On Applying of Generalized Computational Experiment to Numerical Methods Verification

This work is devoted to the application of a generalized computational experiment for a comparative assessment of numerical methods accuracy. A generalized computational experiment allows one to obtain a numerical solution for a class of problems determined by the ranges of defining parameters variation. The approaches to the application of a generalized computational experiment in the presence of a reference solution and in its absence are dis-cussed. An example of constructing error surfaces is given when the solvers of the OpenFOAM software package are compared. The classic inviscid problem of oblique shock wave is used as a basic task. Variations of the key parameters of the problem — the Mach number and angle of attack — are considered. An example of the problem of flow around a cone at an angle of attack with varying Mach number, cone angle and angle of attack is also considered. The concept of an error index is introduced as an integral characteristic of deviations from the exact solution for each solver in the class of problems under consideration.

Analysis of the behavior of OpenFOAM solvers for 3D problem of supersonic flow around a cone at an angle of attack

This paper is devoted to comparative analysis of numerical methods accuracy. Comparative estimation of accuracy is performed for numerical methods presented as solvers integrated to open sotware package OpenFOAM. Three different OpenFOAM solvers are selected to numerically solve the problem of supersonic flow around a cone. The angle of attack, cone half-angle and Mach number were varied in the selected ranges with a certain step. This approach is implemented using a generalized computational experiment that allows, based on parallel technologies, the simultaneous solution of the same basic problem with different input parameters. A number of test calculations were carried out. The deviation fields of gas-dynamic quantities for all solvers are analyzed. The construction of a generalized computational experiment made it possible to compare the accuracy of the considered solvers not only for one, separately taken problem, but for a class of problems specified by the ranges of variation of the determining parameters. Such an assessment of accuracy can be very useful for users of the software package when choosing a solver. Also, the results obtained can be useful for solver developers.

Visualizing Methods of Multi-criteria Alternatives for Pairwise Comparison Procedure

The article deals with the problem of choosing a preferred alternative in a pairwise comparison procedure. The difficulties of applying this procedure in a case of using alternatives with a large number of criteria are noted. It is proposed to supplement the procedure of expert pairwise comparison with visualization tools of multi-criteria alternatives. The paper considers several visualization methods for multi-criteria alternatives for pairwise comparison procedures: histograms, two-dimensional graphs, three-dimensional surfaces, probability distribution diagrams, visualization based on modifications of radar and radial diagrams, as well as combined methods. It described an experimental study of the application of the considered method for the task of determining the preferred alternative by the example of choosing one of two OpenFoam solvers (rhoCentralFoam and pisaCentralFoam), with the help of which estimates of the accuracy of calculating the inviscid flow around a cone were obtained. Each solver is characterized by 288 criteria. It is shown that the use of some of the methods considered does not make it possible for the expert to make a choice. In this case, a good result was obtained using methods for constructing three-dimensional surfaces, probability distribution diagrams, as well as using the combined method based on modified radar diagrams. It is concluded that the rhoCentralFoam solver is more preferable if there are no additional criteria for ranking the criteria. The possibility of using the combined method in combination with the ranking procedure of criteria (or their groups) during decision-making is also noted.

Mathematical Support and Software of Visual Filtering of Alternatives in Multi-criteria Decision Making Problems

The article deals with the problem of multi-criteria decision-making problems, which are characterized by a large number of options and alternatives. It is proposed to use visual filtering of graphic images describing the corresponding alternatives as one of the stages in decision-making in such tasks. The approaches and requirements for the construction of graphic images of alternatives are considered. Describes the steps and algorithms for constructing visual images of alternatives, based on the radial and pie charts, and include the normalization procedure. It describes software that implements the proposed algorithms, as well as providing interactive interaction with an expert for visual filtering of multi-criteria alternatives. Additionally, the capabilities of the developed software are described, which include filtering alternatives based on threshold values, as well as the possibility of conducting a series of experiments in order to obtain the union or intersection of filtered sets of alternatives. A synthetic test for filtering 201 alternatives is described, each of which is described by 15 criteria. As a result of a series of experiments, this choice set was reduced by about 28 times. A description is also given of an experiment on visual filtering of real alternatives that describe estimates of the accuracy of calculating inviscid flow around a cone using several OpenFoam solvers. Each solver is characterized by 288 criteria, and according to the results of visual filtering, the advantage in the accuracy of the calculations of two solvers over the others is clearly established.