Processing Uncertain Microstructural Data

The behavior of natural, artificial and tailored materials is governed by the underlying stochastic microstructure and the uncertain microscopic parameters that form the constituents. This is particularly important in view of predicting the failure characteristics of micro-heterogeneous solids. In this project we define a new class of stochastic random fields to model heterogeneous/discontinuous structures. We propose so called L\’evy fields that have semi-heavy-tailed/heavy-tailed pointwise marginal distributions and display specific discontinuous structures. Besides stochastic properties of the fields we investigate fast numerical simulation methods.  Solutions to a class of challenging partial differential equations with these highly irregular random field coefficients are of central interest. The aim is to show well-posedness and develop numerical methods to approximate the law of the solution and quantify the uncertainty, especially in a rare-event context. In collaboration with our tandem project 3-1 we use the findings to model material parameters of complex uncertain 3D microstructures. Simulations are the basis for a database from which critical regions for possible material failure are learned.