Modelling microstructure

Polycrystalline materials such as steels are made of grains that differ in size, shape and crystallographic orientation, and this microstructure governs their mechanical behaviour. Modelling it requires a compact geometric description of the grains: one that reproduces measured microstructures, for instance from electron backscatter diffraction (EBSD), and generates synthetic ones for simulation. This line of work is carried out in collaboration with Tata Steel.

We describe a microstructure as a generalised Voronoi diagram. Each grain $L_i$ is the set of points closest to a seed $x_i$ in a grain-dependent sense,

where $A_i$ is a positive definite matrix encoding the grain’s shape and orientation, and $w_i$ is a weight. With $A_i$ the identity these are power diagrams, with flat grain boundaries; general $A_i$ give anisotropic power diagrams, with curved boundaries and elongated grains. Choosing the weights so that each grain has a prescribed volume is a semi-discrete optimal transport problem, which we solve efficiently on the GPU.

Polynomial diagrams go one step further. Power diagrams and anisotropic power diagrams are the first- and second-degree instances of linear parametrised minimisation diagrams, in which each cell minimises a function that is linear in its parameters. Allowing polynomials of higher degree gives cells whose boundaries are algebraic curves of prescribed degree. We fit such diagrams to images by maximising a regularised concave objective adapted from logistic regression, with Legendre polynomials as the basis and a GPU implementation. The analysis identifies the scale and gauge invariances of the problem and the limiting objective as the regularisation vanishes, and the method reproduces EBSD images of steel.