Optimized packing soft 3D polyhedra

A novel problem of packing soft convex three-dimensional polyhedral objects in an optimized convex container is introduced. Soft objects can change their shapes within certain limits to better fit into the container, while preserving their volumes. Adjusted non-overlapping and containment conditions considering shape changes of soft objects are presented. Novel geometric tools are constructed to state explicitly these packing constraints, using the phi-function technique. The corresponding large-scale nonlinear nonconvex mathematical programming problem is introduced. To cope with high dimensionality, a decomposition-based multistart heuristic is proposed. Numerical results are presented for packing soft tetrahedra and dodecahedra in minimum-volume cuboidal, spherical, and cylindrical containers for different deformation parameters of objects. Based on real petrophysical data, a case study of digital core design is reported. The proposed approach is used for modeling porosity and particle shapes changes under external pressure. It is demonstrated that structural characteristics of the digital and natural cores are very similar.