The role of computers in modeling technogenic processes. Modeling technologies based on the use of computer technology Types of computer modeling of technical devices and processes

Effective use of simulation modeling is impossible without the use of a computer. The terms “computer modeling” and “simulation modeling” have become almost synonymous.

The use of computers in mathematical modeling opens up the possibility of solving a whole class of problems, and not only for simulation modeling. For other types of modeling, the computer is also very useful. For example, performing one of the main stages of research - constructing mathematical models based on experimental data - is currently simply unthinkable without the use of a computer. IN last years, thanks to the development of the graphical interface and graphic packages, computer, structural and functional modeling has received widespread development. The use of the computer even in conceptual modeling has begun, where it is used, for example, in building systems artificial intelligence.

Thus, the concept of “computer modeling” is much broader than the traditional concept of “computer modeling”. Currently, a computer model is usually understood as:

· description of an object or some system of objects (or processes) using interconnected computer tables, flowcharts, diagrams, graphs, drawings, animation fragments, etc., displaying the structure and relationships between the elements of the object. Computer models of this type are called structural-functional;

· a separate program, a set of programs, a software package that allows, using a sequence of calculations and graphical display of their results, to reproduce (simulate) the processes of functioning of an object, a system of objects, subject to the influence of various, including random, factors on it. Such models are called simulation models.

The concept of “algorithmic model” is closely related to the concept of “computer model”. An algorithmic model is a representation of a mathematical model using means of describing algorithms (algorithmic languages, flowcharts, etc.). An algorithmic model is, first of all, a description of the sequence of actions and the order of calculation for implementing the model, as well as the relationship of individual stages of calculations. The algorithmic model is built on the basis of a mathematical and, as a rule, simulation model. In an algorithmic model, in contrast to a conventional mathematical one, the peculiarities of computer operation and methods of implementing individual mathematical operators and functions on a computer are taken into account. After translating or compiling the algorithmic model into the computer's machine language, a computer model is obtained.

Computer modeling is a method for solving the problem of analysis or synthesis of a complex system based on the use of its computer model, i.e. launching a modeling program for execution at various values ​​of system parameters, impacts and initial conditions and using it to obtain quantitative and qualitative results. Qualitative conclusions obtained from the results of the analysis make it possible to discover previously unknown properties of a complex system: its structure, dynamics of development, stability, integrity, etc. Quantitative conclusions are mainly in the nature of a forecast of some future or explanation of past values ​​of variables characterizing the system.

A type of computer modeling is a computational experiment. It is based on the use of a simulation model and a computer, and allows research to be carried out similarly to full-scale modeling.

The subject of computer simulation can be any real object or process, for example, a static or dynamic cutting process. A computer model of a complex system allows you to display all the main factors and relationships that characterize real situations, criteria and limitations. Quantitative and qualitative benefits from the use of mathematical modeling on a computer are as follows:

1. The need for a long and labor-intensive stage of manufacturing a laboratory model or semi-industrial installation is completely or partially eliminated, and, accordingly, the costs for components, materials and structural elements necessary for the manufacture of models and installations, as well as for measuring instruments and equipment for testing the system .

2. Significantly reduces system characterization and testing time.

3. It becomes possible to develop systems containing elements with known characteristics, but absent in reality; simulate effects or modes of operation of the system, the reproduction of which during full-scale tests is difficult, requires complex additional equipment, is dangerous for the installation or the experimenter, and sometimes is completely impossible; obtain additional characteristics of an object that are difficult or impossible to obtain using measuring instruments (characteristics of parametric sensitivity, frequency, etc.).

LECTURE 4

"Classification of types of system modeling"

The modeling is based on similarity theory, which states that absolute similarity can only occur when one object is replaced by another exactly the same. When modeling, absolute similarity does not exist and one strives to ensure that the model sufficiently well reflects the aspect of the object’s functioning under study.

Classification characteristics. As one of the first signs of classification of types of modeling, you can select the degree of completeness of the model and divide the models in accordance with this sign into full, incomplete And close.

The basis of complete modeling is complete similarity, which manifests itself both in time and in space.

Incomplete modeling is characterized by incomplete similarity of the model to the object being studied.

Approximate modeling is based on approximate similarity, in which some aspects of the functioning of a real object are not modeled at all.

Classification of types of system modeling S shown in Fig. 1.

Depending on the nature of the processes being studied in the systemS all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous.

Deterministic Modeling displays deterministic processes, i.e. processes in which the absence of any random influences is assumed.

Stochastic modeling displays probabilistic processes and events. In this case, a number of realizations of a random process are analyzed and the average characteristics, i.e., a set of homogeneous realizations, are estimated.

Static Simulation serves to describe the behavior of an object at any point in time, and dynamic modeling reflects the behavior of an object over time.

Discrete Simulation serves to describe processes that are assumed to be discrete, respectively continuous simulation allows you to reflect continuous processes in systems, and discrete-continuous simulation used for cases where they want to highlight the presence of both discrete and continuous processes.

Depending on the form of representation of the object (systemS ) can be distinguished mental And real modeling.

Mental simulation is often the only way to model objects that are either practically unrealizable in a given time interval or exist outside the conditions possible for their physical creation. For example, on the basis of mental modeling, many situations in the microworld that are not amenable to physical experiment can be analyzed. Mental modeling can be implemented as visual, symbolic And mathematical. At visual modeling , on the basis of human ideas about real objects, various visual models are created that display the phenomena and processes occurring in the object. The basis hypothetical simulation the researcher lays down a certain hypothesis about the patterns of the process in a real object, which reflects the researcher’s level of knowledge about the object and is based on cause-and-effect relationships between the input and output of the object being studied. Hypothetical modeling is used when knowledge about an object is not enough to build formal models. Analog Modeling is based on the use of analogies at various levels. The highest level is a complete analogy, which occurs only for fairly simple objects. As the object becomes more complex, analogies of subsequent levels are used, when the analog model displays several or only one side of the object’s functioning. An important place in mental visual modeling is occupied by prototyping . A mental model can be used in cases where the processes occurring in a real object are not amenable to physical modeling, or can precede other types of modeling. The construction of mental models is also based on analogies, but usually based on cause-and-effect relationships between phenomena and processes in the object. If you introduce a symbol for individual concepts, i.e. signs, as well as certain operations between these signs, then you can implement iconic modeling and using signs to display a set of concepts - to compose separate chains of words and sentences. Using the operations of union, intersection and addition of set theory, it is possible to give a description of some real object in separate symbols. At the core language modeling there is some thesaurus. The latter is formed from a set of incoming concepts, and this set must be fixed. It should be noted that there are fundamental differences between a thesaurus and a regular dictionary. A thesaurus is a dictionary that is cleared of ambiguity, i.e. in it, each word can correspond to only a single concept, although in a regular dictionary several concepts can correspond to one word.

Symbolic modeling is an artificial process of creating a logical object that replaces the real one and expresses the basic properties of its relationships using a certain system of signs or symbols.

Math modeling . To study the characteristics of the process of functioning of any system S using mathematical methods, including machine methods, a formalization of this process must be carried out, i.e., a mathematical model must be built.

By mathematical modeling we mean the process of establishing a correspondence between a given real object and some mathematical object, called a mathematical model, and the study of this model, which allows us to obtain the characteristics of the real object under consideration.. The type of mathematical model depends both on the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem. Any mathematical model, like any other, describes a real object only with a certain degree of approximation to reality. Mathematical modeling for studying the characteristics of the process of functioning of systems can be divided into analytical, simulation and combined.

Analytical modeling is characterized by the fact that the functioning processes of system elements are written in the form of certain functional relationships (algebraic, integro-differential, finite-difference, etc.) or logical conditions. The analytical model can be studied using the following methods:

analytical, when one strives to obtain, in a general form, explicit dependencies for the desired characteristics;

numerical when, not being able to solve equations in general form, they strive to obtain numerical results with specific initial data;

high quality, when, without having an explicit solution, one can find some properties of the solution (for example, assess the stability of the solution).

The most complete study of the process of system functioning can be carried out if explicit dependencies are known that connect the desired characteristics with the initial conditions, parameters and variables of the system S. However, such dependencies can only be obtained for relatively simple systems. As systems become more complex, studying them using the analytical method encounters significant difficulties, which are often insurmountable. Therefore, wanting to use the analytical method, in this case they go to a significant simplification of the original model in order to be able to study at least the general properties of the system. Such a study using a simplified model using an analytical method helps to obtain indicative results for determining more accurate estimates using other methods. The numerical method makes it possible to study a wider class of systems compared to the analytical method, but the solutions obtained are of a particular nature. The numerical method is especially effective when using a computer.

In some cases, system studies can also satisfy the conclusions that can be drawn using a qualitative method of analyzing a mathematical model. Such qualitative methods are widely used, for example, in the theory of automatic control to evaluate the effectiveness of various options for control systems.

Currently, methods for computer implementation of studying the characteristics of the process of functioning of large systems are widespread. To implement a mathematical model on a computer, it is necessary to construct an appropriate modeling algorithm.

In simulation The algorithm that implements the model reproduces the process of functioning of the system S in time, and the elementary phenomena that make up the process are simulated, preserving their logical structure and sequence of occurrence in time, which allows, from the source data, to obtain information about the states of the process at certain points in time, making it possible to evaluate the characteristics of the system S.

The main advantage of simulation modeling compared to analytical modeling is the ability to solve more complex problems. Simulation models make it possible to quite simply take into account factors such as the presence of discrete and continuous elements, nonlinear characteristics of system elements, numerous random influences, etc., which often create difficulties in analytical studies. Currently, simulation is the most effective method for studying large systems, and often the only practically accessible method for obtaining information about the behavior of the system, especially at the design stage.

The simulation modeling method makes it possible to solve problems of analysis of large systems S, including problems of assessing: options for the system structure, the effectiveness of various system control algorithms, the influence of changes in various system parameters. Simulation modeling can also be used as the basis for the structural, algorithmic and parametric synthesis of large systems, when it is necessary to create a system with specified characteristics under certain restrictions, which is optimal according to certain efficiency assessment criteria.

When solving problems of machine synthesis of systems based on their simulation models, in addition to developing modeling algorithms for analyzing a fixed system, it is also necessary to develop algorithms for searching for the optimal version of the system. Further, in the methodology of machine modeling, we will distinguish two main sections: statics and dynamics, the main content of which is, respectively, questions of analysis and synthesis of systems specified by modeling algorithms.

Combined (analytical-simulation) modeling when analyzing and synthesizing systems, it allows you to combine the advantages of analytical and simulation modeling. When building combined models, a preliminary decomposition of the object’s functioning process into its constituent subprocesses is carried out and for those of them, where possible, analytical models are used, and simulation models are built for the remaining subprocesses. This combined approach allows us to cover qualitatively new classes of systems that cannot be studied using only analytical and simulation modeling separately.

Other types of modeling. In real modeling, the opportunity to study various characteristics either on a real object as a whole or on part of it is used. Such studies can be carried out both on objects operating in normal modes and when special modes are organized to assess the characteristics of interest to the researcher (with other values ​​of variables and parameters, on a different time scale, etc.). Real modeling is the most adequate, but at the same time its capabilities taking into account the characteristics of real objects are limited. For example, carrying out a real modeling of an automated control system by an enterprise will require, firstly, the creation of such an automated control system, and secondly, conducting experiments with the controlled object, i.e., the enterprise, which is impossible in most cases. Let's consider the types of real modeling.

Full-scale modeling called conducting research on a real object with subsequent processing of experimental results based on the theory of similarity. When an object functions in accordance with the set goal, it is possible to identify the patterns of the actual process. It should be noted that such types of full-scale experiments as production experiments and complex tests have a high degree of reliability.

With the development of technology and penetration into the depths of processes occurring in real systems, the technical equipment of modern scientific experiments increases. It is characterized by the widespread use of automation tools, the use of very diverse information processing tools, the possibility of human intervention in the process of conducting an experiment, and in accordance with this, a new scientific direction has emerged - the automation of scientific experiments.

The difference between an experiment and a real process is that individual critical situations may appear in it and the boundaries of process stability may be determined. During the experiment, new factors and disturbing influences are introduced during the operation of the object. One of the types of experiments is complex testing, which can also be classified as full-scale modeling, when, as a result of repeating product tests, general patterns about the reliability of these products, quality characteristics, etc.. In this case, modeling is carried out by processing and summarizing information occurring in a group of homogeneous phenomena. Along with specially organized tests, it is possible to implement full-scale modeling by summarizing the experience accumulated during the production process, i.e. we can talk about a production experiment. Here, on the basis of similarity theory, statistical material on the production process is processed and its generalized characteristics are obtained.

Another type of real modeling is physical, which differs from full-scale in that the research is carried out on installations that preserve the nature of the phenomena and have a physical similarity . In the process of physical modeling, certain characteristics of the external environment are specified and the behavior of either a real object or its model is studied under given or artificially created environmental influences. Physical modeling can take place on real and unreal (pseudo-real) time scales, and can also be considered without taking into account time. In the latter case, the so-called “frozen” processes that are recorded at a certain point in time are subject to study. The greatest complexity and interest from the point of view of the accuracy of the results obtained is physical modeling in real time.

From the point of view of the mathematical description of the object and depending on its nature, models can be divided into models analog (continuous), digital (discrete) and analog-digital (combined).

Under analog model is understood as a model that is described by equations relating continuous quantities.

By digital we mean a model, which is described by equations relating discrete quantities presented in digital form.

By analog-to-digital we mean the model, which can be described by equations relating continuous and discrete quantities.

A special place in modeling is occupied by cybernetic modeling, in which there is no direct similarity of the physical processes occurring in the models to real processes. In this case, they strive to display only a certain function and consider the real object as a “black box” with a number of inputs and outputs, and model some connections between outputs and inputs. Most often, when using cybernetic models, an analysis of the behavioral side of an object is carried out under various influences of the external environment. Thus, cybernetic models are based on the reflection of certain information management processes, which makes it possible to evaluate the behavior of a real object. To build a simulation model in this case, it is necessary to isolate the function of the real object under study, try to formalize this function in the form of some communication operators between input and output, and reproduce this function on the simulation model, and on the basis of completely different mathematical relationships and, naturally, a different physical implementation of the process .

LECTURE 5

“CAPABILITIES AND EFFECTIVENESS OF MODELING SYSTEMS ON VVM”

Providing the required indicators of the quality of functioning of large systems, associated with the need to study the flow of stochastic processes in the systems under study and design S, allows for a complex of theoretical and experimental studies that complement each other. The effectiveness of experimental studies of complex systems turns out to be extremely low, since conducting full-scale experiments with a real system either requires large material costs and considerable time, or is practically impossible (for example, at the design stage, when a real system is absent). The effectiveness of theoretical research from a practical point of view is fully manifested only when their results, with the required degree of accuracy and reliability, can be presented in the form of analytical relationships or modeling algorithms suitable for obtaining the corresponding characteristics of the process of functioning of the systems under study.

1.System modeling tools.

The emergence of modern computers was a decisive condition for the widespread introduction of analytical methods into the study of complex systems. It began to seem that models and methods, such as mathematical programming, would become practical tools for solving control problems in large systems. Indeed, significant progress has been made in the creation of new mathematical methods for solving these problems, but mathematical programming has not become a practical tool for studying the functioning of complex systems, since mathematical programming models turned out to be too crude and imperfect for their effective use. The need to take into account the stochastic properties of the system, the non-determinism of the initial information, the presence of correlations between a large number of variables and parameters characterizing processes in systems lead to the construction of complex mathematical models that cannot be used in engineering practice when studying such systems using the analytical method. Analytical relationships suitable for practical calculations can be obtained only with simplifying assumptions, which usually significantly distort the actual picture of the process under study. Therefore, recently there has been an increasingly noticeable need to develop methods that would make it possible to study more adequate models already at the system design stage. These circumstances lead to the increasingly widespread use of simulation modeling methods in the study of large systems.

Computers have now become the most constructive means of solving engineering problems based on modeling. Modern computers can be divided into two groups: universal ones, primarily intended for performing calculation work, and control computers, allowing not only calculation work, but primarily adapted for controlling objects in real time. Control computers can be used both to control the technological process, experiment, and to implement various simulation models.

Depending on whether it is possible to build a sufficiently accurate mathematical model of a real process, or, due to the complexity of the object, it is not possible to penetrate into the depth of the functional connections of a real object and describe them with some kind of analytical relationships, two main ways of using a computer can be considered:

as a means of calculation based on the obtained analytical models and

as a means of simulation modeling.

For a well-known analytical model, assuming that it fairly accurately reflects the studied aspect of the functioning of a real physical object, the computer is faced with the task of calculating the characteristics of the system using some mathematical relationships when substituting numerical values. In this direction, computers have capabilities that practically depend on the order of the equation being solved and on the requirements for the speed of solution, and both computers and automatic computers can be used.

When using a computer, an algorithm for calculating characteristics is developed, in accordance with which programs are compiled (or generated using a package of application programs) that make it possible to carry out calculations using the required analytical relationships. The main task of the researcher is to try to describe the behavior of a real object using one of the well-known mathematical models.

The use of AVM, on the one hand, speeds up the process of solving the problem for fairly simple cases; on the other hand, errors may arise due to the presence of drift of parameters of individual blocks included in the AVM, the limited accuracy with which the parameters entered into the machine can be set, and also malfunctions of technical equipment, etc.

The combination of computers and AVMs is promising, that is, the use of hybrid computer technology - hybrid computing systems (HCC), which in some cases significantly speeds up the research process.

GVK manages to combine the high speed of operation of analogue tools and the high accuracy of calculations based on digital computer technology. At the same time, due to the presence of digital devices, it is possible to ensure control over operations. Experience in using computer technology in modeling problems shows that as the object becomes more complex, the use of hybrid technology provides greater efficiency in terms of solution speed and cost of operations.

The specific technical means for implementing the simulation model can be a computer, an automated computer, and a computer. If the use of analog technology speeds up the production of final results, while maintaining some clarity of the real process, then the use of digital technology makes it possible to control the implementation of the model, create programs for processing and storing modeling results, and ensure an effective dialogue between the researcher and the model.

Typically, a model is built on a hierarchical principle, when individual aspects of the functioning of an object are sequentially analyzed and when the focus of the researcher’s attention moves, the previously considered subsystems move into the external environment. The hierarchical structure of models can also reveal the sequence in which a real object is studied, namely the sequence of transition from the structural (topological) level to the functional (algorithmic) level and from the functional to the parametric.

The result of modeling largely depends on the adequacy of the initial conceptual (descriptive) model, on the obtained degree of similarity to the description of a real object, the number of implementations of the model and many other factors. In a number of cases, the complexity of an object does not allow not only to build a mathematical model of the object, but also to give a fairly close cybernetic description, and promising here is to isolate the part of the object that is most difficult to mathematically describe and include this real part of the physical object in the simulation model. Then the model is implemented, on the one hand, on the basis of computer technology, and on the other hand, there is a real part of the object. This significantly expands the capabilities and increases the reliability of the simulation results.

The modeling system is implemented on a computer and allows you to study the model M , specified in the form of a certain set of individual block models and connections between them in their interaction in space and time during the implementation of any process. There are three main groups of blocks:

blocks characterizing the simulated process of functioning of the system S;

blocks displaying the external environment E and its impact on the process being implemented;

blocks that play an auxiliary role, ensuring the interaction of the first two, as well as performing additional functions for obtaining and processing simulation results.

In addition, the modeling system is characterized by a set of variables with the help of which it is possible to control the process under study, and a set of initial conditions when it is possible to change the conditions for conducting a machine experiment.

Thus, a modeling system is a means of conducting a machine experiment, and the experiment can be performed many times, planned in advance, and the conditions for its conduct can be determined. In this case, it is necessary to choose a method for assessing the adequacy of the results obtained and to automate both the processes of obtaining and the processes of processing results during a machine experiment.

2.Providing simulation.

The modeling system is characterized by the presence of mathematical, software, information, technical, ergonomic and other types of support.

Software The modeling system includes a set of mathematical relationships that describe the behavior of a real object, a set of algorithms that provide both preparation and work with the model. These may include algorithms: input of initial data, simulation, output, processing.

Software its content includes a set of programs: planning an experiment, modeling a system, conducting an experiment, processing and interpreting the results. In addition, the software must ensure synchronization of processes in the model, i.e., a block is needed that organizes pseudo-parallel execution of processes in the model. Machine experiments with models cannot take place without well-developed and implemented information support.

Information Support includes tools and technology for organizing and reorganizing the modeling database, methods for logical and physical organization of arrays, forms of documents describing the modeling process and its results. Information support is the least developed part, since only now is there a transition to the creation of complex models and the development of a methodology for their use in the analysis and synthesis of complex systems using the concept of a database and knowledge.

Technical support includes, first of all, means of computer technology, communication and exchange between the operator and the computer network, input and output of information, and control of the experiment.

Ergonomic support is a set of scientific and applied techniques and methods, as well as regulatory, technical and organizational and methodological documents used at all stages of interaction between a human experimenter and tools (computers, hybrid complexes, etc.). These documents, used at all stages of the development and operation of modeling systems and their elements, are intended to form and maintain ergonomic quality by justifying and selecting organizational and design solutions that create optimal conditions for highly effective human activity in interaction with the modeling complex.

Thus, the modeling system can be considered as a machine analogue of a complex real process. It allows you to replace an experiment with a real process of system functioning with an experiment with a mathematical model of this process in a computer. Currently, simulation experiments are widely used in the practice of designing complex systems when a real experiment is impossible.

Possibilities and efficiency of system modeling on a computer

Despite the fact that computer simulation is a powerful tool for studying systems, its use is not rational in all cases. There are many known problems that can be solved more effectively by other methods. At the same time, for a large class of problems in the research and design of systems, the simulation method is most suitable. Its correct use is possible only if there is a clear understanding of the essence of the simulation modeling method and the conditions for its use in the practice of studying real systems, taking into account the characteristics of specific systems and the possibilities of their study by various methods.

The following can be specified as the main criteria for the advisability of using the computer simulation method: the absence or unacceptability of analytical, numerical and qualitative methods for solving the problem; the presence of a sufficient amount of initial information about the simulated system S to ensure the possibility of constructing an adequate simulation model; the need to carry out, on the basis of other possible methods of solving, a very large number of calculations that are difficult to implement even using a computer; the ability to search for the optimal version of the system when modeling it on a computer.

Computer simulation, like any research method, has advantages and disadvantages that manifest themselves in specific applications. The main advantages of the simulation method in the study of complex systems include the following: a machine experiment with a simulation model makes it possible to study the features of the process of functioning of the system S under any conditions; the use of a computer in a simulation experiment significantly reduces the duration of testing compared to a full-scale experiment; the simulation model allows you to include the results of full-scale tests of a real system or its parts for further research; the simulation model has a certain flexibility in varying the structure, algorithms and parameters of the simulated system, which is important from the point of view of finding the optimal version of the system; Simulation modeling of complex systems is often the only practically feasible method for studying the functioning of such systems at the stage of their design.

The main disadvantage that appears in the machine implementation of the simulation method is that the solution obtained by analyzing the simulation model M is always of a private nature, since it corresponds to fixed elements of the structure, behavior algorithms and values ​​of system parameters S, initial conditions and external influences. environment E. Therefore, in order to fully analyze the characteristics of the system functioning process, and not obtain only a single point, it is necessary to repeatedly reproduce the simulation experiment, varying the initial data of the problem. In this case, as a consequence, there is an increase in the cost of computer time for conducting an experiment with a simulation model of the process of functioning of the system under study S.

The effectiveness of machine modeling. With simulation modeling, as with any other method of analysis and synthesis of system S, the issue of its effectiveness is very important. The effectiveness of simulation modeling can be assessed by a number of criteria, including the accuracy and reliability of the modeling results, the time it takes to build and work with the model M, the cost of machine resources (time and memory), the cost of developing and operating the model. Obviously, the best assessment of effectiveness is to compare the results obtained with a real study, that is, with modeling on a real object during a full-scale experiment. Since this cannot always be done, the statistical approach makes it possible to obtain, with a certain degree of accuracy and repeatability of a machine experiment, some average characteristics of the behavior of the system. The number of implementations has a significant impact on the accuracy of the simulation, and depending on the required reliability, the required number of implementations of a reproducible random process can be estimated.

An essential indicator of efficiency is the cost of computer time. In connection with the use of computers of various types, the total costs consist of the time for input and output of data for each modeling algorithm, the time for performing computational operations, taking into account access to RAM and external devices, as well as the complexity of each modeling algorithm. Calculations of computer time costs are approximate and can be refined as the programs are debugged and the researcher gains experience when working with the simulation model. Rational planning of such experiments has a great influence on the cost of computer time when conducting simulation experiments. Procedures for processing simulation results, as well as the form of their presentation, can have a certain impact on the cost of computer time.

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. The next step is to develop the theory and construct 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 its 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) the explanation is already known results empirical research, 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. Various types of 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”, the internal structure of 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, a computer program that simulates the game of checkers. 1.2. Structural models– these are models whose structure corresponds to the structure of the modeled object. Examples are tabletop exercises, self-government day, electronic circuit model 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 a system of 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, a chemical or mathematical formula, or a graph. This graphic image characterizes the spatial arrangement of elements, their nesting or subordination, and the chronological sequence of various parts of a complex event. When building a model, it is recommended to draw up structural diagrams of 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 systematic approach is emergence principle, –– when elements (parts, components) are combined into a single whole, a systemic effect occurs: the system acquires qualities that none of its constituent elements possesses. The principle of highlighting the main structure system is that the study of a fairly complex object requires highlighting a certain part of its structure, which is the main or fundamental 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, different results will be obtained, contradictory friends to a friend. 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, we start 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 included in it and form 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 an oscillatory circuit consisting of a capacitor and an inductor. When a charged capacitor is connected to an inductor, it begins to discharge, and current, energy, flows through the inductor electric field transforms into magnetic field energy. 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 of a mechanical system is determined by the coordinates of the particles entering it and the projections of their impulses. The state of the electrical circuit is determined by the charge of the capacitor, the current through the inductor, etc. State economic system is determined by a set of indicators such as the amount of money invested in production, profit, the number of workers involved in the manufacture of 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 for studying a 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

An effective way to study 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. The widespread spread of personal computers, information technologies, and the creation of powerful supercomputers have 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. In addition, information support, software and hardware must allow the model to exchange information with the corresponding database and ensure efficient machine implementation and convenient user experience.

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 at the 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 braking of an asynchronous motor, a transient process in a circuit containing a capacitor, the functioning of a computer network, system queuing. 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 explicitly, then we are talking 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 of mathematical equations or a computer program. Modeling is understood as the representation of the main characteristics of the 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. If complete information is available, modeling is meaningless. 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 of the 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 receives external signals, carries out appropriate 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 a formal description of the logic of 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”, educational programs and animations. For example, a model of a technological process, an airfield, a certain industry, etc.

4. Statistical modeling is used to study stochastic systems and consists of repeated testing followed by statistical processing of the resulting results. Such models make it possible to study the behavior of all kinds of queuing systems, multiprocessor systems, information and computer networks, and various dynamic systems affected 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 the knowledge base of a certain subject area (part 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. TO this species models include semantic networks, logical knowledge models, expert systems, logic games, etc. Logic models used to represent knowledge in expert systems, to create artificial intelligence systems, carry out logical inference, prove theorems, mathematical transformations, build robots, use natural language to communicate with computers, create the effect of virtual reality 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) gaming 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 are used to study discrete systems that 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 on digital and analog 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 enough simple systems.

One of the effective methods for studying simulation models 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.

The improvement of computer technology, the emergence of a computer that allows one to quickly and accurately carry out calculations according to a 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 demographic situation in the country and the world; 9) simulation modeling of the operation of various technical, in particular electronic devices; 10) economic research on the development of an enterprise, industry, country.

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URL: (access date: 01/15/2020).

Computer modeling is a method for solving problems of analysis or synthesis of a complex system based on the use of its computer model.

Computer simulation can be thought of as:

math modeling;

simulation modeling;

stochastic modeling.

The term “computer model” is understood as a conventional image of an object or some system of objects (or processes), described using equations, inequalities, logical relationships, interconnected computer tables, graphs, charts, graphs, drawings, animation fragments, hypertexts, etc. and displaying the structure and relationships between the elements of the object. Computer models described using equations, inequalities, logical relationships, interconnected computer tables, graphs, charts, graphs will be called mathematical. Computer models described using interconnected computer tables, graphs, diagrams, graphs, drawings, animation fragments, hypertexts, etc. and displaying the structure and relationships between the elements of the object, we will call it structural-functional;

Computer models (a separate program, a set of programs, a software package), allowing, using a sequence of calculations and graphical display of the results of its work, to reproduce (simulate) the processes of functioning of an object (system of objects) subject to the influence of various, usually random, factors on the object, we will call them imitative.

The essence of computer modeling is to obtain quantitative and qualitative results using the existing model. Qualitative results of the analysis reveal previously unknown properties of a complex system: its structure, dynamics of development, stability, integrity, etc. Quantitative conclusions are mainly in the nature of an analysis of an existing system or a forecast of future values ​​of some variables. The ability to obtain not only qualitative, but also quantitative results is a significant difference between simulation modeling and structural-functional modeling. Simulation modeling has a number of specific features. In each of them, depending on the complexity of the model, the goals

modeling, the degree of uncertainty of the model characteristics, can

there are different ways of conducting research

(experiments), i.e., research methods. For example, with analytical

Various mathematical methods are used in the study. In physical or full-scale modeling, an experimental research method is used.

Analysis of current and promising methods of machine experimentation allows us to distinguish between computational, statistical, simulation and self-organizing research methods.

Computational (mathematical) modeling is used in the study of mathematical models and comes down to their computer implementation with various numerical input data. The results of these implementations (calculations) are presented in graphical or tabular forms. For example, a classic scheme is a machine implementation of a mathematical model, presented in the form of a system of differential equations, based on the use of numerical methods, with the help of which the mathematical model is reduced to an algorithmic form, the software is implemented on a computer, and calculations are carried out to obtain the results.

Simulation modeling is characterized by a high degree of generality, creates the prerequisites for the creation of a unified model, easily adaptable to a wide class of problems, and acts as a means for integrating models of different classes.

computer modeling as the main method of analysis, forecasting and planning of economic systems.

A computer model, or a numerical model, is a computer program running on a separate computer, supercomputer or many interacting computers (computing nodes), implementing an abstract model of a system. Computer models have become a common tool for mathematical modeling and are used in physics, astrophysics, mechanics, chemistry, biology, economics, sociology, meteorology, other sciences and applied problems in various fields of radio electronics, mechanical engineering, automotive industry, etc. Computer models are used to obtain new knowledge about the modeled object or to approximate the behavior of systems that are too complex for analytical study.

Computer modeling is one of the effective methods for studying complex systems. Computer models are easier and more convenient to study due to their ability to carry out the so-called. computational experiments, in cases where real experiments are difficult due to financial or physical obstacles or may give unpredictable results. The logic and formalization of computer models makes it possible to identify the main factors that determine the properties of the original object under study (or an entire class of objects), in particular, to study the response of the simulated physical system to changes in its parameters and initial conditions.

The construction of a computer model is based on abstraction from the specific nature of the phenomena or the original object being studied and consists of two stages - first the creation of a qualitative and then a quantitative model. Computer modeling consists of conducting a series of computational experiments on a computer, the purpose of which is to analyze, interpret and compare the modeling results with the real behavior of the object under study and, if necessary, subsequent refinement of the model, etc.

Comparative computer animation of two building models

The main stages of computer modeling include:

statement of the problem, definition of the modeling object;

development of a conceptual model, identification of the main elements of the system and elementary acts of interaction;

formalization, that is, the transition to a mathematical model; creating an algorithm and writing a program;

planning and conducting computer experiments;

analysis and interpretation of results.

There are analytical and simulation modeling. In analytical modeling, mathematical (abstract) models of a real object are studied in the form of algebraic, differential and other equations, as well as those involving the implementation of an unambiguous computational procedure leading to their exact solution. In simulation modeling, mathematical models are studied in the form of an algorithm(s) that reproduces the functioning of the system under study by sequentially performing a large number of elementary operations.

Related information.

Mathematical model. Classification of mathematical models.

Mathematical model expresses the essential features of an object or process in the language of equations and other math. funds.

Mathematical modeling does not always require computer support. Every specialist who professionally deals with math. modeling does its best for research. An analytical solution (representation by formulas) is usually more convenient and more informative than numerical ones. The concepts of “analytical solution” and “computer solution” do not oppose each other, because:

1) increasingly computers with mat. modeling are used not only for numerical calculations, but also for analytical transformations.

2) the result of an analytical study of mat. A model is often expressed in such a complex formula that when looking at it, one does not develop the perception of the process it describes.

Classification of mat. models.

1. Descriptive (descriptive) models.

2. Optimization models.

3. Multicriteria models.

4. Gaming.

5. Imitation.

By modeling the movement of a comet that has invaded the Solar System, we describe the trajectory of its flight, the distance at which it will pass from the Earth, i.e. We set descriptive goals. We have no way to influence the movement of the comet or change anything.

At another level of processes, we can influence them, trying to achieve some goal. In this case, the model includes one or more parameters available to our influence. For example, by changing the thermal regime in a granary, we can strive to select one that will achieve maximum grain safety, i.e. we optimize the process.

It is often necessary to optimize a process along several parameters at once, and the goals may be contradictory. For example, knowing the prices of food and a person’s need for food, organize meals for large groups of people as healthy and cheap as possible, i.e. When modeling there will be several criteria between which a balance must be sought.

There is a special, rather complex section of modern mathematics - game theory - that studies methods of decision-making under conditions of incomplete information.

It happens that the model imitates the real process to a greater extent, i.e. imitates him. For example, modeling the motion of molecules in a gas, when each molecule is represented as a ball, the conditions for the behavior of these balls when colliding with each other and with a wall are created, without the need to use any equations of motion. It can be said that most often simulation modeling is used in an attempt to describe the properties of a large system, provided that the behavior of its constituent objects is very simple and clearly formulated.

Computer model– this is a model implemented by means of a software environment.

1. Modeling of physical processes. Physics is a science in which math. Modeling is an extremely important research method.

Numerical modeling (as well as laboratory experiments) are most often a tool for understanding the qualitative laws of nature. Its most important stage, when the calculations have already been completed, is understanding the results, presenting them in the most visual and easy-to-understand form. Cramming a computer screen with numbers or getting a printout of the same numbers does not mean finishing the simulation (even if the numbers are correct). This is where another remarkable feature of the computer comes to the rescue, complementing the ability to quickly calculate - the ability to visualize abstractions. Presentation of results in the form of graphs, diagrams, trajectories of movement of dynamic objects, due to the peculiarities of human perception, enriches the researcher with qualitative information.

2. Computer modeling in ecology. The goals of creating the mat. models in ecology.

1. Models help to highlight or combine and express, using several parameters, important properties of a large number of unique observations, which makes it easier for an ecologist to analyze the process or problem under consideration.

2. Models act as a “common language” through which each unique phenomenon can be described and the relative properties of such phenomena become better understood.

3. A model can serve as an example of an “ideal object” or idealized behavior, by comparison with which real objects and processes can be assessed and measured.

4. Models can actually shed light on the real world, of which they are imperfect imitations.

When building models in mat. ecology uses the experience of mat. modeling mechanical and physical systems, but taking into account the specific features of biological systems:

The complexity of the internal structure of each individual;

Dependence of living conditions of organisms on many environmental factors;

Not closed ecological systems;

A huge range of external characteristics that maintain the viability of systems.

3. Computer mat. modeling in economics- this is mate. description of the object under study. This model expresses the laws of the economic process in abstract form using math. ratios. Use of mat. modeling in economics allows us to deepen quantitative economic analysis and expand the field of economic informatics.