André Liebscher

3D image analysis and stochastic modelling of open foams

Abstract We present methods for the geometric characterisation and stochastic modelling of open foams based on tomographic 3D image data. In the first step, geometric characteristics of the foam microstructure are estimated from the image. Using these characteristics, a random tessellation model is fitted to the cell system of the foam. The edges of this tessellation then serve as a skeleton for the foams strut system. The focus of the paper is on the correct simulation of the foams locally varying strut thickness in the model. For this purpose, the local strut thickness and the strut profiles are estimated from the image and reproduced in the model using locally adaptable morphology.

Stereological reconstruction of polycrystalline materials

Laguerre tessellations are suitable models for many polycrystalline materials. In this work, we present a reconstruction‐based approach to fit a spatial Laguerre tessellation model to a plane section of a cellular material under the condition that one section of the model resembles the observed section of the material. To account for this special requirement, we introduce a novel Euclidean distance‐based criterion for the model fitting. The model fitting itself is based on Simulated Annealing. If the structure under consideration is a Laguerre tessellation, we found a nearly perfect reconstruction of its spatial cell characteristics in the model. Even for a real sample of a sintered alumina the observed section is captured quite well by the model.

Laguerre approximation of random foams

Stochastic models for the microstructure of foams are valuable tools to study the relations between microstructure characteristics and macroscopic properties. Owing to the physical laws behind the formation of foams, Laguerre tessellations have turned out to be suitable models for foams. Laguerre tessellations are weighted generalizations of Voronoi tessellations, where polyhedral cells are formed through the interaction of weighted generator points. While both share the same topology, the cell curvature of foams allows only an approximation by Laguerre tessellations. This makes the model fitting a challenging task, especially when the preservation of the local topology is required. In this work, we propose an inversion-based approach to fit a Laguerre tessellation model to a foam. The idea is to find a set of generator points whose tessellation best fits the foam’s cell system. For this purpose, we transform the model fitting into a minimization problem that can be solved by gradient descent-based optimization. The proposed algorithm restores the generators of a tessellation if it is known to be Laguerre. If, as in the case of foams, no exact solution is possible, an approximative solution is obtained that maintains the local topology.

Random Tessellations and their Application to the Modelling of Cellular Materials

This chapter introduces various tessellation models and discusses their application as models for cellular materials. First, the notion of a random tessellation, the most well-known model types (Voronoi and Laguerre tessellations, hyperplane tessellations, STIT tessellations), and their basic geometric characteristics are introduced. Assuming that a cellular material is a realisation of a suitable random tessellation model, these characteristics can be estimated from 3D images of the material. It is explained how estimates are obtained and how these characteristics can be used to fit tessellation models to the observed structure. All analysis and modelling steps are illustrated using the example of an open cell aluminium foam.

Microstructural analysis of a C/SiC ceramic based on the segmentation of X-ray phase contrast tomographic data

Abstract This paper is dedicated to the analysis of 3D data of carbon fiber reinforced silicon carbide ceramics. In the production process of C/SiC, a porous carbon preform reinforced with bundles of carbon fibers is infiltrated with liquid silicon at approximately 1 500 °C. The reaction between liquid silicon and carbon creates a layer of silicon carbide while the silicon vanishes almost completely. This material is able to withstand extremely high temperatures and it is extremely tough with respect to fracture. To increase the efficiency of the costly and time consuming production process, methods for monitoring the quality of the material are helpful. For instance, the thickness of the silicon carbide layer is a valuable measure. Moreover, due to different coefficients of thermal expansion of the components, typically cracks appear during the production process. For effective analysis 3D high resolution image data are necessary that can be acquired by synchrotron computed tomography. In a first image processing step, we segment the 3D image data with a convex optimization approach incorporating spatial regularity. Further, we work on the detection of cracks using an eigenvalue analysis of the 3D Hessian matrix determined in each pixel.

Statistical Volume Elements for Metal Foams

Open cell metal foams can be represented by a network of beams. Due to the heterogeneity of the geometry, the length scale of the representative volume element is often nearly of the same order as the length scale of structures made of metal foam. Therefore, classical homogenization techniques for the computation of effective properties can not be applied. Statistical volume elements lead to apparent material properties that depend on the boundary conditions. Here, we introduce a model for structures made of metal foam that consists of two domains, an interior region and a boundary region. For both regions, unique random fields are identified by simulations of the microstructure. The model is validated by comparison with Finite Element simulations and experiments.