Informatics, computer technology and management. Рубрика в журнале - Siberian Aerospace Journal
Scientific article
The development of unmanned aircraft systems is becoming one of the key areas of Russia's scientific and technological development. Within the framework of the approved strategy until 2030, it is planned to form a new industry specializing in the creation and use of civilian drones, and priority areas for the development of unmanned technologies have been identified. The great potential for the use of unmanned vehicles is observed in the creation of geospatial databases. In this connection, when using autonomous aerial platforms for aerogeophysical works, the issues of improvement and development of information-measuring systems used on such carriers, as well as the issues of training of operators of aerogeo-physical systems become relevant. The development of algorithmic and software of information-measuring systems on the basis of implementation of previously created electronic content: reference materials, instructions for work, operation of instrumentation products, protocols for processing of obtained results, decisionmaking techniques, etc. is becoming important. However, the application of such electronic content carries risks of reducing the quality of surveying works. One of the ways of risk assessment is to combine the traditional qualitative and categorical approach of modern risk management with multilevel representation of the hierarchy of aggregation of expert-assessed risks in the composition of summary and integral risks by applying the apparatus of soft computing. For systematization and prioritization of risks it is proposed to use gradation scales and terms of linguistic variables of risk components associated with the probability of occurrence of a decrease in the quality of content of information support of operator training systems, with the potential value of possible damage in the event of the fact of a decrease in the quality of information support. As a result, it becomes possible to develop a risk-assessment model, which is a procedure for the specific construction of matrices of consequences and probabilities for individual risk-indicators of the assessment of the integration of electronic content in the information support of training systems for operators of information-measuring systems of aerogeophysical applications. This procedure is a logical basis for calculation of summary and integral values of risk-indicators of assessment, as well as gradation of risk levels.
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Software system for mathematical simulation of the electronic beam welding process
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Within the framework of this study, a software system for modeling the distribution of the temperature field in the steady-state mode of the electron-beam welding process for thin-walled aerospace structures is proposed. The purpose of creating such a software system is to improve the quality of control of the electron-beam welding process and, accordingly, to reduce the number of defects in welded joints of thin-walled structures. The software system has a model structure and implements the energy distribution models proposed earlier by the authors. The MySQL database management system and the Embarcadero RAD Studio programming system were chosen as the means of implementing the program. The central link of the system is a database that allows you to store and process information both on mathematical modeling and on the results of simulation and field experiments. The article describes the structure of the developed software system, and also presents algorithms for the operation of its constituent modules. The system provides the user with the opportunity not only to carry out simulation according to the specified technological parameters (welding speed, accelerating voltage, beam current, boundary conditions, simulation time, product material), but also to visualize the results and save them in a single database. The use of the proposed system allows not only to minimize the costs of the enterprise for the development of technological parameters of the steady state for the electron-beam welding process, but also to create a flexible information base for collecting experimental information with the aim of further automating and intellectualizing the technological process of creating permanent joints in the framework of Industry 4.0.
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In this paper, we consider the solution of the filtering problem using the Kalman filter with the optimal tuning of the radio-reflecting net. A large-sized transformable space-based reflector is considered. In the process of this structure placing in orbit, it is possible that the real form of the radio-reflecting net can deviate from the desired one. To ensure point-to-point adjustment of the active part of the mesh, a cable-cable system is used. The nodal points of the radio-reflecting surface are connected to the back side of the net through cables. They have built-in actuators that allow one to change the length of the cables. A piezo actuator was selected as a control device. By point-by-point adjustment of the piezo actuators the net is stretched to the required shape. This ensures a high-quality radiation pattern and a high signal level when receiving and transmitting data. Specific values of the disturbing influences are given. To measure the supply voltage on the piezo actuator and the cable length, a voltage converter and a laser scanner are used. Possible deviations from the calculated initial position are determined. In accordance with the principle of separation, the estimation problem is solved first, then the control problem. The estimation problem is solved using the Kalman filter. The control problem is solved using the optimal control algorithm according to the hierarchy of target criteria. The results of numerical simulation are presented. The successful solution of the problem is shown with variable values of measurement noise and disturbing influences. Comparison with trajectories obtained applying various optimal control algorithms is given.
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Scientific article
Failures of elements during the operation of technical and many other systems are, as a rule, random in nature. This leads to various models of the recovery process, studied in probability theory and mathematical reliability theory. During the restoration process, failed elements are restored or replaced with new ones, and there is often a change in the costs and quality of the restored elements (time-to-failure distribution functions). The work examines the cost function (average cost of restoration) in the process of restoration of order k1, k2 , in which, according to a certain rule, the costs of each restoration and the distribution functions of operating time change. Considering, that the recovery function (average number of failures) is well studied in reliability theory, a solution to the integral equation for the cost function is obtained through the recovery function of the model under consideration. For the order restoration process 1 2 k , k , a formula is obtained for calculating the cost function through the restoration function of a simple process formed by the convolution of all distribution functions of the periodic part. For practical application, explicit formulas are obtained for the cost function during the restoration process, in which the periodic part is distributed according to an exponential law or an Erlang law of order m with the same exponent α. The resulting formulas can be used to study the properties of the cost function and solve optimization problems in strategies for carrying out the restoration process in terms of price, quality, risk, if, for example, the average number of failures is taken as quality, the average cost of restorations as price, the dispersion of the number of failures as the risk, or cost of restoration.
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Solution to the first boundary value problem of plane elasticity theory using conservation laws
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A huge number of works are devoted to solving boundary value problems for the equations of plane elasticity theory. The largest number of studies in this area are based on the formula found by G. V. Kolosov. He was the first to express the general solution to the problem of plane elastic deforma-tion by finding two independent functions of a complex variable. This made it possible to apply a welldeveloped theory of analytic functions to solving problems of elasticity theory. Later, the solution method based on Kolosov's formula was developed by his student N. I. Muskhelishvili. But the described method also has significant limitations. It is applicable only to those areas that can be conformally mapped onto a circle. Therefore, other methods for solving elasticity theory problems are also needed, since a large number of practically important problems are solved for areas that do not satisfy this condition. The method developed in the work is based on the use of conservation laws that are constructed for equations describing a plane deformable state. The assumptions made in the work make it possible to construct a solution to the first boundary value problem for arbitrary plane areas bounded by a piecewise smooth contour. In this case, finding the components of the stress tensor is reduced to calculating contour integrals along the boundary of the region under consideration. As in the case considered by G. V. Kolosov, the solution to the problem is based on two exact solutions of the Cauchy – Riemann equations, which have singularities at an arbitrary point in the region under consideration.
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Scientific article
The plane problem for elasticity equations is well studied. It can be explained by its importance for applications and by the fact that the equations can be reduced to the Cauchy-Riemann system. In spite of this importance, exact solutions that would describe the stress-strain state of bodies of finite dimensions are not numerous. Conservation laws for differential equations have been appeared more than a hundred years ago, but, as a rule, they were not used to solve specific problems, but were of purely academic interest. The situation changed with the development of the technique of construction of conservation laws for arbitrary systems of differential equations, and then with the use of conservation laws to solve boundary value problems of the theory of plasticity and elastic-plasticity. In this article, new conservation laws are constructed for the equations of the plane theory of elasticity in the stationary case. These laws form an infinite series, which is closely related to the elasticity equations solving. This fact made possible to reduce solving of boundary value problems, in terms of displacements, to the calculation of contour integrals along the boundary of a domain bounded by the studying elastic body. As it follows from the proposed technique, the studied area can be multiply connected, and the considered boundary can be piecewise-smooth.
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Solving the problem of stretching an elastic-plastic strip weakened by cuts and holes
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In this paper, the boundary between elastic and plastic regions in a stretchable strip is con-structed. The band is weakened by side slits and holes. Such tasks are still relevant, since its solution allows us to make an assessment of the limiting state of the structure under consideration. Cuts can have an arbitrary shape, their number is not limited. For simplicity, only sections with rectilinear boundaries are considered in the article. The holes may have an arbitrary shape and be located anywhere in the strip. In operation, only circular holes are considered for simplicity. Numerical methods are currently very often used to solve such a problem, unfortunately, often without much justification. Therefore, analytical methods for solving such problems are becoming more and more relevant. In this paper, the conservation laws of differential equations are used. The conserved current is linear in the first derivatives. The task is solved in two stages. At the first stage, Dirichlet is solved for the Laplace equation, and at the second stage, the technique of conservation laws is used. Their use makes it possible to reduce the finding of the components of the stress tensor at each point to a contour integral along the boundaries of the region un-der consideration. And this makes it possible to build an elastic-plastic boundary.
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Scientific article
The article presents the problem of optimizing the structure of information processing computer appli-ances for real-time control systems used, among other things, in the rocket and space industry. In addition, the features of this problem that affect the choice of optimization methods are studied. It’s concluded that this problem can be effectively solved using evolutionary optimization methods. Existing performance models allow you to determine the minimum hardware configuration of a multi-processor computing system. The approach proposed in this article allows us to find configurations that have hardware redundancy (compared to the minimum configuration), but, due to this, have a greater probability of being in states that provide performance sufficient to achieve the goals of functioning of the designed real-time control system. The described approach is more flexible than simply duplicating all hardware components of the minimum configuration, which can be used to reduce the cost of creating and operating the designed control system. The proposed model can be used to optimize the performance of multiprocessor hardware and software complexes of real-time control systems. At the same time, it should be taken into account that the resources allocated for the creation and operation of the hardware and software complex are always limited. There-fore, it is advisable to consider the problem of performance optimization as a multi-criterion: one criterion will be performance, and the other-the cost of creating a hardware and software complex.
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Scientific article
The S-box is the most important component of modern cryptographic algorithms which largely determines the quality of cryptographic transformation. The modern method of estimating the S-boxes quality employs their representation as component Boolean functions to which cryptographic quality criteria are applied. Such criteria include: nonlinearity, correlation immunity, an error propagation criterion, and a strict avalanche criterion. Nevertheless, it is obvious that a cryptanalyst is not constrained in the ways of representing the cipher components, in particular, using the functions of many-valued logic. The design features of modern cryptographic algorithms allow their representation in the form of 4-logic functions, which determines the need to research cryptographic properties of the S-boxes represented as component 4-functions. In the literature today there are methods for measuring the nonlinearity of 4-functions; nevertheless, there are no similar methods for researching the differential properties of 4-functions, in particular, involving their compliance with the strict avalanche criterion. In this paper the strict avalanche criterion is generalized to the case of 4-functions and the compliance of the S-boxes component 4-functions of the “Magma” cryptoalgorithm to the strict avalanche criterion has been researched. All balanced 4-functions of length N = 16 satisfying the strict avalanche criterion were synthesized using the restricted brute-force method. The basic properties of the constructed class of 4-functions are determined, and bijective S-boxes based on them are constructed. It has been established that S-boxes of length N = 16 satisfying the strict avalanche criterion, both in terms of component Boolean functions and in terms of 4-functions, also possess optimal nonlinear properties. This circumstance allows us to recommend S-boxes satisfying the strict avalanche criterion of component 4-functions for use in modern cryptographic algorithms.
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Structure, network protocols of the internet of things for quality production control
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The subject of research is the model and structure of the Internet of things (IoT) network for product quality control in industry and agriculture. The purpose of the article is to analyze communication protocols and structures of IoT networks. The method of analysis and structural design of IoT networks is applied. The field of application is automation of monitoring products of enterprises of the aerospace industry. The article provides an overview and analysis of existing IoT technology; it considers the protocols and composition of IoT networks, and provides variations in the structures of building such networks. 4 levels of IoT architecture are described, as well as the communication protocols are used. The directions of building the Internet of things network for product quality control are defined. A multi-agent model of such system is presented, for the implementation of which the structure of the IoT network is given. The structure of a multi-agent system (MAS) for monitoring product quality in industry and agriculture includes many agents, such as product quality agents, communication agents, database agents, agents for analyzing information received from sensor agents, and decision-making agents. This MAS implements functions to ensure the required class of product quality and it is based on building a local network of the Internet of things. The research proposes an algorithm for processing information in such an IoT network. Analyzers (sensors) product qualities will be periodically polled and their values will be recorded in the server database. The decision-making subsystem sends data on product quality compliance to the enterprise administrator on a mobile device. The server structure is implemented using cloud IoT platforms, for which a brief overview is provided. The one IoT network implementation is developed using LTE NBIoT technology. This approach can be used in the aerospace industry for product quality control within automation 4.0.
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Studying dynamics and classification of attacks to corporate network web services
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The article presents a study of the dynamics of attacks onto the web services using the classification of cyber threats by types due to the example of the corporate network of the Krasnoyarsk Scientific Center of the Siberian Branch of the Russian Academy of Sciences. The analysis is carried out on the basis of web services logs and it allows to solve urgent problems of ensuring the integrated security of web services, including identifying both existing and potential cybersecurity threats. The article demonstrates a review of the main approaches to the processing and analyzing logs. The authors describe the type and composition of data sources and provide a list of the software used. A feature of the study is a long observation period. The structure of the processing system is proposed and software tools for attack analysis and classification are implemented. The research shows that the use of classified samples allows to detect periodicity and reveal trends of certain types of attacks. Unclassified attacks have similar distribution parameters for different years, while in the case of classification, the distribution parameters have changed significantly, which makes it possible to track risks in automated intrusion prevention systems. A correlation matrix by type of attack is constructed. The analysis shows that most attack types have weak correlation, with the exception of the attacks “command injection”, “directory browsing”, “Java code injection”, which can be aggregated. The authors propose a heuristic method of risk comparison based on cyber threat classification. The method uses statistical parameters of sample distributions and permits to deal with different time intervals. The paper georeferences the IP addresses from which the attacks are carried out, builds attack profiles for different countries, and provides a list of countries with a stable attack profile. The conclusion indicates the features of the proposed method and outlines the prospects for its use in other areas.
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System analysis of dynamic problems of anisotropic plasticity theory
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Dynamic problems are the least studied area of plasticity theory. These problems arise in various fields of engineering and science, but the complexity of the original differential equations do not allow to develop accurate solutions and correctly solve numerical boundary value problems. This is even more typical of dynamic equations of anisotropic plasticity. Anisotropy reduces the group of symmetries allowed by the equations, and therefore narrows the number of invariant solutions. One-dimensional dynamic plasticity problems are well studied, but two-dimensional problems cause insurmountable mathematical difficulties due to the nonlinearity of the basic equations, even in the isotropic case. The study of the symmetries of the plasticity equations allowed us to find some exact solutions. The most known solution was found by B. D. Annin, who described the unsteady compression of a plastic layer made of isotropic material by rigid plates. Annin's solution is linear in two spatial variables, however, it includes arbitrary functions of time. Symmetries are also used in the proposed work. Point symmetries are first calculated for dynamic plasticity equations in the anisotropic case and are presented in the paper. The Lie algebra generated by the found symmetries appeared to be infinite-dimensional. This circumstance made it possible to apply the method of constructing new classes of nonstationary solutions. Symmetry can transform the exact solution of stationary dynamic equations in non-stationary solutions. The framed solutions include arbitrary functions and arbitrary constants. The outline of the article is as follows: according to the method of Lie group of point symmetries allowed by the equations of anisotropic plasticity is calculated. Two classes of new stationary invariant solutions are framed. These stationary solutions, by means of transformations generated by point symmetries, are transformed into new non-stationary solutions. In conclusion, a new selfsimilar solution of unsteady equations of anisotropic plasticity is framed; Annin's solution is generalized for the anisotropic case. The framed solutions can be used to describe the compression of plastic material between rigid plates, as well as to test programs, designed to solve anisotropic plastic problems.
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Technical system simulation with Python
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The results of the development of a scheduler for the joint execution of simulation models of multicomponent systems are presented. The software is implemented in Python, which allows integration with numerous libraries for control and data analysis. Data exchange is carried out via UDP packets that support different programming languages. This simplifies the implementation of hardware-in-the-loop technology, improving the development of control systems. An example of using the scheduler is demon-strated on the model of the attitude determination and condtrol system of a CubeSat spacecraft with a magnetic orientation system. The B-Dot algorithm and the results of simulating the transient process are provided. The source code is available under the BSD license on GitFlic, and the documentation is available on ReadTheDocs.
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Scientific article
Dynamic multi-objective optimization problems are challenging and currently not-well understood class of optimization problems but this class is of great practical value. In such problems, the objective functions, their parameters and restrictions imposed on the search space can change over time. This fact means that solutions of the problem change too. When changes appear in the problem, an optimization algorithm needs to adapt to the changes in such a way that the convergence rate is sufficiently high. The work is devoted to the comparison of the different approaches to formation of a new population when changes in the dynamic multi-objective optimization problem appear: using solution, which obtained in the previous step; using a random generating of the population; partial using solutions which obtained in the previous step. In the first part of the article the classification of the changes in the problems is provided; the currently existing approaches to solving the problems based on evolutionary algorithms are considered. During the research NSGA-2 and SPEA2 algorithms are used to solving the dynamic optimization problems, the benchmark problems set is used to the comparison of the approaches. Obtained results being processed by Mann–Whitney U-test. It was obtained that changes rate in the problem affect the efficiency of the application of the solutions obtained in the previous step of new population the forming.
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Scientific article
In practice problems, which consist in the search of the best (optimal) solution according to the different irredundant and contradictory (conflicting) criteria, called multi-objective problems, are of frequent occurrence. One of the most commonly used methods for solving this kind of problems consists in combination of all criteria into the single one by using some linear relation. However, despite the simplicity of this method, solving problems with its help may cause other problems related to the determination of the mentioned linear combination, namely related to the determination of the weight coefficients for each criterion. The incorrect selection of these coefficients may lead to non-optimal solutions (according to the Pareto theory). In this regard, recently various population-based algorithms have been proposed for solving the described problems, which are the modifications of these population-based algorithms for solving singleobjective optimization problems. This article describes the developed modifications of the Differential Evolution algorithm (DE) for solving multi-objective unconstrained optimization problems based on the well-known NSGA (Nondominated Sorting Genetic Algorithm) and MOEA/D (Multiobjective Evolutionary Algorithm Based on Decomposition) schemes, which use the Pareto theory. The investigation into the efficiency of the Differential Evolution algorithm for solving multi-objective optimization problems in relation to the chosen mutation operator of the original DE algorithm and to the multi-objective scheme was conducted. The developed modifications were tested by using some well-known multi-objective real-valued optimization problems with 30 variables, such as ZDT1, ZDT2, ZDT3, etc. The practical problem of spacecraft control contour variant choice was solved as well. The experimental results show that better results were achieved by the Differential Evolution algorithm with the simplest mutation operators combined with the NSGA scheme. Thus, the applicability of the described modification for solving practical multi-objective optimization problems was demonstrated.
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Scientific article
The objective of this work is the computational and experimental assessment of the effect of twisting of an active-reactive type projectile-piercer around its own axis on the parameters of its motion and the depth of penetration into the soil. Research methods: the equations of motion of a rotating active-reactive type projectile-piercer (SPART) are considered. The features of determining the engine thrust, rotating SPART and the resistance force during penetration of SPART into the soil with rotation are ana-lyzed. A comparative analysis of the penetration depths of rotating and non-rotating SPART into loam is carried out. As a result of the studies, a mathematical model of the process of penetration of an active-reactive type projectile-piercer into the soil is developed. The effect of SPART rotation around its own axis of symmetry on the performance of the propulsion system is shown. The effect of contact friction forces be-tween the rotating SPART and the soil on the parameters of its motion and the depth of penetration is as-sessed. Calculations show that by spinning the SPART around its own axis of symmetry, the depth of pene-tration of rotating active-reactive type projectilе-piercers into the soil can be increased by 8–10 %. Con-clusion: the results presented in the article can be useful for researchers, postgraduate students and engi-neers involved in the creation and operation of aviation and rocket and space technology, and can also be useful for students of technical universities studying at the relevant specialties.
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The paper proposes the method for the numerical solution of Poisson kinematic equations system determining the evolution of the spacecraft position. The system of Poisson kinematic equations is used to designate the transition matrix from the coordinate system associated with the spacecraft at the selected time t1 to the coordinate system associated with the spacecraft at the current time t2. This matrix is used in the process of solving problems of determining a three-axis orientation of the spacecraft from the readings of the magnetometer using information about its angular velocities. The proposed method is based on replacing the derivatives of the desired functions in the Poisson kinematic equations by partial sums of series in the scaled Haar system. The partial sums of these series are generalized polynomials in the scaled Haar system. Hence, these sums are step (piecewise constant) functions. The estimates of the proposed method error are derived, which reveal that in the case of the coefficients of the equations which are functions matching the Lipschitz condition, the absolute error in calculating each of the elements of the transition matrix from one coordinate system to another is the value O(N–1) at N , where N is the number of partitions of the segment [t1, t2] when constructing a grid of nodes involved in this method. It is proved that the complexity of constructing an algorithm for approximating the system of Poisson kinematic properties insignificantly exceeds the complexity of solving this system by Euler method, which has the first order of accuracy. The research presents the results of numerical experiments, showing that in certain cases the Haar sums method gives an error that is much smaller than the Euler method, and is almost identical to the errors of the Euler – Cauchy and Runge – Kutta methods of the 2nd order, the complexity of which is approximately two times greater than the complexity of the Haar sums method.
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Scientific article
Plates, beams and shells with non-uniform and micro-inhomogeneities regular structure are widely used in aviation and rocket and space technology. At the preliminary design stage, it is initially important to know whether the design safety factor meets the specified strength conditions. To determine the margin factor, it is necessary to solve the elasticity problem for the designed structure by the finite element method (FEM), taking into account its inhomogeneous structure, which requires large computer resources. In this paper, we propose a method of equivalent strength conditions (MESC) for calculating the static strength of elastic structures with a inhomogeneous regular structure. The proposed method is reduced to the calculation of the strength of isotropic homogeneous bodies using equivalent strength conditions. The MESC is based on the following statement. For any composite body V0 , there exists such an isotropic homogeneous body Vb and such a number p (equivalence coefficient) that if the body Vb stock coefficient satisfies 0 nb the equivalent strength conditions 0 pn1 nb pn2 , then the body V0 stock coefficient satisfies n0 the given strength conditions n1 n0 n2 , and Vice versa, n1 , n2 – given, the coefficients 0 nb , n0 , meet the exact solutions of elasticity problems constructed for bodies V0 , Vb . The method under consideration is reduced to FEM strength calculation of isotropic homogeneous bodies, which is the easiest to implement and requires less computer memory than a similar calculation of composite bodies taking into account their inhomogeneous structure. The procedure for determining the equivalence coefficients for a number of composite plates, beams and shells of rotation is described. High-precision multigrid finite elements generating discrete models of small dimension and solutions with small error are used in the construction of elastic solutions according to FEM for isotropic homogeneous bodies. The adjusted equivalent strength conditions are of the form pn1(11) nb pn2 (12 ) , where nb is the body Vb reserve coefficient and the values 1 , 2 correspond to the approximate solution constructed for the body Vb . Implementation of FEM for multigrid discrete models requires several 103-106 times less computer memory than for basic models. The calculation of the strength of a beam with a micro-homogeneous regular structure with the help of MESC is given.
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Scientific article
When the strength of elastic composite structures (plates, beams, shells) widely used in aviation, rocket and space technology is calculated with the finite element method (FEM), it is important to know the solu-tion error. To analyze the solution error, it is necessary to use a sequence of approximate solutions con-structed according to the FEM using the grinding procedure for basic discrete models (BMs), which take into account an inhomogeneous microheterogeneous structure of bodies within the microapproach. Dis-crete models obtained by grinding BMs have a high dimension, which makes it difficult to use the FEM for them. In addition, there are BMs of composite solids (CSs), for example, BMs of bodies with a microhet-erogeneous structure, which have such a high dimension that the implementation of the FEM for such BMs is practically impossible due to limited computer resources. To solve these problems, it is proposed to use fictitious discrete models in the calculations of CSs according to the FEM. In this paper we propose a method of fictitious discrete models (MFDM) for calculating the strength of elastic bodies with an inhomogeneous microheterogeneous regular structure. The MFDM is implemented with the help of the FEM using corrected strength conditions, which take into account the error of ap-proximate solutions. The method is based on the following provision. We believe that BMs of CSs generate solutions that slightly differ from the exact ones. Such BMs always exist for CSs due to the convergence of the FEM. The calculation of CSs according to the MFDM is reduced to the construction and calculation of the strength of fictitious discrete models (FMs), the dimensions of which are smaller than the dimension of the BMs. FMs reflect: the shape, characteristic dimensions, fastening, loading and the type of the inhomogeneous structure of CSs and the distribution of the elastic moduli corresponding to the BM of the CS. The sequence consisting of the FM converges to the BM, i.e., the limiting FM coincides with the BM. The convergence of such a sequence ensures uniform convergence of the FM stresses to the corresponding BM stresses. The implementation of the FEM for FMs with the use of multigrid finite elements leads to a large saving of computer resources, which makes it possible to use the MFDM for strength calculations of bodies with a microheterogeneous regular structure. Calculation of the CS strength according to the MFDM requires times less computer memory volume than a similar calculation using the BM of the CS, and does not contain the procedure for grinding the BM. The given example of calculating the strength of a beam with an inhomogeneous regular fibrous structure according to the MFDM shows its high efficiency. Applying the adjusted strength conditions allows using approximate solutions with larger errors in CS strength calculations, which leads to improving the efficiency of the MFDM.
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