Efficient Calculation of Uncertainty Propagation with an Application in Robust Optimization of Forming Processes

Robust optimization is being used in metal forming processes to select the design which is least sensitive to the presence of uncertainty in the input parameters. In most cases, a mathematical surrogate model is built via input data and resulting output obtained from finite element simulations. The influence of uncertainty in input parameters is then considered using a large number of function evaluations via Monte Carlo analysis with a given probabilistic distribution. Although this method is quite fast and simple, it needs a lot of function evaluations to increase accuracy. This random sampling is neither efficient nor reproducible. A new approach is used to calculate the uncertainty propagation analytically. Compared to conventional Monte Carlo approach this method is accurate, fast, stable, and efficient. In addition, it is possible to employ this method with different types of probability distributions and most commonly-used metamodels. To show the applicability of this method in robust optimization process, a stretch-bending process is investigated with two design and two noise variables. Comparing the results obtained by Monte Carlo and the analytical approach shows that different Monte Carlo runs lead to fluctuations around the exact analytical solution. In addition, the analytical approach reduces the evaluation time of finding the robust optimum to a great extent.