# The Uddeholm Problem

Powder metallurgy is becoming increasingly important for Uddeholm. That includes Hot Isostatic Pressing (HIP) as well as production of powder for additive manufacturing (AM).

In the first case the initial packing density determines the shrinkage of the HIPped container. In AM, the precision and quality of each new layer of metal applied on the part will be improved if the packing density of the powder is high and well defined.

Uddeholm would have much use of an algorithm that could estimate the packing density of the powder i.e. the volume fraction of the space occupied by particles. The powder particles can be approximated as spheres but they are of varying size. The size (diameter) distribution of a given powder can be measured.

The first part of the problem would be to find a packing density “function” or algorithm.

The second stage would be to find the optimum size distribution, which would be given by maximizing the density function at a boundary condition specifying min and max diameter.

The solver should not search for close packed configurations. The solution that we seek for is the densest possible random configuration. Size distributions of actual powders are continuous, so our first priority would be to have the method work on continuous distributions.