=Paper=
{{Paper
|id=Vol-2864/paper38
|storemode=property
|title=Bi-objective Circular-Hole Based Topology Optimization Problem in Additive Manufacturing
|pdfUrl=https://ceur-ws.org/Vol-2864/paper38.pdf
|volume=Vol-2864
|authors=Georgiy Yaskov,Andrey Chugay,Tatiyana Romanova,Sergey Shekhovtsov
|dblpUrl=https://dblp.org/rec/conf/cmis/YaskovCRS21
}}
==Bi-objective Circular-Hole Based Topology Optimization Problem in Additive Manufacturing==
https://ceur-ws.org/Vol-2864/paper38.pdf
Bi‐objective Circular‐Hole Based Topology Optimization
Problem in Additive Manufacturing
Georgiy Yaskova,b, Andrey Chugaya,c, Tatiyana Romanovaa,d and Sergey Shekhovtsovc,d
a
Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, 2/10
Pozharskogo st., Kharkiv 61046, Ukraine
b
Kharkiv Aviation Institute, National Aerospace University, 17 Chkalov st., Kharkiv 61070, Ukraine
c
National University of Internal Affairs, L. Landau av., 27, Kharkiv, 61080, Ukraine
d
Kharkiv National University of Radioelectronics, 14 Nauky ave., Kharkiv 61166, Ukraine
Abstract
The paper deals with the circle packing problem which arises in topology optimization for
additive manufacturing. The problem consists in packing a number of circles of radii within a
particular range imposed by technical limitations, the packing factor being maximized. A bi-
objective formulation for the problem compromising the packing factor and the maximum
mechanical stress of parts is suggested. The -constraint method is applied to search for a
trade-off solution of the problem. A new packing approach based on a modified Apollonian
circle packing and nonlinear optimization is developed. Numerical examples and graphical
illustration of results are given.
Keywords 1
Additive manufacturing, topology optimization, mechanical stress, bi-objective optimization,
-constraint method, packing, circle, polygon, Apollonian circle packing, nonlinear
optimization
1. Introduction
One of the most important topics of modern mechanical engineering is reducing weight while
maintaining specific characteristics of structures designed. Such problems with conflicting criteria are
related to decision making in a formidable manufacturing process. A key objective of the problems is
finding optimal geometric shape and topology of the designed product ensuring minimum weight at
specified strength. To this end, topology optimization methods [1] are used to find the best design
parameters satisfying the technological and strength constraints and thus providing the objective
function extremum. When optimizing topology of a structure, the stress level in a particular part of the
structure can be used as an indicator of ineffective material usage. Ideally, the stress level in the
structure should be uniform, close to the limiting, but safe value [2].
Applying topology optimization methods in mechanical engineering is a newish development in
the design procedure. The methods received the greatest impetus in their development when it became
possible to use additive technologies in the manufacturing process [3] instead of classical subtractive
methods. Additive technologies enable to expand the range of designs for the same product.
In the last two decades, topology optimization became an active field for scientific research. This
led to the use of a multidisciplinary approach in the search for solutions to the problems, which use
the methods of solid mechanics, thermodynamics, biology simultaneously [4].
Paper [2] reviews modern topology optimization methods.
CMIS-2021: The Fourth International Workshop on Computer Modeling and Intelligent Systems, April 27, 2021, Zaporizhzhia, Ukraine
EMAIL: yaskov@ukr.net (G. Yaskov); chugay.andrey80@gmail.com (A. Chugay); tarom27@yahoo.com (T. Romanova) ;
tarom7@yahoo.com (S. Shekhovtsov)
ORCID: 0000-0002-1476-1818 (G. Yaskov); 0000-0002-4079-5632 (A. Chugay); 0000-0002-8618-4917 (T. Romanova) ; 0000-0003-2381-
7999 (S. Shekhovtsov)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
Currently, the following main methods of topology optimization can be distinguished: SIMP (solid
isotropic material with penalization), ESO (evolutionary structural optimization), Level-Set (level
setting method) and their various combinations. One of the most effective applications of these
methods is in optimizing the topology of continuous structures, i.e. finding the best placement
location and geometry for holes (cavities) within the modeling area.
Circle packing problems (CPP) have a wide range of industrial and engineering applications, for
example, facility layout design, cutting stock problems in the glass, metal, paper, textile and wood
industries etc.
As a rule, practical optimization problems have several conflicting objectives. Such problems are
called multi-objective. Optimization methods for multi-objective optimization problems and its
applications are reviewed in [5, 6].
In this paper we first propose a bi-objective formulation for circular-hole based topology
optimization which takes into account both the packing factor and the maximum mechanical stress of
parts. A new approach for packing circles based on a modified finite Apollonian circle packing (ACP)
and optimized homothetic transformations of circles is suggested. Both techniques allow to
advantageously arrange circles within polygonal parts minimizing material expenses. The -constraint
method [7] is applied to find out a trade-off circular arrangement.
In accordance with ACP (see, for example [8]), starting from three mutually tangent circles, next
circles are added to be tangent to the three circles forming four mutually tangent circles (or tangent to
the frontier of the container). We modify ACP for taking into account the upper bound of circle radii
and the minimal allowed distance between the circles. Some circles are allowed to be moved within
the container and located to positions which give a possibility to enlarge radii of next circles being
packed.
The main contributions of the paper are as follows:
A bi-objective model of non-standard circular packing problem considering both the
maximum packing factor and mechanical stress of parts for a topology optimization
A new approach for constructing starting points based on ACP
New benchmark instances for packing circles with variable radii.
2. Related papers
The number of papers concerning CPP goes up and up each year. Paper [9] reviews selected works
dealing with packing methods and applications for CPP. We make mention of some recent papers [10,
11, 12]. A powerful tool of solving CPP with unequal circles is treating additional variable metric
characteristics of circles and/or containers (see, for example, [11, 12, 13, 14]). The best results for a
set of benchmark CPP instances are continuously updated in the well known website [15].
Paper [16] proposes a circular packing model and a fast heuristic algorithm to optimize part
geometries subject to the DMLS constraints. A feature of the model is that radii of circles are
preassigned.
In [9] a model and a numerical solution approach to packing egg-shaped objects with arbitrary size
and orientation into optimized convex containers is presented. The area of the container is minimized
The solution strategy is based on the analysis of embedded Lagrange multipliers and nonlinear
optimization. Within the universal model, circles and ellipses are considered to be special cases of
egg.
The ε-constraint method introduced in [7] is often used for solving multi-objective optimization
problems. The method optimizes only one objective while the other objectives are imposed by limits.
In [5] the Pareto optimality of solutions obtained by the ε-constraint method is investigated.
A bi-objective function for packing in a spacecraft module is proposed in [17]. The dimensions of
the module are minimized and other criteria such as desired adjacency between items and packing
costs are also taken into account. Paper [18] uses a genetic algorithm for bi-objective topology
optimization: minimization mass and maximization effective flexural and torsional rigidities. Methods
for analysis of stress state in parts with different geometries are discussed in [16, 19]. Effective
methods of nonsmooth optimization for allocation problems are considered in [20].
3. Problem statement
Let P 0 be a polygon given by its vertices v0i ( x0i , y0i ), i I0 {1,2,..., s0 } . We define a
disconnected polygonal domain of the form
P U Pl , Pl P 0 , l I p , P l I Pj , l j I p {1, 2,..., nP } ,
lI p
where
Pl ={( x, y) R 2 : ml ( x, y) 0, m 1,2,..., M l }
are convex polygons given by vertices vli ( xli , yli ), i Il {1,2,..., sl } ;
ml ( x, y) aml x+bml y +cml 0 , l I p , are normal equations of the edges of Pl , l I p . Let us also
define a collection of circles
Cq Cq (vq , rq ) {( x, y ) R 2 : ( x xq ) 2 ( y yq ) 2 rq2 }
of variable radii rq and translation vectors vq ( xq , yq ) for q I N {1,2,..., N} ,
0 r rq r r , r , r are given lower and upper allowed values for rq , r is the maximal
possible radius of the circle which can be inscribed into Pl , l I p .
Conditions of packing the circles Сq (vq , rq ) , q I N , into the domain P are determined as
follows.
Containment the circle Сq (vq , rq ) into the domain P
Сq (vq , rq ) P int Cq (vq , rq ) I P* , q I N , (1)
where P* Ў 2 \ int P
Minimal allowed distance between the circles Сq (vq , rq ) and Сg (vg , rg )
dist(Сq (vq , rq ), С g (vg , rg )) , ( q, g ) l , l I p , 0 (2)
where
dist(Cq (vq , rq ), C g (vg , rg )) min (a, b) ,
aCq ( vq , rq ), bC g ( v g , rg )
(a, b) is Euclidean distance between points a, b R 2 ,
l {(q, g ) : Сq (vq , rq ) Pl , С g (vg , rg ) Pl , q g} .
A problem of bi-objective circle topology optimization into the polygonal domain is formulated as
follows.
Pack circles Сq (vq , rq ) , q I N , into the domain P providing containment l circles into the
polygon P l , l I p , lI l n N and the distance constraints (2) and maximizing the packing
p
factor while minimizing the maximum mechanical stress.
The packing conditions (1), (2) are analytically described using the phi-function technique [21],
which makes it possible presenting a mathematical model as the following bi-objective problem:
max{(), ()} (3)
subject to W ,
where (v, r ) is a vector of variables; v (v1, v2 ,..., vn ) is a vector of variable packing parameters;
r (r1, r2 ,..., rn ) is a vector of variable radii of circles; function
(4)
() rq2
qI n
is the total sum of the circle areas; () is an implicit function which defines the maximum
mechanical stress depending on the vector v of center coordinates of circles and the vector r of radii
of circles;
)
W = { R 3n : qg (vq , vg , rq , rg ) 0, ( q, g ) l , l I p , q (vq , rq ) 0, q I n , (5)
rq rq 0, q I n , rq rq 0, q I n }
is a feasible region;
)
qg (vq , v g , rq , rg ) ( xq xg ) 2 ( yq y g ) 2 ( rq rg ) 2 ,
is an adjusted phi-function of the circles Cq and C g , ( q, g ) l ;
q (vq , rq ) max{ min {ml (vq ) rq }}
lI p m 1,2,..., M l
is a phi-function of the circle Cq , q I n , and the set P* R 2 \ int P .
)
The inequality qg (vq , vg , rq , rg ) 0 ensures the condition (2) for packing the circles Cq and C g ,
( q, g ) l , on the minimal allowed distance and the inequality q (vq , rq ) 0 provides the
condition (1) for packing the circle Cq into the polygon P .
We note that () and () are compromising objectives. Therefore, our aim is to find a trade-
off solution of the bi-objective problem (3)-(5).
4. Solution approach
We use the multistart strategy [22] for solving the circular problem (3)-(5).
4.1. Solution strategy
On the assumption that an accepted value of the maximum mechanical stress is tolerable we solve
the problem by means of the -constraint method [7] which reduces the problem (3)-(5) to a single-
objective one
max () (6)
subject to W
W = { W ( ) 0} .
The main objective is maximizing sum of squares of circles to be packed. Continual review of
mechanical stress is a formidable problem. Of importance is the threshold value of mechanical
stress.
Let us temporarily relax the inequality
( ) 0 (7)
Then we can evaluate a posteriori the maximum mechanical stress for the obtained topology of the
n
domain P0 \ int Ui 1 Ci (* ) and verify the condition (7) with respect to the local maximum point * .
If the condition (7) holds true, i.e. (* ) 0 , then a feasible point of the problem (6)-(7) is
found, otherwise we calculate another local maximum point ** and repeat the procedure until (7)
will be met.
We decompose the problem (6)-(7) into l subproblems separately for each polygon P l , l I p .
Our multistart strategy involves the following main stages.
Stage 1. Define a lower bound l of the number l of circles which can be packed into the
appropriate polygon Pl , l I p with the maximum packing factor, using Algorithm 1 based on
packing equal circles [14]. Form a point ul* .
Stage 2. Generate feasible points for the problem (6) starting from the point ul* by use of
Algorithm 2 based on ACP (Apollonian Circle Packing) [8]. Form a point wl* .
Stage 3. Search for a local maximum point of the problem (6)-(7) starting from the point
0 ( w1* , w2* , ..., wn* p ) using IPOPT [23].
Stage 4. Choose the best local maximum point satisfying the inequality (7) found at Stage 3.
4.2. Solution algorithm
Let us consider our solution strategy in details.
4.2.1. Algorithm 1. Searching for a lower bound of the number of circles
Packing results depend heavily on the number of circles packed and starting points. We make use
the idea of ACP. According to ACP circles are packed repeatedly touching three other circles [8]. To
start ACP we first construct an initial configuration of tangent circles. To this end we realize a
preliminary packing of equal circles with radius r (if possible). The problem is known as IIPP
(Identical Item Packing Problem) [24] The circle layout algorithm is based on the models described in
[14].
l 1
We consider packing circles into the polygon P l , l I p . Let nl 1 j 1 j be packed into the
polygons Pj , j 1, 2,..., l 1. Now we pack the circle C q , q n 1 1 . By condition, rq is bounded
above by r , that can be, in general, less than the maximal possible radius rl , l I p , the circle
C q (vq , rl ) being inscribed into P l . In this case the center vq of the circle C q (vq , r ) is free to
move within a polygonal set Pˆ l P l (Figure 1) ensuring feasible locations of C q (vq , r ) in P l .
Pl r
Pˆ l
rl
Figure 1: Packing the first circle into P l
A random position of the circle center is chosen and then a step-by-step procedure of sequential
addition of next circles according to the algorithm [14, 25] is carried out. If a circle with radius r
cannot be packed into P l , we go to Stage 2 (ACP) starting from the circle with the maximal possible
radius rl* , l I p . If there are several possible locations for the circle, we choose a location for which
the circle is tangent to the maximal number of the polygon edges.
To this end we solve consequently the problems.
max rq k , k 0,1,2,..., kl* (8)
(k ) (k )
subject to ul Wl
where ul( k ) ( xq , yq , rq ) Wl ( k ) R 3 if k 0 and ul( k ) ( xq , yq ,..., xq k , yq k , rq k ) Wl ( k ) R 2 k 3 for
k 1,2,..., kl* ;
Wl ( k ) { ul( k ) R 2 k 3 : ml ( xq , yq ) rq 0, m 1, 2,..., M l , l I p , (9)
xq j 1 xq t 1 y q j1 y qt1 2r 0,
2 2 2
j , t 1, 2,..., k , j t , k {2,..., k *} ,
xq j 1 xq k yq j 1 yq k rq rq k d 0,
2 2 2
j 1,2,..., k , k {1,.2,.., kl*} }.
A local maximum of the problem (8)-(9) is calculated. If rq* k r , then we continue to solve
problem (8)-(9) for the next k , considering ul( k )* ( xq* , yq* ,..., xq* k 1 , yq* k 1 , xq k , yq k ,0) where
xq k , yq k are randomly chosen ( ul( k )* Wl ( k ) ) as a starting point. If rq* k r , we stop the iterative
process. The point
ul* (vl* , rl* ) ( xq* , yq* ,..., xq* k 1 , yq* k 1 , xq* k , yq* k , r144 r43, rq* k * )
,...,44
42 l
kl* 1
is the algorithm output. Then k kl* , the circle Cq k * touches at least three circles (or edges of P l )
l
and can be considered as a first circle packed according to ACP (Figure 2).
Pl
r
rq* k * Сq
l Сq k *
l
rq
r
r Сq 2 Сq 1
rq rq
Figure 2: Construction of a starting circle configuration for ACP
4.2.2. Algorithm 2. Generating starting feasible points
If rq* k is a strict local maximum of the problem (6)-(7), then the circles C q , Cq 1 ,..., Cq k * are
l
rigidly fixed forming a “jammed” packing [26]. We pack next circles Cq k * 1 , Cq k* 2 ,..., Cq l 1
l l
according to ACP until the current radius rq*k * k 1 where k ASP is the number of additional circles
l ASP
following ACP becomes less than r (Figure 3). A point
wl* ( xq* , yq* ,..., xq* l 1 , yq* l 1 , r144 r43, rq* k * ,..., rq*l 1 )
,...,44
42 l
kl* 1
is obtained. The total number of circles packed into P l is l kl* k ASP , the radii values being within
[r , r ] .
4.2.3. Local optimization
After having constructed the starting packing for each polygon P l , l I p , we have the total
number n lI l of circles to be packed into P .
p
We turn to problem (6)-(7), calculate a local maximum point and then construct a starting point
0
( w1* , w2* , ..., wn* p ) . To solve the nonlinear programming problem we make use of IPOPT solver [23]
together with the decomposition strategy [15].
Сq k * 1
Сq +k * +3 rq l
rq* k *
l
Сq
l
Pl
Сq 2
Сq 1
Сq k * 2 Сq +l -1
l
Сq k * k
l ACP
Figure 3: Circle arrangement according to ACP
Several local maximum points are computed and ordered in decreasing order of () (see
problem (6)-(7)) and then one can choose a point meeting (7). The first point in the ordering for which
the inequality (7) holds true is a trade-off solution of the problem (3)-(5). If there is no a point
satisfying the threshold value a (see (7)), new local maximum points are computed or a compromise
value a is selected. Calculation of mechanical stress values is not considered in this paper. We
refer to paper [10].
As computations show, local maximum points are close to or coincide with the starting points
constructed according to ACP.
5. Computational results
We consider the benchmark geometry presented in [10] and provide new benchmark examples for
packing circles with variable radii. The dependence of the number of circles packed and the packing
density on the minimal allowed distance between circles is studied. All experiments were running on
an Intel Core I5 750 computer. We use the Delphi programming language and the Windows 10
operation system. Software library IPOPT [23, 27] for nonlinear optimization problems exploiting
first and second derivative (Hessians) information is utilized.
Example 1. P 0 is given by 11 vertices v01 (0,0), v02 (0,9), v03 (18,9), v04 (5,28),
v05 (0,33), v05 (0,40), v06 (0,40), v07 (60,40), v08 (76,34), v09 (98,11), v0,10 (100,6),
v0,11 (100,0) . P UlI P l , where I P {1,2,...,5} , vertices of Pl , l I p , are v11 (22,31),
p
v12 (35,27), v13 (6,8); v21 (40,25), v22 (54,9), v23 (12,7); v31 (44,30),
v32 (68,12), v33 (59,8), v34 (42,28); v41 (42,36), v42 (63,36), v43 (89,30), v34 (73,13);
v51 (74,37), v52 (95,37), v53 (91,31), v54 (74,35). The minimal allowed distance between
circles is 0; r 0.5; r 5;
The total number of circles packed into P is n 91 ( 1 15, 2 23, 3 14, 4 31, 5 8 ).
Value of the objective in the starting point (ACP) is (0 ) 361.3287 and one in the local maximum
point is * 362.0946 . Illustration of the circles packed is shown in Figure 4. This example is
relevant to maximizing the packing factor and meaningless in relation to mechanical stress being
infinite at 0 .
Example 2. See Example 1. 0.5. The starting objective value is (0 ) 317.9666 and the
local maximum point is * 319.3224 . Illustration is shown in Figure 5. n 66
(1 11, 2 19, 3 10, 4 21, 5 5) .
P0
P0
P2
P3
P4
P1 P5
Figure 4: Circle arrangement according to Example 1
P0
P2
P3
P4
P1 P5
Figure 5: Circle arrangement according to Example 2
Example 3. See Example 1. 0.75. The starting objective value is (0 ) 301.0078 and the
local maximum point is * 301.7119 . Illustration is shown in Figure 6. n 50 (1 8, 2 12,
3 8, 4 17, 5 5) .
Example 4. See Example 1. 1. The starting objective value is (0 ) 288.9095 and the local
maximum point is * 289.7335 . Illustration is shown in Figure 7. n 47 (1 8, 2 12,
3 8, 4 14, 5 5) .
The runtime is within 30 seconds for all examples.
As computational results show, with an increase in the minimal allowed distance between circles,
the contribution of large circles to the total circle area increases, since it is proportional to the squares
of the circle radii. Obviously, this leads to a decrease in the number of circles with small radii and an
increase in the radii of other circles.
In this regard, the choice of is of importance. A compromise value is chosen: a too low value
can lead to poor quality printing due to the technical features of the process, while a too large one
causes additional material consumption. Furthermore, the secondary objective () can be
influenced by varying the values of , r and r .
P0
P2
P3
P4
P1 P5
Figure 6: Circle arrangement according to Example 3
P0
P2
P3
P4
P1 P5
Figure 7: Circle arrangement according to Example 4
6. Conclusions
Topology optimization is a key point in additive manufacturing. Circular-hole based layout models
help to cut down material expenses while meeting strength requirements.
We formulate non-standard packing problem of circles with variable metric characteristics. The
proposed bi-objective model takes into account both the maximum packing factor and mechanical
stress of parts. An efficient packing algorithm based on Apollonian circle packing, nonlinear
programming and the -constraint method has been developed. The approach allows estimating the
number of circular perforations needed and search for an approximate solution of the problem
maximizing the packing factor and following a threshold value of mechanical stress.
Further research is directed to layout of circular perforations inside a 3D part considering
balancing conditions [22, 28]. Therewith a more nuanced approach to multi-objective optimization
should be constructed.
7. Acknowledgements
The investigation is partially supported by the “Program for the State Priority Scientific Research
and Technological (Experimental) Development of the Department of Physical and Technical
Problems of Energy of the National Academy of Sciences of Ukraine” (#6541230) and National
Research Foundation of Ukraine (#02.2020/167).
8. References
[1] O. Sigmund, K. Maute, Struct topology optimization approaches, Structural and
Multidisciplinary Optimization 48 (2013) 1031–1055. doi:10.1007/s00158-013-0978-6.
[2] J. Liu, Y. Ma, A survey of manufacturing oriented topology optimization methods, Advances in
Engineering Softwar 100 (2016) 161–175. doi:10.1016/j.advengsoft.2016.07.017.
[3] A. Ramya, S. I. Vanapalli, 3D printing technologies in various applications, International Journal
of Mechanical Engineering and Technology 7 (2016) 396–409. Available from:
http://www.iaeme.com/ currentissue.asp?JType=IJMET&VType=7&IType=3.
[4] J. D. Deaton, R. V. Grandhi, A survey of structural and multidisciplinary continuum topology
optimization: post 2000, Structural and Multidisciplinary Optimization 49 (2014) 1–38.
doi:10.1007/s00158-013-0956-z.
[5] K. Miettinen, Nonlinear Multiobjective Optimization, Springer Science & Business Media, 2012.
[6] N. Gunantara, Q. Ai, A review of multi-objective optimization: Methods and its applications,
Cogent Engineering, 5:1 (2018). doi:10.1080/23311916.2018.1502242.
[7] Y. Y. Haimes, L. S. Lasdon, D. A. Wismer, On a bicriterion formulation of the problems of
integrated system identification and system optimization,. IEEE Trans. Syst. Man. Cybern. 1
(1971) 296–297. doi:10.1109/TSMC.1971.4308298.
[8] W. Chen, M. Jiao, C. Kessler, A. Malik, X. Zhang, Spatial Statistics of Apollonian Gaskets,
Experimental Mathematics 28 (2019) 263270. doi:10.1080/10586458.2017.1385037.
[9] F. J. Kampas, J. D. Pintér, I. Castillo, Packing ovals in optimized regular polygons, J Glob Optim
77 (2020) 175–196. doi:10.1007/s10898-019-00824-8.
[10] S. P. Fekete, S. Morr, C. Scheffer, Split Packing: Algorithms for Packing Circles with Optimal
Worst-Case Density, Discrete Comput Geom 61 (2019) 562–594. doi:10.1007/s00454-018-0020-
2.
[11] Y. Stoyan, G. Yaskov, T. Romanova, I. Litvinchev, S. Yakovlev, J. M. Velarde Cantú,
Optimized packing multidimensional hyperspheres: a unified approach, Math Biosci Eng. 17
(2020) 6601–6630. doi:10.3934/mbe.2020344.
[12] S. Yakovlev, O. Kartashov, K. Korobchynskyi, B. Skripka, Numerical Results of Variable Radii
Method in the Unequal Circles Packing Problem, in: Proceedings of 2019 IEEE 15th
International Conference on the Experience of Designing and Application of CAD Systems
(CADSM), Polyana, Ukraine, 2019, pp. 1–4. doi:10.1109/CADSM.2019.8779288.
[13] Y. Stoyan, G. Yaskov, Packing equal circles into a circle with circular prohibited areas,
International Journal of Computer Mathematics 89 (2012) 1355–1369. doi:10.1080/00207160.
2012.685468.
[14] S. Yakovlev, The Expanding Space Method in Sphere Packing Problem, in: S. Babichev,
V. Lytvynenko, W. Wójcik, S. Vyshemyrskaya (Eds.), Lecture Notes in Computational
Intelligence and Decision Making, ISDMCI 2020, Advances in Intelligent Systems and
Computing, volume 1246, Springer, Cham, 2021, pp. 151–163. doi:10.1007/978-3-030-54215-
3_10.
[15] E. Specht, www.packomania.com, 2020. Available from: http://packomania.com.
[16] I. Yanchevskyi, R. Lachmayer, I. Mozgova, R-B. Lippert, G. Yaskov, T. Romanova,
I. Litvinchev, Circular packing for support-free structures, EAI Endorsed Transactions 7 (2020).
doi:10.4108/eai.13-7-2018.164561.
[17] R. A. Alaimo, Overlap Packing Optimization for Spacecraft Layout Design. M.S. thesis. The
University of North Carolina, Charlotte, NC, 2018.
[18] J Lim, C. You, I. Dayyani, Multi-objective topology optimization and structural analysis of
periodic spaceframe structures, Materials & Design 190 (2020) 108552. doi:10.1016/
j.matdes.2020.108552.
[19] T. Romanova, Y. Stoyan, A. Pankratov, I. Litvinchev, I. Yanchevsky, I. Mozgova, Optimal
Packing in Additive Manufacturing, IFAC-PapersOnLine 52 (2019) 2758–2763.
doi:10.1016/j.ifacol.2019.11.625.
[20] E. M. Kiseleva, Y. E. Kadochnikova, Solving a Continuous Single-product Problem of Optimal
Partitioning with Additional Conditions, Journal of Automation and Information Sciences 41
(2009) 48–63. doi: 10.1615/JAutomatInfScien.v41.i7.30.
[21] N. Chernov, Y. Stoyan, T. Romanova, Mathematical model and efficient algorithms for object
packing problem, Computational Geometry: Theory and Applications 43 (2014) 535–553.
doi:10.1016/j.comgeo.2009.12.003.
[22] P. I. Stetsyuk, T. E. Romanova, G. Scheithauer, On the global minimum in a balanced circular
packing problem, Optim Lett 10 (2016) 1347–1360. doi:10.1007/s11590-015-0937-9.
[23] A. Wächter, L. T. Biegler, On the implementation of a primal-dual interior point filter line search
algorithm for large-scale nonlinear programming, Math. Program. 106 (2006) 25–57.
doi:10.1007/s10107-004-0559-y.
[24] G. Waescher, H. Haussner, An improved typology of cutting and packing problems. European
Journal of Operational Research 183 (2007) 1109–1130. doi:10.1016/j.ejor.2005.12.047.
[25] G. Yaskov, A. Chugay, Packing equal spheres by means of the block coordinate descent method,
CMIS (2020) 156–168.
[26] S. Torquato, A. Donev, F. H. Stillinger, Breakdown of elasticity theory for jammed hard-particle
packings: conical nonlinear constitutive theory, International Journal of Solids and Structures 40
(2003) 7143–7153. doi:10.1016/S0020-7683(03)00359-7.
[27] http://www.coin-or.org/download/source/Ipopt.
[28] I. V. Grebennik, A. A. Kovalenko, T. E. Romanova, I. A. Urniaieva, S. B. Shekhovtsov,
Combinatorial Configurations in Balance Layout Optimization Problems, Cybern Syst Anal 54
(2018) 221–231. doi:10.1007/s10559-018-0023-2.