In this work, metamodel-based robust optimization is performed using measured scatter of noise variables. Principal component analysis is used to describe the input noise using linearly uncorrelated principal components. Some of these principal components follow a normal probability distribution, others however deviate from a normal probability distribution. In that case, for more accurate description of material scatter, a multimodal distribution is used. An analytical method is implemented to propagate the noise distribution via metamodel and to calculate the statistics of the response accurately and efficiently. The robust optimization criterion as well as the constraints evaluation are adjusted to properly deal with multimodal response. Two problems are presented to show the effectiveness of the proposed approach and to validate the method. A basketball free throw in windy weather condition and forming of B-pillar component are presented. The significance of accounting for non-normal distribution of input variables using multimodal distributions is investigated. Moreover, analytical calculation of response statistics, and adjustment of the robust optimization problem are presented and discussed.

Production efficiency in metal forming processes can be improved by implementing robust optimization. In a robust optimization method, the material and process scatter are taken into account to predict and to minimize the product variability around the target mean. For this purpose, the scatter of input parameters are propagated to predict the product variability. Consequently, a design setting is selected at which product variation due to input scatter is minimized. If the minimum product variation is still higher than the specific tolerance, then the input noise must be adjusted accordingly. For example this means that materials with a tighter specification must be ordered, which often results in additional costs. In this article, an inverse robust optimization approach is presented to tailor the variation of material and process noise parameters based on the specified product tolerance. Both robust optimization and tailoring of material and process scatter are performed on the metamodel of an automotive part. Although the robust optimization method facilitates finding a design setting at which the product to product variation is minimized, the tighter product tolerance is only achievable by requiring less scatter of noise parameters. It is shown that the presented inverse approach is able to predict the required adjustment for each noise parameter to obtain the specified product tolerance. Additionally, the developed method can equally be used to relax material specifications and thus obtain the same product tolerance, ultimately resulting in a cheaper process. A strategy for updating the metamodel on a wider (noise) base is presented and implemented to obtain a larger noise scatter while maintaining the same product tolerance.