Mathematical morphology and uncertainty quantification for the modeling of dual phase steel

This project is part of a collaboration between the ADAMSS Centre at Università degli Studi di Milano (Italy) and Nippon Steel & Sumitomo Metal, Tokio (Japan), and originates from a previous collaboration between Nippon Steel and WIAS – Berlin, another ECMI Centre, involving other areas of mathematical expertise. Thus it could be considered a virtuous example of collaboration inside ECMI.

Dual Phase steels (DP steels) have shown high potential for the production of steel sheets, due to their remarkable property combination between high strength and good formability.

Steel sheets have found wide-ranging applications that support various industries and the daily lives of people everywhere. These applications include automobiles, home appliances, construction materials, housing materials, beverage cans, and transformers, to name but a few. In line with worldwide economic growth, particularly in emerging nations, demand for steel sheets is steadily expanding.

The mechanical properties of the material are strictly related with the spatial distribution of the two phases composing the steel, ferrite and martensite, and their stochastic geometry. Unfortunately the experimental costs to obtain images of sections of steel samples are very high, thus one important industrial problem is to reduce the number of 2D sections needed to reconstruct or simulate in a realistic way the 3D geometry of the material. This reduction causes an increase of the uncertainty in the parameters estimates of suitable geometric models for the material.

The aim of the project is to create a geometric model (typically a germ-grain model) able to approximate the main geometric characteristics of the martensite. The parameters of the model are estimated on the basis of the morphological characteristics of the images of about 150 tomographic sections of a real sample, with a sort of geometric regression model.

Confidence regions for the estimated morphological measurements of the real sample are then built on the basis of the simulated fitted model.

The same procedure is repeated using a lower number of sections, and the relative amplitude of the confidence regions is compared in order to quantify the increase in uncertainty due to the sample reduction.

The overall procedure for uncertainty quantification that we have thus obtained can be generalized to other study cases and can be used by the industry to set up a suitable experimental plan to fit a model to the data with a desired accuracy.