No abstract

We present a fast algorithm for generating Laguerre diagrams with cells of given volumes, which can be used for creating RVEs of polycrystalline materials for computational homogenisation, or for fitting Laguerre diagrams to EBSD or XRD measurements of metals. Given a list of desired cell volumes, we solve a convex optimisation problem to find a Laguerre diagram with cells of these volumes, up to any prescribed tolerance. The algorithm is built on tools from computational geometry and optimal transport theory which, as far as we are aware, have not been applied to microstructure modelling before. We illustrate the speed and accuracy of the algorithm by generating RVEs with user-defined volume distributions with up to 20,000 grains in 3D. We can achieve volume percentage errors of less than 1% in the order of minutes on a standard desktop PC. We also give examples of polydisperse microstructures with bands, clusters and size gradients, and of fitting a Laguerre diagram to 3D EBSD measurements of an IF steel.

No abstract

Although Poisson-Voronoi diagrams have interesting mathematical properties, there is still much to discover about the geometrical properties of its grains. Through simulations, many authors were able to obtain numerical approximations of the moments of the distributions of more or less all geometrical characteristics of the grain. Furthermore, many proposals on how to get close parametric approximations to the real distributions were put forward by several authors. In this paper we show that exploiting the scaling property of the underlying Poisson process, we are able to derive the distribution of the main geometrical features of the grain for every value of the intensity parameter. Moreover, we use a sophisticated simulation program to construct a close Monte Carlo based approximation for the distributions of interest. Using this, we also determine the closest approximating distributions within the mentioned frequently used parametric classes of distributions and conclude that these approximations can be quite accurate.

Multiscale tools are important for the development of multiphase steel grades within Tata Steel R&D. The spatial distribution and morphology of the hard and soft phases in the microstructure as well as their micromechanical properties influences strongly the macroscopic behaviour. To be able to predict the macroscopic response and be of use in an industrial research environment accurate modelling on microscale has to be coupled to efficient homogenization principles. A new algorithm, which extends the capabilities of voronoi tessellations has been developed capturing relevant microstructure parameters. In this paper we show the versatility of the algorithm in simulating many microstructure features in 2 and 3 dimensions and how it is used for micromechanical simulations.