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Laser polishing is a promising technology for the finishing of fused quartz (fused silica or quartz glass) products, enabling the removal of subsurface defects induced by mechanical processing. However, the complexity and nonlinearity of the physical processes occurring during laser irradiation complicate the selection of optimal technological parameters. The present paper aims to develop, comparatively analyze, and apply high-precision predictive models for forecasting and optimizing the key performance indicators of the laser polishing process for quartz glass. A verified finite element model implemented in the ANSYS software environment produced a dataset of temperature and stress fields for various combinations of process parameters. This dataset was used to develop and validate four types of predictive models: Polynomial Regression, a Fuzzy Logic System, an Adaptive Neuro-Fuzzy Inference System (ANFIS), and a Multilayer Perceptron (MLP) neural network. The models’ quality was evaluated on a test set using the statistical metrics MAE, RMSE, MAPE, $R^2$, and  $R^2_{Adj}$. A comparative analysis of the models revealed the significant superiority of the MLP neural network, which demonstrated the highest prediction accuracy for all output parameters, achieving Adjusted $R^2$ ($R^2_{Adj}$.) values above 0.97 and a Mean Absolute Percentage Error (MAPE) in the range of 0.7–2.8%. This model was effectively utilized as a surrogate function in combination with a genetic algorithm to successfully identify the optimal process parameters. The constructed MLP neural network model functions as a reliable and high-precision tool, facilitating both prediction and the optimization of fused quartz polishing outcomes using a CO₂ laser. This approach effectively approximates the complex nonlinear dependencies inherent in the process and can serve as a foundation for developing intelligent control and optimization systems for this technology.

A proposed approximate model of the rolling of a deforming wheel with a pneumatic tire allows one to account as well forces in tires as the effect of the dry friction on the stability of the rolling upon the shimmy phenomenon prognosis. The model os based on the theory of the dry friction with combined kinematics of relative motion of interacting bodies, i. e. under the condition of simultaneous rolling, sliding, and spinning with accounting for the real shape of a contact spot and contact pressure distribution. The resultant vector and couple of the forces generated by the contact interaction with dry friction are defined by integration over the contact area, whereas the static contact pressure under the conditions of vanishing velocity of sliding and angular velocity of spinning is computed after the finite-element solution for the statical contact of a pneumatic with a rigid road with accounting forreal internal structure and properties of a tire. The solid finite element model of a typical tire with longitudinal thread is used below as a background. Given constant boost pressure, vertical load and static friction factor 0.5 the numerical solution is constructed, as well as the appropriate solutions for lateral and torsional kinematic loading. It is shown that the contact interaction of a pneumatic tire and an absolutely rigid road could be represented without crucial loss of accuracy as two typical stages, the adhesion and the slip; the contact area shape remains nevertheless close to a circle. The approximate diagrams are constructed for both lateral force and friction torque; on the initial stage the diagrams are linear so that corresponds to the elastic deformation of a tire while on the second stage both force and torque values are constant and correspond to the dry friction force and torque. For the last stages the approximate formulae for the longitudinal and lateral friction force and the friction torque are constructed on the background of the theory of the dry friction with combined kinematics. The obtained model can be treated as a combination of the Keldysh model of elastic wheel with no slip and spin and the Klimov rigid wheel model interacting with a road by dry friction forces.

The work is devoted to the study of subgradient methods with different variations of the Polyak stepsize for minimization functions from the class of weakly convex and relatively weakly convex functions that have the corresponding analogue of a sharp minimum. It turns out that, under certain assumptions about the starting point, such an approach can make it possible to justify the convergence of the subgradient method with the speed of a geometric progression. For the subgradient method with the Polyak stepsize, a refined estimate for the rate of convergence is proved for minimization problems for weakly convex functions with a sharp minimum. The feature of this estimate is an additional consideration of the decrease of the distance from the current point of the method to the set of solutions with the increase in the number of iterations. The results of numerical experiments for the phase reconstruction problem (which is weakly convex and has a sharp minimum) are presented, demonstrating the effectiveness of the proposed approach to estimating the rate of convergence compared to the known one. Next, we propose a variation of the subgradient method with switching over productive and non-productive steps for weakly convex problems with inequality constraints and obtain the corresponding analog of the result on convergence with the rate of geometric progression. For the subgradient method with the corresponding variation of the Polyak stepsize on the class of relatively Lipschitz and relatively weakly convex functions with a relative analogue of a sharp minimum, it was obtained conditions that guarantee the convergence of such a subgradient method at the rate of a geometric progression. Finally, a theoretical result is obtained that describes the influence of the error of the information about the (sub)gradient available by the subgradient method and the objective function on the estimation of the quality of the obtained approximate solution. It is proved that for a sufficiently small error $\delta > 0$, one can guarantee that the accuracy of the solution is comparable to $\delta$.

The paper gives a detailed description of the developed methods for simulating the turbulent flow around a helicopter rotor and calculating its aerodynamic characteristics. The system of Reynolds-averaged Navier – Stokes equations for a viscous compressible gas closed by the Spalart –Allmaras turbulence model is used as the basic mathematical model. The model is formulated in a non-inertial rotating coordinate system associated with a rotor. To set the boundary conditions on the surface of the rotor, wall functions are used.

The numerical solution of the resulting system of differential equations is carried out on mixed-element unstructured grids including prismatic layers near the surface of a streamlined body.The numerical method is based on the original vertex-centered finite-volume EBR schemes. A feature of these schemes is their higher accuracy which is achieved through the use of edge-based reconstruction of variables on extended quasi-onedimensional stencils, and a moderate computational cost which allows for serial computations. The methods of Roe and Lax – Friedrichs are used as approximate Riemann solvers. The Roe method is corrected in the case of low Mach flows. When dealing with discontinuities or solutions with large gradients, a quasi-one-dimensional WENO scheme or local switching to a quasi-one-dimensional TVD-type reconstruction is used. The time integration is carried out according to the implicit three-layer second-order scheme with Newton linearization of the system of difference equations. To solve the system of linear equations, the stabilized conjugate gradient method is used.

The numerical methods are implemented as a part of the in-house code NOISEtte according to the two-level MPI–OpenMP parallel model, which allows high-performance computations on meshes consisting of hundreds of millions of nodes, while involving hundreds of thousands of CPU cores of modern supercomputers.

Based on the results of numerical simulation, the aerodynamic characteristics of the helicopter rotor are calculated, namely, trust, torque and their dimensionless coefficients.

Validation of the developed technique is carried out by simulating the turbulent flow around the Caradonna – Tung two-blade rotor and the KNRTU-KAI four-blade model rotor in hover mode mode, tail rotor in duct, and rigid main rotor in oblique flow. The numerical results are compared with the available experimental data.

The problem of developing efficient numerical methods for non-convex (including non-smooth) problems is relevant due to their widespread use of such problems in applications. This paper is devoted to subgradient methods for minimizing Lipschitz $\mu$-weakly convex functions, which are not necessarily smooth. It is well known that subgradient methods have low convergence rates in high-dimensional spaces even for convex functions. However, if we consider a subclass of functions that satisfies sharp minimum condition and also use the Polyak step, we can guarantee a linear convergence rate of the subgradient method. In some cases, the values of the function or it’s subgradient may be available to the numerical method with some error. The accuracy of the solution provided by the numerical method depends on the magnitude of this error. In this paper, we investigate the behavior of the subgradient method with a Polyak step when inaccurate information about the objective function value or subgradient is used in iterations. We prove that with a specific choice of starting point, the subgradient method with some analogue of the Polyak step-size converges at a geometric progression rate on a class of $\mu$-weakly convex functions with a sharp minimum, provided that there is additive inaccuracy in the subgradient values. In the case when both the value of the function and the value of its subgradient at the current point are known with error, convergence to some neighborhood of the set of exact solutions is shown and the quality estimates of the output solution by the subgradient method with the corresponding analogue of the Polyak step are obtained. The article also proposes a subgradient method with a clipped step, and an assessment of the quality of the solution obtained by this method for the class of $\mu$-weakly convex functions with a sharp minimum is presented. Numerical experiments were conducted for the problem of low-rank matrix recovery. They showed that the efficiency of the studied algorithms may not depend on the accuracy of localization of the initial approximation within the required region, and the inaccuracy in the values of the function and subgradient may affect the number of iterations required to achieve an acceptable quality of the solution, but has almost no effect on the quality of the solution itself.

The paper presents the results of applying a scheme of very high accuracy and resolution to obtain numerical solutions of the Navier – Stokes equations of a compressible gas describing the occurrence and development of instability of a two-dimensional laminar boundary layer on a flat plate. The peculiarity of the conducted studies is the absence of commonly used artificial exciters of instability in the implementation of direct numerical modeling. The multioperator scheme used made it possible to observe the subtle effects of the birth of unstable modes and the complex nature of their development caused presumably by its small approximation errors. A brief description of the scheme design and its main properties is given. The formulation of the problem and the method of obtaining initial data are described, which makes it possible to observe the established non-stationary regime fairly quickly. A technique is given that allows detecting flow fluctuations with amplitudes many orders of magnitude smaller than its average values. A time-dependent picture of the appearance of packets of Tollmien – Schlichting waves with varying intensity in the vicinity of the leading edge of the plate and their downstream propagation is presented. The presented amplitude spectra with expanding peak values in the downstream regions indicate the excitation of new unstable modes other than those occurring in the vicinity of the leading edge. The analysis of the evolution of instability waves in time and space showed agreement with the main conclusions of the linear theory. The numerical solutions obtained seem to describe for the first time the complete scenario of the possible development of Tollmien – Schlichting instability, which often plays an essential role at the initial stage of the laminar-turbulent transition. They open up the possibilities of full-scale numerical modeling of this process, which is extremely important for practice, with a similar study of the spatial boundary layer.

We propose a family of Gauss –Newton methods for solving optimization problems and systems of nonlinear equations based on the ideas of using the upper estimate of the norm of the residual of the system of nonlinear equations and quadratic regularization. The paper presents a development of the «Three Squares Method» scheme with the addition of a momentum term to the update rule of the sought parameters in the problem to be solved. The resulting scheme has several remarkable properties. First, the paper algorithmically describes a whole parametric family of methods that minimize functionals of a special kind: compositions of the residual of a nonlinear equation and an unimodal functional. Such a functional, entirely consistent with the «gray box» paradigm in the problem description, combines a large number of solvable problems related to applications in machine learning, with the regression problems. Secondly, the obtained family of methods is described as a generalization of several forms of the Levenberg –Marquardt algorithm, allowing implementation in non-Euclidean spaces as well. The algorithm describing the parametric family of Gauss –Newton methods uses an iterative procedure that performs an inexact parametrized proximal mapping and shift using a momentum term. The paper contains a detailed analysis of the efficiency of the proposed family of Gauss – Newton methods; the derived estimates take into account the number of external iterations of the algorithm for solving the main problem, the accuracy and computational complexity of the local model representation and oracle computation. Sublinear and linear convergence conditions based on the Polak – Lojasiewicz inequality are derived for the family of methods. In both observed convergence regimes, the Lipschitz property of the residual of the nonlinear system of equations is locally assumed. In addition to the theoretical analysis of the scheme, the paper studies the issues of its practical implementation. In particular, in the experiments conducted for the suboptimal step, the schemes of effective calculation of the approximation of the best step are given, which makes it possible to improve the convergence of the method in practice in comparison with the original «Three Square Method». The proposed scheme combines several existing and frequently used in practice modifications of the Gauss –Newton method, in addition, the paper proposes a monotone momentum modification of the family of developed methods, which does not slow down the search for a solution in the worst case and demonstrates in practice an improvement in the convergence of the method.