Optimized Layout of Spherical Objects in a Polyhedral Domain

Abstract

Introduction. The article studies the problem of arranging spherical objects in a bounded polyhedral domain in order to maximize the packing factor. The spherical objects have variable placement parameters and variable radii within the given upper and lower bounds. The constraints on the allowable distance between each pair of spherical objects are taken into account. The phi-function technique is used for analytical description of the placement constraints, involving object non-overlapping and containment conditions. The problem is considered as a nonlinear programming problem. The feasible region is described by a system of inequalities with differentiable functions. To find the local maximum of the problem the decomposition algorithm is used. We employ the strategy of active set of inequalities for reducing the computational complexity of the algorithm. IPOPT solver for solving nonlinear programming subproblems is used. The multistart strategy allows selecting the best local maximum point. Numerical results and the appropriate graphic illustration are given. The purpose of the article is presenting a mathematical model and developing a solution algorithm for arranging spherical objects in a polyhedral region with the maximum packing factor. It allows obtaining a locally optimal solution in a reasonable time. Results. A new formulation of the problem of arranging spherical objects in a polyhedral domain is considered, where both the placement parameters and the radii of the spherical objects are variable. A mathematical model in the form of nonlinear programming problem is derived. A solution approach based on the decomposition algorithm and multistart strategy is developed. The numerical results combined with the graphical illustration are given. Conclusions. The proposed approach allows modeling optimized layouts of spherical objects into a polyhedral domain. Keywords: layout, spherical objects, polyhedral domain, phi-function.