Quasi-Packing Different Spheres with Ratio Conditions in a Spherical Container

Abstract

This paper considers the optimized packing of different spheres into a given spherical container under non-standard placement conditions. A sphere is considered placed in the container if at least a certain part of the sphere is in the container. Spheres are allowed to overlap with each other according to predefined parameters. Ratio conditions are introduced to establish correspondence between the number of packed spheres of different radii. The packing aims to maximize the total number of packed spheres subject to ratio, partial overlapping and quasi-containment conditions. A nonlinear mixed-integer optimization model is proposed for this ratio quasi-packing problem. A heuristic algorithm is developed that reduces the original problem to a sequence of continuous open dimension problems for quasi-packing scaled spheres. Computational results for finding global solutions for small instances and good feasible solutions for large instances are provided.

1. Introduction

In classical sphere packing problems, non-overlapping spherical objects must be placed completely inside a container [1]. These conditions, referred to as non-overlapping and containment, form key constraints in the corresponding optimized packing problems. In these problems, either the container is fixed and the total volume of packed spheres is maximized, or the spheres are fixed and the smallest container is constructed to fit all the spheres. Sphere packing problems are well studied and arise in biology [2,3,4,5], medicine [6,7], physics [8,9,10,11] and engineering [12,13,14,15,16,17], to mention a few.

Numerical methods to solve optimized sphere packing problems can be divided into exact and approximate approaches. Exact solution techniques are typically based on corresponding nonlinear programming models and use global optimization schemes (e.g., branch and bound) for their solution [18]. This approach is efficient for relatively small problem instances. For larger instances, exact schemes are frequently combined with heuristics [19]. Many combined approaches are based on nonlinear optimization models and then use exact or heuristic local techniques [19,20,21,22] to obtain reasonable solutions. Approximation of the container by a finite grid results in a binary assignment problem for centers of spheres to nodes of the grid [23,24,25,26]. Many approaches [27,28,29,30,31,32,33] combine several techniques.

This paper is focused on non-standard packing problems where either non-overlapping or containment or both conditions are allowed to be violated. In what follows, these problems are referred to as quasi-packing. Allowing a controlled overlap is frequently used in modeling packing problems in natural sciences since soft spheres can be modeled as hard spheres with a limited overlap. Examples are the spatial organization of chromosomes in the cell nucleus [2,34], the spatial organization of neurons [3,4] or the arrangement of ganglion cell receptive fields on the retinal surface [35,36]. From a mathematical point of view, introducing a limited overlap may be considered as an intermediate case between sphere packing and sphere covering [37]. In sphere packing, the main objective is to achieve the densest layout, i.e., layout where the spheres fill the maximal possible space in the container. In contrast, sphere covering corresponds to a layout of spheres that covers the whole space. In this case, overlapping is not only allowed but inevitable. Our interest in packing with overlap is motivated by modeling porous media under pressure [38,39,40]. These problems arise frequently in oil and gas extracting industries [41,42,43]. Elements of porous media can be deformed by external force. This corresponds to the partial overlapping of spheres.

In quasi-packing, a sphere is considered placed in the container if at least a certain part of the sphere is in the container. This relaxation of classical containment arises, e.g., in modeling a distribution of spherical particles in porous media [44,45]. The porous material can be analyzed experimentally by extracting a volumetric sample (container) for further investigation or by mathematical simulation of the material structure in the sample. Since some spheres are cut off in the extraction process, only parts of them may remain in the sample. Correspondingly, the mathematical model should allow spheres that are not entirely placed in the container.

In this paper, optimized quasi-packing of different spheres in a spherical container is introduced. The objective is to maximize the total number of spheres placed in the container subject to quasi-packing constraints. It is assumed that a certain correspondence between the number of packed spheres of different radii must be kept. Ratio conditions are introduced to establish lower and upper bounds for the relative impact (percentage) of the number of spheres of a certain radius to the total number of spheres.

The main contributions of this paper are as follows:

(1)

A novel class of sphere packing in a spherical container with partial overlapping, quasi-containment and ratio conditions.

(2)

A mixed-integer nonlinear programming model for the ratio quasi-packing of spheres.

(3)

A heuristic for large problem instances consisting of finding feasible starting points in an open dimension continuous optimization problem to obtain reasonable feasible solutions.

(4)

Numerical experiments for small problem instances with exact solutions obtained using the global solver BARON.

(5)

Numerical experiments for larger problem instances with good feasible solutions obtained by the heuristic using the local solver IPOPT for open dimension problems.

The paper is organized as follows. Section 2 presents the ratio quasi-packing problem for spheres and provides the corresponding nonlinear mixed-integer programming formulation. A heuristic strategy for finding good feasible solutions to the problem is described in Section 3. Numerical results are given in Section 4, while Section 5 concludes the paper.

2. Problem Formulation

Let $S_0$ be a spherical container with a given radius $R$ and centered at $(0,0,0)$. In addition, a family $S=${$S_i(u_i),$ $i=1,\dots ,n$} of spheres $S_i(u_i)=\{u\in {\mathbf{R}}^3:‖u-u_i‖\le r_i\}$ centered at $u_i=(x_i,y_i,z_i)$ is given. It is assumed that there are $K$ different radii of the spheres and $n_k$ spheres of type $k,\hspace{0.17em}(k=1,2,\dots K)$, $n={\displaystyle \sum _{k=1}^K{n}_k}$.

Ratio Quasi-Packing Spheres (RQPS). Find the maximum number of spheres $S_i(u_i)$ from the family $S$ under the following conditions:

• $‖u_i-u_j‖\ge (r_i+r_j-{\delta }_{ij})$ (partial overlapping condition) for $1\le i<j\le n$,

where ${\delta }_{ij}\in (r_i+r_j)*{\delta }_0$, ${\delta }_0\ge 0$ is a given parameter providing allowable pairwise overlapping of spheres;

• $‖u_i‖\le R+{\epsilon }_i$ (quasi-containment condition) for $i=1,\dots ,n$,

where ${\epsilon }_i$, $i=1,\dots ,n$ is a given (or randomly chosen) parameter providing the allowable intersection of the sphere $S_i(u_i)$ with the boundary of $S_0$, ${\epsilon }_i\in [-r_i,r_i]$;

• ${\tau }_k^{-}\le {\tau }_k\le {\tau }_k^{+}$ for $k=1,\dots ,K$ (ratio condition),

where ${\tau }_k=(thenumberofpackedspheres\hspace{0.17em}ofk-thtype)/\hspace{0.17em}(totalnumberofpackedspheres)$, $0\le {\tau }_k^{-}\le {\tau }_k^{+}\le 1$ are given nonnegative constants.

Below, the basic concepts of quasi-packing are commented and graphically illustrated.

The partial overlapping condition $‖u_i-u_j‖\ge (r_i+r_j-{\delta }_{ij})$ assures that the minimal distance between centers $u_i$ and $u_j$ of two spheres $S_i(u_i)$ and $S_j(u_j)$ having radii $r_i$ and $r_j$ is at least $r_i+r_j-{\delta }_{ij}$. Here, ${\delta }_{ij}\ge 0$ is a given (sufficiently small) positive parameter of allowable pairwise overlapping of the spheres. Figure 1a presents the standard non-overlapping corresponding to ${\delta }_{ij}=0$. Condition $‖u_i-u_j‖\ge r_i+r_j$, in this case, means that the intersection of the interiors of spheres $S_i(u_i)$ and $S_j(u_j)$ is empty. Figure 1b illustrates ${\delta }_{ij}$ overlapping (partial overlapping) for ${\delta }_{ij}>0$.

Quasi-containment condition $‖u_i‖\le R+{\epsilon }_i$ assures that the maximal distance between the center $u_i$ of the sphere $S_i$ and the origin (center of the container $S_0$) is at most $R+{\epsilon }_i$. Here, ${\epsilon }_i\in [-r_i,r_i]$ is a given parameter characterizing a permitted “protrusion” of $S_i$ from the container $S_0$. Figure 2a corresponds to ${\epsilon }_i=-r_i$, $‖u_i‖\le R-r_i$ and illustrates the standard containment condition $S_i(u_i)\subseteq S_0$. In Figure 2b, the quasi-containment is presented for ${\epsilon }_i=0$ and $‖u_i‖\le R$, while Figure 2c corresponds to ${\epsilon }_i=r_i$ and $‖u_i‖\le R$ $+r_i$.

Ratio condition ${\tau }_k^{-}\le {\tau }_k\le {\tau }_k^{+}$ for $k=1,\dots ,K$ assures a certain correspondence (proportion) between the number of packed spheres of different radii. Here, ${\tau }_k$ is defined as a relative number of spheres of k-th type, ${\tau }_k=$ (number of packed spheres of k-th type)/(total number of packed spheres), and $0\le {\tau }_k^{-}\le {\tau }_k^{+}\le 1$ is a given nonnegative parameter. We distinguish between strict (${\tau }_k^{-}={\tau }_k^{+}$) or non-strict (${\tau }_k^{-}<{\tau }_k^{+}$) ratio conditions. The ratio condition is introduced to maintain a certain structure of a particulate matter under different packing configurations. Figure 3 illustrates the ratio condition for $n=9$ spheres of $K=3$ different types (blue, green and yellow 2D spheres in Figure 3). The strict ratio condition ${\tau }_k^{}={\tau }_k^{-}={\tau }_k^{+},k=1,2,3$ is considered with ${\tau }_1^{}=2/9$, ${\tau }_2^{}=3/9$, ${\tau }_3^{}=4/9$. Packing in Figure 3a does not satisfy the ratio condition since $n_1=1$ $n_2=4$ $n_3=4$. Figure 3b provides a feasible packing that satisfies the ratio condition since $n_1=2$ $n_2=3$ $n_3=4$.

Mixed-Integer Nonlinear Programming Model for RQPS

Let us define the RQPS problem variables. The vector $u=(u_i,i=1,\dots ,n)\in R^{3n}$ specifies the coordinates of the centers of the spheres $S_i(u_i)$, while the vector $\pi =({\pi }_i,i=1,\dots ,n)\in B^n$ defines the binary variables.

The vector of all variables is denoted as $v=(u,\pi )$.

The mathematical model of the RQPS problem has the following form:

$$\underset{{\pi }_i\in \{0,1\}}{\max }{\displaystyle \sum _{i=1}^n{\pi }_i}$$

(1)

subject to

$${\pi }_i{\psi }_i(u_i)\ge 0,i=1,\dots ,n$$

(2)

$${\pi }_i{\pi }_j{\phi }_{ij}(u_i,u_j)\ge 0,1\le i<j\le n,$$

(3)

$${\tau }_k^{-}{\displaystyle \sum _{i=1}^n}{\pi }_i\le {\displaystyle \sum _{i\in I_k}{\pi }_i}\le {\tau }_k^{+}{\displaystyle \sum _{i=1}^n}{\pi }_i,k=1,\dots ,K$$

(4)

$$\mathrm{for}{I}_1=\{1,\dots ,n_1\},I_2=\{n_1+1,\dots ,n_1+n_2\},\dots ,I_K=\{n-n_K+1,\dots ,n\},I=I_1\cup I_2\cup \dots \cup I_K,{\displaystyle \sum _{k=1}^K}{n}_k=n.$$

Here,

$${\psi }_i(u_i)=R+{\epsilon }_i-‖u_i‖,{\phi }_{ij}(u_i,u_j)=‖u_i-u_j‖-(r_i+r_j-{\delta }_{ij}),$$

and inequalities (2) state the quasi-containment conditions, while inequalities (3) represent the partial overlapping conditions for spheres $S_i(u_i)$. Constraints (4) state the ratio conditions. Constraints (2) assure that if ${\psi }_i(u_i)<0$ (quasi-containment condition is not satisfied), then ${\pi }_i=0$. Constraints (3) guarantee that if ${\phi }_{ij}(u_i,u_j)<0$ (partial overlapping is not satisfied), then ${\pi }_i$ and ${\pi }_j$ cannot be equal to 1, simultaneously. Note that for ${\psi }_i(u_i)\ge 0$, both ${\pi }_i=1$ and ${\pi }_i=0$ satisfy constraint (2). This is similar for the case ${\phi }_{ij}(u_i,u_j)\ge 0$ in (3). However, to maximize the objective (1), ${\pi }_i=1$ must be selected as often as possible. This implies that a solution of model (1)–(4) indeed places the maximal possible number of spheres so that the constraints are satisfied.

The main characteristics of the model (1)–(4) are as follows:

• The model (1)–(4) is a mixed-integer nonlinear mathematical programming problem (MINLP—Mixed-Integer Nonlinear Programming).
• The objective function (1) is linear.
• The total number of variables is $4n$ ($n$ binary and $3n$ continuous variables).
• The solution space is defined by $n$ + $0.5n(n-1)$ nonlinear constraints of the form (2), (3) and $2k$ linear inequalities in (4).

If the ratio conditions (4) are omitted (e.g., in case of uniform spheres), then integrality conditions for the variables ${\pi }_i$ can be relaxed without losing the optimal solution.

Proposition 1.

Consider the quasi-packing sphere problem (1)–(3) with integrality constraints  ${\pi }_i\in \{0,1\}$ relaxed to $0\le {\pi }_i\le 1$. Then, all optimal solutions $({\pi }^{*},u^{*})$ to this continuous nonlinear programming problem have only binary components ${\pi }_i^{*}\in \{0,1\}$.

Demonstration.

The proof is by contradiction. Suppose that there exists a rational component $0<{\pi }_i^{*}<1$ in the optimal solution. Since ${\pi }_i^{*}>0$, then by (2) we have ${\psi }_i(u_i^{*})\ge 0$. Hence, ${\pi }_i^{*}$ can be increased to 1 without violating constraint (2). If ${\pi }_j^{*}>0$, then by (3) we have ${\phi }_{ij}(u_i^{*},u_j^{*})\ge 0$ and ${\pi }_i^{*}$ can be increased to 1 without violating constraint (3). For ${\pi }_j^{*}=0$, we can increase ${\pi }_i^{*}$ to 1 without violating constraint (3) for any sign of ${\phi }_{ij}(u_i^{*},u_j^{*})$. Thus, if $0<{\pi }_i^{*}<1$, then we can increase ${\pi }_i^{*}$ to 1 without violating constraints of the problem and thus increase the value of the objective function (1). This contradicts the optimality of ${\pi }_i^{*}$.

Packing spheres are known to be NP-hard [1]. Global solvers can find provably optimal solutions to model (1)–(4) for small problem instances with dozens of spheres (see Section 4.1). A heuristic approach below is proposed to find reasonable feasible solutions for larger instances.

3. Heuristic Approach

The main idea of the proposed heuristic algorithm is reducing the original problem to a sequence ($l=1,2,\dots ,$) of quasi-packing problems of a given number of scaled spheres in a given spherical container. The objective is to maximize the scaling parameter (and, with this, to grow the spheres up to their original size) subject to the ratio conditions.

Let us consider the optimization problem we are going to use in our heuristic algorithm.

To this end, let $\lambda S_i^{}(u_i^{})$, $i=1,\dots ,l$, $l\le n$ be a given number of scaled spheres, $0\le \lambda \le 1$. Let $S_0$ denote a sphere centered at $(0,0,0)$ with radius $R$ providing

(a)

$‖u_i‖\le R+\lambda {\epsilon }_i$, $i=1,\dots ,l$, $\lambda {\epsilon }_i\in \lambda [-r_i,r_i]$;

(b)

$‖u_i-u_j‖\ge \lambda (r_i+r_j-{\delta }_{ij})$ and $\lambda {\delta }_{ij}\in \lambda (r_i+r_j)*{\delta }_0$.

Each of the ($l=1,2,\dots ,$) quasi-packing problems is stated as a nonlinear programming problem:

$${\lambda }^{*}=\underset{v_l\in {\Re }_l}{\max }\lambda ,$$

(5)

where $v_l=(u_1,\dots ,u_l,\lambda )$ is a vector of variables, and the feasible set ${\Re }_l\subset {\mathbf{R}}^{3l+1}$ is described by the following inequality system:

$${\stackrel{⌢}{\psi }}_i(u_i,\lambda )\ge 0,i=i,\dots ,l,$$

(6)

$${\stackrel{⌢}{\phi }}_{ij}(u_i,u_j,\lambda )\ge 0,1\le i<j\le n,$$

(7)

$$0\le \lambda \le 1,$$

(8)

with

$${\stackrel{⌢}{\psi }}_i(u_i,\lambda )=-‖u_i‖+(R+\lambda {\epsilon }_i),{\stackrel{⌢}{\phi }}_{ij}(u_i,u_j,\lambda )=‖u_i-u_j‖-\lambda (r_i+r_j-{\delta }_{ij}).$$

Inequality (6) provides the quasi-containment of the scaled sphere $\lambda S_i^{}(u_i^{})$ in the spherical domain $S_0(\lambda {\epsilon }_i)=$ $\{u\in R^3:‖u‖\le R+\lambda {\epsilon }_i\}$, i.e., $u_i\in S_0(\lambda {\epsilon }_i)$, while inequality (7) describes the partial overlapping of scaled spheres $\lambda S_i^{}(u_i^{})$ and $\lambda S_j^{}(u_j^{})$, i.e., $‖u_i-u_j‖\ge \lambda (r_i+r_j-{\delta }_{ij})$.

Let us introduce some notations used in our algorithm. Let $I_{\min }$ denote a minimal set of spheres meeting the ratio condition (4), where: $q$ is the cardinality of the set $I_{\min }$; $I_s$ denotes an aggregate set of spheres; $\left\{i_k\right\}$ means a sphere of type $k$.

Our algorithm involves three main stages:

Stage 1. Generation of a minimal set of spheres (minimal block), $I_{\min }$, that satisfies the strict ratio conditions (Steps 1–7).

Stage 2. Multiple sequential placements of the spheres in $I_{\min }$ meeting the quasi-packing and strict ratio conditions. Creation of an aggregate set $I_s$ (accumulation of $I_{\min }$ blocks) (Steps 8–12).

Stage 3. Placement of additional spheres in the quasi-packing found at Stage 2 meeting the quasi-packing and non-strict ratio conditions (Steps 13–19).

Let us consider the algorithm in detail.

Firstly, an integer $t$ specifying the cardinality of the set $I_{\min }$ of spheres is calculated that meets the ratio conditions. Denote ${{\tau }^{\prime }}_k\in [{\tau }_k^{-},{\tau }_k^{+}]$ assuming ${\displaystyle \sum _{k=1}^K{{\tau }^{\prime }}_k}=1$ (Steps 1–3).

Step 1. Set $t:=0$.

Step 2. Set $t:=t+1$.

Step 3. If $t{{\tau }^{\prime }}_k-⌊t{{\tau }^{\prime }}_k⌋=0$, $k=1,\dots ,K$, then set $q:=t$ and go to Step 4; otherwise, go to Step 2. Here, $⌊·⌋$ is the floor function.

Then, the minimum index set $I_{\min }$ is formed (Steps 4–7).

Step 4. Set $I_{\min }:=\varnothing$.

Step 5. Set $k:$$=1$.

Step 6. If $k=K+1$, then go to Step 8.

Step 7. Set $I_{\min }:=I_{\min }\cup$${\displaystyle \underset{j=1}{\overset{t{{\tau }^{\prime }}_k}{\cup }}\left\{i_{k_j}\right\}}$, $k=k+1$ and go to Step 6.

Further, the aggregate index set $I_s$ having cardinality $l$ and meeting the quasi-packing and the ratio conditions is constructed (Steps 8–10).

Step 8. Set $I_s:=\varnothing$, $l:=0$.

Step 9. Set $I_s:=I_s\cup I_{\min }$, $l:=l+q$.

Step 10. Search for a local maximum of the problem (5)–(8) for the set $I_s$ using a local optimization solver combined with the decomposition procedure. This technique substitutes (approximately) a large-scale nonlinear programming problem with a sequence of considerably smaller nonlinear programming problems with non-overlapping (partially non-overlapping) conditions considering only neighbor spheres. The detailed description of the technique can be found, e.g., in [22,46].

At the next step, we verify the quasi-packing conditions and update the set $I_s:=I_s\backslash I_{\min }$ (Step 11).

Step 11. If $\lambda =1$, then go to Step 9; otherwise, ($\lambda <1$) set $I_s:=I_s\backslash I_{\min }$, $l:=l-q$ and go to Step 12.

For the case of the strict ratio conditions, we provide a stopping criterion for our algorithm at Step 12.

Step 12. If ${\tau }_k^{-}={\tau }_k^{+}$, $k=1,\dots ,K$, then stop the algorithm with $n_{}^{*}:=l$; otherwise, go to Step 13.

An addition of single spheres (in a one-by-one scenario) to the set $I_s$ meeting the ratio and quasi-packing conditions is produced at Steps 13–19.

Step 13. Set $k:=$ $K+1$.

Step 14. Set $k:=k$ $-$ $1$.

Step 15. If $k\ge 1$, then set $I_s:=I_s\cup \left\{i_k\right\}$, $l:=l+1$; otherwise, stop the algorithm with $n^{*}$ $:=l$.

Step 16. If $I_s$ meets (4), then go to Step 17; otherwise, set $I_s:=I_s\backslash \left\{i_k\right\}$, $l:=l-1$ and go to Step 14.

Step 17. Search for a local maximum of the problem (5)–(8) for the sphere set $I_s$. If $\lambda =1$, then go to Step 18; otherwise, set $I_s:=I_s\backslash \left\{i_k\right\}$ and stop the algorithm with $n^{*}:=$ $l-1$.

Step 18. If $k<K$, then set $k:=K$.

Step 19. Go to Step 15.

The flowchart of the algorithm is provided in Figure 4 below.

4. Computational Results

To verify the model (1)–(4), the global optimization solver BARON [47,48,49] was used to solve small problem instances. BARON was executed on the NEOS server [50] with AMPL [51] applied for modeling optimization problem (1)–(4). The nonlinear optimization problems (5)–(8) arising in the heuristic algorithm were solved by the local optimization solver IPOPT [52] combined with the decomposition procedure. All computations were executed on Desktop-54TG91N with Intel(R) Core(TM) i3-6100T CPU @ 3.20GHz, RAM 8 Gb.

Three types of instances were used in the computational experiments. In the first case, classical packing problems without violation of non-overlapping and containment conditions (${\epsilon }_i=-r_i$, ${\delta }_{ij}=0$) were generated. The second group is composed of packing problems with non-overlapping (${\epsilon }_i=-r_i$) and quasi-containment (${\epsilon }_i\in [-r_i,r_i]$) conditions. In the last group, partial overlapping ($0\le {\delta }_{ij}\le (r_i+r_j)*{\delta }_0$) and quasi-containment (${\epsilon }_i\in [-r_i,r_i]$) were permitted. Moreover, strict (${\tau }_k^{-}={\tau }_k^{+}$) and non-strict (${\tau }_k^{-}<{\tau }_k^{+}$) ratio conditions (4) were considered.

For each problem, the instance below the maximal number of packed spheres is provided together with the corresponding computation time. For all small problem instances in Section 4.1, global solutions were obtained using BARON, i.e., the optimality-stopping criterion was fulfilled (see [48] for more details). For larger instances in Section 4.2, the best solutions obtained by the heuristic are reported. The value of porosity is also presented. The expression for approximately calculating the porosity can be found in Appendix A.

4.1. Results for Small Instances Using BARON

Example 1.

$R=5$, $K=5$, ${\delta }_0=0.2$. The rest of the input data are given in

Table 1. Input data for Examples 1–2.

Example	$\mathit{k}$	${\mathit{n}}_{\mathit{k}}$	Radius of {ik}	${\mathit{\epsilon }}_{\mathit{k}}$	${\mathit{\tau }}_{\mathit{k}}^{-}$	${\mathit{\tau }}_{\mathit{k}}^{+}$
1	1	11	1.1	−1	0.2	0.2
	2	10	1.2	−1	0.2	0.2
	3	10	1.3	−1	0.2	0.2
	4	12	1.4	−1.1	0.2	0.2
	5	14	1.5	−1.2	0.2	0.2
2	1	10	1.1	−1	0.1	0.3
	2	9	1.2	−1	0.1	0.3
	3	9	1.3	−1	0.1	0.3
	4	13	1.4	−1.1	0.1	0.3
	5	12	1.5	−1.2	0.1	0.3

. The optimal number of spheres obtained in 222.149 s is $n^{*}=50$ ($n_1^{*}=10,$  $n_2^{*}=10,$  $n_3^{*}=10$, $n_4^{*}=10,$  $n_5^{*}=10$). The value of porosity is 0.18500961206. The corresponding quasi-packing is shown in Figure 5a.

Example 2.

$R=5$, $K=5$, ${\delta }_0=0.2$. The rest of the input data are given inTable 1. The optimal number of spheres obtained in 72.3216 s is $n^{*}=53$ ($n_1^{*}=10,$  $n_2^{*}=9,$  $n_3^{*}=9$, $n_4^{*}=13,$  $n_5^{*}=12$) with ${\tau }_1^{*}=0.188679$, ${\tau }_2^{*}=0.169811$, ${\tau }_3^{*}=0.169811$, ${\tau }_4^{*}=0.245283$, ${\tau }_5^{*}=0.226415$. The value of porosity is 0.14106740448. The corresponding quasi-packing is shown in Figure 5b.

Example 3.

$R=5$, $K=3$, ${\delta }_0=0.2$. The rest of the input data are given in

Table 2. Input data for Examples 3–4.

Example	$\mathit{k}$	${\mathit{n}}_{\mathit{k}}$	Radius of {ik}	${\mathit{\epsilon }}_{\mathit{k}}$	${\mathit{\tau }}_{\mathit{k}}^{-}$	${\mathit{\tau }}_{\mathit{k}}^{+}$
3	1	24	1.2	−0.7	0.6	0.6
	2	15	1.3	−1	0.2	0.2
	3	10	1.4	−1.1	0.2	0.2
4	1	24	1.2	−0.7	0.5	0.7
	2	15	1.3	−1	0.1	0.3
	3	10	1.4	−1.1	0.1	0.3

. The optimal number of spheres obtained in 98.4785 s is $n^{*}=40$ ($n_1^{*}=24,$  $n_2^{*}=8,$  $n_3^{*}=8$). The value of porosity is 0.40560543556. The corresponding quasi-packing is shown in Figure 6a.

Example 4.

$R=5$, $K=3$, ${\delta }_0=0.2$. The rest of the input data are given inTable 2. The optimal number of spheres obtained in 79.458 s is $n^{*}=48$ ($n_1^{*}=24,$  $n_2^{*}=14,$  $n_3^{*}=10$) with ${\tau }_1^{*}=0.5$, ${\tau }_2^{*}=0.291667$, ${\tau }_3^{*}=0.208333$. The value of porosity is 0.24992548994. The corresponding quasi-packing is shown in Figure 6b.

4.2. Results Obtained by the Heuristic

Example 5.

$R=9.5$, $K=3$, ${\delta }_0=0.1$. The rest of the input data are given in

Table 3. Input data for Examples 5 and 6.

Example	$\mathit{k}$	${\mathit{n}}_{\mathit{k}}$	Radius of {ik}	${\mathit{\epsilon }}_{\mathit{k}}$	${\mathit{\tau }}_{\mathit{k}}^{-}$	${\mathit{\tau }}_{\mathit{k}}^{+}$
5	1	10	2	−1.8	1/7	1/7
	2	20	1.5	−1.35	2/7	2/7
	3	40	1	−0.9	4/7	4/7
6	1	10	2	−2	1/7 − 0.01	1/7 + 0.01
	2	20	1.5	−1.5	2/7 − 0.01	2/7 + 0.01
	3	40	1	−0.9	4/7 − 0.01	4/7 + 0.01

. The number of spheres obtained in 5 s is $n^{*}=49$ ($n_1^{*}=7$, $n_2^{*}=14,$  $n_3^{*}=28$). The corresponding quasi-packing of 2D spheres is shown in Figure 7a.

Example 6.

$R=9.5$, $K=3$, ${\delta }_0=0.1$. The rest of the input data are given inTable 3. The number of spheres obtained in 10 s is $n^{*}=50$ ($n_1^{*}=7$, $n_2^{*}=14,$  $n_3^{*}=29$). The corresponding quasi-packing of circles is shown in Figure 7b.

Example 7.

$R=6.1$, $K=3$, ${\delta }_0=0$. The rest of the input data are given in

Table 4. Input data for Examples 7–20.

Example	$\mathit{k}$	${\mathit{n}}_{\mathit{k}}$	Radius of {ik}	${\mathit{\epsilon }}_{\mathit{k}}$	${\mathit{\tau }}_{\mathit{k}}^{-}$	${\mathit{\tau }}_{\mathit{k}}^{+}$
7, 14	1	50	2	−2	1/7	1/7
	2	100	1.5	−1.5	2/7	2/7
	3	200	1	−1	4/7	4/7
8, 15	1	50	2	−2	1/7 − 0.01	1/7 + 0.01
	2	100	1.5	−1.5	2/7 − 0.01	2/7 + 0.01
	3	200	1	−1	4/7 − 0.01	4/7 + 0.01
9, 11, 13, 16, 18, 20	1	50	2	−1.8	1/7	1/7
	2	100	1.5	−1.35	2/7	2/7
	3	200	1	−0.9	4/7	4/7
10, 12, 17, 19	1	50	2	−1.8	1/7 − 0.01	1/7 + 0.01
	2	100	1.5	−1.35	2/7 − 0.01	2/7 + 0.01
	3	200	1	−0.9	4/7 − 0.01	4/7 + 0.01

. The number of spheres obtained in 5 s is $n^{*}=42$ ($n_1^{*}=6,$  $n_2^{*}=12,$  $n_3^{*}=24$). The value of porosity is 0.504364. The corresponding quasi-packing is shown in Figure 8a.

Example 8.

$R=6.1$, $K=3$, ${\delta }_0=0$. The rest of the input data are specified inTable 4. The number of spheres obtained in 8 s is $n^{*}=43$ ($n_1^{*}=6,$  $n_2^{*}=12,$  $n_3^{*}=25$). The value of porosity is 0.499958. The corresponding quasi-packing is shown in Figure 8b.

Example 9.

$R=6.1$, $K=3$, ${\delta }_0=0$. The rest of the input data are given inTable 4. The number of spheres obtained in 5 s is $n^{*}=49$ ($n_1^{*}=7,$  $n_2^{*}=14,$  $n_3^{*}=28$). The value of porosity is 0.433188. The corresponding quasi-packing is shown in Figure 9a.

Example 10.

$R=6.1$, $K=3$, ${\delta }_0=0$. The rest of the input data are given inTable 4. The number of spheres obtained in 11 s is $n^{*}=50$ ($n_1^{*}=7,$  $n_2^{*}=14,$  $n_3^{*}=29$). The value of porosity is 0.41836950385. The corresponding quasi-packing is shown in Figure 9b.

Example 11.

$R=6.1$, $K=3$, ${\delta }_0=0.1$. The rest of the input data are specified inTable 4. The number of spheres obtained in 7 s is $n^{*}=63$($n_1^{*}=9,$  $n_2^{*}=18,$  $n_3^{*}=36$). The value of porosity is 0.2814923736. The corresponding quasi-packing is shown in Figure 10a.

Example 12.

$R=6.1$, $K=3$, ${\delta }_0=0.1$. The rest of the input data are specified inTable 4. The number of spheres obtained in 15 s is $n^{*}=64$ ($n_1^{*}=9,n_2^{*}=18,n_3^{*}=37$). The value of porosity is 0.27854473893. The corresponding quasi-packing is shown inFigure 10b.

Example 13.

$R=6.1$, $K=3$, ${\delta }_0=0.2$. The rest of the input data are given inTable 4. The number of spheres obtained in 20 s is $n^{*}=84$ ($n_1^{*}=12,$  $n_2^{*}=24,$  $n_3^{*}=48$). The value of porosity is 0.155574. The corresponding quasi-packing is shown in Figure 11a.

Example 14.

$R=8$, $K=3$, ${\delta }_0=0$. The rest of the input data are given inTable 4. The number of spheres obtained in 12 s is $n^{*}=105$ ($n_1^{*}=15,$  $n_2^{*}=30,$  $n_3^{*}=60$). The value of porosity is 0.446782. The corresponding quasi-packing is shown in Figure 11b.

Example 15.

$R=8$, $K=3$, ${\delta }_0=0$. The rest of the input data are given inTable 4. The number of spheres obtained in 14 s is $n^{*}=109$ ($n_1^{*}=15,$  $n_2^{*}=31,$  $n_3^{*}=63$). The value of porosity is 0.438232. The corresponding quasi-packing of spheres is shown in Figure 12a.

Example 16.

$R=8$, $K=3$, ${\delta }_0=0$.The rest of the input data are given inTable 4. The number of spheres obtained in 50 s is $n^{*}=112$ ($n_1^{*}=16,$  $n_2^{*}=32,$ $n_3^{*}=64$). The value of porosity is 0.414063. The corresponding quasi-packing is shown in Figure 12b.

Example 17.

$R=8$, $K=3$, ${\delta }_0=0$. The rest of the input data are specified inTable 4. The number of spheres obtained in 18 s is $n^{*}=118$ ($n_1^{*}=16,n_2^{*}=34,n_3^{*}=68$). The value of porosity is 0.393993. The corresponding spheres are shown inFigure 13a.

Example 18.

$R=8$, $K=3$, ${\delta }_0=0$. The rest of the input data are specified inTable 4. The number of spheres obtained in 29 s is $n^{*}=147$ ($n_1^{*}=21,$  $n_2^{*}=42,$  $n_3^{*}=84$). The value of porosity is 0.254555. The corresponding quasi-packing is shown in Figure 13b.

Example 19.

$R=8$, $K=3$, ${\delta }_0=0.1$. The rest of the input data are specified inTable 4. The number of spheres obtained in 147 s is $n^{*}=152$($n_1^{*}=21,$$n_2^{*}=43,$$n_3^{*}=88$). The value of porosity is 0.24759. The corresponding quasi-packing is shown inFigure 14a.

Example 20.

$R=8$, $K=3$, ${\delta }_0=$$0.2$. The rest of the input data are specified inTable 4. The number of spheres obtained in 47 s is $n^{*}=210$ ($n_1^{*}=30,$  $n_2^{*}=60,$  $n_3^{*}=120$). The value of porosity is 0.129044. The corresponding quasi-packing is shown in Figure 14b.

Example 21.

$R=9$, $K=3$, ${\delta }_0=0$. The rest of the input data are given in

Table 5. Input data for Examples 21–22.

Example	$\mathit{k}$	${\mathit{n}}_{\mathit{k}}$	Radius of {ik}	${\mathit{\epsilon }}_{\mathit{k}}$	${\mathit{\tau }}_{\mathit{k}}^{-}$	${\mathit{\tau }}_{\mathit{k}}^{+}$
21	1	50	2	0.2	1/7	1/7
	2	100	1.5	0.15	2/7	2/7
	3	200	1	0.1	4/7	4/7
22	1	50	2	0.2	1/7 − 0.01	1/7 + 0.01
	2	100	1.5	0.15	2/7 − 0.01	2/7 + 0.01
	3	200	1	0.1	4/7 − 0.01	4/7 + 0.01

. The number of spheres obtained in 68 s is $n^{*}=238$ ($n_1^{*}=34,$  $n_2^{*}=68,$  $n_3^{*}=136$). The value of porosity is 0.244979. The corresponding quasi-packing is shown in Figure 15a. If the number of IPOPT runs is increased, the better value $n^{*}=245$ ($n_1^{*}=35,$  $n_2^{*}=70,$  $n_3^{*}=140$) can be obtained in 230 s with corresponding porosity 0.209132.

Example 22.

$R=9$, $K=3$, ${\delta }_0=0$. The rest of the input data are given inTable 5. The number of spheres obtained in 412 s is $n^{*}=244$, ($n_1^{*}=34,$  $n_2^{*}=70,$  $n_3^{*}=140$). The value of porosity is 0.222265. The corresponding quasi-packing is shown in Figure 15b. If the number of IPOPT runs is increased, the better value $n^{*}=246$ $(n_1^{*}=35,n_2^{*}=70,n_3^{*}=141)$ is obtained in 471 s with corresponding porosity 0.211930.

5. Conclusions

A novel sphere quasi-packing model that allows controlled violation of non-overlapping and containment conditions was developed. Ratio constraints were introduced to establish correspondence between the number of packed spheres of different radii. Corresponding MINLP formulation as well as the heuristic strategy for obtaining good feasible solutions were proposed. Proven optimal global solutions for the MINLP formulation were obtained for small instances with dozens of spheres using the global solver BARON on the NEOS server. For larger instances with hundreds of spheres, the proposed heuristic was successfully implemented providing reasonable feasible solutions. Testing the heuristic for large instances with thousands of spheres is an interesting area for future research. To simplify nonlinear optimization problems, grid approximations [23,24,25,26,53] can be considered to linearize optimized packing problems.

Spherical approximations for particles in porous media are frequently used in material sciences models [8,9,10,11,54,55,56]. However, to make analyses of granular structures more realistic, more sophisticated shapes can be used, e.g., ellipsoids, tetrahedrons, convex polyhedra and their mixtures [57,58]. Mathematical tools for placing convex and irregular non-spherical objects can be found in [59,60,61,62,63,64]. Some results on quasi-packing non-spherical objects are currently being prepared.