Application of computer modeling in street traffic. Modeling and systems approach. Species diversity of simulated systems

Physical science has been inextricably linked with mathematical modeling since the time of Isaac Newton (XVII–XVIII centuries). I. Newton discovered the fundamental laws of mechanics, the law of universal gravitation, describing them in the language of mathematics. I. Newton (along with G. Leibniz) developed differential and integral calculus, which became the basis of the mathematical apparatus of physics. All subsequent physical discoveries (in thermodynamics, electrodynamics, atomic physics, etc.) were presented in the form of laws and principles described in mathematical language, i.e. in the form of mathematical models.

We can say that the solution to any physical problem theoretically is mathematical modeling. However, the possibility of a theoretical solution to the problem is limited by the degree of complexity of its mathematical model. The more complex the physical process described with its help is, the more complex a mathematical model is, and the more difficult it becomes to use such a model for calculations.

In the simplest situation, the solution to the problem can be obtained “manually” analytically. In most practically important situations, it is not possible to find an analytical solution due to the mathematical complexity of the model. In this case, numerical methods are used to solve the problem, the effective implementation of which is possible only on a computer. In other words, physical research based on complex mathematical models is carried out using computer mathematical modeling. In this regard, in the twentieth century, along with the traditional division of physics into theoretical and experimental, a new direction arose - “computational physics”.

The study of physical processes on a computer is called a computational experiment. Thus, computational physics builds a bridge between theoretical physics, from which it draws mathematical models, and experimental physics, implementing a virtual physical experiment on a computer. Usage computer graphics when processing calculation results, it provides clarity of these results, which is the most important condition for their perception and interpretation by the researcher.

Physics how academic discipline, provides the widest range of applications of electronic technologies as a teaching tool. This includes modeling of physical processes (demonstration and laboratory), training systems, computer control, simulators, generators individual tasks when solving problems. These can also be reference and information systems, experiment control systems and, finally, carrying out various calculations (in particular, when processing the results of a laboratory workshop).

The computer allows you to build dynamic models, because it reacts to user actions similar to the reaction of a real object. Computer models provide greater flexibility when conducting experiments while solving experimental problems; they allow you to slow down or speed up the passage of time, compress or stretch space, supplement the model with a graph, table, animation, repeat or change the situation.

A computer, as a means of controlling a technical object, which occupies a special place in improving the technology and methodology of physical experiments, can perform the following functions:

Measuring instrument;

Control over physical processes or behavior of an object;

Control of a physical experiment or technical object;

Various processing of experimental results.

Efficiency computer training is determined by a number of factors: the didactic capabilities of the computer, the educational potential of multimedia technologies and the organization of the educational process in which the capabilities of new information technologies reveal themselves most fully.

Multimedia technologies can be used within the framework of the implementation of such models of educational activity as independent and controlled discovery of knowledge. Existing electronic tools for developing multimedia applications can be used in educational process for creating multimedia teaching aids. The use of such a didactic tool as a multimedia educational presentation in the educational process makes it possible to increase the degree to which students assimilate the educational information they receive.

As a similar multimedia application flash technologies can be used, the use of which is currently relevant.

Flash is the most popular technology that allows you to create various multimedia and interactive applications for various fields of activity. Flash is a package for creating and saving format for two-dimensional animated computer graphics.

There is absolutely no doubt that computer modeling of various physical processes has significantly accelerated the process of developing technical products, while saving developers a lot of money on assembling test models. With the help of modern computing power and software, engineers can simulate the operation of individual components and assemblies complex systems, which will reduce the number of physical tests required before launching a new product. Manufacturers can also calculate the cost of development after CAD modeling, rather than waiting until the end of physical testing of the product.

Modern industry, when launching new products, faces problems such as time to develop a new product and development costs. And in the automotive industry aerospace industry It is almost impossible to do without CAD modeling, since modeling helps to significantly speed up development and reduce costs, which is very important in modern market. Historically, the emergence of modern computing systems that are capable of simulating the dynamic properties of objects under various influences has pushed into the background the modernization of physical test benches, as well as the development of test methods. Many organizations try to choose modeling because it requires minimal cost and minimal development time. However, in some studies, only the process of physically testing the product can provide an accurate answer. Without closer interaction between electronic models and physical testing, many organizations can become overly dependent on computer models for development, which, if used incorrectly, can subsequently lead to unexpected failures in the operation of expensive equipment.

In the automotive industry, computer modeling is becoming an integral part as modern vehicle designs have become much more complex and computer modeling systems have improved significantly. However, unfortunately, many manufacturers reduce physical testing of products to a minimum, relying on computer simulation results.

Physical testing processes have not kept pace with computer modeling in improving techniques. Testing engineers usually try to perform the minimum necessary tests on a product. The result is more frequent test repetitions to obtain more reliable results or their confirmation. Relying purely on computer modeling without physical testing can lead to very serious consequences in the future, since the mathematical model of the product, on the basis of which the process of calculating dynamic properties is carried out, is created with certain assumptions, and in real work the product may behave slightly differently than what was displayed on the monitor.

Computer modelling has a symbiotic relationship with physical testing of equipment, which allows (as opposed to a computer model) to obtain experimental data. Therefore, lags in technologies for testing finished devices, with such an increase in the capabilities of computer technology, can lead to unnecessary savings on experimental samples with subsequent problems in finished products. The accuracy of the models directly depends on the input data about the behavior of the model (mathematical description) under various conditions.

Of course, the elements of the models cannot include all possible options and conditions for the behavior of certain components, since the complexity of the calculations and the cumbersomeness of the mathematical model would become simply enormous. To simplify the mathematical model, certain assumptions are made that “should not” have a significant impact on the operation of the mechanism. But, unfortunately, the reality is always much harsher. For example, a mathematical model will not be able to calculate how the device will behave if there are microcracks in the material, or if there is a sudden change in weather, which can lead to a completely different load distribution in the structure. Experimental data and calculated data quite often differ from each other. And this must be remembered.

There is another important advantage to the physical testing of equipment. This is the ability to point out flaws to engineers when drawing up mathematical models, and also provides a good opportunity for discovering new phenomena and improving old calculation methods. After all, you must agree that if you put variables into a mathematical formula, the result will depend on the variables, and not on the formula. The formula will always remain constant, and only a real physical test can supplement or change it.

The emergence of new materials in all branches of modern industry creates additional problems for computer modeling. If engineers continued to use time-tested materials and their improved mathematical descriptions, then yes, the problems with modeling would be much less. But the emergence of new materials requires mandatory carry out physical tests of finished products with these materials. However, new elements are increasingly appearing on the market and growth trends are only going up.

For example, the aeromobile and automotive industries have rapidly adopted composite materials due to their good strength-to-weight ratio. One of the main problems with computer modeling is the inability of the model to accurately predict the behavior of a material that suffers from certain performance disadvantages compared to the aluminum, steel, plastic and other materials that have long been used in this industry.

Validation of computer models for composite materials is critical during the design phase. After carrying out the calculations, it is necessary to assemble a test stand on a real part. When conducting physical tests to measure deformation and load distribution, engineers focus on critical points determined by a computer model. Strain gauges are used to collect information about critical points. This process is only monitored for expected issues that may create blind spots in the testing process. Without comprehensive research, the authenticity of a model may be confirmed when in fact it is not.

There is also a problem with gradually outdated measurement technologies, for example, strain gauges and thermocouples do not allow covering the entire required measurement range. For the most part, traditional sensors are only able to measure the required value by separate areas, not allowing you to deeply penetrate into the essence of what is happening. As a result, scientists are forced to rely on pre-modeled processes that show vulnerabilities and force testers to pay increased attention to one or another node of the system under test. But as always there is one thing. This approach works well for time-tested and well-studied materials, but for designs that include new materials, it can be harmful. Therefore, design engineers in all industries are trying to update old measurement methods as much as possible, as well as introduce new ones that will allow more detailed measurements than older sensors and techniques.

Strain gauge technology has remained virtually unchanged since its invention decades ago. New technologies such as , are capable of measuring full field strength and temperature. Unlike legacy strain gauge technologies, which can only collect information at critical points, fiber optic sensors can collect continuous strain and temperature data. These technologies are much more beneficial when conducting physical testing, as they allow engineers to observe the behavior of the structure under study at and between critical points.

For example, fiber optic sensors can be embedded inside composite materials during downtime to better understand curing processes. A common disadvantage, for example, may be the process of wrinkling in one of the layers of the material, which causes mechanical stress inside. These processes are still very poorly understood and there is very little information about the stress and deformation inside composite materials, which makes it almost impossible to apply computer modeling to them.

Outdated strain gauge technologies are quite capable of detecting residual strain in composite materials, but only when the strain field reaches the surface and the sensor is installed exactly in the right place. On the other hand, spatially continuous measurement technologies such as fiber optics can measure all field strength data at and between critical points. It was also previously mentioned that fiber optic sensors can be embedded in composite materials to study internal processes.

The development process is considered complete when the product has passed all tests and has begun to be shipped to consumers. However, the current level allows manufacturers to receive the first reports on their products immediately after users begin using them. As a rule, immediately after leaving serial product work begins on its modernization.

Computer models and physical tests go hand in hand. They simply cannot exist without each other. Further development technology requires maximum interaction between these design tools. Investments in the advancement of physical research data require initially large investments, but the “return” will also please. But, unfortunately, most developers try to get benefits here and now and do not care at all about long-term prospects, the benefits of which, as a rule, are much greater.

Those looking to secure the long-term future of their products will seek to implement more innovative and reliable methodologies and elements of product testing, such as fiber optic measurements. The combination of computer modeling and physical testing technologies will only grow stronger in the future, because they complement each other.

Kobelnitsky Vladislav

Computer modelling. Simulation physical and mathematical processes on the computer.

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Research

"COMPUTER MODELLING"

COMPLETED:

KOBELNITSKY VLADISLAV

9TH CLASS STUDENT

MKOU secondary school No. 17

Supervisor:

mathematics and computer science teacher

Tvorozova E.S.

KANSK, 2013

1. INTRODUCTION………………………………………………………………………………3
2. COMPUTER SIMULATION…………………………………...5
3. PRACTICAL PART……………………………………………………………..10
4. CONCLUSION……………………………………………………………...18
5. REFERENCES……………………………………………………………...20

INTRODUCTION

In most areas of human activity it is currently used computer technology. For example, in a hairdresser you can use a computer to select in advance the hairstyle that the client will like. For this, the client is photographed, the photo is electronically entered into a program containing a wide variety of hairstyles, and a photo of the client is displayed on the screen, to whom you can “try on” any hairstyle. You can also easily choose hair color and makeup. Using a computer model, you can see in advance whether a particular hairstyle will suit the client. Of course, this option is better than actually conducting an experiment; in real life, correcting an undesirable situation is much more difficult.

While studying a topic in computer science, “Computer Modeling,” I became interested in the question: “Can any process or phenomenon be simulated using a PC?” This was the choice for my research.

Topic of my research:"Computer modelling".

Hypothesis: any process or phenomenon can be simulated using a PC.

Goal of the work - study the possibilities of computer modeling and its use in various subject areas.

To achieve this goal, the work solves the following: tasks:

- give theoretical information about modeling;

– describe the stages of modeling;

– give examples of models of processes or phenomena from various subject areas;

Draw a general conclusion about computer modeling in subject areas.

I decided to take a closer look at computer modeling in MS Excel and Living Mathematics. The paper discusses the advantages of MS Excel. Using these programs, I built computer models from various subject areas, such as mathematics, physics, and biology.

Building and studying models is one of the most important methods of cognition; the ability to use a computer to build models is one of the requirements of today, so I consider this work relevant. It is important for me, since I want to continue my further studies in this direction, as well as consider other programs when developing computer models, this is the goal for further continuation of this work.

COMPUTER MODELLING

Analyzing the literature on the research topic, I found out that in almost all natural and social sciences, the construction and use of models is a powerful research tool. Real objects and processes are so multifaceted and complex that the best way their study turns out to be the construction of a model that reflects only some part of reality and is therefore many times simpler than this reality.

Model (Latin modulus - measure) is a substitute object for the original object, providing the study of some properties of the original.

Model - a specific object created for the purpose of receiving and (or) storing information (in the form of a mental image, description by means of signs or a material system), reflecting the properties, characteristics and connections of the object - the original of an arbitrary nature, essential for the problem solved by the subject.

Modeling – the process of creating and using a model.

Modeling Goals

1. Knowledge of reality
2. Conducting experiments
3. Design and management
4. Predicting the behavior of objects
5. Training and education of specialists
6. Data processing

Classification by presentation form

1. Material - reproduce the geometric and physical properties of the original and always have a real embodiment (children's toys, visual teaching aids, models, models of cars and airplanes, etc.).

1. a) geometrically similar scale, reproducing the spatial and geometric characteristics of the original regardless of its substrate (models of buildings and structures, educational models, etc.);
2. b) based on the theory of similarity, substrate-like, reproducing with scaling in space and time the properties and characteristics of the original of the same nature as the model (hydrodynamic models of ships, purging models of aircraft);
3. c) analog instruments that reproduce the studied properties and characteristics of the original object in a modeling object of a different nature based on some system of direct analogies (a type of electronic analog modeling).

1. Information - a set of information characterizing the properties and states of an object, process, phenomenon, as well as their relationship with the outside world).

1. 2.1. Verbal - verbal description in natural language).
2. 2.2. Iconic - information model expressed special signs(by means of any formal language).

1. 2.2.1. Mathematical - mathematical description of the relationships between the quantitative characteristics of the modeling object.
2. 2.2.2. Graphic - maps, drawings, diagrams, graphs, diagrams, system graphs.
3. 2.2.3. Tabular - tables: object-property, object-object, binary matrices and so on.

1. Ideal – a material point, an absolutely rigid body, a mathematical pendulum, an ideal gas, infinity, a geometric point, etc....

1. 3.1. Unformalizedmodels are systems of ideas about the original object that have developed in the human brain.
2. 3.2. Partially formalized.

1. 3.2.1. Verbal - a description of the properties and characteristics of the original in some natural language (text materials of project documentation, verbal description of the results of a technical experiment).
2. 3.2.2. Graphic iconic - features, properties and characteristics of the original that are actually or at least theoretically accessible directly to visual perception (art graphics, technological maps).
3. 3.2.3. Graphical conditionals - data from observations and experimental studies in the form of graphs, diagrams, diagrams.

1. 3.3. Quite formalized(mathematical) models.

Model properties

1. Limb : the model reflects the original only in a finite number of its relations and, in addition, modeling resources are finite;
2. Simplification : the model displays only the essential aspects of the object;
3. Approximation: reality is represented roughly or approximately by the model;
4. Adequacy : how successfully the model describes the system being modeled;
5. Information content: the model must contain sufficient information about the system - within the framework of the hypotheses adopted when constructing the model;
6. Potentiality: predictability of the model and its properties;
7. Complexity : ease of use;
8. Completeness : all necessary properties are taken into account;
9. Adaptability.

It should also be noted:

1. The model is a “quadruple construct”, the components of which are the subject; problem solved by the subject; the original object and description language or method of reproducing the model. The problem solved by the subject plays a special role in the structure of the generalized model. Outside the context of a problem or class of problems, the concept of a model has no meaning.
2. Each material object, generally speaking, corresponds to an infinite number of equally adequate, but essentially different models associated with different tasks.
3. The task-object pair also corresponds to many models that contain, in principle, the same information, but differ in the forms of its presentation or reproduction.
4. A model, by definition, is always only a relative, approximate similarity to the original object and in informationally fundamentally poorer than the latter. This is its fundamental property.
5. The arbitrary nature of the original object, which appears in the accepted definition, means that this object can be material, can be of a purely informational nature, and, finally, can be a complex of heterogeneous material and information components. However, regardless of the nature of the object, the nature of the problem being solved and the method of implementation, the model is an information formation.
6. A particular, but very important for theoretically developed scientific and technical disciplines is the case when the role of a modeling object in a research or applied problem is played not by a fragment of the real world considered directly, but by some ideal construct, i.e. in fact, another model, created earlier and practically reliable. Such secondary, and in general n-fold, modeling can be carried out theoretical methods with subsequent verification of the results obtained using experimental data, which is typical for fundamental natural sciences. In less theoretically developed areas of knowledge (biology, some technical disciplines), the secondary model usually includes empirical information that is not covered by existing theories.

The process of building a model is called modeling.

Due to the polysemy of the concept “model” in science and technology, there is no unified classification of types of modeling: classification can be carried out according to the nature of the models, the nature of the objects being modeled, and the areas of application of modeling (in engineering, physical sciences, cybernetics, etc.). For example, you can highlight the following types modeling:

1. Information Modeling
2. Computer modelling
3. Math modeling
4. Mathematical cartographic modeling
5. Molecular modeling
6. Digital modeling
7. Logic modeling
8. Pedagogical modeling
9. Psychological modeling
10. Statistical Modeling
11. Structural Modeling
12. Physical modeling
13. Economic and mathematical modeling
14. Simulation modeling
15. Evolutionary modeling
16. Graphic and geometric modeling
17. Full-scale modeling

Computer modellingincludes the process of implementing an information model on a computer and researching a modeling object using this model - conducting a computational experiment. Many scientific and industrial issues are solved with the help of computer modeling.

Isolating the essential aspects of a real object and abstracting from its secondary properties from the point of view of the task at hand allows one to develop analytical skills. Implementing an object model on a computer requires knowledge application programs, as well as programming languages.

In the practical part, I built models according to the following scheme:

1. Statement of the problem (description of the problem, modeling goals, formalization of the problem);
2. Model development;
3. Computer experiment;
4. Analysis of simulation results.

PRACTICAL PART

Modeling of various processes and phenomena

Work 1 “Determination of the specific heat capacity of a substance.”

Purpose of the work: to experimentally determine the specific heat capacity of a given substance.

First stage

Second phase

1. Entering the values ​​of the measured quantities.
2. Introduction of formulas for calculating the specific heat capacity of a substance.
3. Calculation of specific heat capacity.

Third stage . Compare the tabulated and experimental values ​​of heat capacity.

Determination of the specific heat capacity of a substance

The exchange of internal energy between bodies and the environment without performing mechanical work is called heat exchange.

During heat exchange, the interaction of molecules of bodies having different temperatures leads to the transfer of energy from a body with a higher temperature to a body with a lower temperature.

If heat exchange occurs between bodies, then the internal energy of all heating bodies increases by as much as the internal energy of cooling bodies decreases.

Work order:

Weigh the inner aluminum vessel of the calorimeter. Pour water into it, up to about half of the vessel and weigh again to determine the mass of water in the vessel. Measure the initial temperature of the water in the vessel.

From a vessel with boiling water common to the whole class, carefully, so as not to burn your hand, remove a metal cylinder with a wire hook and lower it into the calorimeter.

Monitor the increase in water temperature in the calorimeter. When the temperature reaches its maximum value and stops increasing, record its value in the table.

Remove the cylinder from the vessel, dry it with filter paper, weigh it and record the mass of the cylinder in the table.

From the heat balance equation

c 1 m 1 (T-t 1 )+c 2 m 2 (T-t 1 )=cm(t 2 -T)

Calculate the specific heat capacity of the substance from which the cylinder is made.

m 1 – mass of the aluminum vessel;

c 1 – specific heat capacity of aluminum;

m 2 - mass of water;

from 2 - specific heat capacity of water;

t 1 - initial water temperature

m - cylinder mass;

t 2 - initial temperature of the cylinder;

T - general temperature

Work 2 “Study of oscillations of a spring pendulum”

Purpose of the work: to determine experimentally the stiffness of the spring and determine the frequency of oscillation of the spring pendulum. Find out the dependence of the oscillation frequency on the mass of the suspended load.

First stage . A mathematical model is compiled.

Second phase . Working with the compiled model.

1. Enter formulas to calculate the spring constant value.
2. Introduction to the cells of formulas for calculating the theoretical and experimental values ​​of the oscillation frequency of a spring pendulum.
3. Conducting experiments by suspending loads of various masses from a spring. Enter the results in the table.

Third stage . Draw a conclusion about the dependence of the oscillation frequency on the mass of the suspended load. Compare the theoretical and experimental frequency values.

Description of work in the laboratory workshop:

A load suspended on a steel spring and brought out of equilibrium moves under the influence of gravity and the elasticity of the spring harmonic vibrations. The natural frequency of oscillation of such a spring pendulum is determined by the expression

where k – spring stiffness; m – body weight.

The task of laboratory work is to experimentally verify the theoretically obtained pattern. To solve this problem, you first need to determine the stiffness k springs used in a laboratory installation, mass m load and calculate the natural frequency 0 pendulum oscillations. Then, hanging a load of mass m on the spring, experimentally verify the theoretical result obtained.

Completing of the work.

1. Fasten the spring in the tripod leg and hang a load weighing 100 g from it. Next to the load, attach a measuring ruler vertically and mark the initial position of the load.

2. Hang two more weights of 100 g each to the spring and measure its elongation caused by the action of force F2Н. Enter the force value F and extension x into the table and you will get the hardness value k springs, calculated by the formula

3. Knowing the spring stiffness, calculate the natural frequency 0 oscillations of a spring pendulum weighing 100, 200, 300 and 400 g.

4. For each case, experimentally determine the oscillation frequency pendulum. To do this, measure the time intervalt, during which the pendulum will make 10-20 complete oscillations, and you will receive the frequency value calculated by the formula

where n – number of oscillations.

5. Compare the calculated natural frequency values 0 oscillations of a spring pendulum with a frequency , obtained experimentally.

Work 3 “Law of conservation of mechanical energy”

Purpose of the work: to experimentally test the law of conservation of mechanical energy.

First stage . Drawing up a mathematical model.

Second phase . Working with the compiled model.

1. Entering data into a spreadsheet.
2. Enter formulas to calculate the value of potential and kinetic energy.
3. Conducting experiments. Enter the results in the table.

Third stage . Compare the kinetic energy of the ball and the change in its potential energy and draw a conclusion.

Description of work in laboratory workshop

CHECKING THE LAW OF CONSERVATION OF MECHANICAL ENERGY.

In the work, it is necessary to experimentally establish that the total mechanical energy of a closed system remains unchanged if only gravitational and elastic forces act between the bodies.

The setup for the experiment is shown in Figure 1. When rod A deviates from a vertical position, the ball at its end rises to a certain height h relative to entry level. In this case, the Earth-ball system of interacting bodies acquires an additional reserve of potential energyΔEp=mgh.

If the rod is released, it will return to the vertical position up to a special stop. Considering the frictional forces and changes in the potential energy of elastic deformation of the rod to be very small, we can assume that during the movement of the rod only gravitational forces and elastic forces act on the ball. Based on the law of conservation of mechanical energy, we can expect that the kinetic energy of the ball at the moment it passes the initial position will be equal to the change in its potential energy:

To determine the kinetic energy of the ball, it is necessary to measure its speed. To do this, fix the device in the tripod leg at a height H above the table surface, move the rod with the ball to the side and then release it. When the rod hits the stop, the ball jumps off the rod and, due to inertia, continues to move at speed v in the horizontal direction. Measuring the range of the ball l when it moves along a parabola, you can determine the horizontal speed v:

where t - time of free fall of a ball from a height H.

Having determined the mass of the ball m using scales, you can find its kinetic energy and compare it with the change in potential energyΔEp.

In the practical part of this work, I built models of physical processes, as well as mathematical models, and described laboratory work.

As a result of the work, I built the following models:

Physical models of body motion (Ms Excel, physics subject)

Uniform rectilinear motion, uniformly accelerated motion (Ms Excel, physics subject);

Movements of a body thrown at an angle to the horizon (Ms Excel, physics subject);

Movements of bodies taking into account the force of friction (Ms Excel, physics subject);

Movements of bodies taking into account many forces acting on the body (Ms Excel, physics subject);

Determination of the specific heat capacity of a substance (Ms Excel, physics subject);

Oscillations of a spring pendulum (Ms Excel, physics subject);

Mathematical model for calculating arithmetic and algebraic progression; (Ms Excel, subject algebra);

Computer model of modification variability (Ms Excel, biology subject);

Construction and study of function graphs in the “Living Mathematics” program.

After building the models, we can conclude: in order to correctly build a model, it is necessary to set a goal, I adhered to the scheme presented in the theoretical part.

Conclusion

I have identified the advantages of using Excel:

A) functionality Excel programs obviously cover all the needs for automation of experimental data processing, construction and research of models; b) has understandable interface; c) learning Excel is provided for in general education programs in computer science, therefore, effective using Excel; G) this program it is accessible to study and easy to manage, which is fundamentally important for me as a student; e) the results of activities on the Excel worksheet (texts, tables, graphs, formulas) are “open” to the user.

Among all known software Excel tools has perhaps the richest tools for working with charts. The program allows you to use autofill techniques to present data in tabular form, quickly convert it using a huge library of functions, build graphs, edit them for almost all elements, enlarge the image of any fragment of the graph, select functional scales along the axes, extrapolate graphs, etc. .

To summarize the work, I would like to conclude: the goal set at the beginning of this study was achieved. My research has shown that it is indeed possible to simulate any process or phenomenon. The hypothesis I posed is correct. I was convinced of this when I built a sufficient number of such models. To build any model, you need to adhere to certain rules, which I described in the practical part of this work.

This research will be continued, other programs that allow modeling processes will be studied.

BIBLIOGRAPHY

1. Degtyarev B.I., Degtyareva I.B., Pozhidaev S.V. , Solving problems in physics on programmable calculators, M., Prosveshchenie, 1991.
2. Demonstration experiment in physics in high school. Ed. Pokrovsky A.A., M. Education, 1972
3. Dolgolaptev V. Working in Excel 7.0. for Windows 95.M., Binom, 1995
4. Efimenko G.E. Solving environmental problems using spreadsheets. Informatics, No. 5 – 2000.
5. Zlatopolsky D.M., Solving equations using spreadsheets. Informatics, No. 41 – 2000
6. Ivanov V. Microsoft Office System 2003. Russian version. Publishing house "Peter", 2005
7. Izvozchikov V.A., Slutsky A.M., Solving problems in physics on a computer, M., Prosveshchenie, 1999.
8. Nechaev V.M. Spreadsheets and databases. Informatics, No. 36-1999
9. Programs for general education institutions. Physics grades 7-11, M., Bustard, 2004
10. Saikov B.P. Excel: charting. Computer Science and Education No. 9 – 2001
11. Collection of problems in physics. Ed. S.M. Kozela, M., Science, 1983
12. Semakin I.G. , Sheina T.Yu, Teaching basic course computer science in high school., M., publishing house Binom, 2004.
13. Physics lesson in a modern school. Ed. V.G.Razumovsky, M.Prosveshchenie, 1993

COMPUTER MODELLING(eng. computational simulation), construction using computers and computer devices (3D scanners, 3D printers, etc.) symbolic [see. Symbolic modeling(s-modeling)] and physical models of objects studied in science (physics, chemistry, etc.), created in technology (for example, in aircraft manufacturing, robotics), medicine (for example, in implantology, tomography), art (for example, ., in architecture, music) and other areas of human activity.

Computer modeling makes it possible to significantly reduce the cost of developing models in comparison with non-computer modeling methods and carrying out full-scale tests. It makes it possible to build symbolic computer models of objects for which it is impossible to build physical models (for example, models of objects studied in climatology). Serves effective means modeling complex systems in technology, economics and other fields of activity. It is the technological basis of computer-aided design (CAD) systems.

Physical computer models are made on the basis of symbolic models and are prototypes of simulated objects (parts and assemblies of machines, building structures, etc.). To manufacture prototypes, 3D printers can be used that implement technologies for layer-by-layer formation of non-planar objects. Symbolic prototype models can be developed using CAD machines, 3D scanners or digital cameras and photogrammetric software.

A computer system is a human-machine complex in which the construction of models is carried out using computer programs that implement mathematical (see. Mathematical modeling) and expert (eg, simulation) modeling methods. In the computational experiment mode, the researcher has the opportunity, by changing the initial data, to obtain and save in a computer modeling system a large number of variants of the object model in a relatively short time.

Clarification of ideas about the object under study and improvement of methods for its modeling may make it necessary to change the software of the computer modeling system, while the hardware may remain unchanged.

The high efficiency of computer modeling in science, technology and other fields of activity stimulates the development of hardware (including supercomputers) and software [including instrumental systems (see. Instrumental system in computer science) development of parallel programs for supercomputers].

These days, computer models are a rapidly growing part of the arsenal.

Mayer R.V. Computer modelling

Mayer R.V., Glazov Pedagogical Institute

COMPUTER MODELLING:

MODELING AS A METHOD OF SCIENTIFIC KNOWLEDGE.

COMPUTER MODELS AND THEIR TYPES

The concept of a model is introduced, various classes of models are analyzed, and the connection between modeling and general systems theory is analyzed. Numerical, statistical and simulation modeling and its place in the system of other methods of cognition are discussed. Various classifications of computer models and areas of their application are considered.

1.1. The concept of a model. Modeling Goals

In the process of studying the surrounding world, the subject of knowledge is confronted with the studied part of objective reality –– object of knowledge. A scientist, using empirical methods of cognition (observation and experiment), establishes data, characterizing the object. Elementary facts are summarized and formulated empirical laws. Next step consists of developing theory and constructing theoretical model, which explains the behavior of the object and takes into account the most significant factors influencing the phenomenon being studied. This theoretical model must be logical and consistent with established facts. We can assume that any science is a theoretical model of a certain part of the surrounding reality.

Often in the process of cognition, a real object is replaced by some other ideal, imaginary or material object
, bearing the studied features of the object under study, and is called model. This model is subjected to research: it is subjected to various influences, parameters and initial conditions are changed, and it is found out how its behavior changes. The results of the model research are transferred to the research object, compared with available empirical data, etc.

Thus, a model is a material or ideal object that replaces the system under study and adequately reflects its essential aspects. The model must in some way repeat the process or object under study with a degree of correspondence that allows us to study the original object. In order for the simulation results to be transferred to the object under study, the model must have the property adequacy. The advantage of replacing the object under study with its model is that models are often easier, cheaper and safer to study. Indeed, to create an airplane, you need to build a theoretical model, draw a drawing, perform the appropriate calculations, make a small copy of it, study it in a wind tunnel, etc.

Object model should reflect it most important qualities, neglecting the secondary ones. Here it is appropriate to recall the parable of the three blind wise men who decided to find out what an elephant is. One wise man held an elephant by the trunk and said that the elephant is a flexible hose. Another touched the elephant's leg and decided that the elephant was a column. The third wise man pulled the tail and came to the conclusion that the elephant is a rope. It is clear that all the wise men were mistaken: none of the named objects (hose, column, rope) reflect the essential aspects of the object being studied (elephant), therefore their answers (proposed models) are not correct.

When modeling, various goals can be pursued: 1) knowledge of the essence of the object under study, the reasons for its behavior, the “structure” and the mechanism of interaction of elements; 2) explanation of already known results of empirical studies, verification of model parameters using experimental data; 3) predicting the behavior of systems under new conditions under various external influences and control methods; 4) optimization of the functioning of the systems under study, search for the correct control of the object in accordance with the selected optimality criterion.

1.2. Different kinds models

The models used are extremely varied. System analysis requires classification and systematization, that is, structuring an initially unordered set of objects and turning it into a system. There are various ways to classify the existing variety of models. Thus, the following types of models are distinguished: 1) deterministic and stochastic; 2) static and dynamic; 3) discrete, continuous and discrete-continuous; 4) mental and real. In other works, models are classified on the following grounds (Fig. 1): 1) by the nature of the modeled side of the object; 2) in relation to time; 3) by the method of representing the state of the system; 4) according to the degree of randomness of the simulated process; 5) according to the method of implementation.

When classifying according to the nature of the modeled side of the object The following types of models are distinguished (Fig. 1): 1.1. Cybernetic or functional models; in them the modeled object is considered as a “black box”, internal organization which is unknown. The behavior of such a “black box” can be described by a mathematical equation, graph or table that relates the output signals (reactions) of the device to the input signals (stimuli). The structure and principles of operation of such a model have nothing in common with the object under study, but it functions in a similar way. For example, computer program, simulating the game of checkers. 1.2. Structural models– these are models whose structure corresponds to the structure of the modeled object. Examples are command post exercises, self-government day, model electronic circuit in Electronics Workbench, etc. 1.3. Information models, representing a set of specially selected quantities and their specific values ​​that characterize the object under study. There are verbal (verbal), tabular, graphical and mathematical information models. For example, a student's information model may consist of grades for exams, tests, and labs. Or an information model of some production represents a set of parameters characterizing the needs of production, its most essential characteristics, and the parameters of the product being produced.

In relation to time highlight: 1. Static models–– models whose condition does not change over time: a model of the development of a block, a model of a car body. 2. Dynamic models are functioning objects whose state is constantly changing. These include working models of an engine and generator, a computer model of population development, an animated model of computer operation, etc.

By way of representing the system state distinguish: 1. Discrete models– these are automata, that is, real or imaginary discrete devices with a certain set of internal states that convert input signals into output signals in accordance with given rules. 2. Continuous models– these are models in which continuous processes occur. For example, the use of an analog computer to solve a differential equation, simulate radioactive decay using a capacitor discharging through a resistor, etc. According to the degree of randomness of the simulated process isolated (Fig. 1): 1. Deterministic models, which tend to move from one state to another in accordance with a rigid algorithm, that is, there is a one-to-one correspondence between the internal state, input and output signals (traffic light model). 2. Stochastic models, functioning like probabilistic automata; the output signal and the state at the next time are specified by a probability matrix. For example, a probabilistic model of a student, a computer model of transmitting messages over a communication channel with noise, etc.

Rice. 1. Various ways to classify models.

By implementation method distinguish: 1. Abstract models, that is, mental models that exist only in our imagination. For example, the structure of an algorithm, which can be represented using a block diagram, a functional dependence, a differential equation that describes a certain process. Abstract models also include various graphic models, diagrams, structures, and animations. 2. Material (physical) models They are stationary models or operating devices that function somewhat similar to the object under study. For example, a model of a molecule made of balls, a model of a nuclear submarine, a working model of an alternating current generator, an engine, etc. Real modeling involves building a material model of an object and performing a series of experiments with it. For example, to study the movement of a submarine in water, a smaller copy of it is built and the flow is simulated using a hydrodynamic tube.

We will be interested in abstract models, which in turn are divided into verbal, mathematical and computer. TO verbal or text models refer to sequences of statements in natural or formalized language that describe the object of cognition. Mathematical models form a wide class of iconic models that use mathematical operations and operators. They often represent a system of algebraic or differential equations. Computer models are an algorithm or computer program that solves a system of logical, algebraic or differential equations and simulates the behavior of the system under study. Sometimes mental simulation is divided into: 1. Visual,–– involves the creation of an imaginary image, a mental model, corresponding to the object under study based on assumptions about the ongoing process, or by analogy with it. 2. symbolic,–– consists in creating a logical object based on the system special characters; is divided into linguistic (based on the thesaurus of basic concepts) and symbolic. 3. Mathematical,–– consists in establishing correspondence to the object of study of some mathematical object; divided into analytical, simulation and combined. Analytical modeling involves writing a system of algebraic, differential, integral, finite-difference equations and logical conditions. To study the analytical model can be used analytical method and numerical method. Recently, numerical methods have been implemented on computers, so computer models can be considered as a type of mathematical ones.

Mathematical models are quite diverse and can also be classified on different grounds. By degree of abstraction when describing system properties they are divided into meta-, macro- and micro-models. Depending on the presentation forms There are invariant, analytical, algorithmic and graphical models. By the nature of the displayed properties object models are classified into structural, functional and technological. By method of obtaining distinguish between theoretical, empirical and combined. Depending on the nature of the mathematical apparatus models can be linear and nonlinear, continuous and discrete, deterministic and probabilistic, static and dynamic. By way of implementation There are analogue, digital, hybrid, neuro-fuzzy models, which are created on the basis of analogue, digital, hybrid computers and neural networks.

1.3. Modeling and systems approach

The modeling theory is based on general systems theory, also known as systems approach. This is a general scientific direction, according to which the object of research is considered as a complex system interacting with the environment. An object is a system if it consists of a set of interconnected elements, the sum of whose properties are not equal to the properties of the object. A system differs from a mixture by the presence of an ordered structure and certain connections between elements. For example, a TV set consisting of a large number of radio components connected to each other in a certain way is a system, but the same radio components lying randomly in a box are not a system. There are the following levels of description of systems: 1) linguistic (symbolic); 2) set-theoretic; 3) abstract-logical; 4) logical-mathematical; 5) information-theoretic; 6) dynamic; 7) heuristic.

Rice. 2. System under study and environment.

The system interacts with the environment, exchanges matter, energy, and information with it (Fig. 2). Each of its elements is subsystem. A system that includes the analyzed object as a subsystem is called supersystem. We can assume that the system has inputs, to which signals are received, and exits, issuing signals on Wednesday. Treating the object of cognition as a whole, made up of many interconnected parts, allows you to see something important behind a huge number of insignificant details and features and formulate system-forming principle. If the internal structure of the system is unknown, then it is considered a “black box” and a function is specified that links the states of the inputs and outputs. This is cybernetic approach. At the same time, the behavior of the system under consideration, its response to external influences and environmental changes are analyzed.

The study of the composition and structure of the object of cognition is called system analysis. His methodology is expressed in the following principles: 1) the principle physicality: the behavior of the system is described by certain physical (psychological, economic, etc.) laws; 2) principle modelability: the system can be modeled in a finite number of ways, each of which reflects its essential aspects; 3) principle focus: the functioning of fairly complex systems leads to the achievement of a certain goal, state, preservation of the process; at the same time, the system is able to withstand external influences.

As stated above, the system has structure – a set of internal stable connections between elements, determining the basic properties of a given system. It can be represented graphically in the form of a diagram, chemical or mathematical formula or count. This graphic image characterizes the spatial arrangement of elements, their nesting or subordination, and chronological sequence various parts complex event. When building a model, it is recommended to draw up block diagrams the object being studied, especially if it is quite complex. This allows us to understand the totality of all integrative properties of an object that its constituent parts do not possess.

One of the most important ideas of the systems approach is emergence principle, –– when combining elements (parts, components) into a single whole, a system effect: the system acquires qualities that none of its constituent elements possesses. The principle of highlighting the main structure system is that studying is enough complex object requires highlighting a certain part of its structure, which is the main or main one. In other words, there is no need to take into account all the variety of details, but one should discard the less significant and enlarge the important parts of the object in order to understand the main patterns.

Any system interacts with other systems that are not part of it and form the environment. Therefore, it should be considered as a subsystem of some larger system. If we limit ourselves to analyzing only internal connections, then in some cases it will not be possible to create a correct model of the object. It is necessary to take into account the essential connections of the system with the environment, that is external factors, and thereby “close” the system. This is principle of closure.

The more complex the object under study, the more different models (descriptions) can be built. Thus, looking at a cylindrical column from different sides, all observers will say that it can be modeled as a homogeneous cylindrical body of certain dimensions. If, instead of a column, observers begin to look at some complex architectural composition, then everyone will see something different and build their own model of the object. In this case, as in the case of the sages, various results will be obtained that contradict each other. And the point here is not that there are many truths or that the object of knowledge is fickle and multifaceted, but that the object is complex and the truth is complex, and the methods of knowledge used are superficial and did not allow us to fully understand the essence.

When studying large systems come from principle of hierarchy, which is as follows. The object under study contains several related subsystems of the first level, each of which is itself a system consisting of subsystems of the second level, etc. Therefore, the description of the structure and the creation of a theoretical model must take into account the “location” of elements at various “levels,” that is, their hierarchy. The main properties of the systems include: 1) integrity, that is, the irreducibility of the properties of the system to the sum of the properties of individual elements; 2) structure, – heterogeneity, the presence of a complex structure; 3) plurality of description, –– the system can be described different ways; 4) interdependence of system and environment, –– elements of the system are connected with objects that are not part of it and form the environment; 5) hierarchy, –– the system has a multi-level structure.

1.4. Qualitative and quantitative models

The task of science is to build a theoretical model of the surrounding world that would explain known and predict unknown phenomena. The theoretical model can be qualitative or quantitative. Let's consider quality explanation of electromagnetic oscillations in oscillatory circuit consisting of a capacitor and an inductor. When a charged capacitor is connected to an inductor, it begins to discharge, current flows through the inductor, and the energy of the electric field is converted into the energy of the magnetic field. When the capacitor is completely discharged, the current through the inductor reaches its maximum value. Due to the inertia of the inductor, caused by the phenomenon of self-induction, the capacitor is recharged, it is charged in the opposite direction, etc. This qualitative model of the phenomenon allows one to analyze the behavior of the system and predict, for example, that as the capacitor capacity decreases, the natural frequency of the circuit will increase.

An important step on the path of knowledge is transition from qualitative-descriptive methods to mathematical abstractions. The solution to many problems in natural science required the digitization of space and time, the introduction of the concept of a coordinate system, the development and improvement of methods for measuring various physical, psychological and other quantities, which made it possible to operate with numerical values. As a result, quite complex mathematical models were obtained, representing a system of algebraic and differential equations. Currently, the study of natural and other phenomena is no longer limited to qualitative reasoning, but involves the construction of a mathematical theory.

Creation quantitative models of electromagnetic oscillations in an RLC circuit involves the introduction of accurate and unambiguous methods for determining and measuring quantities such as current , charge , voltage , capacity , inductance , resistance . Without knowing how to measure the current in a circuit or the capacitance of a capacitor, it is pointless to talk about any quantitative relationships. Having unambiguous definitions of the listed quantities, and having established the procedure for their measurement, you can begin to build a mathematical model and write a system of equations. The result is a second-order inhomogeneous differential equation. Its solution allows, knowing the charge of the capacitor and the current through the inductor at the initial moment, to determine the state of the circuit at subsequent moments of time.

The construction of a mathematical model requires the determination of independent quantities that uniquely describe state the object under study. For example, the state mechanical system determined by the coordinates of the particles entering it and the projections of their momenta. State electrical circuit is given by the charge of the capacitor, the current through the inductor, etc. The state of the economic system is determined by a set of indicators such as the number Money, invested in production, profit, number of workers engaged in manufacturing products, etc.

The behavior of an object is largely determined by its parameters, that is, quantities that characterize its properties. Thus, the parameters of a spring pendulum are the stiffness of the spring and the mass of the body suspended from it. The electrical RLC circuit is characterized by the resistance of the resistor, the capacitance of the capacitor, and the inductance of the coil. The parameters of a biological system include the reproduction rate, the amount of biomass consumed by one organism, etc. Another important factor influencing the behavior of an object is external influence. It is obvious that the behavior of a mechanical system depends on the external forces acting on it. The processes in the electrical circuit are affected by the applied voltage, and the development of production is associated with the external economic situation in the country. Thus, the behavior of the object under study (and therefore its model) depends on its parameters, initial state and external influence.

Creating a mathematical model requires defining a set of system states, a set of external influences (input signals) and responses (output signals), as well as setting relationships connecting the system response with the influence and its internal state. They allow you to study a huge number of different situations, setting other system parameters, initial conditions and external influences. The required function characterizing the response of the system is obtained in tabular or graphical form.

All existing methods studies of the mathematical model can be divided into two groups .Analytical solving an equation often involves cumbersome and complex mathematical calculations and, as a result, leads to an equation expressing the functional relationship between the desired quantity, system parameters, external influences and time. The results of such a solution require interpretation, which involves analyzing the obtained functions and constructing graphs. Numerical methods researching a mathematical model on a computer involves creating a computer program that solves a system of corresponding equations and displays a table or graphic image. The resulting static and dynamic pictures clearly explain the essence of the processes under study.

1.5. Computer modelling

Effective way studying the phenomena of the surrounding reality is scientific experiment, consisting in reproducing the studied natural phenomenon under controlled and controlled conditions. However, often carrying out an experiment is impossible or requires too much economic effort and can lead to undesirable consequences. In this case, the object under study is replaced computer model and study its behavior under various external influences. Ubiquitous personal computers, information technology, the creation of powerful supercomputers has made computer modeling one of the effective methods for studying physical, technical, biological, economic and other systems. Computer models are often simpler and more convenient to study; they make it possible to carry out computational experiments, the real implementation of which is difficult or may give an unpredictable result. The logic and formalization of computer models makes it possible to identify the main factors that determine the properties of the objects under study and to study the response of a physical system to changes in its parameters and initial conditions.

Computer modeling requires abstracting from the specific nature of phenomena, building first a qualitative and then a quantitative model. This is followed by a series of computational experiments on a computer, interpretation of the results, comparison of modeling results with the behavior of the object under study, subsequent refinement of the model, etc. Computational experiment in fact, it is an experiment on a mathematical model of the object under study, carried out using a computer. It is often much cheaper and more accessible than a full-scale experiment, its implementation requires less time, and it provides more detailed information about the quantities characterizing the state of the system.

Essence computer modeling system consists in creating a computer program (software package) that describes the behavior of the elements of the system under study during its operation, taking into account their interaction with each other and the external environment, and conducting a series of computational experiments on a computer. This is done with the aim of studying the nature and behavior of the object, its optimization and structural development, and predicting new phenomena. Let's list t requirements, which the model of the system under study must satisfy: 1. Completeness models, that is, the ability to calculate all characteristics of the system with the required accuracy and reliability. 2. Flexibility models, which allows you to reproduce and play out various situations and processes, change the structure, algorithms and parameters of the system under study. 3. Duration of development and implementation, characterizing the time spent on creating the model. 4. Block structure, allowing the addition, exclusion and replacement of some parts (blocks) of the model. Besides, Information Support, software and hardware must allow the model to exchange information with the corresponding database and ensure efficient machine implementation and comfortable work user.

To the main stages of computer modeling include (Fig. 3): 1) formulation of the problem, description of the system under study and identification of its components and elementary acts of interaction; 2) formalization, that is, the creation of a mathematical model, which is a system of equations and reflects the essence of the object under study; 3) algorithm development, the implementation of which will solve the problem; 4) writing a program in a specific programming language; 5) planning And performing calculations on a computer, finalizing the program and obtaining results; 6) analysis And interpretation of results, their comparison with empirical data. Then all this is repeated on next level.

The development of a computer model of an object is a sequence of iterations: first, a model is built based on the available information about the system S
, a series of computational experiments is carried out, the results are analyzed. When receiving new information about an object S, additional factors are taken into account, and a model is obtained
, whose behavior is also studied on a computer. After this, models are created
,
etc. until a model is obtained that corresponds to the system S with the required accuracy.

Rice. 3. Stages of computer modeling.

In general, the behavior of the system under study is described by the law of functioning, where
–– vector of input influences (stimuli),
–– vector of output signals (responses, reactions),
–– vector of environmental influences,
–– vector of system eigenparameters. The operating law can take the form of a verbal rule, table, algorithm, function, set of logical conditions, etc. In the case when the law of functioning contains time, we talk about dynamic models and systems. For example, acceleration and deceleration of an asynchronous motor, transient process in a circuit containing a capacitor, operation computer network,queuing systems. In all these cases, the state of the system, and hence its model, changes over time.

If the behavior of the system is described by the law
, not containing time obviously then we're talking about about static models and systems, solving stationary problems, etc. Let's give a few examples: calculating a nonlinear direct current circuit, finding a stationary temperature distribution in a rod at constant temperatures of its ends, the shape of an elastic film stretched over a frame, the velocity profile in a steady flow of a viscous fluid, etc.

The functioning of the system can be considered as a sequential change of states
,
, … ,
, which correspond to some points in the multidimensional phase space. Set of all points
, corresponding to all possible states of the system, are called object state space(or models). Each implementation of the process corresponds to one phase trajectory passing through some points from the set . If a mathematical model contains an element of randomness, then a stochastic computer model is obtained. In a particular case, when the system parameters and external influences uniquely determine the output signals, we speak of a deterministic model.

Principles of computer modeling. Connection with other methods of cognition

So, A model is an object that replaces the system under study and imitates its structure and behavior. A model can be a material object, a set of data ordered in a special way, a system mathematical equations or a computer program. Modeling is understood as representing the main characteristics of an object of study using another system (material object, set of equations, computer program). Let us list the principles of modeling:

1. Principle of adequacy: The model must take into account the most significant aspects of the object under study and reflect its properties with acceptable accuracy. Only in this case can the simulation results be extended to the object of study.

2. The principle of simplicity and economy: The model must be simple enough for its use to be effective and cost-effective. It should not be more complex than is required for the researcher.

3. The principle of information sufficiency: In the complete absence of information about the object, it is impossible to build a model. In the presence of complete information modeling makes no sense. There is a level of information sufficiency, upon reaching which a model of the system can be built.

4. Feasibility principle: The created model must ensure the achievement of the stated research goal in a finite time.

5. The principle of plurality and unity of models: Any specific model reflects only some aspects real system. For a complete study, it is necessary to build a number of models that reflect the most significant aspects of the process under study and have something in common. Each subsequent model should complement and clarify the previous one.

6. Systematic principle. The system under study can be represented as a set of subsystems interacting with each other, which are modeled by standard mathematical methods. Moreover, the properties of the system are not the sum of the properties of its elements.

7. Principle of parameterization. Some subsystems of the modeled system can be characterized by a single parameter (vector, matrix, graph, formula).

The model must satisfy the following requirements: 1) be adequate, that is, reflect the most essential aspects of the object under study with the required accuracy; 2) contribute to the solution of a certain class of problems; 3) be simple and understandable, based on a minimum number of assumptions and assumptions; 4) allow oneself to be modified and supplemented, to move on to other data; 5) be convenient to use.

The connection between computer modeling and other methods of cognition is shown in Fig. 4. The object of knowledge is studied by empirical methods (observation, experiment), established facts are the basis for constructing a mathematical model. The resulting system of mathematical equations can be studied by analytical methods or with the help of a computer - in this case we are talking about creating a computer model of the phenomenon being studied. A series of computational experiments or computer simulations is carried out, and the resulting results are compared with the results of an analytical study of the mathematical model and experimental data. The findings are taken into account to improve the methodology for experimental study of the research object, develop a mathematical model and improve the computer model. The study of social and economic processes differs only in the inability to fully use experimental methods.

Rice. 4. Computer modeling among other methods of cognition.

1.6. Types of computer models

By computer modeling in the broadest sense we will understand the process of creating and studying models using a computer. The following types of modeling are distinguished:

1. Physical modeling: a computer is part of an experimental setup or simulator; it perceives external signals, carries out the corresponding calculations and issues signals that control various manipulators. For example, a training model of an aircraft, which is a cockpit mounted on appropriate manipulators connected to a computer, which reacts to the pilot’s actions and changes the tilt of the cockpit, instrument readings, view from the window, etc., simulating the flight of a real aircraft.

2. Dynamic or numerical modeling, which involves the numerical solution of a system of algebraic and differential equations using methods of computational mathematics and conducting a computational experiment under various system parameters, initial conditions and external influences. It is used to simulate various physical, biological, social and other phenomena: pendulum oscillations, wave propagation, population changes, populations of a given animal species, etc.

3. Simulation modeling consists of creating a computer program (or software package) that simulates the behavior of a complex technical, economic or other system on a computer with the required accuracy. Simulation modeling provides formal description the logic of the functioning of the system under study over time, which takes into account the significant interactions of its components and ensures the conduct of statistical experiments. Object-oriented computer simulations are used to study the behavior of economic, biological, social and other systems, to create computer games, the so-called virtual world”, training programs and animations. For example, a model of a technological process, an airfield, a certain industry, etc.

4. Statistical modeling used to study stochastic systems and consists of repeated tests followed by statistical processing the resulting results. Such models make it possible to study the behavior of all kinds of queuing systems, multiprocessor systems, information and computer networks, various dynamic systems, which are influenced by random factors. Statistical models are used in solving probabilistic problems, as well as in processing large amounts of data (interpolation, extrapolation, regression, correlation, calculation of distribution parameters, etc.). They are different from deterministic models, the use of which involves the numerical solution of systems of algebraic or differential equations, or the replacement of the object under study with a deterministic automaton.

5. Information modeling consists in creating an information model, that is, a set of specially organized data (signs, signals) reflecting the most significant aspects of the object under study. There are visual, graphic, animation, text, and tabular information models. These include all kinds of diagrams, graphs, graphs, tables, diagrams, drawings, animations made on a computer, including a digital map of the starry sky, a computer model of the earth's surface, etc.

6. Knowledge modeling involves the construction of an artificial intelligence system, which is based on a knowledge base of some subject area(parts of the real world). Knowledge bases consist of facts(data) and rules. For example, a computer program that can play chess (Fig. 5) must operate with information about the “abilities” of various chess pieces and “know” the rules of the game. This type of model includes semantic networks, logical knowledge models, expert systems, logic games, etc. Logic models are used to represent knowledge in expert systems, for creating artificial intelligence systems, carrying out logical inference, proving theorems, mathematical transformations, building robots, using natural language to communicate with a computer, creating a virtual reality effect in computer games, etc.

Rice. 5. Computer model of chess player behavior.

Based modeling purposes, computer models are divided into groups: 1) descriptive models, used to understand the nature of the object being studied, identifying the most significant factors influencing its behavior; 2) optimization models, allowing you to choose the optimal way to control a technical, socio-economic or other system (for example, a space station); 3) predictive models, helping to predict the state of an object at subsequent points in time (a model of the earth’s atmosphere that allows one to predict the weather); 4) training models, used for teaching, training and testing students, future specialists; 5) game models, allowing you to create a game situation that simulates control of an army, state, enterprise, person, airplane, etc., or playing chess, checkers and other logic games.

Classification of computer models

according to the type of mathematical scheme

In the theory of system modeling, computer models are divided into numerical, simulation, statistical and logical. In computer modeling, as a rule, one of the standard mathematical schemes is used: differential equations, deterministic and probabilistic automata, queuing systems, Petri nets, etc. Taking into account the method of representing the state of the system and the degree of randomness of the simulated processes allows us to construct Table 1.

Table 1.

According to the type of mathematical scheme, they are distinguished: 1 . Continuously determined models, which are used to model dynamic systems and involve solving a system of differential equations. Mathematical schemes of this type are called D-schemes (from the English dynamic). 2. Discrete-deterministic models used for research discrete systems, which can be in one of many internal states. They are modeled by an abstract finite automata, specified by the F-scheme (from the English finite automata): . Here
, –– a variety of input and output signals, –– a variety of internal states,
–– transition function,
–– function of outputs. 3. Discrete-stochastic models involve the use of a scheme of probabilistic automata, the functioning of which contains an element of randomness. They are also called P-schemes (from the English probabilistic automat). The transitions of such an automaton from one state to another are determined by the corresponding probability matrix. 4. Continuous-stochastic models As a rule, they are used to study queuing systems and are called Q-schemes (from the English queuing system). For the functioning of some economic, industrial, technical systems inherent random occurrence of requirements (applications) for service and random service time. 5. Network models are used to analyze complex systems in which several processes occur simultaneously. In this case, they talk about Petri nets and N-schemes (from the English Petri Nets). The Petri net is given by a quadruple, where – many positions,
– many transitions, – input function, – output function. The labeled N-scheme allows you to simulate parallel and competing processes in various systems. 6. Combined schemes are based on the concept of an aggregate system and are called A-schemes (from the English aggregate system). This universal approach, developed by N.P. Buslenko, allows us to study all kinds of systems that are considered as a set of interconnected units. Each unit is characterized by vectors of states, parameters, environmental influences, input influences (control signals), initial states, output signals, transition operator, output operator.

The simulation model is studied using digital and analogue computers. The simulation system used includes mathematical, software, information, technical and ergonomic support. The effectiveness of simulation modeling is characterized by the accuracy and reliability of the resulting results, the cost and time of creating a model and working with it, and the cost of machine resources (computation time and required memory). To assess the effectiveness of the model, it is necessary to compare the resulting results with the results of a full-scale experiment, as well as the results of analytical modeling.

In some cases, it is necessary to combine the numerical solution of differential equations and simulation of the functioning of one or another rather complex system. In this case they talk about combined or analytical and simulation modeling. Its main advantage is the ability to study complex systems, take into account discrete and continuous elements, nonlinearity of various characteristics, and random factors. Analytical modeling allows you to analyze only fairly simple systems.

One of effective methods simulation model research is statistical test method. It involves repeated reproduction of a particular process with various parameters changing randomly according to a given law. A computer can conduct 1000 tests and record the main characteristics of the system’s behavior, its output signals, and then determine their mathematical expectation, dispersion, and distribution law. The disadvantage of using a machine implementation of a simulation model is that the solution obtained with its help is of a private nature and corresponds to specific parameters of the system, its initial state and external influences. The advantage is the ability to study complex systems.

1.8. Areas of application of computer models

The improvement of information technology has led to the use of computers in almost all areas of human activity. The development of scientific theories involves putting forward basic principles, constructing a mathematical model of the object of knowledge, and obtaining consequences from it that can be compared with the results of an experiment. The use of a computer allows, based on mathematical equations, to calculate the behavior of the system under study under certain conditions. Often this is the only way to obtain consequences from a mathematical model. For example, consider the problem of the motion of three or more particles interacting with each other, which is relevant when studying the motion of planets, asteroids and other celestial bodies. In the general case, it is complex and does not have an analytical solution, and only the use of computer modeling allows one to calculate the state of the system at subsequent points in time.

Improvement of computer technology, the emergence of a computer that allows you to quickly and accurately carry out calculations according to given program, marked a qualitative leap in the development of science. At first glance, it seems that the invention of computers cannot directly influence the process of cognition of the surrounding world. However, this is not so: solving modern problems requires the creation of computer models, carrying out a huge number of calculations, which became possible only after the advent of electronic computers capable of performing millions of operations per second. It is also significant that calculations are performed automatically, in accordance with a given algorithm, and do not require human intervention. If a computer belongs to the technical basis for conducting a computational experiment, then its theoretical basis is made up of applied mathematics and numerical methods for solving systems of equations.

The successes of computer modeling are closely related to the development of numerical methods, which began with the fundamental work of Isaac Newton, who back in the 17th century proposed their use for the approximate solution of algebraic equations. Leonhard Euler developed a method for solving ordinary differential equations. Among modern scientists, a significant contribution to the development of computer modeling was made by Academician A.A. Samarsky, the founder of the methodology of computational experiments in physics. It was they who proposed the famous triad “model – algorithm – program” and developed computer modeling technology, successfully used to study physical phenomena. One of the first outstanding results of a computer experiment in physics was the discovery in 1968 of a temperature current layer in the plasma created in MHD generators (T-layer effect). It was performed on a computer and made it possible to predict the outcome of a real experiment conducted several years later. Currently, the computational experiment is used to carry out research in the following areas: 1) calculation of nuclear reactions; 2) solving problems of celestial mechanics, astronomy and astronautics; 3) study of global phenomena on Earth, modeling of weather, climate, study of environmental problems, global warming, consequences of a nuclear conflict, etc.; 4) solving problems of continuum mechanics, in particular, hydrodynamics; 5) computer modeling of various technological processes; 6) calculation of chemical reactions and biological processes, development of chemical and biological technology; 7) sociological research, in particular, modeling elections, voting, dissemination of information, changes in public opinion, military operations; 8) calculation and forecasting of the demographic situation in the country and the world; 9) simulation modeling of the work of various technical, in particular, electronic devices; 10) economic research on the development of an enterprise, industry, country.

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