According to this feature, models are divided into two broad classes:

• abstract (mental) models;
• material models.

Rice. 1.1.

Often in the practice of modeling there are mixed, abstract-material models.

abstract patterns are certain constructions of generally accepted signs on paper or other material carrier or in the form computer program.

Abstract models, without going into too much detail, can be divided into:

• symbolic;
• mathematical.

Symbolic model is a logical object that replaces real process and expressing the main properties of its relations with the help of a certain system of signs or symbols. It's either the words natural language, or words of the corresponding thesaurus, graphs, diagrams, etc.

The symbolic model may have independent meaning, but, as a rule, its construction is initial stage any other simulation.

Mathematical modeling- this is the process of establishing correspondence to the modeled object of some mathematical construction, called a mathematical model, and the study of this model, which allows obtaining the characteristics of the modeled object.

Mathematical modeling - the main objective and the main content of the studied discipline.

Mathematical models can be:

• analytical;
• imitation;
• mixed (analytical and simulation).

Analytical Models- these are functional relationships: systems of algebraic, differential, integro-differential equations, logical conditions. Maxwell's equations - analytical model electromagnetic field. Ohm's law is a model of an electrical circuit.

Transformation of mathematical models according to known laws and rules can be seen as experiments. A solution based on analytical models can be obtained as a result of a single calculation, regardless of the specific values ​​of the characteristics ("in general view"). This is visual and convenient for identifying patterns. However, for complex systems It is not always possible to build an analytical model that fully reflects the real process. Nevertheless, there are processes, for example, Markov ones, the relevance of modeling of which by analytical models has been proven by practice.

Simulation. Creation computers led to the development of a new subclass of mathematical models - simulation.

Simulation modeling involves the representation of the model in the form of some algorithm - a computer program - the execution of which imitates the sequence of changing states in the system and thus represents the behavior of the simulated system.

The process of creating and testing such models is called simulation modeling, and the algorithm itself is called a simulation model.

What is the difference between simulation and analytical models?

In the case of analytical modeling, a computer is a powerful calculator, adding machine. Analytical model solved on a computer.

In the case of simulation modeling, the simulation model - the program - implemented on a computer.

Simulation models quite simply take into account the influence of random factors. For analytical models, this serious problem. In the presence of random factors, the necessary characteristics of the simulated processes are obtained by multiple runs (realizations) of the simulation model and further statistical processing accumulated information. Therefore, simulation modeling of processes with random factors is often called statistical modeling.

If the study of the object is difficult using only analytical or simulation modeling, then mixed (combined), analytical and simulation modeling is used. When constructing such models, the processes of the object functioning are decomposed into constituent subprocesses, and for which, perhaps, analytical models are used, and simulation models are built for the remaining subprocesses.

material modeling based on the use of models representing real technical structures. It can be the object itself or its elements (natural modeling). This may be a special device - a model that has either a physical or geometric similarity to the original. It may be a different device. physical nature than the original, but the processes in which are described by similar mathematical relationships. This is the so-called analog simulation. Such an analogy is observed, for example, between oscillations of a satellite communication antenna under wind load and oscillation electric current in a specially selected electrical circuit.

Often created material abstract models. That part of the operation that cannot be described mathematically is modeled materially, the rest is abstract. Such, for example, are command-and-staff exercises, when the work of headquarters is a full-scale experiment, and the actions of troops are reflected in documents.

Classification according to the criterion considered - the method of implementing the model - is shown in fig. 1.2.

1.3. Modeling steps

Mathematical modeling like any other, it is considered an art and a science. A well-known specialist in the field of simulation modeling Robert Shannon called his book widely known in the scientific and engineering world: " Simulation- art and science". Therefore, in engineering practice there is no formalized instruction on how to create models. And, nevertheless, an analysis of the techniques used by model developers allows us to see a fairly transparent stage of modeling.

First stage: clarification of the goals of modeling. Actually this main stage any activity. The goal essentially determines the content of the remaining stages of modeling. Note that the difference between a simple system and a complex one is generated not so much by their essence, but also by the goals set by the researcher.

Typically, the goals of modeling are:

• forecast of the object's behavior under new modes, combinations of factors, etc.;
• selection of a combination and values ​​of factors that provide optimal value process performance indicators;
• analysis of the sensitivity of the system to changes in certain factors;
• testing various hypotheses about characteristics random parameters the process under study;
• definition functional connections between the behavior ("response") of the system and the influencing factors, which can contribute to the prediction of behavior or sensitivity analysis;
• clarification of the essence, a better understanding of the object of study, as well as the formation of the first skills for operating a simulated or operating system.

Second phase: building a conceptual model. conceptual model(from lat. conception) - a model at the level of the defining idea, which is formed when studying the modeled object. At this stage, the object is investigated, the necessary simplifications and approximations are established. Significant aspects are identified, secondary ones are excluded. Units of measurement and ranges of change are set model variables. If possible, then conceptual model presented in the form of well-known and well-developed systems: queuing, control, auto-regulation, different kind vending machines, etc. conceptual model fully sums up the study of design documentation or experimental examination of the object being modeled.

The result of the second stage is a generalized scheme of the model, fully prepared for mathematical description- building mathematical model.

Third stage: choice of a programming or modeling language, development of an algorithm and a model program. The model can be analytical or simulation, or a combination of both. In the case of an analytical model, the researcher must master the solution methods.

In the history of mathematics (and this, by the way, is the history of mathematical modeling) there are many examples of when the need to model various kinds of processes led to new discoveries. For example, the need to simulate motion led to the discovery and development of differential calculus(Leibniz and Newton) and the corresponding solution methods. The problems of analytical modeling of the stability of ships led Academician A. N. Krylov to create the theory of approximate calculations and an analog computer.

The result of the third stage of modeling is a program compiled in the most convenient language for modeling and research - universal or special.

Fourth stage: planning an experiment. Mathematical model is the object of the experiment. The experiment should be as informative as possible, satisfy the constraints, provide data from required accuracy and credibility. There is a theory of experiment planning, we will study the elements of this theory that we need in the appropriate place in the discipline. GPSS World, AnyLogic, etc.) and can be applied automatically. It is possible that in the course of the analysis of the obtained results, the model can be refined, supplemented, or even completely revised.

After analyzing the simulation results, they are interpreted, that is, the results are translated into terms subject area. This is necessary because usually subject matter specialist(the one who needs the results of research) does not have the terminology of mathematics and modeling and can perform his tasks, operating only with concepts that are well known to him.

This concludes the consideration of the simulation sequence by making a very important conclusion about the need to document the results of each stage. This is necessary for the following reasons.

Firstly, modeling is an iterative process, that is, from each stage, a return can be made to any of the previous stages to clarify the information needed at this stage, and the documentation can save the results obtained at the previous iteration.

Secondly, in the case of studying a complex system, it involves large teams developers, and various stages carried out by different teams. Therefore, the results obtained at each stage must be transferable to subsequent stages, that is, have unified form presentation and understandable content to other interested specialists.

Thirdly, the result of each of the stages should be a valuable product in itself. For example, conceptual model may not be used for further transformation into a mathematical model, but be a description that stores information about the system, which can be used as an archive, as a learning tool, etc.

In order to understand the essence of mathematical modeling, consider the basic definitions, features of the process.

The essence of the term

Modeling is the process of creating and applying a model. It is considered any abstract or material object, which replaces the real object of simulation in the process of studying. An important point is the preservation of the properties necessary for a full-fledged analysis of the subject.

Computer modelling is a variant of knowledge based on a mathematical model. It implies a system of inequalities, equations, logical sign expressions that fully reflect all the characteristics of a phenomenon or object.

Mathematical modeling involves specific calculations, the use of computer technology. More research is needed to explain the process. This task is successfully solved by computer simulation.

Specificity of computer simulation

This way of studying complex systems is considered effective and efficient. It is more convenient and easier to analyze computer models, since various computational actions can be performed. This is especially true in cases where physical or material reasons real experiments do not allow you to get the desired result. The logic of such models makes it possible to determine the main factors that determine the parameters of the studied original.

This application of mathematical modeling makes it possible to identify the behavior of an object in various conditions to identify the influence of various factors on his behavior.

Fundamentals of computer modeling

What is the basis for this modeling? What Scientific research based on ICT? Let's start with the fact that any computer simulation is based on certain principles:

• mathematical modeling to describe the process under study;
• application of innovative mathematical models for detailed consideration of the processes under study.

Varieties of modeling

Currently allocate different methods mathematical modeling: simulation and analytical.

The analytical option is associated with the study of abstract models real object in the form of differential algebraic equations, which provide for the implementation of a clear computer technology that can give an accurate solution.

Simulation modeling involves the study of a mathematical model in the form of a specific algorithm that reproduces the functioning of the analyzed system through the sequential execution of a system of simple calculations and operations.

Features of building a computer model

Let's take a closer look at how this simulation works. What are the stages of computer research? Let's start with the fact that the process is based on moving away from a clear object or phenomenon being analyzed.

Such modeling consists of two main stages: the creation of a qualitative and quantitative model. computer learning consists in carrying out a system of computational actions on personal computer aimed at analyzing, systematizing, comparing the results of the study with the real behavior of the analyzed object. If necessary, additional refinement of the model is carried out.

Modeling steps

How is modeling carried out? What are the stages of computer research? So, we distinguish the following algorithm of actions regarding the construction computer model:

Stage 1. Setting the goal and objectives of the work, identifying the object of modeling. It is supposed to collect data, formulate a question, identify the goals and forms of research, and describe the results obtained.

Stage 2. Analysis and study of the system. The description of the object, the creation of an information model, the selection of software and technical means, examples of mathematical modeling are selected.

Stage 3. Transition to a mathematical model, development of a design method, selection of an algorithm of actions.

Stage 4. Selection of a programming language or environment for modeling, discussion of analysis options, recording the algorithm on certain language programming.

Stage 5 It consists in carrying out a complex of computational experiments, debugging calculations, and processing the results obtained. If necessary, on this stage modeling is corrected.

Stage 6 Interpretation of results.

How is the simulation analyzed? What software products for research? First of all, the use of text graphic editors, spreadsheets, mathematical packages that allow you to receive maximum result from the research done.

Conducting a computational experiment

All methods of mathematical modeling are based on experiments. Under them, it is customary to understand experiments conducted with a model or object. They consist in the implementation of certain actions that allow you to determine the behavior of the experimental sample in response to the proposed actions.

A computational experiment cannot be imagined without carrying out calculations that are associated with the use of a formalized model.

The basics of mathematical modeling involve research with a real object, but computational actions are carried out with it an exact copy(model). When choosing a specific set of initial indicators of the model, after completing the computational steps, you can get optimal conditions for the full functioning of the real object.

For example, having Mathematical equation, which describes the flow of the analyzed process, when changing the coefficients, initial and intermediate conditions, we can assume the behavior of the object. In addition, it is possible to create a reliable forecast of the behavior of this object or natural phenomenon under certain conditions. In the case of a new set of initial data, it is important to carry out new computational experiments.

Comparison of received data

In order to carry out an adequate verification of a real object or a created mathematical model, as well as to evaluate the results of research on computer science with the results of an experiment carried out on a full-scale prototype, a comparison of the research results is carried out.

The decision to build depends on the discrepancy between the information obtained during the research. finished sample or about adjusting the mathematical model.

Such an experiment makes it possible to replace natural expensive research with calculations on computer technology, to analyze the possibilities of using an object in the shortest possible time, to identify the conditions for its actual operation.

Modeling in environments

For example, in a programming environment, three stages of mathematical modeling are used. At the stage of creating an algorithm and an information model, values ​​are determined that will be input parameters, research results, and their type is revealed.

If necessary, draw up special mathematical algorithms in the form of flowcharts written in a specific programming language.

A computer experiment involves the analysis of the results obtained in the calculations, their correction. Among the important stages of such a study, we note the testing of the algorithm, the analysis of the performance of the program.

Its debugging involves finding and eliminating errors that lead to an undesirable result, the appearance of errors in calculations.

Testing involves checking the correct functioning of the program, as well as assessing the reliability of its individual components. The process consists in checking the operability of the program, its suitability for studying a certain phenomenon or object.

Spreadsheets

Modeling using spreadsheets allows you to cover a large amount of tasks in various subject areas. They are considered universal tool, which allows solving the laborious task of calculating the quantitative parameters of the object.

In the case of such a simulation option, some transformation of the algorithm for solving the problem is observed, there is no need to develop a computational interface. At the same time, there is a debugging stage, which includes the removal of data errors, the search for a connection between cells, and the identification of computational formulas.

As the work progresses, additional tasks, for example outputting results to paper media, rational representation information on a computer monitor.

Sequencing

Modeling is carried out in spreadsheets according to a certain algorithm. First, the objectives of the study are determined, the main parameters and relationships are identified, and a specific mathematical model is compiled based on the information received.

For qualitative consideration of the model, initial, intermediate, as well as final characteristics are used, supplemented with drawings, diagrams. With the help of graphs and charts, they get a visual representation of the results of the work.

Modeling in a DBMS environment

It allows you to solve the following tasks:

• store information, carry out its timely editing;
• organize the available data according to specific characteristics;
• create different criteria for data selection;
• present the information in a convenient way.

As the model is developed on the basis of the initial data, optimal conditions are created for describing the characteristics of the object using special tables.

At the same time, information is sorted, data is searched and filtered, and algorithms for calculations are created. Using the computer information panel, you can create different screen forms, as well as options for obtaining printed paper reports on the progress of the experiment.

If the results obtained do not coincide with the planned options, the parameters are changed, additional studies are carried out.

Application of a computer model

Computational experiment and computer simulation are new scientific research methods. They make it possible to modernize the computing apparatus used to build a mathematical model, to concretize, refine, and complicate experiments.

Among the most promising practical use, conducting a full-fledged computational experiment highlight the design of reactors for powerful nuclear power plants. In addition, this includes the creation of magnetohydrodynamic transducers electrical energy, as well as a balanced perspective plan for the country, region, industry.

It is with the help of computer and mathematical modeling that it is possible to design the devices necessary for studying thermonuclear reactions, chemical processes.

Computer modeling and computational experiments make it possible to reduce far "non-mathematical" objects to the formulation and solution of a mathematical problem.

It opens great opportunities to apply the mathematical apparatus in a system with a modern computer technology to address issues related to the development outer space, "conquest" of atomic processes.

It is modeling that has become one of the most important options for understanding various surrounding processes and natural phenomena. This knowledge is a complex and time-consuming process, involves the use of a system various kinds modeling, starting with the development of reduced models of real objects, ending with the selection of special algorithms for complex mathematical calculations.

Depending on what processes or phenomena will be analyzed, certain algorithms of actions are selected, mathematical formulas for computing. Computer simulation allows minimal cost get the desired result important information about the properties and parameters of an object or phenomenon.

In this paper, we propose to analyze in detail the topic of modeling in computer science. This section has great importance for the training of future specialists in the field of information technology.

To solve any problem (industrial or scientific), computer science uses the following chain:

It is worth paying special attention to the concept of "model". Without the presence of this link, the solution of the problem will not be possible. Why is the model used and what is meant by this term? We will talk about this in the next section.

Model

Modeling in computer science is the compilation of an image of a real-life object that reflects all essential features and properties. A model for solving a problem is necessary, since it is, in fact, used in the process of solving.

AT school course Informatics, the topic of modeling begins to be studied as early as the sixth grade. At the very beginning, children need to be introduced to the concept of a model. What it is?

• Simplified similarity of the object;
• Reduced copy of a real object;
• Scheme of a phenomenon or process;
• Image of a phenomenon or process;
• Description of the phenomenon or process;
• Physical analogue of the object;
• Information analogue;
• A placeholder object that reflects the properties of the real object, and so on.

The model is a very broad concept, as it has already become clear from the above. It is important to note that all models are usually divided into groups:

• material;
• ideal.

A material model is understood as an object based on a real existing facility. It can be any body or process. This group further subdivided into two types:

• physical;
• analog.

Such a classification is conditional, because it is very difficult to draw a clear boundary between these two subspecies.

The ideal model is even more difficult to characterize. She is associated with:

• thinking;
• imagination;
• perception.

It includes works of art (theater, painting, literature, and so on).

Modeling Goals

Modeling in computer science is very milestone because it has a lot of goals. Now we invite you to get to know them.

First of all, modeling helps to understand the world around us. From time immemorial, people have accumulated the acquired knowledge and passed it on to their descendants. Thus, a model of our planet (globe) appeared.

In past centuries, non-existent objects were modeled, which are now firmly entrenched in our lives (umbrella, windmill, and so on). Currently, modeling is aimed at:

• identification of the consequences of any process (increase in the cost of travel or disposal of chemical waste underground);
• ensuring the effectiveness of decisions made.

Simulation tasks

information model

Now let's talk about another type of models studied in the school computer science course. Computer modeling, which every future IT specialist needs to master, includes the process of implementing an information model using computer facilities. But what is it, an information model?

It is a list of information about any object. What does this model describe, and what useful information carries:

• properties of the object being modeled;
• his condition;
• connections with the outside world;
• relationships with external entities.

What can serve as an information model:

• verbal description;
• text;
• picture;
• table;
• scheme;
• drawing;
• formula and so on.

A distinctive feature of the information model is that it cannot be touched, tasted, and so on. It does not carry a material embodiment, as it is presented in the form of information.

A systematic approach to creating a model

In what class school curriculum studying modeling? Informatics grade 9 introduces students to this topic in more detail. It is in this class that the child learns about the systematic approach of modeling. Let's talk about this in a little more detail.

Let's start with the concept of "system". It is a group of interrelated elements that work together to complete a task. Often used to build a model systematic approach, since the object is considered as a system functioning in some environment. If any complex object is modeled, then the system is usually divided into smaller parts - subsystems.

Purpose of use

Now we will consider the goals of modeling (computer science grade 11). Earlier it was said that all models are divided into certain types and classes, but the boundaries between them are conditional. There are several features by which it is customary to classify models: purpose, area of ​​expertise, time factor, presentation method.

As for the goals, it is customary to distinguish the following types:

• educational;
• experienced;
• imitation;
• gaming;
• scientific and technical.

The first type includes educational materials. To the second, reduced or enlarged copies of real objects (a model of a structure, an airplane wing, and so on). allows you to predict the outcome of an event. Simulation modeling is often used in medicine and social sphere. For example, does the model help to understand how people will react to this or that reform? Before doing a serious operation on a person to transplant an organ, many experiments were carried out. In other words, the simulation model allows you to solve the problem by trial and error. A game model is a kind of economic, business or military game. With the help of this model, it is possible to predict the behavior of an object in different situations. A scientific and technical model is used to study a process or phenomenon (a device that simulates lightning discharge, planetary motion model solar system etc).

Field of knowledge

In which class do students get more familiar with modeling? Grade 9 computer science focuses on preparing its students for exams for admission to higher educational establishments. Since in USE tickets and GIA meet questions on modeling, now it is necessary to consider this topic in as much detail as possible. And so, how is the classification by area of ​​knowledge? By given feature distinguish the following types:

• biological (for example, artificially induced diseases in animals, genetic disorders, malignant neoplasms);
• firm behavior, market price formation model, and so on);
• historical ( genealogical tree, models historical events, model of the Roman army and the like);
• sociological (model of personal interest, the behavior of bankers when adapting to new economic conditions) etc.

Time factor

According to this characteristic, two types of models are distinguished:

• dynamic;
• static.

Already, judging by the name alone, it is not difficult to guess that the first type reflects the functioning, development and change of an object in time. Static, on the contrary, is able to describe an object at a particular moment in time. This view is sometimes called structural, since the model reflects the structure and parameters of the object, that is, it provides a slice of information about it.

Examples are:

• a set of formulas that reflect the movement of the planets of the solar system;
• graph of air temperature change;
• video recording of a volcanic eruption and so on.

Examples of a statistical model are:

• list of planets in the solar system;
• area map and so on.

Presentation method

To begin with, it is very important to say that all models have a shape and form, they are always made of something, somehow presented or described. On this basis, it is accepted as follows:

• material;
• intangible.

The first type includes material copies of existing objects. They can be touched, smelled and so on. They reflect the external or internal properties, actions of an object. What are material models for? They are used for experimental method knowledge (experimental method).

We also addressed non-material models earlier. They use theoretical method knowledge. Such models are called ideal or abstract. This category is divided into several subspecies: imaginary models and informational.

Information models provide a list of various information about the object. The information model can be tables, figures, verbal descriptions, diagrams and so on. Why this model called intangible? The thing is that it cannot be touched, since it does not have a material embodiment. Among information models distinguish between signs and visuals.

The imaginary model is one of the creative process, passing in the imagination of a person, which precedes the creation of a material object.

Modeling steps

The 9th grade computer science topic "Modeling and Formalization" has big weight. It is required to be studied. In grades 9-11, the teacher is obliged to introduce students to the stages of creating models. This is what we will do now. So, the following stages of modeling are distinguished:

• meaningful statement of the problem;
• mathematical formulation of the problem;
• developments with the use of computers;
• model operation;
• getting a result.

It is important to note that when studying everything that surrounds us, the processes of modeling and formalization are used. Computer science is a subject dedicated to modern methods studying and solving problems. Therefore, the emphasis is on models that can be implemented using a computer. Special attention in this topic should be given to the point of developing a solution algorithm using electronic computers.

Links between objects

Now let's talk a little about relationships between objects. There are three types in total:

• one to one (such a connection is indicated by a one-way arrow in one or the other direction);
• one-to-many (multiple relationships are indicated by a double arrow);
• many-to-many (such a relationship is indicated by a double arrow).

It is important to note that relationships can be conditional and unconditional. An unconditional relationship involves the use of each instance of an object. And in the conditional, only individual elements are involved.