\n", "Solution\n", "
\n", "Unfortunately the math equations (defined using Latex) don't work on the website. Hopefully they are readable; if you download and open the notebook in Jupyter or VS Code the equations will display correctly.\n", "\n", "Task 0:\n", "Tasks to complete during the workshop are provided in boxes like this.\n", "
\n", "\n", "Explanations in the solution are provided in boxes like this.\n", "
\n", " In-class notes: \n", "
\n", " On Friday our solution notebook had an error in the standard deviation for $M_2$, so the results you saw in class were slightly different.On Friday our solution notebook had an error in the standard deviation for $M_2$, so the results you saw in class were slightly different.\n", "\n",
" Note: c.o.v. is
\n", "Task 1:\n", "Define the marginal distributions of the random variables using OpenTURNS.\n", "
\n", "\n", "The solution is in the code cell below.\n", "
\n", "Task 2:\n", "Define the limit-state function.\n", "
\n", "\n", "The solution is in the code cell below. OpenTURNS will send arguments to the function such that a row defines the value of all random avariables, and expects a single value in return. If there are $N$ samples and $N_{rv}$ random variables, the input and output should be $N$x$N_{rv}$ and $N$x1, respecitvely.\n", "
\n", "Task 3:\n", "Define the correlation matrix for the multivariate probability distribution.\n", "
\n", "\n", "The solution is in the code cell below. As you can see we specify multivariate normal, which requires only a matrix of correlation coefficients. Since the first line generates a correlation matrix of independent random variables by default, and only the first two random variables are correlated, you simply need to add the coefficient to two indices. Remember the matrix is symmetric with elements $\\rho_{ij}$.\n", "
\n", "Task 4:\n", "Interpret the FORM analysis. Be sure to consider: pf, beta, the design point in the x and u space.\n", "
\n", "\n", "Task 5:\n", "Interpret the MCS and compare it to FORM. What can you learn about the limit-state function?\n", "
\n", "\n", "Task 6:\n", "Interpret the importance factors. Which random variables act as loads or resistances? What is the order of importance?\n", "
\n", "\n", "Sensitivity is covered in HW4 and Week 6; it is included here to give you a hint for how it is used in the interpretation of FORM.\n", "
\n", "\n", "Task 7:\n", "Complete the code below by filling in the missing values of the design point for the plot. Then take a look at the plots and see if you can learn anything about the limit-state function, and whether it is aligned with your FORM and MCS results. If you were designing a structure using the beam in this proble, would you feel comfortable using the design point from this analysis?\n", "
\n", "\n", "Task 8:\n", "You do not need to fill out any code in the bonus part. Take a look at the figures and compare them to the original case, which included linear dependence (i.e., bivariate Gaussian) between the two load variables. Pay particular attention to the correlation structure in both plots, and try to understand where it comes from. Note in particular the commented piece of code, which uses an (incorrect) alternative for computing the PDF. Why is it different, and why is it incorrect? This is especially important for the first plot, which compares two random variables which, based on the correlation matrix, independent.\n", "
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