applied sciences
Article An Optimized Covering Spheroids by Spheres
Alexander Pankratov 1, Tatiana Romanova 1, Igor Litvinchev 2 and Jose Antonio Marmolejo-Saucedo 3,* 1 Department of Mathematical Modeling and Optimal Design, Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine Kharkov 61046, Ukraine, Kharkiv National University of Radioelectronics, Kharkov 61166, Ukraine; [email protected] (A.P.); [email protected] (T.R.) 2 Faculty of Mechanical and Electrical Engineering, Graduate Program in Systems Engineering, Nuevo Leon State University (UANL), Monterrey 66450, Mexico; [email protected] 3 Facultad de Ingeniería, Universidad Panamericana, Augusto Rodin 498, Ciudad de Mexico 03920, Mexico * Correspondence: [email protected]
Received: 29 December 2019; Accepted: 3 March 2020; Published: 7 March 2020
Abstract: Covering spheroids (ellipsoids of revolution) by different spheres is studied. The research is motivated by packing non-spherical particles arising in natural sciences, e.g., in powder technologies. The concept of an ε-cover is introduced as an outer multi-spherical approximation of the spheroid with the proximity ε. A fast heuristic algorithm is proposed to construct an optimized ε-cover giving a reasonable balance between the value of the proximity parameter ε and the number of spheres used. Computational results are provided to demonstrate the efficiency of the approach.
Keywords: covering; spheroids; spheres; mathematical model; nonlinear optimization
1. Introduction Covering a certain region by simple shapes has various applications. Our interest in covering problems is motivated by packing particles arising in science and engineering applications. Packing problems are widely used in modeling liquid and glass structures [1,2], representing granular materials [3], packing beds, and cermet [4], as well as in many other applications (see, e.g., [5–10]). Many algorithms have been proposed for packing spherical particles (see, e.g., [11–16] and the references therein). However, to have a better and more adequate representation of particulate microstructure, packing non-spherical particles must be considered. The approaches proposed to handle non-spherical shapes can be divided in three large groups [17]. Techniques in the first group use analytical equations for the shapes of the particles, e.g., ellipsoids, and the main modeling problem is to state analytically non-overlapping and containment conditions [18–23]. The second approach is based on tessellating the container/particles shapes with a grid and then approximating them by corresponding collections of grid nodes (see, e.g., [15,24] and the references therein). This way, detecting the overlap is reduced to verify if two shapes share the same node. However, to get a reasonable approximation, fine grids must be used resulting in large-scale and memory-consuming problems. In the third approach, the shape of the particle is represented approximately by a collection of spheres having different sizes and positions (see, e.g., [25–27]). Then, detecting the overlap is reduced to verifying the overlapping for two spheres from different particle collections. This problem is much simpler than detecting overlapping in the first approach. The third approach can be considered as a compromise between the simplicity of the tessellating techniques and the rigor of the methods based on analytical shapes presentation. The efficiency of the third approach depends on the quality of the multi-spherical approximation of the shape of the particle. The number of the spheres representing the shape of the particle must be small enough to simplify the non-overlapping (pairwise) test. At the same time, the union of the spheres in the collection must represent the original shape of the particle closely enough.
Appl. Sci. 2020, 10, 1846; doi:10.3390/app10051846 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 1846 2 of 13
Among the different shapes used to represent the microstructure of non-spherical particles, the spheroid (ellipsoid of revolution) is one of the most frequently used (see, e.g., [5–8,10]). One of the reasons to use the spheroid is its relative simplicity comparing to the general ellipsoid. Moreover, in many cases, analyses of 3D spheroids are reduced to the examination of the 2D ellipses used to generate the spheroid. In this paper, the problem of covering the spheroid by different spheres is studied. An ε-cover is introduced as an outer multi-spherical approximation of the spheroid within the error ε. A fast heuristic algorithm is proposed to construct the optimized ε-cover giving a reasonable balance between the value of the proximity parameter ε and the number of spheres used. Computational results are provided to demonstrate efficiency of the approach. The main contributions of the paper are as follows: The concept of the ε-cover is introduced for the outer multi-spherical approximation of the spheroid. • A fast two-stage approach is proposed to get a reasonable (optimized) ε-cover. • Numerical results are provided to illustrate the main constructions. • The paper is organized as follows. Section2 presents the basic definitions and formulations used in the paper as well as the two-stage conceptual approach. Solution techniques are presented in detail in Section3. Numerical results are presented in Section4, while Section5 presents the conclusions. Expressions for the parameters used in Section3 are derived in the AppendixA.
2. Basic Constructions The following definitions are used throughout the paper. Let ( ) x2 y2 z2 E = (x, y, z) + + 1 0 (1) | a2 b2 c2 − ≤ be a given spheroid (ellipsoid of revolution) with c = b.
Definition 1. A set Λ R3 is called a cover (outer approximation) for the spheroid E if Λ E. ⊂ ⊇ The following extended spheroid, referred to as an ε-spheroid, 2 2 2 x y z E(ε) = (x, y, z) + + 1 0 (2) | (a + ε)2 (b + ε)2 (b + ε)2 − ≤ can be considered as an outer approximation of the original spheroid E with a given error ε 0. ≥ Definition 2. set Λ(ε) R3 is called an ε-cover if E(ε) Λ(ε) E, where ε is an error of the cover. ⊂ ⊇ ⊇
In this paper, the ε-cover of the spheroid E by spheres Sk, k = 1, ... , M is studied. M The multi-spherical ε-cover is denoted by Λ(ε, M), i.e., E Λ(ε, M) = Sk. ⊂ k∪=1 From a practical point of view, a “good” ε-cover Λ(ε, M) must use a few circles M and provide a small approximation error ε. However, we may expect that for sufficiently small ε, decreasing ε in Λ(ε, M) results in increasing M, and vice versa. Thus, we cannot minimize ε and M simultaneously. Instead, our problem is formulated as follows: Find Λ(ε, M), i.e., positions, radii and total number M of spheres, providing a reasonable balance between ε and M. The following heuristic two-stage approach is proposed to construct the optimized ε-cover. Stage 1. For a given ε, find the minimum number M∗ of the spheres covering the given spheroid E; i.e., find Λ(ε, M∗). Here, the decision variables are the positions and radii of the circles and their total number M. Appl. Sci. 2020, 10, 1846 3 of 13
Stage 2. For M∗ obtained at Stage 1, find a cover of the given spheroid E with the minimal value ε ε; i.e., find Λ(ε , M ). Here, the decision variables are the positions and radii of the circles and the ∗ ≤ ∗ ∗ proximity parameter ε. In what follows, constructing the cover Λ(ε∗, M∗) is referred to as an Optimized Spherical Covering (OSC). Detailed descriptions of problems arising at Stages 1,2 and their solutions are presented in the next section.
3. Solution Algorithm Let an ellipse E with semi-axes a and b be the projection of the spheroid E on the plane XOY. It is assumed that the centers vk = (xk, yk, zk) of the spheres Sk of variable radii rk belong to the axis OX, i.e., vk = (xk, 0, 0). The parametric description of the ellipse E presented in [28] is used: E = (x(s, t), y(s, t)) : 0 s b, 0 t 2π , (3) ≤ ≤ ≤ ≤ cos t x(s, t) = (a2 b2 + s b), y(s, t) = s sin t. (4) a − · · Note that the point x(s, t) = cos t (a2 b2) belongs to the axis OX for s = 0. Denote by Λ(ε, M) the a − cover that is the projection of Λ(ε, M) on the plane XOY. By the definition, the cover Λ(ε, M) is an outer M approximation of the ellipse E, i.e., Λ(ε, M) = Ck(xk, 0). Suppose that the odd number of the circles is k∪=1 used for the cover, and the first circle has its radius b + ε and is centered at the point (0, 0, 0), as shown at Figure1a. Then, due to the symmetry, the ellipse E can be covered by a collection of M = 2m + 1 circles. Here, the right-hand side of the E can be covered by the circles Ck(xk, 0), k = 0, 1, ... , 2m with x + > x , x 0, and similarly the left-hand side. k 1 k k ≥ For the case of an even number of circles, the ellipse E be covered by a collection of 2m circles Ck(xk, 0), k = 1, ... , 2m. Let the first circle be centered at the point (x0, 0, 0), as shown at Figure1b. To find the parameters x , r of the first circle for 0 t π , the following optimization problem is solved: 0 0 ≤ ≤ 2
t0 = min t (r,t) V ∈ 2 aε 2 2 2 2 r ( (aε bε )) b 0 − pcos t · − − ≥ . bε 2 bε2 cos2 t + aε2 sin t r 0 aε · − ≥ For the given t0 corresponding parameters x0, r0 are defined in the form of
a q = ε ( 2 2) = 2 + 2 x0 aε bε , r0 x0 b . cos t0 · −
Using the parametric description of the ellipse [28], the parameter x0 can be obtained analytically as follows (we would like to thank the anonymous referee for pointing out the possibility of getting the explicit solution): p (aε2 bε2)(bε2 b2) x0 = − − . aε The details are presented in the AppendixA. Then, due to the symmetry, the right-hand side of the ellipse E can be covered by a collection of m circles Ck(xk, 0), k = 1, ... , m, for xk+1 > xk, xk > 0 and similarly for the left-hand side. In what follows, the solution algorithm for the OSC problem is presented only for the odd number of spheres, since the difference between the odd and even cases is only in the parameters of the first sphere. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 13
2 2 2 2 ()()a b b b x0 . a The details are presented in the appendix. Then, due to the symmetry, the right-hand side of the ellipse E can be covered by a collection Appl. Sci. 2020, 10, 1846 4 of 13 of m circles Ckk( x ,0), k 1,..., m , for xxkk1 , xk 0 and similarly for the left-hand side.
(a) (b)
FigureFigure 1. 1. TheThe center center of of the the first first approximating approximating circle: circle: ( (aa)) for for odd odd circular circular covering covering and and ( (bb)) for for even even circularcircular covering covering..
3.1. SolutionIn what Algorithm follows, the for Stagesolution 1 algorithm for the OSC problem is presented only for the odd numberLet of the spheres value of, sinceε > 0the be difference given. between the odd and even cases is only in the parameters of the first sphere. At Stage 1, the objective is to find the minimum number of circles Ck(xk, 0), k = 0, 1, ... , 2m covering the ellipse E centered at (0, 0) for the given value of ε. 3.1. Solution Algorithm for Stage 1 Step 1. Set r0 = b + ε, x0 = 0, k = 0, aε = a + ε, bε = b + ε, where aε, bε are the sizes of the ε ellipseLetE the. value of 0 be given. = ( ) AtStep Stage 2. Get 1, the the objective intersection is to point findp 0thek minimump0kx, p0ky numberof the ellipse of circlesE and Cx thekk( circle , 0), Ckmksolving0,1,..., 2 the following system: covering the ellipse E centered at (0, 0) for the given value of . 2 2 2 (xk pkx) + p = r , Step 1. Set rb , x 0, k −0 , aa ky , bbk , where a , b are the sizes of the 0 0 2 p2 , p ky kx + = 1, p 0, p 0 ellipse E . a2 b2 kx ≥ ky ≥ whereSteprk is 2.the Get radius the intersection of the circle pointCk centered pk (,) p atkx the p ky point of the(xk ellipse, 0). E and the circle Ck solving the = ( ) followingUsing system: the parametric description of the ellipse [28], the point p0k p0kx, p0ky is presented analytically by the following formula (see the AppendixA for details): 2 2 2 (),xk p kx p ky r k q 2 + 2 2 2( 2 22)( 2 2 2) , a xk a a xk p b pkya rk xk b q −kx − − − 2 2 p0 = 1,ppkx 0, ky, p0 0 = r (xk p ) . kx aba222 b2 ky k − − 0kx − whereStep rk 3is. Setthe kradius= k + of1. the circle centered at the point (xk , 0) . UsingStep 4 . the Solve parametric the following description nonlinear of programming the ellipse [28] problem:, the point is presented analytically by the following formula (see the appendix for details): tk∗ = min tk (r ,t ) V a2 x a a 2 x 2 ()() b 2 a 2 r 2 xk k 2∈ b 2 p k k k k , p r22 () x p . kx ( ) 22 ky k k kx and get its optimal solution rk∗, tk∗ab. Here, the feasible set V is defined by the following system of inequalities:Step 3. Set kk1. a 2 2 ( ε ( 2 2) ) 2 Step 4. Solve the followingrk nonlinearaε programmingbε p(k 1 )problemx p(k: 1)y 0, (5) − cos tk · − − − − − ≥ tt* min b q kk ε 2 2 (,)rkk t2 V 2 bε cos tk + aε sin tk rk 0, (6) aε · − ≥ ** and get its optimal solution (,)rtkk. Here, the feasible set V is defined by the following system of tk tk∗ 1, (7) inequalities: ≤ − rk > 0, (8)
where inequality (5) assures that the point pk 1 is inside the circle Ck of radius rk centered at (xk, 0); − ε inequality (6) guarantees that the circle Ck of radius rk centered at (xk, 0) is inside the ellipse E with semi-axes aε = a + ε, bε = b + ε; inequality (7) reflects the monotonous decrease of the corresponding angle tk; and the last inequality describes the natural constraint for the radius of the circle Ck. Find Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 13
2a 2 2 2 2 rk( ( a b ) p( k 1) x ) p ( k 1) y 0 , (5) cos tk
b 2 2 2 2 bcos tk a sin t k r k 0 , (6) a
* ttkk 1 , (7)
rk 0 , (8) where inequality (5) assures that the point pk1 is inside the circle Ck of radius rk centered at (xk , 0) ; inequality (6) guarantees that the circle Ck of radius rk centered at (xk , 0) is inside the ellipse E with semi-axes aa , bb ; inequality (7) reflects the monotonous decrease of the corresponding angle tk ; and the last inequality describes the natural constraint for the radius of Appl. Sci. 10 a 2020C, , 1846 x (). a22 b 5 of 13 the circle k . Find k * cos tk Usingaε the2 parametric2 description of the ellipse [28], the parameters x and are presented xk = cos t (aε bε ). Using the parametric description of the ellipse [28], thek parameters xk and rk k∗ · − areanalytically presented in analytically the following in theform following (see the formappendix (see thefor Appendixdetails): A for details): 2 2 2 2 2 2 2 2 2 (a b ) pkx ( a q b )( a ( b p ky ) p kx b ) (a2 b2 )p + (a2 b2 )(a2 (b2 p2 ) p2 b2 ) x ε ε kx ε ε ε ε ky kx ε , k x = − a−2 − − , k a2 r ε 2 2 x2 ab22aε bε x 2k if xk a− then rk = bε 1 2k 2 , otherwise rk = aε xk. if x ≤ ε then rb1 − aε bε , otherwise r a− x . k a k 22− kk ab Step 5. If the point (a, 0) is inside the circle Ck of radius rk centered at (xk, 0), then the solution is Step 5. If the point (a , 0) is inside the circle C of radius r centered at (x , 0) , then the obtained and the algorithm stops. Otherwise, go to Stepk 2. k k solutionFigures is obtained2 and3 illustrate and the analgor iterativeithm stops. procedure Otherwise, for the go algorithm to Step 2. described above. Figures 2 and 3 illustrate an iterative procedure for the algorithm described above.
(a)
(b)
FigureFigure 2.2. IllustrationsIllustrations ofof thethe firstfirst andand thethe secondsecond iterationsiterations ofof thethe algorithmalgorithm forfor StageStage 1:1: ((aa)) thethe firstfirst Appl. Sci.iteration;iteration; 2020, 10 ((b, bx)) FOR thethe second secondPEER REVIEW iteration.iteration . 6 of 13
Figure 3. Illustration ofof thethe solutionsolution obtainedobtained at at Step Step 6 6 of of the the last last iteration iteration of of the the algorithm algorithm for for Stage Stage 1. 1. 3.2. Solution Algorithm for the Stage 2 3.2. SolutionAt this stage,Algorithm the minimumfor the Stage value 2 of ε is obtained for a given number 2m + 1 of covering circles. ε LetAt thisE be stage the, ellipse the minimum centered at value(0, 0 ) ofwith is semi-axes obtaineda ε for= a + givenε, bε number= b + ε (see21m Figure of 4 covering). circles. Let E be the ellipse centered at (0, 0) with semi-axes aa , bb (see Figure 4). 2 The points pk(,) p kx p ky R , pCkk , pEk int , pkx 0, pky 0, km 0,1,2,..., , which are referred to as the pilot points, are introduced for the “gluing” circles Cxkk( , 0) , km 0,1,2,..., . cos tk 22 Let panx , pny 0 , t0 , x0 x 0(0, t 0 ) 0 , xkk(0, t ) ( a b ) , and 2 a km1,2,..., . Here, tk is the parameter (angle) introduced in Equations (3)–(4) and obtained at Stage 1.
The following nonlinear programing problem is used to optimize : * = min . vV¡ 41n
Here, v(, tk , k 1,..., m , p kx , p ky , r k , k 0,1,2,..., m ) is the (4n 1) -dimensional vector of unknown variables
Figure 4. Illustration of Stage 2.
and the feasible set V is defined by the following system of inequalities:
0 , rb0 (9)
tk 0 , ttkk 1 , km1,2,..., (10)
rk 0 , km 0,1,2,..., (11) Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 13
Figure 3. Illustration of the solution obtained at Step 6 of the last iteration of the algorithm for Stage 1.
3.2. Solution Algorithm for the Stage 2 At this stage, the minimum value of is obtained for a given number 21m of covering circles. Let E be the ellipse centered at (0, 0) with semi-axes aa , bb (see Figure 4). 2 The points pk(,) p kx p ky R , pCkk , pEk int , pkx 0, pky 0, km 0,1,2,..., , which are referred to as the pilot points, are introduced for the “gluing” circles Cxkk( , 0) , km 0,1,2,..., . cos tk 22 Let panx , pny 0 , t0 , x0 x 0(0, t 0 ) 0 , xkk(0, t ) ( a b ) , and 2 a km1,2,..., . Here, tk is the parameter (angle) introduced in Equations (3)–(4) and obtained at Stage 1.
The following nonlinear programing problem is used to optimize : * = min . vV¡ 41n
Appl.Here Sci. ,2020 v,10(,, 1846 tk , k 1,..., m , p kx , p ky , r k , k 0,1,2,..., m ) is the (4n 1) -dimensional vector 6 of 13 unknown variables
Figure 4. Illustration of Stage 2. Figure 4. Illustration of Stage 2. 2 The points pk = (pkx, pky) R , pk Ck, pk < intE, pkx 0, pky 0, k = 0, 1, 2, ... , m, which are V ∈ ∈ ≥ ≥ referredand the to feasible as the pilot set points, is defined are introduced by the following for the “gluing” system of circles inequalities:Ck(xk, 0) , k = 0, 1, 2, ... , m. π cos tk 2 2 Let pnx = a, pny = 0, t0 = , x0 = x0(0,0t0, )rb =0, x k (0, tk) = (aε bε ), and k = 1, 2, ... , m. 2 0 aε · − (9) Here, tk is the parameter (angle) introduced in Equations (3)–(4) and obtained at Stage 1. The following nonlinear programingtk 0 , ttkk problem 1 , iskm used1,2,..., to optimize ε: (10)
rk 0ε,∗ =km 0,1,2,...,min ε. (11) v V R4n+1 ∈ ⊂
Here, v = (ε, tk, k = 1, ... , m, pkx, pky, rk, k = 0, 1, 2, ... , m) is the (4n + 1)-dimensional vector of unknown variables and the feasible set V is defined by the following system of inequalities:
ε 0, r b + ε (9) ≥ 0 ≤
tk 0, tk tk 1, k = 1, 2, ... , m (10) ≥ ≤ − r 0, k = 0, 1, 2, ... , m (11) k ≥ q bε 2 2 2 2 bε cos tk + aε sin tk rk 0, k = 1, 2, ... , m (12) aε · − ≥ p 0, p 0, k = 0, 1, 2, ... , m (13) kx ≥ ky ≥ r2 (x p )2 p2 0, k = 0, 1, 2, ... , m (14) k − k − kx − ky ≥ 2 2 2 r (x + + p ) p 0, k = 0, 1, ... , m 1 (15) k+1 − k 1 kx − ky ≥ − p2 p2 kx + kx 1 0, k = 0, 1, 2, ... , m. (16) a2 b2 − ≥ Here, inequality (9) assures C (0, 0) Eε; inequality (10) reflects the monotonous decrease of the 0 ⊂ corresponding angle tk; inequality (11) describes the natural constraint for the radius of the circle Ck; constraints (12) guarantee C Eε; inequalities (13) present the nonnegativity of all p ; constraints (14) k ⊂ k and (15) assure p C C + , while (16) guarantees p < intE. k ∈ k ∩ k 1 k 4. Computational Results Computational experiments were run on an AMD Athlon 64 X2 5200+ computer. Local optimization was performed by the IPOPT solver [29], which is available at an open access noncommercial software depository (https://projects.coin-or.org/Ipopt). To illustrate the algorithm performance, examples for ellipses and spheroids are considered. For each problem instance, the computational time required to construct the approximated cover is less than a second. Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 13
b 2 2 2 2 bcos tk a sin t k r k 0 , km1,2,..., (12) a
ppkx0, ky 0, km 0,1,2,..., (13)
2 2 2 rk( x k p kx ) p ky 0 , km 0,1,2,..., (14)
2 2 2 rk11( x k p kx ) p ky 0 , km0,1,..., 1 (15)
22 ppkx kx 10 , km 0,1,2,..., . (16) ab22 Here, inequality (9) assures CE0 (0, 0) ; inequality (10) reflects the monotonous decrease of the corresponding angle tk ; inequality (11) describes the natural constraint for the radius of the circle Ck ; constraints (12) guarantee CEk ; inequalities (13) present the nonnegativity of all pk ; constraints (14) and (15) assure pk C kI C k1 , while (16) guarantees pEk int .
4. Computational Results Computational experiments were run on an AMD Athlon 64 X2 5200+ computer. Local
Appl.optimization Sci. 2020, 10, 1846 was performed by the IPOPT solver [29], which is available at an open7 of access 13 noncommercial software depository (https://projects.coin-or.org/Ipopt). To illustrate the algorithm performance, examples for ellipses and spheroids are considered. InstanceFor each 1. The problem ellipse Einstance with semi-axes, the computational a =1.3, b =1 time is given. required to construct the approximated cover is less than a second. (a) For ε = 0.3, the solution with one covering circle is presented in Figure5a: Instance 1. The ellipse E with semi-axes a 1.3, b 1 is given. (a) For 0.3 , the solution with one covering circle is presented in Figure 5a: Λ(ε∗, M∗) = C1(x1 = 0, 0, 0), ε∗ = 0.3, M∗ = 1. ** * * ( ,M ) C11 ( x 0, 0, 0) , 0.3 , M 1. (b) For ε = 0.2, the solution with an even number of covering circles is shown in Figure5b: (b) For 0.2 , the solution with an even number of covering circles is shown in Figure 5b: 22 ** * * Λ(( ε∗ ,,MM∗) ) = CCk (( x k, 0, 0, 0 0)),, ε∗ = 0.087477,0.087477M, ∗M= 2, 2 , Uk∪=1 kk k1 x , k = 1, 2 ={0.265385, 0.265385}, and r , k = 1, 2 ={1.034615, 1.034615}. {xk{kk , 1,2} } {0.265385,−0−.265385}, and {{rkkk , 1,2}} {1.034615, 1.034615}. (c) For ε = 0.05, the solution with an odd number of covering circles is given in Figure5c: (c) For 0.05, the solution with an odd number of covering circles is given in Figure 5c: 3 ** 3 * * Λ(( ε∗ ,,M∗) ) = U CCkkk ( xk, , 0, 0, 0 0)),, ε∗= 0.040356,0.040356M, ∗M= 3, 3 , kk∪=11 x ,{kxk= ,1, 2, 1,2, 3 = 3}{0.357061,{0.357061,0.357061, −0.357061, 0.000}, 0.000},r ,{krk= ,1, 2, 1,2, 3 = 3}{0.942939,{0.942939, 0.942939, 0.942939,1.040356 1.040356}. }. { k k } − { k k }
(a) (b) (c)
Figure 5. The optimized cover of the ellipse E:(a) by M∗ = 1 circle; (b) by M∗ = 2 circles; and (c) by M∗ = 3 circles.
Instance 2. The ellipse E with semi-axes a =2.3, b =1 is given and ε = 0.1.
(a) The optimized cover with an even number of circles is presented in Figure6a:
6 Λ(ε∗, M∗) = Ck(xk, 0, 0), ε∗ = 0.072085, M∗ = 6, k∪=1
x , k = 1, ... , 6 = {1.596217, 1.596217, 1.008362, 1.008362, 0.344754, 0.344754}, { k } − − − r , k = 1, ... , 6 = {0.703783, 0.703783, 0.942524, 0.942524 1.057759, 1.057759}. { k } (b) The optimized cover with an odd number of circles is shown in Figure6b:
7 Λ(ε∗, M∗) = Ck(xk, 0, 0), ε∗ = 0.05372, M∗ = 7, k∪=1
x , k = 1, ... , 7 = {1.638879, 1.638879, 1.146544, 1.146544, 0.589831, 0.589831, 0.0000}, { k } − − − r , k = 1, ... , 7 = {0.661121, 0.661121, 0.883641, 0.883641, 1.011495, 1.011495, 1.053720}. { k } Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13
* * Figure 5. The optimized cover of the ellipse E : (a) by M 1 circle; (b) by M 2 circles; and (c) * by M 3 circles.
Instance 2. The ellipse E with semi-axes a 2.3, b 1 is given and 0.1. (a) The optimized cover with an even number of circles is presented in Figure 6a: 6 ** * * ( ,M ) U Ckk ( x , 0, 0) , 0.072085 , M 6, k1
{xkk , 1,..., 6} {1.596217, −1.596217, 1.008362, −1.008362, 0.344754, −0.344754}, {rkk , 1,..., 6} {0.703783, 0.703783, 0.942524, 0.942524 1.057759, 1.057759}. (b) The optimized cover with an odd number of circles is shown in Figure 6b: 7 ** * * ( ,M ) U Ckk ( x , 0, 0) , 0.05372 , M 7 , k1
{xkk , 1,..., 7} {1.638879, −1.638879, 1.146544, −1.146544, 0.589831, −0.589831, 0.0000}, Appl. Sci. 2020, 10, 1846 8 of 13 {rkk , 1,..., 7} {0.661121, 0.661121, 0.883641, 0.883641, 1.011495, 1.011495, 1.053720}.
(a)
(b) * * FigureFigure 6. The 6.optimizedThe optimized cover cover of the of theellipse ellipse E E:: (a (a) )by by MM∗ =66 circles; circles; (b) ( byb)M by∗ =M7 circles. 7 circles.
InstanceInstance 3.3.The The spheroid spheroidE with E semi-axeswith semi a =-1.9,axes b =a c=1.9,1 is givenbc and1 εis= given0.1. and 0.1. (a) The optimized cover with an even number of spheres is presented in Figure 7a: (a) The optimized cover with an even number of spheres is presented in Figure7a: 4 ** * * ( ,M ) U Skk (4 x , 0, 0) , 0.092225 , M 4 , Λ(ε∗, Mk∗)1 = Sk(xk, 0, 0), ε∗ = 0.092225, M∗ = 4, k∪=1 {xk , 1,..., 4} {1.054209, −1.054209, 0.367367, −0.367367}, k x , k = 1, ... , 4 = {1.054209, 1.054209, 0.367367, 0.367367}, { k } − − {rkk , 1,..., 4} {0.845791, 0.845791, 1.065344, 1.065344}. r , k = 1, ... , 4 = {0.845791, 0.845791, 1.065344, 1.065344}. { k } (b)(b) TheThe optimized optimized cover cover (for a (for=2) with an2) oddwith number an odd of spheresnumber is shownof spheres in Figure is 7shownb: in Figure 7b: 5 ** * * ( ,M ) U Skk5 ( x , 0, 0) , 0.070009?, M 5 , Λ(ε∗, M∗k)1 = Sk(xk, 0, 0), ε∗ = 0.070009 , M∗ = 5, k∪=1
x , k = 1, ... , 5 = {1.229404, 1.229404, 0.644245, 0.644245, 0.0000}, { k } − − r , k = 1, ... , 5 = {0.770596, 0.770596, 0.996786, 0.996786, 1.070009}. { k } Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13
{xkk , 1,..., 5} {1.229404, −1.229404, 0.644245, −0.644245, 0.0000}, {rk , 1,..., 5} Appl. kSci. 2020, 10, x FOR{0.770596, PEER REVIEW 0.770596, 0.996786, 0.996786, 1.070009}. 9 of 13
{Appl.xkk , Sci. 1,...,2020, 5}10, 1846 {1.229404, −1.229404, 0.644245, −0.644245, 0.0000}, 9 of 13 {rkk , 1,..., 5} {0.770596, 0.770596, 0.996786, 0.996786, 1.070009}.
(a)
(a)
(b)
* * Figure 7. The optimized cover of the spheroid E : (a) by M 4 spheres; (b) by M 5 spheres.
Instance 4. The spheroid E with semi-axes (ab) 2, bc1 is given and 0.03. The optimized cover with an odd number of spheres* is presented in Figure *8: FigureFigure 7. The 7. oTheptimized optimized cover cover of the of spheroid the spheroid E : (Ea:() bya) byMM∗=4 4 spheres; spheres; ( (bb)) by by MM∗ =55 spheres. spheres. 9 ** * * ( ,M ) U Skk ( x , 0, 0) , 0.022435 , M 9 , InstanceInstance 4. The4. The spheroid spheroidE with Esemi-axes with ksemi1 a =-axes2, b =a c=2,1 bc is given1 andis givenε = 0.03. and 0.03. TheThe optimized optimized covercover with with an an odd odd number number of spheres of spheres is presented is presented in Figure in 8Figure: 8: {xkk , 1,..., 9} {0.000000, 0.366202, 0.721511, 1.055358, 1.357813, −0.366202, −0.721511, 9 ** * * −1.055358, −1.357813}, ( ,M ) U S9kk ( x , 0, 0) , 0.022435 , M 9 , Λ(ε∗, M∗) = Sk(xk, 0, 0), ε∗ = 0.022435, M∗ = 9, {rkk , 1,..., 9} {1.022435, 0.999666,k1k∪=1 0.930939, 0.814240, 0.642187, 0.999666, 0.930939, 0.814240, 0.642187}.{xkk , 1,..., 9} {0.000000, 0.366202, 0.721511, 1.055358, 1.357813, −0.366202, −0.721511, −1.055358, −1.357813}, {rkk , 1,..., 9} {1.022435, 0.999666, 0.930939, 0.814240, 0.642187, 0.999666, 0.930939, 0.814240, 0.642187}.
=* FigureFigure 8. 8. TheThe optimized optimized cover cover of of the the spheroid spheroid EE bybyM M∗ 9 spheres.9 spheres.
xk, k = 1, ... , 9 = {0.000000, 0.366202, 0.721511, 1.055358, 1.357813, 0.366202, 0.721511, {Instance 5. The} spheroid E with semi-axes a 10, bc1 is given and− 0.3 . − 1.055358, 1.357813}, − The optimized− cover with an odd number of spheres is shown *in Figure 9: rk, k = 1, ...Figure, 9 = 8.{1.022435, The optimized 0.999666, cover 0.930939, of the spheroid 0.814240, E by 0.642187, M 9 0.999666, spheres. 0.930939, 0.814240, { } 21 0.642187}. ** * * ( ,M ) U Skk ( x , 0, 0) , 0. 223518 , M 21, Instance 5. The spheroid E with semi-axes a 10, bc1 is given and 0.3 . Instance 5. The spheroid E with semi-axesk1 a =10, b = c =1 is given and ε = 0.3. The optimized cover with an odd number of spheres is shown in Figure 9: The{xkk optimized , 1,...,21} cover with {0.000000, an odd number 1.395933 of spheres, 2.764207 is shown, 4.077711 in Figure, 9:5.310411, 6.437857, 7.437658, 21 8.289887, 8.977362, 9.485512( ** ,M, ) 9.798415, S ( x ,−1 0,.395933 0) * , 0.−2 .764207223518,, −M4.077711* 21 , −5.310411, −6.437857, U 21kk , , −7.437658, −8.289887, −8Λ.977362,(ε∗, M∗ )−9k =.485512,1 Sk(x −9k,.798415, 0, 0), ε∗ },= 0.223518, M∗ = 21, k∪=1 {xkk , 1,...,21} {0.000000, 1.395933, 2.764207, 4.077711, 5.310411, 6.437857, 7.437658, x , k = 1, ... , 21 = {0.000000, 1.395933, 2.764207, 4.077711, 5.310411, 6.437857, 7.437658, 8.289887, 8.289887,{ k 8.977362, 9.485512} , 9.798415, −1.395933, −2.764207, −4.077711, −5.310411, −6.437857, 8.977362, 9.485512, 9.798415, 1.395933, 2.764207, 4.077711, 5.310411, 6.437857, 7.437658, −7.437658, −8.289887, −8.977362, −−9.485512, −9−.798415,}, − − − − 8.289887, 8.977362, 9.485512, 9.798415,}, − − − − Appl. Sci. 2020, 10, 1846 10 of 13 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13
r{,rkk ,= 1, 1,...,21}... , 21 ={1.223518,{1.223518, 1.211410 1.211410,, 1.175328 1.175328,, 1.115988 1.115988,, 1.034567, 1.034567, 0.932687, 0.932687, 0.812378, 0.812378, 0.676054, 0.676054, { k } 0.526503,0.526503, 0.367023, 0.367023, 0.203604,0.203604, 1.211410, 1.211410, 1.175328,1.175328, 1.115988,1.115988, 1.034567, 0.932687, 0.932687, 0.812378, 0.812378, 0.676054, 0.676054, 0.526503,0.526503, 0.367023, 0.367023, 0.203604}. 0.203604}.
* FigureFigure 9. 9. TheThe optimized optimized cover cover of ofthe the spheroid spheroid EE byby MM∗ =2121 spheres. spheres.
5.5. Conclusions Conclusions TheThe problem problem of of optimizedoptimized multi-sphericalmulti-spherical covering covering for for spheroid spheroidss is isintroduced. introduced. This This problem problem is ismotivated motivated by by the the packing packing non non-spherical-spherical particles particles arising arising in naturals in naturals sciences sciences and engineering. and engineering. The Thesimple simple heuristicheuristic approach approach is proposed is proposed to toconstruct construct an an optimized optimized covering covering,, providing providing a areasonable reasonable balancebalance between between the the number number of of spheresspheres andand thethe errorerror of approximation.approximation. Computational Computational experiments experiments indicateindicate that that the the proposed proposed approach approach constructs constructs good good feasible feasible coverings coverings very very fast: in fast less: in than less a than second. a Thesecond. multi-spherical The multi-spherical approximations approximations obtained obtain in theed paper in the provide paper provide a basis fora basis fast optimizedfor fast optimized packing spheroidspacking spheroid in differents in containers,different containers, using algorithms using algorithms proposed, proposed, e.g., in [ 30e.g.,–32 in]. [30 The–32 two-stage]. The two scheme-stage presentedscheme presented in Section in2 can Section be applied 2 can be for applied the general for the optimized general optimized multi-spherical multi- covering.spherical covering. An extension An ofext theension covering of the approach covering to the approach case of to more the complex case of more objects complex [33–36 ] objects is an interesting [33–36] is direction an interesting for the futuredirection research. for the Some future results research. on this Some topic results are onon thethis way. topic are on the way.
AuthorAuthor Contributions: Contributions:Investigation, Investigation, A.P., T.R., A.P., I.L., T.R., J.A.M.-S.; I.L., A.M.; Methodology, Methodology, A.P., T.R., A.P., I.L., T.R. J.A.M.-S.;, I.L., A.M.; Project administration,Project administration, T.R., I.L.; Writing—original T.R., I.L.; Writing draft,— T.R.,original I.L.; Writing—review draft, T.R., I.L.; and Writing editing,—review T.R., I.L.; and Programming editing, algorithms, A.P.; Numerical experimentation, A.P., T.R. All authors have read and agreed to the published version ofT.R., the manuscript. I.L.; Programming algorithms, A.P.; Numerical experimentation, A.P., T.R. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments:Funding: This researchThe authors received would no external like to thank funding. anonymous referees for careful reading the paper and constructiveAcknowledgement comments.: The The firstauthors two authorswould like were to partially thank anonymous supported by referees the “Program for careful for the reading State Priority the Scientific Research and Technological (Experimental) Development of the Department of Physical and Technical Problemspaper and of Energy constructive of the National comments. Academy The first of Sciences two authors of Ukraine” were (#6541230).partially supported by the “Program for the State Priority Scientific Research and Technological (Experimental) Development of the Conflicts of Interest: The authors declare no conflict of interest. Department of Physical and Technical Problems of Energy of the National Academy of Sciences of AppendixUkraine” A(#6541230). Conflicts of Interest: The authors declare no conflict of interest. We would like to thank the anonymous referee for pointing out that the explicit expressions can be obtained for rk, xk and p0 = (p0 , p0 ) in our algorithm at Stage 1. Appendix A k kx ky The parameters have been derived based on constructions proposed in [28]. We would like to thank the anonymous referee for pointing out that the explicit expressions can 1. Deriving r . be obtained fork rk , xk and pk (,) p kx p ky in our algorithm at Stage 1. The parameters have been derived based on constructions proposed in [28]. x(s t) = cos t (a2 b2 + s b ) y(s t) = s t Observe that the point , aε ε ε ε , , sin belongs to the coordinate 1. Deriving . e − e · axis OX for s = 0, while the point (x = aε cos t, y = bε sin t) belongs to the frontier of the ellipse. Since x = cos t (a2 b2 ), then cos t 22 Observe a thatε ε theε point xst(,) ( absbysts ),(,) sin t belongs to the − r a a x a2 x2 cos t = ε , sin t = 1 ε for a > b and 2 2 2 2 2 ε εee aε bε − (aε bε) coordinate axis− OX for s 0 , while− the point (x a cos t , y b sin t ) belongs to the frontier of 2 2 2 2 2 2 2 aεx 2 (aε bε) (aε bε) b2 the ellipse. (a2 b2 ) a2 x2 x2 − x − x a 1 2 2 2 0, ε ε ε 0, 2 , aε , ε aε . − (aε bε) ≥ − − ≥ ≤ aε ≤ ≤ − − r 2 2 2 (x y ) x = aεx y = b aεx The tangent point 0, 0 , 0 2 2 , 0 1 2 2 2 belongs to the frontier of the ellipse. aε bε − (aε bε) − − Appl. Sci. 2020, 10, 1846 11 of 13
2 a2 x Since r2 = (x x )2 + (0 y )2, then r2 = x ε + y 2 and thus 0 0 a2 b2 0 − − ! − ε− ε ! 2 2 2 2 2 2 2 2 2 2 aεx aεx (aε bε)x +(aε bε) x2 r2 = x + b2 = b2 − − − = b2 2 2 ε 1 2 2 2 ε 2 2 2 ε 1 2 2 . − aε bε − (aε bε) (aε bε) − aε bε − − − − 2 2 2 2 2 2 2 2 4 2 x(aε bε) a x b x b x b x − ε = − ε = ε = ε 2 2 2 2 2 2 2 2 2 2 2 . Here we used aε bε − aε bε aε bε aε bε (aε bε) − − − − − r x2 Therefore r = b 1 k at the k-th iteration. k ε a2 b2 − ε− ε
2. Deriving p0k = (p0kx, p0ky).
The tangent point p0k = (p0kx, p0ky) is defined from the system 2 p2 p kx + ky = a2 b2 1 2 2 2 (pkx xk) + p = r − ky k Therefore p2 = r2 (p x )2, b2p2 + a2(r2 (p x )2) = a2b2, ky k − kx − k kx k − kx − k 2 2 2 2 2 2 2 2 (b a )p + 2a xkp + a (r xk b ) = 0 and thus − kx q kx k − − 2 2 2 2 2 2 2 2 a xk a a xk (b a )(r xk b ) q p = − ± − − k − − p = r2 (p x )2 0kx (b2 a2) , 0ky k kx k . − − −
3. Deriving xk+1.
From the system at Step2 we have (x p )2 + p2 = r2 k+1 kx ky k r − x2 . k+1 bε 1 = r a2 b2 k − ε− ε Therefore ! x2 (x p )2 + p2 = b2 1 k+1 , k+1 kx ky ε a2 b2 − − ε− ε a2 ε x2 2p x + p2 + p2 b2 = 0 and thus a2 b2 k+1 kx k+1 kx ky ε ε− ε − r − 2 (a2 b2 )p + p2 (a2 b2 ) a2 (a2 b2 )(p2 +p2 b2 ) ε− ε kx kx ε− ε − ε ε− ε kx ky− ε x = , k+1 a2 q ε (a2 b2 )p + (a2 b2 )(a2 (b2 p2 ) p2 b2 ) ε− ε kx ε− ε ε ε− ky − kx ε xk+1 = 2 . aε
References
1. Bernal, J.D. A geometrical approach to the structure of liquids. Nature 1959, 183, 141–147. [CrossRef] 2. Stillinger, F.H.; Marzio, E.A.D.; Kornegay, R.L. Systematic approach to explanation of the rigid disk phase transition. J. Chem. Phys. 1964, 40, 1564–1576. [CrossRef] 3. Makse, H.; Kurchan, J. Testing the thermodynamic approach to granular matter with a numerical model of a decisive experiment. Nature 2002, 415, 614–617. [CrossRef][PubMed] 4. Costamagna, P.; Costa, P.; Antonucci, V. Micro-modelling of solid oxide fuel cell electrodes. Electrochim. Acta 1998, 43, 375–394. [CrossRef] 5. Zerhouni, O.; Tarantino, M.G.; Danas, K. Numerically-aided 3D printed random isotropic porous materials approaching the Hashin-Shtrikman bounds. Compos. Part B Eng. 2019, 156, 344–354. [CrossRef] 6. Gan, J.; Yu, A.; Zhou, Z. DEM simulation on the packing of fine ellipsoids. Chem. Eng. Sci. 2016, 156, 64–76. [CrossRef] 7. Gately, R.D.; Panhuis, M. Filling of carbon nanotubes and nanofibres. Beilstein J. Nanotechnol. 2015, 6, 508–516. [CrossRef] Appl. Sci. 2020, 10, 1846 12 of 13
8. Ustach, V.; Faller, R. The raspberry model for protein-like particles: Ellipsoids and confinement in cylindrical pores. Eur. Phys. J. Spec. Top. 2016, 225, 1643–1662. [CrossRef] 9. Wang, X.; Zhao, L.; Fuh, J.Y.H.; Lee, H.P. Effect of Porosity on Mechanical Properties of 3D Printed Polymers: Experiments and Micromechanical Modeling Based on X-ray Computed Tomography Analysis. Polymers 2019, 11, 1154. [CrossRef] 10. Li, C.X.; Zou, R.P.; Pinson, D.; Yu, A.B.; Zhou, Z.Y. An experimental study of packing of ellipsoids under vibrations. Powder Technol. 2020, 361, 45–51. [CrossRef] 11. Akeb, H. A Look-Ahead-Based Heuristic for Packing Spheres into a Bin: The Knapsack Case. Procedia Comput. Sci. 2015, 65, 652–661. [CrossRef] 12. Stoyan, Y.; Scheithauer, G.; Yaskov, G. Packing Unequal Spheres into Various Containers. Cybern. Syst. Anal. 2016, 52, 419–426. [CrossRef] 13. Stetsyuk, P.; Romanova, T.; Scheithauer, G. On the global minimum in a balanced circular packing problem. Optim. Lett. 2016, 10, 1347–1360. [CrossRef] 14. Hifi, M.; Yousef, L. A local search-based method for sphere packing problems. Eur. J. Oper. Res. 2019, 274, 482–500. [CrossRef] 15. Litvinchev, I.; Infante, L.; Ozuna, L. Packing circular-like objects in a rectangular container. J. Comput. Syst. Sci. Int. 2015, 54, 259–267. [CrossRef] 16. Torres-Escobar, R.; Marmolejo-Saucedo, J.A.; Litvinchev, I. Binary monkey algorithm for approximate packing non-congruent circles in a rectangular container. Wirel. Netw. 2018.[CrossRef] 17. Bertei, A.; Chueh, C.-C.; Pharoah, J.G.; Nicolella, C. Modified collective rearrangement sphere-assembly algorithm for random packings of nonspherical particles: Towards engineering applications. Powder Technol. 2014, 253, 311–324. [CrossRef] 18. Kallrath, J. Packing ellipsoids into volume-minimizing rectangular boxes. J. Glob. Optim. 2017, 67, 151–185. [CrossRef] 19. Birgin, E.; Lobato, R. A matheuristic approach with nonlinear subproblems for large-scale packing of ellipsoids. Eur. J. Oper. Res. 2019, 272, 447–464. [CrossRef] 20. Kampas, F.; Castillo, I.; Pintér, J. Optimized ellipse packings in regular polygons. Optim. Lett. 2019, 13, 1583–1613. [CrossRef] 21. Pankratov, A.; Romanova, T.; Litvinchev, I. Packing ellipses in an optimized rectangular container. Wirel. Netw. 2018.[CrossRef] 22. Pankratov, A.; Romanova, T.; Litvinchev, I. Packing ellipses in an optimized convex polygon. J. Glob. Optim. 2019, 75, 495–522. [CrossRef] 23. Romanova, T.; Stoyan, Y.; Pankratov, A.; Litvinchev, I.; Avramov, K.; Chernobryvko, M.; Yanchevskyi, I.; Mozgova, I.; Bennell, J. Optimal layout of ellipses and its application for additive manufacturing. Int. J. Prod. Res. 2019.[CrossRef] 24. Litvinchev, I.; Infante, L.; Ozuna, L. Approximate Circle Packing in a Rectangular Container: Integer Programming Formulations and Valid Inequalities. In Lecture Notes in Computer Science; Gonzalez-Ramirez, R.G., Ed.; Springer: Berlin/Heidelberg, Germany, 2014; Volume 8760, pp. 47–60. 25. Jones, D.R. A fully general, exact algorithm for nesting irregular shapes. J. Glob. Optim. 2014, 59, 367–404. [CrossRef] 26. Rocha, P.; Gomes, A.M.; Rodrigues, R.; Toledo, F.M.B.; Andretta, M. Constraint aggregation in non-linear programming models for nesting problems. In Computational Management Science: State of the Art 2014; Fonseca, J.R., Weber, G.-W., Telhada, J., Eds.; Springer International Publishing: Cham, Switzerland, 2016; pp. 175–180. 27. Yuan, Y.; Liu, L.; Deng, W.; Li, S. Random-packing properties of spheropolyhedra. Powder Technol. 2019, 351, 186–194. [CrossRef] 28. Birgin, E.; Bustamante, L.; Callisaya, H.; Martnez, J. Packing circles within ellipses. Int. Trans. Oper. Res. 2013, 20, 365–389. [CrossRef] 29. Wachter, A.; Biegler, L. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Programm. 2006, 106, 25–57. [CrossRef] 30. Stoyan, Y.; Romanova, T. Mathematical models of placement optimisation: Two- and three-dimensional problems and applications. In Modeling and Optimization in Space Engineering; Fasano, G., Pintér, J., Eds.; Springer Optimization and its Applications: New York, NY, USA, 2013; Volume 73, pp. 363–388. Appl. Sci. 2020, 10, 1846 13 of 13
31. Stoyan, Y.; Pankratov, A.; Romanova, T. Cutting and packing problems for irregular objects with continuous rotations: Mathematical modeling and nonlinear optimization. J. Oper. Res. Soc. 2016, 67, 786–800. [CrossRef] 32. Romanova, T.; Bennell, J.; Stoyan, Y.; Pankratov, A. Packing of concave polyhedra with continuous rotations using nonlinear optimization. Eur. J. Oper. Res. 2018, 268, 37–53. [CrossRef] 33. Kiseleva, E.M.; Lozovskaya, L.I.; Timoshenko, E.V. Solution of continuous problems of optimal covering with spheres using optimal set-partition theory. Cybern. Syst. Anal. 2009, 45, 421–437. [CrossRef] 34. Scheithauer, G.; Stoyan, Y.G.; Romanova, T.; Krivulya, A. Covering a polygonal region by rectangles. Comput. Optim. Appl. Springer Neth. 2011, 48, 675–695. 35. Yakovlev, S.; Kartashov, O.; Komyak, V.; Shekhovtsov, S.; Sobol, O.; Yakovleva, I. Modeling and simulation of coverage problem in geometric design systems. In Proceedings of the 15th International Conference on the Experience of Designing and Application of CAD Systems, CADSM 2019, Lviv, Ukraine, 26 February–2 March 2019; pp. 20–23. 36. Torres-Escobar, R.; Marmolejo-Saucedo, J.A.; Litvinchev, I.; Vasant, P. Monkey algorithm for packing circles with binary variables. In Intelligent Computing & Optimization. ICO 2018; Vasant, P., Zelinka, I., Weber, G.-W., Eds.; Springer International Publishing: Cham, Switzerland, 2019; Volume 866, pp. 547–559.
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