## Volume 24, # 3

### Mathematics

1. Maksim I. Petrushko On the basics of common theory for coordination polyhedron composition (polyhedron fundamental equations)
Abstract. We study the features of the polyhedron geometric structure related to their belonging to different symmetry groups. We derive basic equations for the vertices, faces, and edges of the polyhedron depending on their location on the axial, planar, and primitive orbits.
Keywords: polyhedron, coordination polyhedron, fundamental cells, symmetry axes, symmetry planes, equations.

2. Kazbek U. Khubiev On a non-local problem for mixed hyperbolic-parabolic equations
Abstract. A non-local boundary value problem with inner boundary conditions in its hyperbolic part is studied for loaded mixed hyperbolic-parabolic equations.
Keywords: mixed-type equation, hyperbolic-parabolic equation, loaded equation, Tricomi problem, nonlocal problem.

### Mathematical modeling

3. Valentin N. Alekseev, Maria V. Vasilyeva, Georgiy A. Prokopiev, Aleksei A. Tyrylgin Models of thermoelasticity for porous materials with fractures taken into account
Abstract. We propose a mathematical model and a computational algorithm for solving thermoelasticity problems in fractured porous media. To simulate heat transfer in porous media, a mathematical model is constructed using double diffusion models. The heat transfer in the cracks is taken into account by setting the interface condition which allows modeling the temperature jump at the crack boundary. To calculate the stress-strain state, a linear elasticity model is used with an additional condition on the crack. For the numerical solution of the problem an approximation is constructed using the Galerkin discontinuous method which allows taking into account the interface condition in the variational formulation. We present the results of the numerical realization of the model problem using the proposed model of thermoelasticity.
Keywords: thermoelasticity, fracture, porous medium, fuel element, double diffusion model, interface condition, discontinuous Galerkin method, numerical simulation.

4. Vasily I. Vasil’ev, Anatoly M. Kardashevsky, Vasily V. Popov Iterative method for the Dirichlet problem and its modifications
Abstract. A series of works of S. I. Kabanikhin’s scientific school are devoted to study of the existence, uniqueness, and numerical methods for the inverse Dirichlet problem for the second-order hyperbolic equations. We consider a numerical solution to the non-classical Dirichlet problem and its modifications for the two-dimensional hyperbolic second-order equations. The method of iterative refinement of the missing initial condition is applied by means of an additional condition specified at the final time. Moreover, the direct problem is numerically realized at each iteration. The efficiency of
the proposed computational algorithm is confirmed by calculations for two-dimensional model problems, including additional conditions with random errors.
Keywords: hyperbolic equation, inverse problem, Dirichlet problem, finite difference method, iterative method, conjugate gradients method, random error.

5. Maria V. Vasilyeva, Petr E. Zakharov, Petr V. Sivtsev, and Denis A. Spiridonov Numerical modeling of thermoelasticity problems for constructions with inner heat source
Abstract. We consider the numerical simulation of the thermomechanical state of a structure consisting of a heat source, a gas gap and a shell. The mathematical model is described by a nonlinear system of equations for temperature and displacements. The heat is released in the subdomain of the heat source. The resulting displacements due to the temperature gradient are calculated in the heat source region and separately in the shell, and can be described by both linear elasticity models and nonlinear plasticity models. The numerical implementation is based on the finite element method. The results of numerical modeling of a nonlinear model problem in two- and three-dimensional domains are presented.
Keywords: thermoelasticity problems, thermal expansion, heat transfer, linear elasticity problem, plasticity models, nonlinear problems, finite element method, mathematical modeling.

6. Uygulaana S. Gavrilieva, Valentin N. Alekseev, Maria V. Vasilyeva Flow and transport in perforated and fractured domains with Robin boundary conditions
Abstract. We consider transport and ﬂow problems in perforated and fractured domains. The system of equations is described by the Stokes equation for modeling ﬂuid ﬂow and the equation for the concentration transfer of a certain substance. Concentration is supplemented by inhomogeneous boundary conditions of the third type which simulate the occurring reaction on the faces of the modeled object. For the numerical solution of the problem, a finite-element approximation of the equation is constructed. To obtain a sustainable solution to the transport problem, the SUPG (streamline upwind Petrov–Galerkin) method is used to stabilize the classical Galerkin method. The computational implementation is based on the Fenics computational library. The results of solving the model problem in perforated and fractured domains are presented. Numerical studies of various regimes of heterogeneous boundary conditions were carried out.
Keywords: transport equation, Stokes problem, perforated domain, fractured domain, numerical modeling, Robin boundary condition, numerical stabilization, SUPG, finite element method.

7. Vasiliy V. Grigoriev, Petr E. Zakharov, Alexey S. Kondakov, Irina G. Larionova Calculation of conditions for joint laying of overground pipelines
Abstract. Long-term experience in the operation of water supply systems in permafrost conditions shows that the most rational method is the above-ground joint laying of pipelines for various purposes. In the Far North regions, with low winter and high summer temperatures and great difference between day and night temperatures, it is a problem to establish conditions for the joint laying of cold and hot water pipes with heat supply pipes in one bundle. The problem is that because in the cold water pipe the water is under pressure and moves only when it is used, so it can freeze at night and thus destroy the pipe. A numerical solution of the two-dimensional temperature problem by the finite element method is considered to determine the optimum intertube distance for the joint laying of cold water supply pipes, heat supply and hot water supply ensuring the permissible water temperature in the cold water supply pipe. In conclusion, optimal intertube distances are obtained for two different schemes of laying pipes so that cold water did not freeze in winter and did not overheat in summer.
Keywords: polypropylene tubes, temperature insulation, mathematical modeling, tube bundle, intertubular clearance.

8. Dulus Kh. Ivanov Numerical recovering of leading coefficient of nonlinear parabolic equation
Abstract. We numerically recover the leading coefficient of one parabolic equation in a multidimensional region. We consider the case when the leading coefficient depends only on solution itself and observations are taken in some interior points of the region as an additional condition. finite element method implemented by FEniCS library is used for numerical solution of the problem. Several examples of identification of the leading coefficient of a two-dimensional parabolic equation are given.
Keywords: inverse coefficient problem, parabolic equation, finite element method, FEniCS.

9. Denis A. Spiridonov, Maria V. Vasilyeva Simulation of filtration problems in fractured porous media with mixed finite element method (Embedded Fracture Model)
Abstract. We present a mathematical model of mixed dimension for modeling filtration problems in fractured porous media (built-in model of cracks). The mathematical model is described by a system of parabolic equations: d-dimensional for a porous medium and (d − 1)-dimensional for a system of cracks. The system of equations is connected by specifying a special ﬂow function. This model allows the use of grids for a matrix of a porous medium not dependent on the grid for cracks. For the numerical solution, an approximation using the mixed finite element method is constructed. The results of the numerical solution of the model problem that show the operability of the proposed method for modeling the ﬂow in fractured porous media are presented.
Keywords: finite element method, built-in model of cracks, fractured medium, single-phase liquid, liquid ﬂow, law of conservation of mass, Darcy law.