Alena V. Favorskaya

Seismic resistance of nuclear power plants is an important subject of research, since reactor damage can lead to significant environmental pollution, extensive destruction and casualties among the population. Since seismic resistance experiments using shaker tables are expensive and cannot fully capture the complex velocity model of surrounding geological rocks and the detailed design of a nuclear power plant, computer simulation is widely used. This presents researchers with the challenge of developing new numerical methods for calculating seismic wave propagation within a nuclear power plant, depending on its design. We apply grid-characteristic method on Cartesian and curvilinear computational meshes to assess the seismic resistance of nuclear power plants in this manuscript. The proposed method allows for arbitrary definition of the geometry of the object under consideration, varying the degree of structural reinforcement, and the type of seismic load (wave type, amplitude, frequency). We conducted a comparative analysis of three reinforcement variations for the concrete structure of a nuclear power plant to demonstrate the capabilities of the proposed numerical method. The obtained results can be used to create a digital twin of a nuclear power plant during design.

In the tasks of non-destructive testing of rails, a combination of physical and numerical modeling methods is used. Finite element approaches have become widespread as numerical methods, for which the profile and the computational domain can be created by different software. However, in the case of using finite difference methods, such as grid- characteristic methods, it is necessary to create other approaches to account for the complex geometry of the rails. The grid-characteristic method on overset Chimera meshes involves interpolation between a background Cartesian mesh and a curvilinear structured overset mesh. During calculations on the curvilinear mesh, a coordinate transformation is used, modifying it to a Cartesian mesh with unit cells. The time step acceptable for stability conditions depends on the specifics of this transformation. Constructing the overset Chimera curvilinear structured mesh along a rail boundary is a difficult issue due to the complex shape of the rail boundary. Methods for constructing a mesh using perpendicular to the surface prove ineffective. Ensuring a minimal reduction in the permissible time step is difficult. This manuscript proposes a method for constructing the overset mesh along the boundary of an arbitrary shaped rail based on analytical calculations. The obtained approach will be useful for reducing the time required to create the computational mesh for finite difference methods in the case of approximating various rail profiles, which will accelerate the numerical simulation using the grid-characteristic method in the considered class of problems.

A modified grid-characteristic method for modeling elastic wave propagation in heterogeneous media with an explicit description of material interfaces is presented. The approach is based on implicit and explicit-implicit numerical schemes that ensure stability and accuracy at large time steps and small spatial steps in elongated domains. Contact conditions between the materials are properly formulated in the form of reflected and refracted waves or by modifying the system of equations for an implicit scheme. Numerical results are presented for one- and two-dimensional problems, including the modeling of waves propagating through multilayer structures and laminated glasses. The order of convergence is estimated for various schemes. The results show that the developed method is able to provide high modeling accuracy and describe complex wave patterns in heterogeneous media.

Nowadays, the solution of applied problems of seismic exploration and ultrasonic nondestructive testing is accompanied by the use of computer simulation. This poses a challenge for scientists to develop new modifications of numerical methods that can increase calculation accuracy while minimizing the cost of computing resources. In contrast to numerical methods on unstructured computational meshes, the use of Chimera computational meshes (or superimposed or adaptive meshes) makes it possible to describe boundaries and contact boundaries of arbitrary shape, but at the same time spending less RAM and time on calculations. This is especially important for the active use of neural networks in solving inverse problems, since when training samples are generated, the accuracy of modeling, speed of calculations, and amount of RAM spent are crucial. The paper considers and compares various modifications of the grid-characteristic method on Chimera computational meshes. Examples of test calculations are given.

The grid-characteristic method on regular structured meshes is combined with a discontinuous Galerkin method on tetrahedral meshes to solve the direct problem of three-dimensional elastic wave propagation through a geological medium consisting of four linear elastic layers with different parameters and arbitrary curved boundaries. The numerical methods are matched using a special algorithm taking into account the features of the transition from unstructured tetrahedral to regular structured meshes in three-dimensional space. A comparative convergence analysis of the combined method and the grid-characteristic method on curvilinear structured meshes is performed depending on the step sizes in spatial directions. The wave field of the velocity magnitude is obtained for a perturbation propagating from the source.

Weather conditions in the Far North require the use of pile foundations for multi-story buildings. Unlike mid- and equatorial latitudes, it is impossible to strengthen the soil between the piles; on the contrary, the properties and geometry of the subsidence of this soil depend on weather conditions. In this paper, the grid-characteristic method on systems of individual pairwise non-conformal curvilinear and Cartesian grids is used to solve inverse problems of determining the strength properties of piles and foundations of buildings operated in the Far North. To solve the inverse problem, a software package has been developed that uses an iterative solution of direct problems. The method allows one to vary such values as pile length, wavelength, number of piles per room, pile thickness, soil subsidence and its shape, elastic properties of the surrounding geological environment, number of floors, ceiling height, wall thickness, etc. In order to demonstrate the capabilities of the proposed method, a number of dependencies of strength characteristics on variable parameters were obtained.

When designing railways in the permafrost zone, many applied problems arise related to the formation of frost heaven in frozen soils. In the present study, the vibration analysis of the typical design of the ballast railway, under which the ice lens lies in the soil, is carried out on the basis of the obtained spatiotemporal distributions of the displacement velocity modulus and vertical stresses using numerical simulation with the grid-characteristic method. The results obtained can be used together with calculations of soil sediments by finite element methods to substantiate the geometry, materials of the track, and the necessity of artificial ground freezing technology.