In the first example we show that our method is capable of generating very large packings (with over 1 million spheres) in reasonable computation times. In a second example, quantitative correlation functions are imposed as a restriction in the optimization process, which leads to sphere-packings with desired spatial structure (e.g., layering or agglomeration of like-sized particles). In a third example, we demonstrate the simulation of sphere packings inside specifc boundary geometries. These latter simulations are useful in the modeling of a number of applications in chemical engineering. For instance, packed bed reactors can be simulated by random sphere packings in a cylindrical geometry, while a packing in an aperture bounded by surfaces with irregular geometry is a good representation of hydraulically fractured oil reservoirs containing proppant particles.
The novelty of this new algorithm is that these widely varying constraints are all contained in a single objective function in the optimization procedure, which allows for a powerful and highly general algorithm. The algorithm can be extended to packings of non-spherical particles.