Construction of a Generalized Computational Experiment and Visual Analysis of Multidimensional Data A.E. Bondarev1, V.A. Galaktionov1 bond@keldysh.ru|vlgal@gin.keldysh.ru 1 Keldysh Institute of Applied Mathematics Russian Academy of Sciences, Miusskaya sq. 4, 125047 Moscow, Russia The work is devoted to the problems of constructing a generalized computational experiment in the problems of computational aerodynamics. The construction of a generalized computational experiment is based on the possibility of carrying out parallel calculations of the same problem with different input data in multitasking mode. This allows carrying out parametric studies and solving problems of optimization analysis. The results of such an experiment are multidimensional arrays, for the study of which visual analytics methods should be used. The construction of a generalized experiment allows one to obtain dependences for valuable functionals on the determining parameters of the problem under consideration. The implementation of a generalized experiment allows one to obtain a solution for a class of problems in the ranges under consideration, and not just for one problem. Examples of constructing a generalized computational experiment for various classes of problems of computational aerodynamics are presented. The article also provides an example of constructing such an experiment for a comparative assessment of the accuracy of numerical methods. Keywords: generalized computational experiment, visual analysis, multidimensional data, parallel computations. Before the advent of parallel computing technologies, such 1. Introduction calculations were difficult to implement and they were quite rare. However, they were the prototype of the generalized The physical experiment was the main and often the only computational experiment. source of information on the problems of gas dynamics long The generalized computational experiment is based on before the advent of the computer age. In practice, the main goal solving direct and inverse problems of mathematical modeling. of a physical experiment was almost always not to model the These tasks can be considered in a parametric and optimization physical phenomenon itself, but to clarify the circumstances setting. Problem solving is carried out using parallel technologies under which it occurs, i.e. obtaining the dependence of the in multitasking mode. Numerical solutions are volumes of appearance of the phenomenon on the determining parameters of multidimensional data. To process and analyze this data, it is the problem, such as Mach numbers, Reynolds numbers, Prandtl necessary to apply the methods of Data Analysis and Visual numbers, and the geometric parameters of the problem. Such Analytics. The construction of a generalized computational large-scale experimental work made it possible to obtain key experiment makes it possible to obtain a solution not for one, relationships for the dependence of the gasdynamic functions of separately taken problem, but for a whole class of problems. The interest or the conditions for the appearance of a physical effect class of problems is determined in the ranges of variation of the on the key determining parameters. In fact, the establishment of defining parameters of the problem, such as characteristic such physical laws for shock waves, separated flows, numbers (Mach, Reynolds, Strouhal numbers, etc.) and characteristic configurations of streamlined bodies was the main geometric characteristics. In a practical sense, this makes it task of fluid and gas mechanics. possible to reveal hidden dependences of valuable functionals on As an example of such a dependence, one can cite the famous the determining parameters of the problem, similar to the above formula of G.I. Petrov, representing the fundamental law on the formulas. This work continues a series of works devoted to the ultimate pressure drop in the shock, which the turbulent development and implementation of a generalized computational boundary layer is able to withstand without detachment from the experiment for various classes of computational aerodynamics wall [18]: problems [2, 3, 9-17]. P2 / P1 = 0.713Me + 0.213. Despite the fact that there are very few works devoted to the Here P2 / P1 is the pressure drop, Me is the Mach number development of a generalized computational experiment, the before the separation point, varying from 1.5 to 4. development of such experiments is gradually taking place in Another example is the famous Kozlov formula [25], which many areas. First of all, tools are being developed to implement represents the dependence of surface friction on Mach numbers, such experiments in many software packages for solving Reynolds numbers and the temperature factor: −0,29+0,01lg𝑅𝑒𝑤 ̅ 0,39 ̅ 0,2 optimization problems. Here we can cite as an example the work 𝑐𝑓𝑤 = 0,085𝑅𝑒𝑤 𝑇𝑤𝑒 𝑇𝑒 . [22], where algorithms are implemented that allow a generalized Here 𝑐𝑓𝑤 , Rew is the coefficient of surface friction and the computational experiment in such fields as seismic exploration, Reynolds number calculated with reference to the wall plasma physics and turbid media optics, solving fundamental and temperature, Te is the temperature at the outer boundary of the applied problems of studying magnetic materials and creating boundary layer, and Twe is the temperature factor. spintronics devices, simulation of field development for the oil The advent of computer technology allowed solving the reservoir that contains kerogen with in-situ combustion taken problems of mathematical modeling of currents, which sharply into account, simulation of poroelastic medium problems and reduced the need for large-scale physical experiments. However, hydraulic fracture problems. in the problems of mathematical modeling, the main tendency of carrying out series of calculations with the variation of the 2. Prerequisites for the creation of a generalized defining parameters of the problem also remained. The main goal computational experiment was the same – to determine the conditions for the appearance of a physical phenomenon when the external conditions of the The development of technologies and software tools for problem are varied. An example of such approach is described in constructing a generalized computational experiment occurs as article [8], which presents a series of numerical experiments on the modern development of mathematical methods and high- the flow of a backward ledge by a viscous gas flow. As a result performance computing tools. Two main reasons should be of the experiments, a generalized formula is obtained that pointed out as the main factors determining the possibility of represents the characteristic time of the establishment of the flow efficiently constructing a generalized computational experiment. as a function of the Mach and Reynolds numbers of the external The first of these is the emergence of high-performance flow. computing clusters and parallel technologies. It is generally Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). accepted that parallel technologies provide a) the ability of fast view the most perspective way for parallelizing is applying the computing and b) the ability to use detailed computational grids. approach of multitask parallelism using the principle “one task – However, parallel technologies also provide researchers with one process”. Due to minimal quantity of internal exchanges another crucial opportunity. This is an opportunity of parallel between the processes we are able to create an effective practical calculation of the same problem with different input data in tool for generalized numerical experiment. We assume that k multitask mode. From the point of view of the author, this processes are provided for parallel computation. The control possibility is not yet fully appreciated. This possibility allows process P0 creates the grid in the multidimensional space of one to effectively solve parametric and optimization problems determining parameters, then P0 forms tasks and sends the tasks and construct a generalized computational experiment. to others processes and to itself also. After task completion P0 The second reason is the intensive development of methods collects the results and implements all procedures defined by and approaches for processing and visualization of user, such as data processing and transformation. Due to the multidimensional data. The results of a generalized absence of internal exchanges between the processes the computational experiment in the form of discrete procedure of parallelizing amounts to creation of control multidimensional arrays need processing and analysis in order to interface for tasks distribution and data collecting in one obtain hidden interdependencies between the determining factors multidimensional array. in the class of problems that interest the researcher. There are some effective and easy ways to create such interface for parallel computations. These ways use such 3. Generalized numerical experiment computational technologies as MPI (Message Passing Interface) [26] and DVM technology [1, 4-7]. DVM technology [1, 4-7] A generalized numerical experiment involves splitting each was elaborated in Keldysh Institute of Applied Mathematics of the defining parameters of a problem within a certain range. A RAS. DVM-system provides unified toolkit to develop parallel grid decomposition is formed for some multidimensional programs of scientific-technical calculations in C and Fortran. parallelepiped composed of the defining parameters of the Unified parallel model is built in C and Fortran languages on the considered problem of gas dynamics. For each point of this grid, base of the constructions, that are "transparent" for standard the problem is calculated in the space of the determining compilers, that allows to have single version of the program for parameters. According to [9, 13], this can be written as follows. sequential and parallel execution. This way of code parallelizing Suppose that there is a reliable numerical method for solving allows one to save a lot of human resources for coding and two-dimensional and three-dimensional nonstationary problems debugging. For both types of parallel technologies special control of computational gas dynamics. Then we can obtain a numerical interfaces for parameter optimization and analysis were designed solution 𝐹(𝑥, 𝑦, 𝑧, 𝑡, 𝐴1 , … , 𝐴𝑁 ) for any point in the space of a [11, 12]. computational domain, where x, y, z are the spatial coordinates, With the help of the constructed interfaces, a series of t is the time, 𝐴1 , … , 𝐴𝑁 are the defining parameters of the calculations were carried out, realizing the concept of a problem. As defining parameters of the problem, we will keep in generalized numerical experiment for various classes of mind the characteristic numbers describing the properties of the problems. The results of the calculations will be shown in the flow under consideration, such as the Mach numbers, Reynolds, following sections. Both developed interfaces are very versatile. Prandtl, Strouhal, etc., and the characteristic geometric They can be applied to almost any software code for solving the parameters. Each of the characteristic parameters is limited in a CFD problem chosen as the base one. certain range According to [9-12], as a result of implementing the 𝐴𝑚𝑖𝑛 𝑖 ≤ 𝐴𝑖 ≤ 𝐴𝑚𝑎𝑥 𝑖 , 𝑖 = 1, … , 𝑁. construction of a generalized numerical experiment and We divide each of the parameters 𝐴𝑖 into k-1 parts, so we performing parallel calculations, we obtain a large data set obtain for each parameter a partition consisting of k points. The representing a set of numerical solutions F(x, y, z, t, A1 , … , AN ) volume of an N-dimensional space formed by a set of defining for each point (A∗1 , … , A∗N ) of the partition of the parameters 𝐴𝑖 is filled with a set of 𝑘 𝑁 points. multidimensional volume of the defining parameters (A1 , … , AN ) Denoting the point from the given set, as (𝐴1∗ , … , 𝐴∗𝑁 ), we of the problem under consideration. This volume in its original arrive at the fact that for each point of the collection it is form is rather difficult to use, although its availability for further necessary to obtain a numerical solution of the gas-dynamic purposes is necessary. In order to get useful information from a problem 𝐹(, 𝑥, 𝑦, 𝑧, 𝑡, 𝐴1∗ , … , 𝐴∗𝑁 ). calculated multidimensional data array, first of all we need to It is easy to see that this will require solving 𝑘 𝑁 gasdynamic reduce its dimension. By lowering its dimension, we are able to problems, which is impossible without the use of parallel apply the methods of visualization and visual analytics calculations in a multitask mode. In practice, the number N [23,24,27,28] in order to understand the internal structure of the usually does not exceed 5, which corresponds to the computing array and to reveal hidden interdependencies between its capabilities at the current time. defining factors. The revealed dependencies can be further It should also be noted that we formulated the classical approximated by geometric primitives in order to obtain a problem of parametric study. Parametric numerical studies allow generalizing dependence, which will represent the solution of one to obtain a solution not for one particular mathematical interest for a class of problems. Examples of the implementation modeling problem, but for a class of problems defined in a of this approach are presented in [9-14]. multidimensional space of defining parameters. Also, such a Also, to reduce the dimension of a multidimensional array, formal formulation allows numerical study of optimization methods of mapping into embedded manifolds of smaller analysis problems, when the inverse problem is solved at each dimension are very effective [20, 21, 29]. Among them, the most point of the grid partition of the multidimensional space of the common method is the principal component method (PCA). The determining parameters. Both types of similar problems are essence of the method consists in the transition from the initial considered in a series of papers [9-14]. coordinate system to the new orthogonal basis in the The only way to effectively carry out a generalized numerical multidimensional space under consideration, whose axes are experiment is applying of parallel computations. The problem of oriented along the directions of maximum dispersion. The the optimal and effective way of parallelization was thoroughly possible scheme of working with an array in this case is the discussed in the papers [11, 12]. There were considered parts of approximation by primitives of the data array in the space of the the whole algorithm for parameter optimization and analysis. For first three main components and the subsequent transition to the these parts the main criterion of applicability for parallelizing is initial space of the determining parameters. independence of specific numerical method. From this point of 4. Some examples of generalized numerical coefficients of the difference scheme, the Reynolds number of experiment the problem. As a result of the generalized computational experiment, a limit surface was constructed for the dependence This section contains the examples of the generalized of the weight coefficient on the other determining parameters of numerical experiment application to some practical problems. It the problem. An example of the limiting surface is presented in is applied in some variations due to different aims for each class figure 2. When choosing the value of the weighting factor below of problems. the surface, in the numerical solution, non-physical oscillations The first example of generalized numerical experiment is arise, which can lead to the collapse of the solution. Such devoted to the problem of tuning the properties of hybrid finite- surfaces are constructed for non-viscous and viscous flow. In the difference schemes [16]. The paper [16] contains the description case of viscous flow, laminar and turbulent regimes are of developed program tool Burgers2. This program tool is considered. intended for tuning and optimization of computational properties for hybrid finite-difference schemes applied to Burgers equation. One-dimensional model Burgers equation describes propagation of disturbances for dissipative medium. The equation has exact solution, so it is widely used for tuning-up of computational tools. Described program tool is based on combining of optimization problem solution and visual data presentation. Visual presentations of maximal error surface and error function are implemented as program tool features. Users have possibility of creating hybrid finite-difference schemes and analyzing computational properties for chosen grid template provided by program tool. Visual presentation of optimization problem solution allows finding of suitable weight coefficients for hybrid finite-difference scheme under consideration. The user is able to make simultaneous calculations varying weight coefficients in the scheme and viscosity coefficient in Burgers equation. The user can make the calculations simultaneously different sets of weight coefficients in accordance with the concept of generalized numerical experiment. Figure 1 presents the surface of absolute error for one of the hybrid scheme variants. The negative data Fig. 2. Surface of absolute error for far wake problem. area indicates where the oscillations occur. The next example considers the problem of the evaluation of the accuracy for different numerical methods. The problem of inviscid compressible flow around a cone at zero angle of attack is used as a base one. The results obtained with the help of various OpenFOAM solvers are compared with the known numerical solution of the problem with the variation of cone angle and flow velocity [17]. Cone angle β changes from 10° to 35° in steps of 5°. Mach number varies from 2 to 7. For comparison, four solvers were selected from the OpenFOAM software package: RhoCentralFoam, SonicFoam, RhoPimpleFoam, RhoPimpleFoam. The results of such kind of numericsl experiment were presented as errors in the form of an analog of the L2 norm for all solvers. Figure 3 illustrates the results in a form of a change in deviation from the exact solution for pressure depending on the cone angle and the velocity for the solver rhoCentralFoam. Such changes were obtained for all solvers. Fig. 1. Surface of absolute error for Burgers equation test [16]. Figure 3 shows a multidimensional dataset for pressure obtained as a result of parametric calculations in the space of the The following example is also devoted to improving the first three principal components. Yellow shows the results for computational properties of finite-difference schemes. The rhoCentralFoam solver, red for pisoCentralFoam, green for problem of mathematical modelling of the flow in the far wake sonicFoam and blue for rhoPimpleFoam. behind the body is solved. In the general case, in a rectangular Figure 3 shows that the errors for rhoCentralFoam and for computational domain, a viscous compressible heat-conducting pisoCentralFoam can be roughly approximated by a plane gas flow is considered, described by a complete system of time- reflecting the dependence of the error on the Mach number and dependent Navier-Stokes equations. At the input boundary, the cone angle. The results for sonicFoam and especially for distributions of gas-dynamic parameters are given, obtained from rhoPimpleFoam are significantly separated from the results for calculations of the flow around an axisymmetric body and a the first two solvers due to their particular numerical portion of the track behind it. The main goal of the generalized characteristics. This methodical research can serve as a basis for computational method was to thoroughly study the properties of selecting the OpenFoam solver for calculating the inviscid artificial viscosity incorporated in the hybrid difference scheme. supersonic flow around the elongated bodies of rotation. The For this purpose, we studied the properties of the weight results of solvers comparison can also be useful for developers coefficients of the hybrid scheme on the example of the problem of OpenFoam software content. The results obtained made it of flow in the far wake and determined the limitations for the possible to get a general idea of the calculation errors for all weight coefficients. In this task, the following defining solvers. parameters were varied, such as the steps of the grid decomposition in the x and y directions, the weighting the field of computational gas dynamics. Parametric studies can serve as such an experiment, where the basis is the ability to solve the direct problem of mathematical modeling. An example of an optimization analysis problem is given, where the generalized computational experiment is based on solving the inverse problem in an optimization statement. Thus, to create such an experiment is quite realistic for almost any mathematical modeling problem. A separate area of application of a generalized computational experiment can be a comparative assessment of the accuracy of numerical methods. Similar attempts are presented in [2, 15] devoted to the problem of accuracy estimation with the help of the ensemble of solutions. According to [2, 15], if a researcher is able to calculate the same problem using several numerical methods with different computational properties, in particular, Fig. 3. Errors for different OpenFOAM solvers in the space of different approximation orders, then in some cases one can principal components. estimate the neighborhood of the approximate solution containing the exact solution (exact solution enclosure). If an The next example of application of general numerical ensemble of numerical solutions can be divided into clusters of experiment considers optimization problem. The example “accurate” and “inaccurate” solutions, then the error ranking of presents a search for optimal shape of three-dimensional blade values can be performed using an a posteriori analysis of the assembly intended for power plant [3]. This experiment is based distances between the numerical solutions. This can serve as a on developed computational technology for the computation of computational proof of the existence of an exact solution in the power loads on the 3D blade assembly of a power plant in a wind case of nonlinear problems. This approach can be considered as flow. The calculation for various combinations of the key perspective. Nevertheless, it has evident draw-back. For using of geometric parameters of the assembly using parallel this approach one should have a set of solvers with different computations makes it possible to find the optimal shape of the accu-racy order. assembly with respect to its power characteristics. A virtual experimental facility for simulating the flow around the power 6. Conclusions plant based on the solution of the Navier–Stokes equations was created. Computations aimed at determining the optimal shape of The concept of generalized numerical experiment presented the blade assembly taking into account constraints on its design in the article has a wide range of possible applications. For the were carried out, and the results were thoroughly analyzed using problems of computational fluid dynamics such an approach the proposed optimization procedure. The solution of the makes it possible to obtain a solution not only for one, separately optimization problem is based on the parameterization of the taken, problem, but for a whole class of problems defined in a design using three key parameters. On the discrete set of values certain range of the complex of determining parameters. Practical of these parameters, the maximums of two objective functions— implementation of the approach becomes possible with the use the magnitude of the total aerodynamic force and the magnitude of parallel calculations in multitask mode. The results of of the rotation torque—determining the lift-to-drag ratio of the calculations are multidimensional volumes of data that can be power plant are found. Figure 4 presents the shape of 3D blade processed using data analysis tools and visual analytics. The assembly and pressure distribution on its surface. application of these methods reveals hidden interdependencies between the determining parameters of the class of problems. Also, these methods allow in many cases to build the dependence of the valuable functional on the determining parameters, which makes it possible to further approximate it with geometric primitives and present it in an analytical form. The examples presented in the article illustrate generalized computational experiments for various types of tasks, such as improving the computational properties of difference schemes, finding the optimal body shape in a stream, a comparative assessment of the accuracy of algorithms. 7. 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