This study presents a comprehensive comparative analysis of four OpenFOAM solvers for simulating supersonic flow around a Spherically Blunted Cone. Utilizing the generalized computational experiment technique, we evaluated solver performance across varying Mach numbers and cone half-angles. The research aimed to provide clear recommendations for solver selection in high-speed flow simulations. Results show that rhoCentralFoam exhibited the lowest deviation from the exact solution, followed closely by pisoCentralFoam. The sonicFoam solver demonstrated limitations at higher Mach numbers. The QGDFoam solver, while showing promise, requires further optimization of its smoothing parameter for improved accuracy. This study offers valuable insights for OpenFOAM users, enabling them to make informed decisions when choosing solvers for compressible gas dynamics problems.

For two-dimensional Euler equations, flow control using spatially distributed stationary heat sources is considered. The initial flow, characterized by the interaction of Edney - type shock waves, was changed with the help of heat sources in such a way as to reduce the maximum pressure on the surface of the body. The problem was solved in an optimization formulation using conjugate equations to calculate the gradient of the target functional. In addition to visualizing flow parameters in this problem, it is of interest to visualize the distribution of adjoint parameters and heat sources.

The article focuses on a comparative assessment of the accuracy of various solvers within the OpenFOAM software package. An axisymmetric flow problem over a spherically blunted cone, utilizing a non-viscous compressible gas at a zero angle of attack, serves as the benchmark case for evaluation. The computational outcomes from different solvers are juxtaposed against established tabulated solutions. In addressing the benchmark problem, each solver is subjected to variations in the cone's half-angle and the incoming flow velocity at specified increments. The comparative analysis spans a broad spectrum of these critical parameters. Detailed analyses and graphical representations of the acquired results yield insights into the relative accuracy of the numerical methods applied to this specific class of aerodynamic problems.

The numerical solution in sense of Prager Synge is defined as a hypersphere containing a true solution of a system of partial differentiation equations (PDE). In the original variant Prager Synge method is based on special orthogonal properties of PDE and may be applied only to several equations. Herein, the Prager Synge solution (center and radius of the hypersphere) is estimated using the ensemble of numerical solutions obtained by independent algorithms. This approach is not problem dependent and may be applied to arbitrary system of PDE. Several options for computation of the Prager Synge solution are considered. The first one is based on the search for the orthogonal truncation errors and their transformation. The second is based on the orthogonalization of approximation errors obtained using the defect correction method. It applies the superposition of numerical solutions. The third option uses the width of the ensemble of numerical solutions. The numerical tests for the two dimensional inviscid flows are presented that demonstrate the acceptable effectivity of the approximation error estimates based on the solution in the Prager Synge sense.

The article presents the results of computational experiments on displaying the points of the original multidimensional information array on the elastic map scan to assess the relative positions of semantic proximity areas in order to improve the processing of text information. Elastic maps are considered as a tool for providing analytical work with text information. As previous works show, in order to obtain the required distances corresponding to the cluster picture of the studied multidimensional volume, it is necessary to use the distances on the elastic map, which reflects the cluster portrait of the studied multidimensional data volume. The paper presents the cluster structures of points of the studied multidimensional volume obtained in this way on the elastic map scan in the plane of the first two principal components. An analysis of the relative positions of clusters of different configurations at different points in time is presented.

This work presents the experience of constructing and conducting the research project "Scientific Visualization and Visual Analytics" at RTU-MIREA in 2024. The research project was conducted for 3rd-year students of the Department of Higher Mathematics of the Institute of Artificial Intelligence at RTU-MIREA. The construction and organization of the research project are described. Examples of tasks and their implementation are given. This work may be of interest for teaching similar disciplines in this subject area.

The approximation of the tensor appearing at a discretization of the multidimensional function is considered from the viewpoint of storing and treating of the results of parametric computations obtained in computational aerogasdynamics. The new algorithm for the computation of the canonical decomposition using gradient descent and approximately decomposable goal functional is described. This algorithm applies the random set of points on the hyperplane orthogonal to the computed core of the canonical decomposition (“umbrella”) that ensures its flexible application for an approximation of the tensors with a priori unknown rank and may be naturally transferred on such tensor decomposition as the tensor train. The results of the numerical tests are presented for the model six-dimensional functions and for an ensemble of the numerical solutions for the two-dimensional Euler equations. These equations describe the flow of the compressible gas with two crossing shock waves. The Mach number and angles of the flow deflection serve as the flow parameters. The results are provided for the dimensionality 3 (simple numerical solution) and 4 (the ensemble of the numerical solutions in dependence on the Mach number).

The work is devoted to the application of previously developed algorithms and methods of stereo animations construction in the field of biological research on the example of stereo images for functional tomogram of the brain. The construction of stereo images on the autostereoscopic monitor provides an opportunity to obtain an in-depth understanding of the studied object. The autostereoscopic monitor allows viewing stereo images without glasses, while providing a quality not inferior to that of a classical 3D projection stereo system. This work continues the cycle of research conducted at the Keldysh Institute of Applied Mathematics. The specificity of the displayed data allowed to improve authoring libraries for creating stereo images. The research is aimed at developing technologies for constructing stereo images and animations for presenting the results of scientific calculations on classical stereo devices and autostereoscopic monitors. Constructing a stereoscopic visual representation of the results of biological research will enable researchers in the field to gain a deeper understanding of the object under study and its properties.

The epistemic uncertainty quantification concerning the estimation of the approximation error using the differences between numerical solutions treated in the Inverse Problem statement is addressed and compared with the Richardson extrapolation. The Inverse Problem is posed in the variational statement with the zero order Tikhonov regularization. The ensemble of numerical results, obtained by the OpenFOAM solvers for the inviscid compressible flow with a shock wave is analyzed. The approximation errors, obtained by the Richardson extrapolation and the Inverse Problem are compared with the exact error, computed as the difference of numerical solutions and the analytical solution. The Inverse problem based approach is demonstrated to be an inexpensive alternative to the Richardson extrapolation.

An approximation of the tensor obtained at a discretization of the multidimensional function on a uniform grid is addressed from the viewpoint of the storage and treating the result of the parametric computations in the CFD problems. The tensor decompositions are considered for this purpose. The new algorithm of the calculation of the canonical decomposition using the gradient descent for an approximately decomposable goal functional. This algorithm applies an ensemble of points on the hyperplane orthogonal to the computed core of the canonical decomposition (”umbrella”) that enables its flexible application for the approximation of the tensors with a priori unknown rank. This algorithm is naturally transferred on such tensor decomposition as the tensor train. The results of the numerical tests are presented for model six-dimensional functions and for the ensemble of the numerical solutions for two-dimensional Euler equations. These equations describe the flow of the compressible gas with two crossing shock waves. Mach number and the the flow deflection angles are considered as the flow parameters.

When organizing a mass practical solution of computational problems of gas dynamics with the help of mathematical modeling, information on the comparative accuracy of the numerical methods used is now increasingly in demand. As a rule, calculators need information not only for a particular combination of the defining gas-dynamic parameters of the problem (characteristic Mach, Reynolds numbers, etc.), but also for the variation of these parameters in certain ranges. This work presents numerical studies devoted to a comparative assessment of the accuracy of numerical methods for a number of problems with reference solutions. The calculation of these problems is carried out for ranges of characteristic numbers using various numerical methods. The results obtained are compared with the reference solution and make it possible to estimate the error for each of the numerical methods. Calculations are carried out using the construction of a generalized computational experiment. A generalized computational experiment is a computational technology that combines the solution of mathematical modeling problems, parallel technologies and visual analytics technologies. The results of the generalized computational experiment are multidimensional arrays, where the dimension of the arrays corresponds to the defining parameters. Analysis and visual representation of the obtained results provide information on the comparative accuracy of the numerical method for the selected class of problems.

This paper deals with modification of the discontinuous shapeless particle method for two-dimensional problems of gas dynamics.In the previous version of the method the particles' shape determined their interaction and this defined high quality of final r esult, especially in one-dimensional case.In order to make away with attachment to the shape of particles, in two-dimensional setting along with particles' heights (solution of differential problem) and their positions in the space, the another invariant was introduced, and it's represented by area of trapezoid, where the particles heights are the bases of trapezoid, and the segment which connects their centers is a lateral side.This invariant can be interpreted as a trace of mass conservation in the space between two particles, the masses of those also don't change (exact conservation is a fundamental feature of the particle methods).Article describes the comparison of numerical solutions received with the help of modified method of particles, and numerical solutions obtained through the use of open software package OpenFOAM, with reference analytical solution in the L2 norm using the example of problem on supersonic flow around a wedge and with formation of an angle shock wave.The speed of overtaking flow and flow incidence angle are varying.The introduced visualization of results provides a clear overview of features peculiar to particles method for problems the solution of which comes with heavy gradients.Comparative verification is performed in the framework of implementation of generalized computational experiment, which helps to get the solution for a class of problems when there is a variation of defining parameters.The present paper is a part of the research on comparative verification of numerical methods in the space of defining parameters.

The results of numerical simulation of a vertical-axis wind turbine (VAWT) based on the solution of three-dimensional Reynolds-averaged Navier–Stokes equations with the Spalart–Allmaras turbulence model are presented. The results of parametric calculations of a viscous compressible flow under conditions simulating urban infrastructure for a helicoid-type wind turbine with three spirally twisted blades are presented.

The work is devoted to the application of previously developed algorithms and methods of stereo imaging in the field of maxillofacial surgery.Construction of stereo images can be useful for solving the problem of visual aids in medicine.Such problems arise when teaching students of medical universities, at doctors' advanced training courses, at councils when discussing specific plans and methods of treatment with patients, when holding international medical conferences, when solving problems of telemedicine.The work is based on the developed algorithms and software tools for representing 3D objects in stereo mode on modern autostereoscopic monitors.Test examples of three-dimensional 3D animations of the results of real CT scans of patients are constructed and presented as illustrations.A similar approach can be used for virtual volumetric 3D visualization of human organs in various fields of medicine.

This paper deals with estimation of local (pointwise) approximation error on an ensemble of numerical solutions obtained by using independent algorithms. A variational inverse problem is posed for approximation error estimation. This problem is ill-posed due to translation-invariance of the governing equations. Zero order Tikhonov regularization is applied to obtain stable solutions. Numerical tests for two-dimensional equations describing inviscid compressible flow are performed to verify the efficiency of the algorithm. The approximation error estimates obtained by using the inverse problem are in satisfactory agreement with those obtained by Richardson extrapolation, but with significantly less computational costs.

The estimation of the approximation errors using the ensemble of numerical solutions is considered in the Inverse Problem statement. The Inverse Problem is set in the nonlinear variational formulation that provides additional opportunities. The ensemble of numerical results is analyzed, which is obtained by the OpenFOAM solvers for the inviscid compressible flow containing an oblique shock wave. The numerical tests demonstrated feasibility to compute the approximation errors without any regularization. The refined solution, corresponding the mean of numerical solutions with the approximation error correction, is also computed and compared with the analytic one.

The paper continues a series of publications of the authors' research materials in the field of developing an approach to dynamic planning and control of a generalized computational experiment based on visualization methods and visual analytics.A generalized computational experiment involves multiple solution of a numerical simulation problem for different sets of values of model defining parameters, which makes it possible to obtain a solution immediately for a certain class of mathematical modeling problems specified in a multidimensional parameter space.The paper considers an extension of the existing authors' approach to analysis of a generalized computational experiment state using visual maps, based on visualization metaphors that can display not only individual images but also their relationships.A method is proposed for constructing visual maps of a generalized computational experiment focused on visualizing relationships between single computational experiments in three-dimensional space.The method is based on the mechanism of formalization of the relationships between single computational experiments, as well as the concept of a graph model visualization metaphor that defines a visual map prototype.A description is given of a software system for constructing and analyzing three-dimensional visual maps of a generalized computational experiment.The paper also considers examples of its application in estimating the accuracy of numerical models of the OpenFOAM software platform for a three-dimensional problem of inviscid flow around a cone.

The approximation of the multidimensional function using the high order tensor and canonical decomposition is considered for the purpose of visualization. Formally, from the standpoint of computational resources (operational memory volume and the time of computation) the canonical decomposition is beyond comparison. However, at present, the main algorithm that calculates the canonical decomposition is based on the Khatri-Rao product, which implies matrization. This circumstance restricts the domain of applicability by tensors of the relatively small order. In order to overcome this drawback, the algorithm of the computation for the canonical decomposition formmatrices is described that is composed by combination of alternating least squares and stochastic gradient descent. This algorithm has no restrictions from the standpoint of the order of the tensor under the consideration. The numerical tests are presented for the approximation of functions in six-dimensional space that demonstrate the high computational efficiency and high quality of results. The instabilities arising at the estimation of the rank of the decomposition are another trouble of the canonical decomposition. The results of the numerical tests are presented that illustrate the search for the optimal rank providing the minimum of the discrepancy of the exact solution and its approximation.

The geometric properties of the ensemble of numerical solutions obtained by the algorithms of different inner structure are addressed from the prospects for a posteriori error estimation. The numerical results are presented for the two-dimensional inviscid supersonic flows, containing shock waves. The truncation errors are computed using a postprocessor, the approximation errors are calculated by the subtraction of the numerical and the analytic solutions. The angles between the approximation errors are found to be far from zero that enables a posteriori estimation of the error norm. The correlation of the angles between the approximation errors and the corresponding angles between the computable truncation errors is observed in numerical tests and discussed from the viewpoint of the measure concentration effect and the algorithmic randomness. The analysis of the truncation errors’ geometry and the distances between solutions enables the estimation of the approximation error norm on the ensemble of numerical solutions obtained by the independent algorithms.