Engineering

### Problem Set: Metamodeling Polynomial Regression

HW3-1: (30 points) For the design problem of the hollowed circular beam (shown below),

• a. Create a three-level factorial design matrix in HiPPO and map to the real ranges for R = [0.02 to 0.2] m and t = [0.001, 0.01] m. Show your design matrix including both the normalized and real design variables along with the function values of M, G, and V for each design in your design matrix.
• b. Create the quadratic PR models (with interactions) of M, G, and V in HiPPO and create the 3D plots of these three metamodels in Matlab. Briefly comment on the accuracy of the three metamodels by comparing to the true function plots in HW2-3.
• c. Create the GimOPT input file for this optimization problem using the three metamodels and obtain the optimum solution. Report your solution and briefly comment on it by comparing to the true solution in

HW2-2. Note that these metamodels and their gradients can be saved to file in HiPPO. Also note that the two functions for G and V are not the constraint functions by themselves; you need to apply their limit values when creating the constraint functions for GimOPT.

HW3-2: (40 points) For the design problem of the hollowed circular beam (shown in the figure of HW3-1),

• a. Create a five-level factorial design matrix in HiPPO and map to the real ranges for R = [0.02 to 0.2] m and t = [0.001, 0.01] m. Show your design matrix including both the normalized and real design variables along with the function values of M, G, and V for each design in your design matrix.
• b. Create the quadratic PR models (with interactions) of M, G, and V in HiPPO and create the 3D plots of these three metamodels in Matlab. Briefly comment on the accuracy of the three metamodels by comparing to the true function plots in HW2-3.
• c. Create the GimOPT input file for this optimization problem using the three metamodels and obtain the
optimum solution. Report your solution and briefly comment on it by comparing to the true solution in

HW2-2. Note that these metamodels and their gradients can be saved to file in HiPPO. Also note that the two functions for G and V are not the constraint functions by themselves; you need to apply their limit values when creating the constraint functions for GimOPT.