The authors have developed an approach to modeling the layer-by-layer filling of the particular 3D volume with a combination of non-spherical and spherical powder particles of different fractional compositions. They have used the phi-function technique and constructed the mathematical model of the problem of packing regular and irregular freely moving objects. They also proposed the heuristic algorithm that uses nonlinear optimization for calculating packing density/porosity factor. The authors compared the results of numerical modeling with experimental data obtained for the mixture of spherical and polyhedral powders of titanium alloys. They have established that the percentage ratio of powder particles within the investigated fraction, obtained using the developed algorithm, corresponds with high accuracy to the experimental results. This finding indicates the possibility of using numerical modeling results instead of costly experimental studies. The use of mathematical modeling and optimization techniques in additive manufacturing makes it possible to improve the efficiency of each stage of the technological process, reduce the number of defective products, and rationally plan the consumption of energy and material resources.

The authors have constructed a digital model of reservoir rock cores based on the problem of packing spherical particles in a cylindrical container. They proposed a new approach to mathematical modeling of the reservoir rock structure and calculating its porosity. A mathematical model of the problem of placing the maximum number of spheres with different diameters in a cylindrical container is presented. A solution algorithm is developed based on the optimization by groups of variables and the lattice decomposition strategy. The results of experimental petrophysical studies of real well cores are used as input data. The modeling results provide a good approximation of the absolute porosity of natural prototype. Applying this approach will help improve hydrocarbon extraction technologies and increase their efficiency.

Packing soft convex polygons in an optimized convex container is considered. It is assumed that the soft polygonal object can change its shape in certain limits, while its area remains constant. Non-overlapping, containment, and area conservation constraints are formulated for soft polygonal objects, and a corresponding nonlinear programming model is presented. Numerical experiments for packing soft triangles and pentagons in optimized circular and quadratic containers are presented to demonstrate efficiency of the proposed approach.

Smart systems and data-driven services have the potential to answer medical needs in nanomedicine to get faster to the clinic and manufacturing stage. As an example from galenics, we explored the molecular packing within nanoparticles composed of fullerene C60 (C60) and the anthracycline antibiotic doxorubicin (Dox); the nanoparticle was previously proven to be a promising candidate for photodynamic and chemotherapeutic treatment of cancer. The Dox-C60 hybrid was evaluated to be around 135 nm and forms an aqueous monodisperse colloid solution. We consider a non-standard packing problem for the geometric design of the Dox-C60 nanocomplex. Each placement object was a disconnected set with a core sphere and two identical spherical components that were allowed moving only along the core sphere orbit at the given distance. Allowable distances between each pair of components, as well as between objects, were given. All disconnected objects could be freely moving within the given volume (cuboid). The packing problem was aimed to maximize the number of disconnected objects that could be fully arranged inside the given volume, considering distance constraints. The problem was formulated as MIP (mixed integer problem). A solution strategy was proposed that combines multistart strategy, nonlinear optimization, and decomposition algorithm. Computational results for 2D and 3D objects were provided. Our findings may contribute to solve molecular packaging problems which play a role in synthesis, upscaling, production as well as formulation, and medication of the complexed drug.

A sparse layout problem for clusters formed by irregular 3D objects is introduced. The shape of a 3D cluster is represented as a convex hull of the objects inside the cluster. The objects in the cluster may have different irregular shapes, can be freely translated and rotated and must be placed in the cluster without mutual overlapping. Each irregular 3D object in the cluster is composed by a union of basic convex 3D objects. The clusters must be placed in a container without mutual overlapping. The objective is to maximize the distance between the 3D clusters. New geometric tools to describe analytically nonoverlapping, containment and distance constraints for 3D clusters are introduced. The sparse layout problem is formulated as a nonlinear nonconvex continuous programming problem. A solution algorithm is proposed, and computational results are provided.

An integrated intelligent approach for solving geometric design problems is studied. A general optimization placement problem of objects with arbitrary shapes in a bounded container is constructed as a mathematical programming problem in terms of the phi-function technique. Various technological requirements (geometric and mechanical) are considered, including continuous translations and rotations of the objects, allowable distances between objects, prohibited zones in the container, balancing conditions, and mechanical strength constraints. Grouping the optimization placement problems based on the typology of the geometric design problems is provided. Solution strategies and various approaches to solve different variants of the model are discussed. A methodology of solving optimization placement problems is developed and illustrated with different examples.

The authors consider a problem of packing circles of given types in a circular container. Circles are allowed to cross the container boundary in a predefined neighborhood that depends on the circle type (pseudo-inclusion condition). A family of circles is placed in the container under the conditions of their non-intersection, pseudo-inclusion, and compliance with the given proportions of circle types (proportionality condition) in order to maximize the total number of circles. A mathematical model is constructed as a problem of mixed integer nonlinear programming. A heuristic algorithm is proposed that applies a nonlinear programming problem to pack a given number of circles in a circular container, maximizing the variable radii of the circles. The results of the computational experiments are given.

The problem of packing a given set of freely translated and rotated convex polygons in a minimum-perimeter convex polygon (in particular the minimum-perimeter convex hull) is introduced. A mathematical model of the problem using the phi-function technique is provided. Problem instances with up to 6 convex polygons are solved by the global NLP solver BARON to get a minimum-perimeter convex hull. Numerical experiments for larger instances are reported using the local NLP solver IPOPT.

The problem of partial lattice coverage of a cuboid of given dimensions with a minimum number of identical hemispheres with a given coverage factor is considered. A mathematical model in the form of a mixed integer nonlinear programming problem is constructed. A solution algorithm is proposed. The problem of three-dimensional coverage is reduced to the problem of covering a rectangular area by a family of identical circles of radius that depends on the height of the cuboid, the radius of the hemispheres, and the distance between the centers of neighboring hemispheres. The results of computational experiments for the problem of optimizing the placement of sensors in a given three-dimensional domain are provided.

Using the biomicroscopy method, we studied the reaction of arterial and venous vessels of the broad ligament of the uterus in outbred female rats to irradiation with helium-neon laser (λ=632.8 nm; power output 2 mW). Small arteries were found to be most sensitive to laser irradiation. The veins of the broad ligament of the uterus demonstrated lower reactivity to laser irradiation of the same duration than arterial vessels, which can be explained by morphological, functional, and hemodynamic differences.

The article considers the problem of optimizing the topology of products in additive manufacturing due to the optimal placement of circular holes. The task is to pack several circles of variable radii, set within the limits set by 3D printing standards. A two-criteria formulation is proposed, which takes into account the packing factor and the maximum mechanical stress of the products. The method of the main criterion is used to find a compromise solution to the problem. A new approach has been developed, which is based on the modified method of Apollonian packing of circles and nonlinear optimization. Numerical examples and graphical illustration of the results are given.

One of the most important challenges in modeling structures of materials is development and application of new intelligent technologies. To study mechanical properties, e.g., density, a small cuboidal core of the material is extracted from a large volume for further analyses. Due to the cutting edges, material particles may not belong completely to the core volume. This gives rice to a new class of packing problems where the standard containment conditions (all particles are entirely in the container) are substituted by a relaxed containment (all centers of the particles are in the container). A 2D version of this non-standard problem is presented and formulated as a nonlinear programming problem considering non-overlapping and relaxed containment constraints. A new solution technique is proposed combining a fast algorithm for generating feasible starting points and a local optimization procedure based on nonlinear programming. Computational results are provided and illustrated with several examples.

Optimized layout of variable-sized spheres in a disconnected polyhedral domain is considered. The problem is motivated by optimized design of void structures in additive manufacturing. The spheres must be arranged in the container; however, a certain protruding is permitted subject to the corresponding center inside the container. The distance between the objects must be at least a certain given threshold. The objective is to find coordinates of the centers and radii of the spheres maximizing the total volume of the spheres for two cases: with and without balancing conditions. Two nonlinear programming models are provided. Corresponding nonlinear optimization problem is formulated and solved. Numerical results are presented to illustrate the main constructions.

In modern additive manufacturing technologies, parts are printed in a special container named the printer working chamber. The efficiency of volume filling with geometric objects is an important factor in the overall efficiency of the manufacturing process. To improve the efficiency of the production process, it is necessary to develop effective methods for packing objects in the working chamber of a 3D printer. The problem of filling the working chamber with parts is formulated as a three-dimensional irregular packing problem and presented as a nonlinear optimization problem. The article presents an origin optimization approach for tackling irregular packing problems for additive manufacturing. The approach involves construction of an exact mathematical model of irregular packing problem (using the phi-function technique) and development of a nonlinear optimization method based on state-of-arts solvers. The solution strategy uses the preliminary clustering of bodies to be packed and variable metric characteristics of bodies and a chamber. An exact mathematical model of the problem is constructed and a solution method is developed. Test results are shown.

Introduction. Optimization placement problems are NP-hard. In most cases related to cutting and packing problems, heuristic approaches are used. The development of analytical methods for mathematical modeling of the problems is of paramount important for expanding the class of placement problems that can be solved optimally using state of the art NLP-solvers. The problem of placing two irregular two-dimensional objects in a convex polygonal region of the minimum size, which is a convex polygonal hull of the minimum area or perimeter, is considered. Continuous rotations and translations of non-overlapping objects are allowed. To solve the problem of optimal compaction of a pair of objects, two algorithms are proposed. The first is a sequentially search for local extrema on all feasible subdomains using a solution tree. The second algorithm searches for a locally optimal extremum on a single subdomain using a "good" feasible starting point. Purpose of the paper. Show how to construct a minimal convex polygonal hull for two continuously moving irregular objects bounded by circular arcs and line segments. Results. A mathematical model is constructed in the form of a nonlinear programming problem using the phi-function technique. Two algorithms are proposed for solving the problem of placing a pair of objects in order to minimize the area and perimeter of the enclosing polygonal area. The results of computational experiments are presented. Conclusions. The construction of a minimal convex polygonal hull for a pair of two-dimensional objects having an arbitrary spatial shape and allowing continuous rotations and translations makes it possible to speed up the process of finding feasible solutions for the problem of placing a large number of objects with complex geometry. Keywords: convex polygonal hull, irregular objects, phi-function technique, nonlinear optimization.