Abstract Sintering is a complex physical and mechanical process, which is one of the most important technological processes in powder metallurgy and ceramic industry. In this paper, the study of free sintering of a ceramic object is carried out using the rheological theory of sintering. Numerical solution of the problem in the three-dimensional case is implemented using the Finite Element Method with the freely distributed FreeFem++ software. Experiments on sintering of aluminium oxide ceramic paste were conducted for several temperature regimes. The validation of the realized model is confirmed by comparing the numerical and experimental data of porosity evolution.

Abstract The paper describes a new numerical method for solving the equations of ideal magnetohydrodynamics (MHD) based on the Godunov method, a combination of the Roe and Rusanov schemes, and a piecewise parabolic representation of the solution. The hybrid scheme for solving the Riemann problem is associated with the possibility to reproduce the numerical solution without singularities along the directions, which is especially important when the velocity and magnetic field components are reconstructed in the transverse direction. The numerical method is implemented as a software package for massively parallel supercomputers. Studies of parallel implementation and computational experiments were carried out on the NKS-1P cluster of the SSCC. A problem with an analytical solution was used as a test for the method verification. A numerical solution of the problem of the interaction of a molecular hydrogen cloud with the oncoming interstellar medium is considered.

Abstract For block-linear dynamical systems of dimensions 3 and 4 considered as models of simplest circular gene networks, we find sufficient conditions for the existence of invariant surfaces in their phase portraits. These surfaces contain periodic trajectories of the dynamical systems.

Abstract A hereditary class is a set of simple graphs closed under deletion of vertices; every such class is defined by the set of its minimal forbidden induced subgraphs. If this set is finite, then the class is said to be finitely defined. The concept of a boundary class is a useful tool for the analysis of the computational complexity of graph problems in the family of finitely defined classes. The dominating set problem for a given graph is to determine whether it has a subset of vertices of a given size such that every vertex outside the subset has at least one neighbor in the subset. Previously, exactly four boundary classes were known for this problem (if \( \mathbb {P}\neq \mathbb {NP} \)). The present paper considers a countable set of concrete classes of graphs and proves that each its element is a boundary class for the dominating set problem (if \( \mathbb {P}\neq \mathbb {NP} \)). We also prove the \( \mathbb {NP} \)-completeness of this problem for graphs that contain neither an induced 6-path nor an induced 4-clique, which means that the set of known boundary classes for the dominating set problem is not complete (if \( \mathbb {P}\neq \mathbb {NP} \)).

Abstract The paper presents one construction of a Godunov-type method based on the separation of operators describing the work of pressure forces and advective transfer. Separate consideration of advective transfer permits describing the motion of both gas and dust components within the framework of a single numerical scheme. In the case of describing gas dynamics, the work of pressure forces is taken into account at a separate stage, independently of transfer. This permits using the numerical scheme in solving star formation problems, where it is necessary to jointly solve the equations of hydrodynamics and dust motion equations. A piecewise parabolic representation of physical variables in all directions is used to reduce the dissipation of the numerical method. The numerical method has been verified on Riemann problems for a hydrodynamic and dust discontinuity, the Sedov problem of point explosion, and the problem of dust cloud collapse, which have an analytical solution.

Abstract We consider a retrospective inverse heat conduction problem with nonstationary inhomogeneous Dirichlet boundary conditions. It is approximated by a Crank–Nicolson scheme that has the second order of approximation both in the spatial variable and in time. It is proposed to use the iterative method of conjugate gradients to determine the solution of the resulting system of linear algebraic equations. Examples are given of reconstructing smooth, nonsmooth, and discontinuous initial conditions, including the introduction of a “noise” characteristic, typical of additional conditions of inverse problems, and its smoothing using the Savitzky–Golay filter.

Abstract The results of computer simulation of the process of collision of rotating molecular clouds in the interstellar medium are presented. As matter is compressed, the gas density in the area of their collision increases; this leads to local changes in the shape and fragmentation of clouds. The gas density in the resulting condensations increases by many orders of magnitude, and gravitationally bound domains appear where star clusters can form. The process of star formation is accompanied by considerable spatial and temporal changes in the interstellar gas in these domains, turbulence of the interstellar medium, gravity, and a sharp change in magnetic and radiation fields at the prestellar stage of the evolution of new formations. The rotation of colliding molecular clouds has a great influence on the ongoing processes. The evolution of the matter of protostellar regions from the moment when they begin to form until the moment when they reach stellar density covers a huge range of scales. Simulation of such astrophysical processes on ultra-high resolution computational grids requires a substantial increase in computer power, and optimization of parallel computing on heterogeneous computing systems is required.

Abstract We obtain effective viscoelasticity coefficients for a heterogeneous medium based on a generalized derivative formalism that reflects internal boundaries in the heterogeneous medium. A solution is sought for the homogenized Green’s function of the resulting modified operator taking into account the homogenization and subsequent analysis of the operator. Based on the obtained solution of the many-body problem in a heterogeneous medium, the effective viscoelasticity coefficients integrally take into account the microstructure of the system (physical properties and characteristic phase sizes) in explicit form.

Abstract We introduce generalizations of the curl operator acting on three-dimensional symmetric \( m \)-tensor fields and establish their properties. For the spaces of three-dimensional tensor fields, we obtain new detailed decompositions in which each term is constructed using one function. Decompositions of this kind play a special role, in particular, when studying tomographic integral operators acting on symmetric \( m \)-tensor fields, \( m\geqslant 1 \), and constructing algorithms for solving the arising inverse problems.

Abstract The paper considers the contact bending problem for a multilayer composite plate. Each layer of the composite is a material reinforced with thin parallel fibers. The mathematical model is constructed based on the assumptions of the existence of a neutral surface in the plate and the validity of Kirchhoff’s hypotheses. Using the Lagrange variational principle, we obtain a bending equation generalizing the Sophie Germain equation. An elastic energy functional taking into account the different resistance of the material to tension and compression is obtained. The contact problem of plate bending with the aid of a rigid die is considered. To solve the contact problem of plate bending by a rigid die, a Lagrangian with an inequality constraint is constructed. The finite element method using a triangular Bell element is applied for the numerical solution of the problem. The results of calculations for the bending of laminated rectangular plates with various directions of fiber laying and various die shapes are presented.

Abstract A class of objects called \( \Pi \)-partitions is defined. In a certain well-defined sense, these objects are the equivalents of formulas in a basis consisting of disjunction, conjunction, and negation in which negations are possible only over variables (normalized formulas). \( \Pi \)-partitions are viewed as representations of formulas, just as \( \Pi \)-schemes can be viewed as equivalents and graphical representations of the same formulas. Some theory of such representations is developed, which is essentially a mathematical apparatus focused on describing a class of minimal normalized formulas implementing linear Boolean functions. REMOVE— \( \Pi \)-scheme, \( \Pi \)-partition—REMOVE

Abstract We prove the uniqueness of a solution of boundary value problems for the static equations of elasticity theory for Cauchy elastic materials with a nonsymmetric (or symmetric but not necessarily positive definite) matrix of elastic moduli. Using eigenstates (eigenbases), we write the linear stress-strain relation in invariant form. There are various ways of writing the constitutive relations, including those using symmetric matrices. The specific strain energy for all cases is represented canonically as a positive definite quadratic form.

Abstract We consider a solution of the problem of constructing a series of families of circulant networks of degree six specified analytically with the help of two parameters one of which is the network diameter. Based on the analysis and generalization of the properties of a new description of an extremal family of circulants, a general series of families of circulant graphs of degree six of arbitrary diameters is constructed that includes extremal circulant graphs of degree six and new infinite families of circulants with an even number of vertices. In the found series of families, descriptions of a series of circulant graphs of any given diameter are analytically determined. Optimality ranges of series graphs are algorithmically identified, where ‘optimal’ is understood as a circulant graph of degree six with the minimum possible diameter for a given number of vertices. The resulting series of families of circulant networks is promising as a scalable topology model for networks on a chip.

Abstract A mathematical model is presented that describes the dynamics of the population of CD4 \( ^{+} \) T-lymphocytes in a single lymph node. The model is based on a high-dimensional system of nonlinear delay differential equations supplemented with initial data. The equations of the model are given, and their well-posedness is investigated. The results of computational experiments with the model illustrating the characteristic cell population dynamics under the conditions of antigen-specific stimulation of the cell reproduction process for cells of various types are presented.

Abstract We derive an asymptotic formula for the number of labeled connected series-parallel \( k \)-cyclic graphs of a large order with a fixed number \( k \). With a uniform probability distribution, we find the probability that a random labeled connected \( n \)-vertex \( k \)-cyclic graph with a fixed \( k \) as \( n\to \infty \) is a series-parallel graph. In addition, we determine the probability that, with a uniform probability distribution, a random labeled connected series-parallel \( n \)-vertex \( k \)-cyclic graph with a fixed \( k \) as \( n\to \infty \) is a cactus.

Abstract Metaheuristic algorithms for a global optimization problem have unbound strategy parameters that affect solution accuracy and algorithm efficiency. The task of determining the optimal values of unbound parameters is called the parameter setting problem and can be solved by static parameter setting methods (performed before the algorithm run) and dynamic parameter control methods (performed during the run). The paper introduces a novel hierarchical parameter setting method for the class of population-based metaheuristic optimization algorithms. A distinctive feature of this method is the use of a hierarchical algorithm model. The lower level represents a sequential algorithm from this class, and the upper level represents an algorithm with the parallel island model. Parameter setting is performed by the hierarchical method, which combines parameter tuning for the sequential algorithm and adaptive parameter control for the parallel algorithm. Parameter control is based on vector fitness criteria which consist of a convergence rate and a solution value. An approach for estimating the convergence rate for a multistep optimization method is proposed. Experimental results for CEC benchmark problems are presented and discussed.

Abstract The process of changing the wave pattern in the problem of nonstationary deformation of a heteromodular elastic bar under uniaxial piecewise linear tension and subsequent compression is described as a connected sequence of local solutions on successive time intervals. All possible variants and results of collisions of primary and secondary strong discontinuities are indicated for a given piecewise linear function of boundary displacements. An efficient algorithm for solving one-dimensional boundary value problems of the dynamics of deformation of a heteromodular elastic medium under a piecewise linear boundary condition is proposed. The algorithm is based on finding a path in a local decision tree.

Abstract The paper presents the equations of the linear moment theory of elasticity for the case of arbitrary anisotropy of material tensors of the fourth rank. Symmetric and skew-symmetric components are distinguished in the defining relations. Some simplified versions of linear defining relations are considered. The possibility of Cauchy elasticity is allowed when material tensors of the fourth rank do not have the main symmetry. For material tensors that determine force and couple stresses, we introduce eigenmoduli and eigenstates that are invariant characteristics of an elastic moment medium. For the case of plane deformation and constrained rotation, an example of a complete solution of the two-dimensional problem is given when there are only shear stresses. The solutions turn out to be significantly different for anisotropic and isotropic elastic media.

Abstract A triode is a tree with three leaves and a single vertex of degree \( 3 \). The independent set problem is solvable in polynomial time for graphs that do not contain a triode as a subgraph with any fixed number of vertices. If the induced triode with \( k \) vertices is forbidden, then for \( k>5 \) the complexity of this problem is unknown. We consider intermediate cases where an induced triode with any fixed number of vertices and some of its spanning supergraphs are forbidden. For an arbitrary triode with a fixed vertex number, we prove the solvability of the independent set problem in polynomial time in the following cases: 1. A triode and all its spanning supergraphs with bounded vertex degrees are forbidden. 2. A triode and all its spanning supergraphs having large deficiency (the number of edges in the complementary graph) are forbidden. 3. A triode and all its supergraphs from which this triode can be obtained using the graph intersection operation are forbidden, provided the graph has a vertex of bounded antidegree.

Abstract The cavitation flow in a channel that prototypes a control valve cage is studied. The average velocity, pressure, and vapor volume fraction fields obtained by the RANS method by means of the OpenFOAM open source CFD software are in good agreement with the data obtained with the Ansys Fluent proprietary CFD solver. A computer code is implemented permitting one to obtain a large number of configurations of the control valve cage geometry, for which RANS calculations are performed so as to compile a comprehensive database.