Vol 63, No 10 (2023)

A grid-characteristic numerical method for solving a multidimensional transport equation on an unstructured grid with an order higher than one is proposed; this method does not use auxiliary points on edges and faces. The avoidance of auxiliary points on edges and faces simplifies the topology of the computational grid during its motion, which is important for solving dynamic problems of mechanics of deformable solids. To increase the approximation order, an analog of the grid stencil extension implemented for an unstructured grid is used. Results of testing the proposed numerical scheme for continuously differentiable, continuous, discontinuous solutions are presented.

The paper considers the problem of determining the natural frequencies and eigenwaves of transverse vibrations for the Bernoulli–Euler equation with variable coefficients. Such problems arise both in the case of complex geometry of a vibrating solid and in the case of functionally graded materials or the accumulation of damage in a classical elastic material. Solutions of boundary value problems are constructed using the expansion in Peano series. Under broad assumptions, the uniform convergence of Peano series is shown and estimates of the residual terms are obtained. Examples of the numerical implementation of the proposed procedure are given for bending vibrations of a rod with certain parameters of a variable cross section (geometric heterogeneity) and elastic modulus distribution (physical heterogeneity). Numerical examples are focused on assessing the geometric and elastic properties of samples in an experimental study of the fatigue strength of alloys during high-frequency cyclic tests based on the general principle of point resonant loading. The method proposed for solving problems of resonant vibrations for the Bernoulli–Euler equation can be used in the design of new promising cyclic test schemes and mathematical modeling of fatigue failure processes under high-frequency resonant vibrations.

The Wilkins model for an elastoplastic medium is considered. A theoretical analysis of discontinuous solutions under the assumption of one-dimensional uniaxial strain is performed. In this approximation, the material equations for the deviator stress tensor components are integrated exactly, and only the conservative part of the governing equations remains, which makes it possible to derive a class of exact analytical solutions for the model. To solve the full nonconservative system of equations (without assuming the uniaxial strain), a Godunov-type numerical method is developed, which uses an approximate Riemann solver based on integrating the system of equations along a path in the phase space. A special choice of path is proposed that reduces the two-wave HLL approximation to the solution of a linear equations. Numerical and exact analytical solutions are compared for a number of problems with various regimes of shockwave processes.

A three-dimensional direct problem of ultrasonic nondestructive testing of a railroad rail treated as a linear elastic medium is solved by applying a grid-characteristic method on curved structured Chimera and Cartesian background meshes. The algorithm involves mutual interpolation between Chimera and Cartesian meshes that takes into account the features of the transition from curved to Cartesian meshes in three-dimensional space. An analytical algorithm for generating Chimera meshes is proposed. The convergence of the developed numerical algorithms under mesh refinement in space is analyzed. A comparative analysis of the full-wave fields of the velocity modulus representing the propagation of a perturbation from its source is presented.

Three-dimensional models of suspension transport in coastal marine systems are considered. The associated processes have a number of characteristic features, such as high concentrations of suspensions (e.g., when soil is dumped on the bottom), much larger areas of suspension spread than the reservoir depth, complex granulometric (multifractional) content of suspensions, and mutual transitions between fractions. Suspension transport can be described using initial-boundary value diffusion–convection–reaction problems. According to the authors' idea, on a time grid constructed for the original continuous initial-boundary value problem, the right-hand sides are transformed with a “delay” so that the right-hand side concentrations of the components other than the underlying one (for which the initial-boundary value problem of diffusion–convection is formulated) are determined at the preceding time level. This approach simplifies the subsequent numerical implementation of each of the diffusion–convection equations. Additionally, if the number of fractions is three or more, the computation of each of the concentrations at every time step can be organized independently (in parallel). Previously, sufficient conditions for the existence and uniqueness of a solution to the initial-boundary value problem of suspension transport were determined, and a conservative stable difference scheme was constructed, studied, and numerically implemented for test and real-world problems. In this paper, the convergence of the solution of the delay-transformed problem to the solution of the original suspension transport problem is analyzed. It is proved that the differences between these solutions tends to zero at an O(τ) rate in the norm of the Hilbert space L2 as the time step t  approaches zero.

Earlier, a method for constructing an initial approximation for solving the inverse problem of acoustics by a gradient method based on a convolutional neural network trained to predict the distribution of velocities in a medium from wave response was proposed [9]. It was shown that the neural network trained on responses from simple layered media can be successfully used for solving the inverse problem for a significantly more complex model. In this paper, we present algorithms for processing data about epidemics and an example of applying a neural network for modeling the propagation of COVID-19 in Novosibirsk region (Russia) based only on data. A neural network NN-COVID-19 that uses data about the epidemics is constructed. It is shown that this neural network predicts the propagation of COVID-19 for five days by an order of magnitude better than SEIR-HCD. When a new variant (Omicron) appeared, this neural network was able to predict (after retraining) the propagation of the epidemics more accurately. Note that the proposed neural network uses not only epidemiological data but also social ones (such as holidays, restrictive measures, etc.). The proposed approach makes it possible to refine mathematical models. A comparison of the curves constructed by SEIR-HCD model and by the neural network shows that the plots of solutions of the direct problem almost coincide with the plots constructed by the neural network. This helps refine coefficients of the differential model.