· The random variable x is normally distributed with mean of 1 and standard deviation 1 and the uncorrelated random variable y is normally distributed with mean of 4 and standard deviation 2. The failure condition is x+y>10. Use FORM to find the reliability index.

· If y=x², and x is normally distributed with mean of 100 and standard deviation of 1, what would be the approximate distribution of y?

· How do you expect the reliability index in the FORM problem to change if we reduce the standard deviation of the two variables by 10%?

Material for first exam

I. Optimization formulation and optimality criteria:

· Convert word problem into mathematical formulation.

• Identify minima, maxima and other stationary points (e.g. saddle points) from definition and from first and second order optimality conditions.

Sample problems:

· What kind of point is the origin for the function f=x₁x₂?

· Formulate the problem of minimizing the perimeter and maximizing the area of a rectangle with an open top.

· Give examples of three possible solutions of the open rectangle, such that two are non-dominated and one is dominated.

· For the problem of maximizing the area of the open-top rectangle subject to the constraint that the perimeter is not more than 20, write the Kun-Tucker conditions.

II. Linear programming and Lagrange multipliers

• Solve graphically LP problems
• Formulate dual of an LP
• Use branch and bound method to solve discrete problems
• Calculate Lagrange multipliers for linear and nonlinear problems

Sample problems: You have a knapsack with a total volume of 2 cubic feet. You have five items that you may want to take with you. You may take as many of each item as you want. The following list gives the volume, weight (in pounds), and utility of each item: (0.5,5,1); (0.7,7,2); (0.2,3,0.5); (0.8,10,3); (0.5,10,3).

· Formulate the problem of which items you want to take with you if you want to maximize the utility without exceeding 25 pounds of weight.

· Formulate the dual problem, solve graphically.

· Perform one branching of the branch and bound method for this problem.

· estimate using Lagrange multipliers what is the effect on the optimum utility of changing the weight limit to 26 lbs.

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III. Topology optimization

· Definition and formulation of topology optimization and related terms (e.g. compliance, checker-boarding)

· Outline of methods used for solving topology optimization: SIMP, ESO, filter method.

Sample problems:

· Formulate the equations needed for solving a topology optimization problem using ESO

· Show an example of checkerboarding.

IV. Global optimization algorithms

· DIRECT Algorithm

· Genetic Algorithms

· Particle Swarm Optimization

Sample problems:

· What is the principle of “no free lunch”?

· Code the knapsack problem for a genetic algorithm

· What Pareto front is used in the DIRECT algorithm?

· What constants can be tuned for PSO?

· What methods are used to deal with constraints in global optimization algorithms? list at least three.