Sparse Balanced Layout of Spherical Voids in Three-Dimensional Domains

The authors consider the optimization problem of layout of spherical voids in three-dimensional domains bounded by cylindrical and spherical surfaces and planes. The problem is reduced to arranging spherical objects in a composite container, with regard for the constraints on their “sparseness” and balance conditions (location of the gravity center of the system). A mathematical model in the form of a nonlinear programming problem is constructed. A method of fast search for feasible solutions based on the balanced homothetic transformations of 3D objects and methods of finding locally optimal solutions using the decomposition algorithm and r-algorithm are proposed. The results of numerical experiments are provided.