Optimized Packings in Space Engineering Applications: Part I

Abstract

Packing optimization problems have a wide spectrum of real-word applications, including transportation, logistics, chemical/civil/mechanical/power/aerospace engineering, shipbuilding, robotics, additive manufacturing, materials science, mineralogy, molecular geometry, nanotechnology, electronic design automation, very large system integration, pattern recognition, biology, and medicine. In space engineering, ever more challenging packing optimization problems have to be solved, requiring dedicated cutting-edge approaches.

Two chapters in this volume investigate very demanding packing issues that require advanced solutions. The present chapter provides a bird’s eye view of challenging packing problems in the space engineering framework, offering some insight on possible approaches. The specific issue of packing a given collection of arbitrary polyhedra, with continuous rotations and minimum item-to-item admissible distance, into a convex container of minimum size, is subsequently analyzed in depth, discussing an ad hoc mathematical model and a dedicated solution algorithm. Computational results show the efficiency of the approach proposed. The following (second) chapter examines a class of packing optimization problems in space with consideration to balancing conditions.

Appendices

Appendix: Phi-Functions and Quasi-Phi-Functions

Phi-Functions

Phi-functions allow us to distinguish among the following three cases: sets A and B are intersecting so that A and B have common interior points; A and B do not have common points; A and B are in contact, i.e., A and B have only common boundary points.

Let A ⊂ R ³ and B ⊂ R ³ be two objects. The sizes of objects can change according to homothetic coefficients (scaling parameters) λ _A, λ _B > 0. The position of object A is defined by a vector of placement parameters (v _A, θ _A), where v _A = (x _A, y _A, z _A) is a translation vector and ${\theta}_A=\left({\theta}_A^1,{\theta}_A^2,{\theta}_A^3\right)$ is a vector of rotation angles. We denote the vector of variables for the object A by u _A = (v _A, θ _A, λ _A) and the vector of variables for the object B by u _B = (v _B, θ _B, λ _B). The object A, rotated by angles ${\theta}_A^1,{\theta}_A^2,{\theta}_A^3$, translated by vector v _A, and rescaled by homothetic coefficient λ _A,will be denoted by A(u _A).

Definition A1

A continuous function Φ^AB(u _A, u _B) is called a phi-function for objects A(u _A) and B(u _B) if

• Φ^AB > 0, if A(u _A) ∩ B(u _B) = ∅;

• Φ^AB = 0, if intA(u _A) ∩  int B(u _B) = ∅ and frA(u _A) ∩ frB(u _B) ≠ ∅;

• Φ^AB < 0, if intA(u _A) ∩  int B(u _B) ≠ ∅;

provided that λ _A, λ _Bare fixed.

Here frA means the boundary (frontier) and intA means the interior of object A.

Thus, inequality Φ^AB ≥ 0 represents the non-overlapping relationship intA(u _A) ∩  int B(u _B) =  ∅ , i.e., Φ^AB ≥ 0 ⇔  int A(u _A) ∩  int B(u _B) =  ∅ .

We use phi-functions for the description of the containment relation A ⊆ B as follows: ${\Phi}^{AB^{\ast }}\ge 0$, where B ^* = R ³\ int B.

We emphasize that according to Definition 1, the phi-function Φ^AB for a pair of objects A and B can be constructed by many different formulas [55], and we can choose the most convenient ones for our optimization algorithms.

We can take into account minimum allowable distance constraints by replacing the phi-functions in the non-overlapping and containment constraints with adjusted phi-functions.

Let ρ > 0 be a given minimum allowable distance between objects A(u _A) and B(u _B).

Definition A2

A continuous and everywhere defined function ${\overset{{\frown}}{\Phi}}^{AB}\left({u}_A,{u}_B\right)$ is called an adjusted phi-function for objects A(u _A) and B(u _B), if

• ${\overset{{\frown}}{\Phi}}^{AB}>0,$ if dist(A, B) > ρ; ${\overset{{\frown}}{\Phi}}^{AB}=0,$ if dist(A, B) = ρ;

• ${\overset{{\frown}}{\Phi}}^{AB}<0,$ if dist(A, B) < ρ.

Now we can describe the distance constraint for objects A(u _A) and B(u _B) in the form: ${\overset{{\frown}}{\Phi}}^{AB}\ge 0\iff$dist(A,B)≥ρ.

Quasi-Phi-Functions

We introduce a function Φ^′AB(u _A, u _B, u ^′) that must be defined for all values of u _A and u _B. In addition to the placement parameters of objects used with phi-functions, quasi-phi-functions depend on auxiliary variables u′. These extra variables u′ take values in some domain U ⊂ R ^η. The number and the nature of variables u′ depend on the shapes of objects A(u _A) and B(u _B), as well as on the restrictions of a layout problem.

Definition A3

A continuous function Φ^′AB(u _A, u _B, u ^′) is called a quasi-phi-function for two objects A(u _A) and B(u _B) if $\underset{u\prime\in U}{\max }{\Phi^{\prime}}^{AB}\left({u}_A,{u}_B,{u}^{^\prime}\right)$ is a phi-function for the objects.

The main property of a quasi-phi-function is the following:

• if Φ^′AB(u _A, u _B, u ^′) ≥ 0 for some u′, then intA(u _A) ∩ int B(u _B) = ∅,

where Φ^′AB(u _A, u _B, u ^′) is a quasi-phi-function for two objects A(u _A) and B(u _B).

Let ρ > 0 be a given minimum allowable distance between objects A(u _A) and B(u _B).

Definition A4

Function ${\overset{{\frown}}{\Phi^{\prime}}^{AB}}\left({u}_A,{u}_B,{u}^{\prime}\right)$ is called an adjusted quasi-phi-function for objects A(u _A) and B(u _B), if function $\underset{u^\prime}{\max }{\overset{{\frown}}{\Phi^\prime}^{AB}}\left({u}_A,{u}_B,{u}^{\prime}\right)$ is an adjusted phi-function for the objects.

We can define the distance constraint for objects A(u _A) and B(u _B) in the form: ${{\overset{{\frown}}{\Phi}}^{\prime}}^{AB}\ge 0.$ The inequality implies dist(A,B)≥ρ.