Pub Date : 2020-09-08DOI: 10.33842/2313-125X/2020/19/124/129
A. Nesvidomina
The process of modeling the temperature distribution on surfaces, applying an image to curved areas with minimal distortion requires the formation of isometric grids on the plane and on the surface. One of the common ways to form planar isometric networks is to use the functions of a complex variable and planar isotropic curves, followed by separation of the real and imaginary parts. The development of computer models for the interactive search and analysis of isometric networks according to various initial geometric conditions provides a generalized method for their formation with the possibility of varying their shape and position. It is proposed to use an isotropic vector for the formation of flat isotropic curves, which ensured a single sequence of analytical calculations according to the following initial conditions: 1) selection of an arbitrary function of a real argument; 2) a given parametric equation of a plane curve; 3) a given polar equation of a plane curve. Since the analytical calculations of the derivation of the parametric equation of a plane isotropic curve and the corresponding isometric grid are rather laborious, their execution is carried out in the environment of the Maple symbolic algebra. To this end, the corresponding software has been created, which interactively allows you to select the function of a real argument, a parametric or polar equation of a plane guide curve. All subsequent stages of analytical transformations to form an isotropic curve and the corresponding isometric grid are carried out automatically. An interactive model for the formation and analysis of plane isotropic curves with various initial conditions has been created, which has shown its effectiveness, which is confirmed by the given examples of plane isometric grids for specific functions of the real parameter, plane curves in the parametric and polar form of their job.
{"title":"GENERALIZED METHOD FOR FORMING PLANE ISOTROPIC CURVES","authors":"A. Nesvidomina","doi":"10.33842/2313-125X/2020/19/124/129","DOIUrl":"https://doi.org/10.33842/2313-125X/2020/19/124/129","url":null,"abstract":"The process of modeling the temperature distribution on surfaces, applying an image to curved areas with minimal distortion requires the formation of isometric grids on the plane and on the surface. One of the common ways to form planar isometric networks is to use the functions of a complex variable and planar isotropic curves, followed by separation of the real and imaginary parts. The development of computer models for the interactive search and analysis of isometric networks according to various initial geometric conditions provides a generalized method for their formation with the possibility of varying their shape and position. It is proposed to use an isotropic vector for the formation of flat isotropic curves, which ensured a single sequence of analytical calculations according to the following initial conditions: 1) selection of an arbitrary function of a real argument; 2) a given parametric equation of a plane curve; 3) a given polar equation of a plane curve. Since the analytical calculations of the derivation of the parametric equation of a plane isotropic curve and the corresponding isometric grid are rather laborious, their execution is carried out in the environment of the Maple symbolic algebra. To this end, the corresponding software has been created, which interactively allows you to select the function of a real argument, a parametric or polar equation of a plane guide curve. All subsequent stages of analytical transformations to form an isotropic curve and the corresponding isometric grid are carried out automatically. An interactive model for the formation and analysis of plane isotropic curves with various initial conditions has been created, which has shown its effectiveness, which is confirmed by the given examples of plane isometric grids for specific functions of the real parameter, plane curves in the parametric and polar form of their job.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"21 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120859676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-08DOI: 10.33842/2313-125X/2019/19/50/56
N. Hrytsyna, V. Ragulin
Currently, the construction industry is in the process of actively introducing BIM technologies - information modeling technologies for buildings and structures. Unlike working in classical computer-aided design (CAD) systems, working in BIM-oriented software systems allows you to simulate not only the building objects themselves, but also manage their characteristics, as well as their various changes over time. In addition, one of the basic principles of information modeling is the desire to combine in one information model all stages of the construction life cycle and all sections of the design. In this work, we consider a version of the developed task in the discipline "Using the Revit package in the tasks of building bridges" for the educational program "Transport construction and civil engineering". At the Department of Engineering and Computer Graphics has computer Autodesk
{"title":"MODELING THE SPAN STRUCTURE OF THE BRIDGE IN THE AUTODESK REVIT SYSTEM","authors":"N. Hrytsyna, V. Ragulin","doi":"10.33842/2313-125X/2019/19/50/56","DOIUrl":"https://doi.org/10.33842/2313-125X/2019/19/50/56","url":null,"abstract":"Currently, the construction industry is in the process of actively introducing BIM technologies - information modeling technologies for buildings and structures. Unlike working in classical computer-aided design (CAD) systems, working in BIM-oriented software systems allows you to simulate not only the building objects themselves, but also manage their characteristics, as well as their various changes over time. In addition, one of the basic principles of information modeling is the desire to combine in one information model all stages of the construction life cycle and all sections of the design. In this work, we consider a version of the developed task in the discipline \"Using the Revit package in the tasks of building bridges\" for the educational program \"Transport construction and civil engineering\". At the Department of Engineering and Computer Graphics has computer Autodesk","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122875984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-08DOI: 10.33842/2313-125X/2020/19/130/137
Y. Orel, A. Magalov
This paper considers several of the approaches to determine the unit cost of constructing pipelines in various parts of the studied area of construction when modeling water supply networks that are effective from the point of view of building and operation. The result of the simulation is the optimized geometric shape of the discrete image of the piping network of the water supply system, the location of the branching nodes of which are determined by solving a system of nonlinear equations. The main complication in the formation of the corresponding equations, as well as in the modeling process as a whole, is the determination of specific indicators of the economic efficiency of a particular location of each of the pipeline sections. This is due to the fact that at different sites of the studied area of development, as a rule, various indicators of the cost of construction and operation of water supply networks are set. At the same time, some of the segments of pipelines can extend immediately to several sections of this area, which means that the above indicator will change over the entire length of the corresponding sections. It is proposed to apply a discrete or integral approach, providing for the determination of averaged values of special objective functions that are introduced for an objective assessment of the economic efficiency of the construction and maintenance of the respective systems, on each straight section of pipelines. Such evaluation functions allow you to display the uneven level of labor costs in the territory covered by the study area of modeling. The optimization problem itself is proposed to be solved by systemic tools of applied discrete geometry. This task is very important, because its solution allows even at the stage of design work to significantly reduce further construction and maintenance costs for water supply networks.
{"title":"DETERMINATION OF SPECIFIC INDICATORS OF THE COST OF CONSTRUCTION AND OPERATION OF PIPELINES OF EFFECTIVE WATER SUPPLY SYSTEMS","authors":"Y. Orel, A. Magalov","doi":"10.33842/2313-125X/2020/19/130/137","DOIUrl":"https://doi.org/10.33842/2313-125X/2020/19/130/137","url":null,"abstract":"This paper considers several of the approaches to determine the unit cost of constructing pipelines in various parts of the studied area of construction when modeling water supply networks that are effective from the point of view of building and operation. The result of the simulation is the optimized geometric shape of the discrete image of the piping network of the water supply system, the location of the branching nodes of which are determined by solving a system of nonlinear equations. The main complication in the formation of the corresponding equations, as well as in the modeling process as a whole, is the determination of specific indicators of the economic efficiency of a particular location of each of the pipeline sections. This is due to the fact that at different sites of the studied area of development, as a rule, various indicators of the cost of construction and operation of water supply networks are set. At the same time, some of the segments of pipelines can extend immediately to several sections of this area, which means that the above indicator will change over the entire length of the corresponding sections. It is proposed to apply a discrete or integral approach, providing for the determination of averaged values of special objective functions that are introduced for an objective assessment of the economic efficiency of the construction and maintenance of the respective systems, on each straight section of pipelines. Such evaluation functions allow you to display the uneven level of labor costs in the territory covered by the study area of modeling. The optimization problem itself is proposed to be solved by systemic tools of applied discrete geometry. This task is very important, because its solution allows even at the stage of design work to significantly reduce further construction and maintenance costs for water supply networks.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117017410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-07DOI: 10.33842/22195203/2020/18/69/82
V. Vereshchaha, Y. Adoniev, O. Pavlenko, K. Lysenko
The definition of compositional matrices (compomatrixes) is given, mathematical objects are defined that can be elements of compomatrixes. Shown are the requirements for indexing one-, two-, and three-dimensional compomatrixes and their purpose. The symbol designation of the real compromomats is established and it is indicated that they are intended for the analytical formalization of the description of geometric figures. It is indicated that the need to introduce the concept of “composite matrices” is caused by the nature of the formation of geometric figures (GF). It is determined what is the unification of the GF and why it is needed in compositional geometric modeling. The rules for the formation of a point compomatrix and a parametric compomatrix are provided, and their conventions are defined. It is proved that compomatrixes are used for geometric modeling of objects, each point of which is K-valued, that is, defined by the k-coordinates of the parameter space. It has been established that the number of elements and the form of their recording in the compatrix are in complete accordance with the number of points and their location on the original GF. and the parametric is combinative. The analysis of the parametric compatrix for point polynomials is carried out, its trace and determinant are given importance, the features of the transposed compomatrix to the original parametric are indicated. The main feature of the transposed parametric compomatrix is that it is equal to the original compomatrix, which allows the mobile simplex method to be used without restrictions for the geometric modeling method.
{"title":"ONE AND TWO-DIMENSIONAL VALID COMPOSITION MATRICES","authors":"V. Vereshchaha, Y. Adoniev, O. Pavlenko, K. Lysenko","doi":"10.33842/22195203/2020/18/69/82","DOIUrl":"https://doi.org/10.33842/22195203/2020/18/69/82","url":null,"abstract":"The definition of compositional matrices (compomatrixes) is given, mathematical objects are defined that can be elements of compomatrixes. Shown are the requirements for indexing one-, two-, and three-dimensional compomatrixes and their purpose. The symbol designation of the real compromomats is established and it is indicated that they are intended for the analytical formalization of the description of geometric figures. It is indicated that the need to introduce the concept of “composite matrices” is caused by the nature of the formation of geometric figures (GF). It is determined what is the unification of the GF and why it is needed in compositional geometric modeling. The rules for the formation of a point compomatrix and a parametric compomatrix are provided, and their conventions are defined. It is proved that compomatrixes are used for geometric modeling of objects, each point of which is K-valued, that is, defined by the k-coordinates of the parameter space. It has been established that the number of elements and the form of their recording in the compatrix are in complete accordance with the number of points and their location on the original GF. and the parametric is combinative. The analysis of the parametric compatrix for point polynomials is carried out, its trace and determinant are given importance, the features of the transposed compomatrix to the original parametric are indicated. The main feature of the transposed parametric compomatrix is that it is equal to the original compomatrix, which allows the mobile simplex method to be used without restrictions for the geometric modeling method.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116542606","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-07DOI: 10.33842/2313-125X/2020/18/18/27
A. Baranyuk, A. Rachinsky
The paper proposes a method of approximate calculation of the temperature field of the turbine blade in non-stationary mode by means of CFD-modeling. The academic version of the ANSYS-Fluent software package was used as a modeling tool. The local values of the heat transfer coefficient along the contour of the blade profile were calculated in the work. Previously, the gas flow velocity in the curved interscapular channel was calculated to determine the distribution of the gas flow velocity along the convex and concave surfaces of the blade. Comparison of the data thus obtained with the result of processing the experience conducted on the turbine, shows that in the conditions of the real rotating installation, the heat transfer to the surface of the blade was approximately 2 times higher. The most probable reason for this discrepancy is the earlier beginning of the transition to the turbulent boundary layer on the surface of the blade in real conditions of high turbulence of the oncoming flow. Calculations show that in the case of a sharp change in gas temperature, the largest temperature difference in the blade occurs after a few seconds and reaches 50 ... 90% of the change in gas temperature, and in blades with thin edges - close to this value. Increasing the thickness of the edges can significantly reduce the uneven temperature in the blades with sharp changes in gas temperature. In order to identify the nature of the stress state of the blades at different modes of operation of the gas turbine unit, the temperature stresses in the blade were calculated. The calculation was performed for Mises averaged stresses in the cross section of an unevenly heated blade. When starting the installation, the maximum force occurred 30 ... 40 seconds after the start of the rotation of the rotor almost simultaneously with the maximum temperature difference between the edges and the central part of the section. The highest stress values occurred in the region of the inlet edge, where they exceeded 680 MPa per compression. In the future, the temperature stresses decreased and even changed sign due to the cooling process of the blade, overheated when throwing the gas temperature.
{"title":"SIMULATION OF UNSTATIONAL THERMO SHOCKED STATE GTU","authors":"A. Baranyuk, A. Rachinsky","doi":"10.33842/2313-125X/2020/18/18/27","DOIUrl":"https://doi.org/10.33842/2313-125X/2020/18/18/27","url":null,"abstract":"The paper proposes a method of approximate calculation of the temperature field of the turbine blade in non-stationary mode by means of CFD-modeling. The academic version of the ANSYS-Fluent software package was used as a modeling tool. The local values of the heat transfer coefficient along the contour of the blade profile were calculated in the work. Previously, the gas flow velocity in the curved interscapular channel was calculated to determine the distribution of the gas flow velocity along the convex and concave surfaces of the blade. Comparison of the data thus obtained with the result of processing the experience conducted on the turbine, shows that in the conditions of the real rotating installation, the heat transfer to the surface of the blade was approximately 2 times higher. The most probable reason for this discrepancy is the earlier beginning of the transition to the turbulent boundary layer on the surface of the blade in real conditions of high turbulence of the oncoming flow. Calculations show that in the case of a sharp change in gas temperature, the largest temperature difference in the blade occurs after a few seconds and reaches 50 ... 90% of the change in gas temperature, and in blades with thin edges - close to this value. Increasing the thickness of the edges can significantly reduce the uneven temperature in the blades with sharp changes in gas temperature. In order to identify the nature of the stress state of the blades at different modes of operation of the gas turbine unit, the temperature stresses in the blade were calculated. The calculation was performed for Mises averaged stresses in the cross section of an unevenly heated blade. When starting the installation, the maximum force occurred 30 ... 40 seconds after the start of the rotation of the rotor almost simultaneously with the maximum temperature difference between the edges and the central part of the section. The highest stress values occurred in the region of the inlet edge, where they exceeded 680 MPa per compression. In the future, the temperature stresses decreased and even changed sign due to the cooling process of the blade, overheated when throwing the gas temperature.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127562014","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-07DOI: 10.33842/22195203/20210/18/52/60
V. Vanin, G. Virchenko, V. Yurchuk, P. Yablonskyi
This article provides the basics of integrated geometric modeling of technical objects using the example of disk working parts of tillage tools. The methods and techniques of shaping technical objects used in this process significantly determine the overall efficiency of the disk working parts of tillage tools during their entire life cycle. The defects of the "traditional" design of the working parts of tillage machines, based on experimental selection, manufacturing of a large number of prototypes and their further testing in various technological modes, are analyzed. The relevance of the search for new and / or adaptation (improvement, generalization, etc.) of the existing methods for designing the working surfaces of tillage tools is shown. Igor Sikorsky on issues of both improving new and developing existing methods of geometric modeling. In particular, special attention is paid to the development of the methodology of structural-parametric geometric modeling, the use of involute-evolute models, the use of surface conjugation theory, namely the formation of a unified methodology for geometric modeling of various technical objects and the processes of their manufacture and operation. A technique for automated geometric modeling of a certain group of technical objects on the example of disk parts of tillage tools used modern computer information technologies is proposed. The main advantage of this approach compared to the existing ones is the creation of the productive formation of a large number of structural-parametric product variants. Also important is the invariant nature of this method of shaping, which extends not only to the field of engineering, but also to other industries, in particular, instrumentation, the aerospace industry, robotics, etc. The given direction makes prospects for further scientific research.
{"title":"INTEGRATED COMPLEX APPROACH TO GEOMETRIC MODELING OF DISC WORKING PARTS OF SOIL-PROCESSING TOOLS","authors":"V. Vanin, G. Virchenko, V. Yurchuk, P. Yablonskyi","doi":"10.33842/22195203/20210/18/52/60","DOIUrl":"https://doi.org/10.33842/22195203/20210/18/52/60","url":null,"abstract":"This article provides the basics of integrated geometric modeling of technical objects using the example of disk working parts of tillage tools. The methods and techniques of shaping technical objects used in this process significantly determine the overall efficiency of the disk working parts of tillage tools during their entire life cycle. The defects of the \"traditional\" design of the working parts of tillage machines, based on experimental selection, manufacturing of a large number of prototypes and their further testing in various technological modes, are analyzed. The relevance of the search for new and / or adaptation (improvement, generalization, etc.) of the existing methods for designing the working surfaces of tillage tools is shown. Igor Sikorsky on issues of both improving new and developing existing methods of geometric modeling. In particular, special attention is paid to the development of the methodology of structural-parametric geometric modeling, the use of involute-evolute models, the use of surface conjugation theory, namely the formation of a unified methodology for geometric modeling of various technical objects and the processes of their manufacture and operation. A technique for automated geometric modeling of a certain group of technical objects on the example of disk parts of tillage tools used modern computer information technologies is proposed. The main advantage of this approach compared to the existing ones is the creation of the productive formation of a large number of structural-parametric product variants. Also important is the invariant nature of this method of shaping, which extends not only to the field of engineering, but also to other industries, in particular, instrumentation, the aerospace industry, robotics, etc. The given direction makes prospects for further scientific research.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122366505","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-07DOI: 10.33842/2313-125X/2010/18/182/189
O. Sobol, S. Kravtsiv, O. Stelmakh
The relevance of the article lies in the lack of models and methods of maximum and complete coverage of specified areas, taking into account the limitations of a special kind. The main problem of coverage problems is to find the minimum and optimal solutions with certain constraints, which will be determined by the models and methods of coverage. In order to solve the problem of covering a given area in two-dimensional space, there are different methods of coverage, namely covering a given area with circles of the same radius, circles of variable radius, rectangles, polygons, and objects with variable metric characteristics. The purpose of the article is to formulate the problem statement and develop a model of complete coverage of a given area (non-convex polygon with a set of sub-areas of coverage), taking into account the limitations of a special type. In this work, a mathematical model of complete coverage of a given area by non-convex polygons with variable metric characteristics was developed, taking into account the following limitations: minimum area of mutual intersection of coverage objects; the minimum area of intersection of objects of coverage and addition of the set area to two-dimensional space; the parameters of placement of coverage objects should belong to the points in the specified sub-areas, taking into account the priority sub-areas; belonging of priority areas to the objects of the coverage area; belonging of priority points of subregions to objects of coverage; restrictions of a special kind that affect the metric characteristics of the objects of coverage. The obtained model allows to develop a substantiated method of geometric modeling of full coverage and to carry out computer modeling of coverage of a given area, taking into account the limitations of a special type. Further research will focus on solving other problems arising from the general formulation, and on the development of methods of geometric optimization.
{"title":"MODEL OF THE FULL COVERAGE OF A PARTICULAR AREA TAKING INTO ACCOUNT OF SPECIAL TYPE RESTRICTIONS","authors":"O. Sobol, S. Kravtsiv, O. Stelmakh","doi":"10.33842/2313-125X/2010/18/182/189","DOIUrl":"https://doi.org/10.33842/2313-125X/2010/18/182/189","url":null,"abstract":"The relevance of the article lies in the lack of models and methods of maximum and complete coverage of specified areas, taking into account the limitations of a special kind. The main problem of coverage problems is to find the minimum and optimal solutions with certain constraints, which will be determined by the models and methods of coverage. In order to solve the problem of covering a given area in two-dimensional space, there are different methods of coverage, namely covering a given area with circles of the same radius, circles of variable radius, rectangles, polygons, and objects with variable metric characteristics. The purpose of the article is to formulate the problem statement and develop a model of complete coverage of a given area (non-convex polygon with a set of sub-areas of coverage), taking into account the limitations of a special type. In this work, a mathematical model of complete coverage of a given area by non-convex polygons with variable metric characteristics was developed, taking into account the following limitations: minimum area of mutual intersection of coverage objects; the minimum area of intersection of objects of coverage and addition of the set area to two-dimensional space; the parameters of placement of coverage objects should belong to the points in the specified sub-areas, taking into account the priority sub-areas; belonging of priority areas to the objects of the coverage area; belonging of priority points of subregions to objects of coverage; restrictions of a special kind that affect the metric characteristics of the objects of coverage. The obtained model allows to develop a substantiated method of geometric modeling of full coverage and to carry out computer modeling of coverage of a given area, taking into account the limitations of a special type. Further research will focus on solving other problems arising from the general formulation, and on the development of methods of geometric optimization.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121178320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-07DOI: 10.33842/22195203/2020/18/190/196
Y. Kholodnyak, E. Gavrilenko, D. Spirintsev, V. Fomenko
The formation of complex functional surfaces based on an array of points is an urgent task of geometric modeling. Creating a geometric model of such a surface involves the formation of a discrete ruled framework. The linear elements of the frame are one-dimensional contours. The paper solves the problem of modeling flat one-dimensional contours with a monotonic change in curvature. The source data for modeling the contour is an ordered point series that represents a discretely presented curve (DPC). The contour is formed by thickening the initial point series of an arbitrary configuration in areas where it is possible to provide a monotonic change in the values of characteristics. After assigning the positions of the tangents in the initial points, we get a chain of basic triangles (BT) bounded by the tangents passing through two consecutive points and the chord that connects these points. After that, the ranges of radiuses of curvature are determined, which can be obtained on the basis of the formed BT chain. Within the obtained ranges, the radiuses of curvature in the initial points are assigned. Assigned characteristics are provided as a result of local thickening of the curve section. Inside the BT, the position of the tangent condensation and the condensation point on it are assigned. As a result, we get two new BT. The positions of the point and the tangent of the condensation are assigned within the ranges providing a second order of smoothness and a monotonic change of radiuses of curvature along the contour. The formed sections of monotonous DPC are joined with the second order of smoothness at the points of change of increase and decrease of the radius of curvature and inflection points. The developed algorithm will allow the formation of contours with a regular change in the curvature of various fixation orders.
{"title":"FORMATION OF BASIS TRIANGLES WHEN SIMULATING A CIRCUIT ACCORDING TO THE GIVEN CONDITIONS","authors":"Y. Kholodnyak, E. Gavrilenko, D. Spirintsev, V. Fomenko","doi":"10.33842/22195203/2020/18/190/196","DOIUrl":"https://doi.org/10.33842/22195203/2020/18/190/196","url":null,"abstract":"The formation of complex functional surfaces based on an array of points is an urgent task of geometric modeling. Creating a geometric model of such a surface involves the formation of a discrete ruled framework. The linear elements of the frame are one-dimensional contours. The paper solves the problem of modeling flat one-dimensional contours with a monotonic change in curvature. The source data for modeling the contour is an ordered point series that represents a discretely presented curve (DPC). The contour is formed by thickening the initial point series of an arbitrary configuration in areas where it is possible to provide a monotonic change in the values of characteristics. After assigning the positions of the tangents in the initial points, we get a chain of basic triangles (BT) bounded by the tangents passing through two consecutive points and the chord that connects these points. After that, the ranges of radiuses of curvature are determined, which can be obtained on the basis of the formed BT chain. Within the obtained ranges, the radiuses of curvature in the initial points are assigned. Assigned characteristics are provided as a result of local thickening of the curve section. Inside the BT, the position of the tangent condensation and the condensation point on it are assigned. As a result, we get two new BT. The positions of the point and the tangent of the condensation are assigned within the ranges providing a second order of smoothness and a monotonic change of radiuses of curvature along the contour. The formed sections of monotonous DPC are joined with the second order of smoothness at the points of change of increase and decrease of the radius of curvature and inflection points. The developed algorithm will allow the formation of contours with a regular change in the curvature of various fixation orders.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116873734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-07DOI: 10.33842/22195203/2020/18/11/17
Y. Badayev, L. Lagodina
Relevance. Rational Bezier surfases and NURBS-surfases are widely used in modeling curviliniar objects due to the great flexibility and efficiency of the method. Therefore, it is sense to develop an approximation method by these surfases Method. The work is devoted to the development of a new approach to approximation surfases, represented by a set of discret points. The analytical description of the desired surfases is implemented a rational Bezier surfases and a NURBS-surfases. To solve this problem, two approaches are propozed. The first approach is that the weights of the control points are set in advance and then the coordinates of the points of the approximation rational Bezier surfase as well as the NURBS-surfase are calculated. The second approach is that the coordinates of the control points are set in advance and then the weights of the control points of Bezier surfase as well as the NURBS-surfase are calculated. At the beginning of the process, are set only coordinates, but also parameters are set to a discret points, that is, each points has the following definition: T(x,y,z,u,v) in the three-dimensional space, where u,v – parameters. To solve the approximation problem, the least squares method is used. In the beginning, a sum of squared functional of the term of the differences between the analytic formula of the surface and the coordinate of the given point is created. The optimization problem of minimizing this functional is solved. For this, a system of linear equations is created, each equation of which is derivatev of the functional with respect to a given parameter and equated to zero. In the first approach, the desired parameters are coordinates of points, and the second weights of given. Results. Tho methods of approximation of a point series by rational Bezier surfase were developed. Conclusions. The test cases carried out of using computer programs fnd calculation of results confirm the validity of the proposed methods.
{"title":"APPROXIMATION BY RATIONAL SURFASES OF BEZIER AND NURBS-SURFASES","authors":"Y. Badayev, L. Lagodina","doi":"10.33842/22195203/2020/18/11/17","DOIUrl":"https://doi.org/10.33842/22195203/2020/18/11/17","url":null,"abstract":"Relevance. Rational Bezier surfases and NURBS-surfases are widely used in modeling curviliniar objects due to the great flexibility and efficiency of the method. Therefore, it is sense to develop an approximation method by these surfases Method. The work is devoted to the development of a new approach to approximation surfases, represented by a set of discret points. The analytical description of the desired surfases is implemented a rational Bezier surfases and a NURBS-surfases. To solve this problem, two approaches are propozed. The first approach is that the weights of the control points are set in advance and then the coordinates of the points of the approximation rational Bezier surfase as well as the NURBS-surfase are calculated. The second approach is that the coordinates of the control points are set in advance and then the weights of the control points of Bezier surfase as well as the NURBS-surfase are calculated. At the beginning of the process, are set only coordinates, but also parameters are set to a discret points, that is, each points has the following definition: T(x,y,z,u,v) in the three-dimensional space, where u,v – parameters. To solve the approximation problem, the least squares method is used. In the beginning, a sum of squared functional of the term of the differences between the analytic formula of the surface and the coordinate of the given point is created. The optimization problem of minimizing this functional is solved. For this, a system of linear equations is created, each equation of which is derivatev of the functional with respect to a given parameter and equated to zero. In the first approach, the desired parameters are coordinates of points, and the second weights of given. Results. Tho methods of approximation of a point series by rational Bezier surfase were developed. Conclusions. The test cases carried out of using computer programs fnd calculation of results confirm the validity of the proposed methods.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124835524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-07DOI: 10.33842/22195203/2020/18/83/89
I. Vygodner, T. Malomuzh, N. Starun, G. Tuluchenko
The state of methodological support for the preparation of students of higher educational institutions of non-core specialties for participation in the olympiad in the discipline "mathematics" is examined in the article. The Olympiad problems of geometric content are most difficult to solve traditionally. Difficulties arise due to the need to perform additional constructions and establish complex relationships between elements of geometric figures and bodies. In addition, geometric problems of the olympiad level, as a rule, require methods of several branches of mathematics for their solution. In this case the coordinate method reduces the cognitive complexity of the solving process. Such a process is easier to algorithmize. It brings the coordinate method to algebraic methods. The efficiency of solving geometric problems by the coordinate method substantially depends on the appropriate placement of the studied figure or body in the coordinate system. In problems where it comes to figures inscribed in a circle, it is advisable to use the relationship between the Cartesian and polar coordinate systems. For the calculating of figure area, one can use formulas containing determinants with the coordinates of the vertices of the triangles that are parts of them. This technique, combined with the coordinates of the vertices, which are expressed in terms of the polar radius and polar angle, allows the use of trigonometric identities to simplify the resulting expressions. Additional opportunities for the development of students' academic search abilities are provided by conditional optimization problems, where cases of coincidence and difference between global and conditional extremes are possible. The article considers the geometric problem that was proposed at the international Olympiad for students. For this problem, the author's solution by the coordinate method is presented. In published solutions for this problem, this approach was not used. The studied problem can be formulated in terms of the problem of conditional optimization. A feature of its solution is the fact that one of the points of global maxima satisfies the existing restrictions.
{"title":"COORDINATE METHOD IN THE PROBLEMS OF HIGHER COMPLEXITY OF GEOMETRIC CONTENT","authors":"I. Vygodner, T. Malomuzh, N. Starun, G. Tuluchenko","doi":"10.33842/22195203/2020/18/83/89","DOIUrl":"https://doi.org/10.33842/22195203/2020/18/83/89","url":null,"abstract":"The state of methodological support for the preparation of students of higher educational institutions of non-core specialties for participation in the olympiad in the discipline \"mathematics\" is examined in the article. The Olympiad problems of geometric content are most difficult to solve traditionally. Difficulties arise due to the need to perform additional constructions and establish complex relationships between elements of geometric figures and bodies. In addition, geometric problems of the olympiad level, as a rule, require methods of several branches of mathematics for their solution. In this case the coordinate method reduces the cognitive complexity of the solving process. Such a process is easier to algorithmize. It brings the coordinate method to algebraic methods. The efficiency of solving geometric problems by the coordinate method substantially depends on the appropriate placement of the studied figure or body in the coordinate system. In problems where it comes to figures inscribed in a circle, it is advisable to use the relationship between the Cartesian and polar coordinate systems. For the calculating of figure area, one can use formulas containing determinants with the coordinates of the vertices of the triangles that are parts of them. This technique, combined with the coordinates of the vertices, which are expressed in terms of the polar radius and polar angle, allows the use of trigonometric identities to simplify the resulting expressions. Additional opportunities for the development of students' academic search abilities are provided by conditional optimization problems, where cases of coincidence and difference between global and conditional extremes are possible. The article considers the geometric problem that was proposed at the international Olympiad for students. For this problem, the author's solution by the coordinate method is presented. In published solutions for this problem, this approach was not used. The studied problem can be formulated in terms of the problem of conditional optimization. A feature of its solution is the fact that one of the points of global maxima satisfies the existing restrictions.","PeriodicalId":188754,"journal":{"name":"Modern problems of modeling","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123249575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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