Chapter I. Computer simulation. Abstract: Computer modeling and its features

, 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.

Construction computer model is based on abstraction from the specific nature of 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.

The main stages of computer modeling include:

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.

Practical Application

Computer modeling is used for a wide range of tasks, such as:

• analysis of the distribution of pollutants in the atmosphere
• designing noise barriers to combat noise pollution
• vehicle design
• flight simulators for pilot training
• weather forecasting
• emulation of the work of others electronic devices
• forecasting prices in financial markets
• study of the behavior of buildings, structures and parts under mechanical load
• predicting the strength of structures and their destruction mechanisms
• design production processes, for example chemical
• strategic management of the organization
• study of the behavior of hydraulic systems: oil pipelines, water pipelines
• modeling of robots and automatic manipulators
• modeling of urban development scenarios
• transport systems modeling
• simulated crash tests
• modeling the results of plastic surgery

Different areas of application of computer models have different requirements for the reliability of the results obtained with their help. Modeling of buildings and aircraft parts requires high precision and confidence, while models of the evolution of cities and socio-economic systems are used to obtain approximate or qualitative results.

Computer simulation algorithms

• Component circuit method
• State variable method

See also

Links

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Mayer R.V. Computer simulation

Mayer R.V., Glazov Pedagogical Institute

COMPUTER SIMULATION:

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 facts, 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) explanation of already known results of empirical studies, verification of model parameters using experimental data; 3) predicting the behavior of systems under new conditions under various external influences and control methods; 4) optimization of the functioning of the systems under study, search for the correct control of the object in accordance with the selected optimality criterion.

1.2. 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”, internal structure 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 the 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 a generator AC, 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 broad class iconic models, which use mathematical operations and operators. They often represent a system of algebraic or differential equations. Computer models are an algorithm or computer program, solving system logical, algebraic or differential equations and simulating 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 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 nature of the mathematical apparatus models can be linear and nonlinear, continuous and discrete, deterministic and probabilistic, static and dynamic. By way of implementation distinguish between analog, digital, hybrid, neuro-fuzzy models, which are created on the basis of analog, digital, hybrid computers and neural networks.

1.3. Modeling and systems approach

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

Rice. 2. System under study and environment.

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

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

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

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

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

The more complex the object under study, the more different models (descriptions) can be built. Thus, looking at a cylindrical column from different sides, all observers will say that it can be modeled as a homogeneous cylindrical body of certain dimensions. If, instead of a column, observers begin to look at some complex architectural composition, then everyone will see something different and build their own model of the object. In this case, as in the case of the sages, various results will be obtained that contradict each other. And the point here is not that there are many truths or that the object of knowledge is fickle and many-sided, 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 in various ways; 4) interdependence of system and environment, –– elements of the system are connected with objects that are not part of it and form the environment; 5) hierarchy, –– the system has a multi-level structure.

1.4. Qualitative and quantitative models

The task of science is to build a theoretical model of the surrounding world that would explain known and predict unknown phenomena. The theoretical model can be qualitative or quantitative. Let's consider quality explanation electromagnetic vibrations V 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 makes it possible 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 makes no sense 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 determined by a set of indicators such as the number cash invested in production, profit, number of workers involved in manufacturing products, etc.

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

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

All existing methods studies of the mathematical model can be divided into two groups .Analytical solving an equation often involves cumbersome and complex mathematical calculations and, as a result, leads to an equation expressing the functional relationship between the desired quantity, system parameters, external influences and time. The results of such a solution require interpretation, which involves analyzing the obtained functions and constructing graphs. Numerical methods research of a mathematical model on a computer involves the creation computer program, which solves the 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 simulation

An effective way to study the phenomena of the surrounding reality is scientific experiment, which consists of reproducing the natural phenomenon being studied under controlled and controlled conditions. However, often carrying out an experiment is impossible or requires too much economic expense 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 gives more detailed information about quantities characterizing the state of the system.

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

To the main stages of computer modeling include (Fig. 3): 1) problem statement, 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 where the law of functioning contains time, we speak of 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, and a queuing system. 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 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 responds 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 by methods of computational mathematics and the conduct of 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”, training 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. This type of model includes 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, building robots, using natural language to communicate with a computer, creating a virtual reality effect in computer games, etc.

Rice. 5. Computer model of chess player behavior.

Based on 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 select the best way management of 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 education, training and testing of 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 systems 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 This type is 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 abstract finite state machine, 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 one to study all kinds of systems, which 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 applied mathematics, numerical methods for solving systems of equations.

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

Literature

Boev V.D., Sypchenko R.P., Computer modeling. –– INTUIT.RU, 2010. –– 349 p. Bulavin L.A., Vygornitsky N.V., Lebovka N.I. Computer modeling of physical systems. –– Dolgoprudny: Publishing House “Intelligence”, 2011. – 352 p. Buslenko N.P. Modeling of complex systems. –– M.: Nauka, 1968. –– 356 p. Dvoretsky S.I., Muromtsev Yu.L., Pogonin V.A. Systems modeling. –– M.: Publishing house. Center “Academy”, 2009. –– 320 p. Kunin S. Computational physics. –– M.: Mir, 1992. –– 518 p. Panichev V.V., Solovyov N.A. Computer modeling: textbook. –– Orenburg: State Educational Institution OSU, 2008. – 130 p. Rubanov V.G., Filatov A.G. Modeling systems tutorial. –– Belgorod: BSTU Publishing House, 2006. –– 349 p. Samarsky A.A., Mikhailov A.P. Mathematical Modeling: Ideas. Methods. Examples. –– M.: Fizmatlit, 2001. –– 320 p. Sovetov B.Ya., Yakovlev S.A. Modeling of systems: Textbook for universities –– M.: Vyssh. School, 2001. – 343 p.

10. Fedorenko R.P. Introduction to computational physics: Proc. manual: For universities. –– M.: Publishing house Mosk. Phys.-Techn. Institute, 1994. –– 528 p.

11. Shannon R. Simulation modeling of systems: art and science. –– M.: Mir, 1978. –– 302 p.

Mayer R.V. COMPUTER SIMULATION: SIMULATION AS A METHOD OF SCIENTIFIC COGNITION. COMPUTER MODELS AND THEIR TYPES // Scientific electronic archive.
URL: (access date: 03/28/2019).

Computer modeling in physics.

Kalenov M.Yu.

Balakin M.A.

Khudyakov A.B.

MBOU Lyceum No. 38

Nizhny Novgorod

3. Thematic planning of the elective - “computer modeling in physics”.

5. The first results obtained during the course “computer modeling in physics”.

1. The role of computer modeling in physics.

Bologna the convention signed in 2003 by the Minister of Education of the Russian Federation significantly changes the situation physicists , as a subject studied in secondary school and non-physical faculties universities Following the provisions of the Sorbonne Declaration, the Russian state, by 2010, undertakes to transform physics from the most important general cultural and educational component of the individual into one of the subjects chosen by the student in accordance with his personal educational trajectory.

The chosen course of education reform causes fair and justified concern among the teaching community. At the same time, it must be admitted that it is consistent with the administrative, financial, legislative and other reforms carried out in the country: the necessary volume and depth of knowledge on physics should determine the needs of the market, not plans to create an abstract person future

At the same time, it should be noted that no reforms in physics education can change the objective status of physics as the fundamental basis of all areas of modern scientific knowledge. The very first attempts of ancient philosophers to explain the structure of the world were nothing more than classes physics, and modern civilization, existing in a single global information space, acquired its characteristic features also thanks to the development of physical science. The history of physics is the history of humanity cognizing the Universe and creating an unnatural reality; the study of physics develops the intellect and shapes the worldview.

In addition to the requirements for the modernization of education, determined by modern trends in the development of education, traditionally relevant is the need to ensure meaningful and methodological continuity in the study of physical phenomena, processes and patterns when they are considered in general physics courses. Formalized presentation of educational material and algorithmization educational research activities of students, characteristic both for the course of general physics and for the disciplines developing its provisions, lead to the fact that understanding of the physical essence of the subject gives way to assimilation ready-made knowledge and acquisition of a limited number skills . At the same time, modern trends in the development of physical education are aimed at developing students’ skills non-standard think, use intellectual andcommunicativeabilities for the successful organization of professional and social activities in continuously changing multifactorial situations.

Computer simulation, which is integral part and tool computer training, contains potential opportunities to increase the efficiency of studying physical foundations in general physics courses. These features include:

Increased visibility , variability, interactivity and information capacity provided educational material, compensation, through this, reduction in the number of hours classroom training;

Carrying out experimental activities that are difficult, impossible or unsafe in a training laboratory, ensuring multiplicity and variability of experiments;

Modernization of full-scale laboratory research through the use of computer models for visual representation;

Increased efficiencyindependentstudents’ work through providing the opportunity to choose and implement an individual routeindependenttraining appropriate to the level of knowledge, temperament and characteristics students' thinking;

Developing students' skills for independent work with the most important form of information representation - a model, developing skills in using a mathematical model when planning, staging and interpreting the results of an educational full-scale experiment, the ability to assess the scope of application of the model;

Creating conditions for the implementation of a person-centered approach to learning;

Rationalization of student work and teacher through the transfer of routine calculation and verification functions and a focus on the creative aspect of educational research.

2. Objectives, goals and methods of the project - “computer modeling in physics”.

Goals:

Develop students' skill in creating programs in languagePascal.

Develop students' modeling skills physical processes, solving problems necessary to create models.

Motivate students for research activities.

Strengthen and develop students' knowledge base in physics and computer science.

Enrich the database of demonstration experiments used in physics lessons.

Tasks:

Creating a plan for elective classes with students on the topic “Computer modeling in physics.”

Preparation necessary materials to implement the course and attract students to it.

Organization of teaching students the basics of computer programming in the languagePascal.

Organization of students' research activities in computer modeling.

Selection of problems for use in physics lessons.

Methods.

We chose students' research work as a method for solving set problems to achieve given goals. In this case, the teacher plays the role of an assistant and only corrects the students’ mental activity. This does not relieve the teacher of his responsibilities, but it does give students greater freedom to express their creativity.

However practical exercises I will also rotate lectures for students to achieve best results and increasing the theoretical base.

The solution to each of the educational tasks is carried out according to the following plan:

Introduction to the problem.

The essence of the task and its practical meaning are explained.

Theory of the issue.

All issues related to the theory of the physical phenomenon/process under consideration are discussed.

Discussion.

Discussion of solutions and modeling methods.

The theory of program creation.

All necessary issues are discussed for students to successfully write a computer program in the language Pascal .

Practical part.

Creation of a computer model by students.

Conclusions.

Discussion of the results obtained.

The course begins with problems on numerical integration and differentiation, in order to further apply these developments when creating physical models. In the future, students become familiar with modeling the motion of bodies in the field of gravity (grade 10), become familiar with the Kepler problem, oscillatory motion (grade 11) and wave phenomena (grade 11). These topics were chosen for study based on the fact that, in our opinion, they are the simplest for students and the most visual. The complexity of the course introduces an age limit: so only students in grades 10 and 11 are invited to participate in elective classes.

For the theoretical basis of the course on computer modeling in physics, we took books by the authors H. Gould, J. Tobochnik. “Computer modeling in physics.”;

3. Thematic planning of the elective - “computer modeling in physics”. 68 hours.

Subject

Number of hours

The importance of computers in physics. The importance of graphics. Programming languagePascal

Reviewing the basics of the languagePascal. Procedures and functions. Constants and variables. Basic algorithmic structures.

Numerical integration

The concept of integral. Simple one-dimensional methods of numerical integration.

Numerical example.

Numerical integration of many integrals.

Calculation of integrals using the Monte Carlo method.

Error analysis of the Monte Carlo method.

Coffee cooling problem.

Basic concepts. Euler's algorithm.

Program for solving the problem.

Stability and precision.

The simplest graphics.

Falling bodies

Basic concepts. Force acting on a falling body.

Numerical solution of equations.

One-dimensional movement.

Two-dimensional trajectories.

Kepler's problem.

Introduction. Equation of planetary motion.

Circular movement.

Elliptical orbits.

Astronomical units. Programming notes.

Numerical simulation of the orbit.

Disturbance.

Space of speeds.

Solar system in miniature.

Oscillations.

A simple harmonic oscillator.

Numerical modeling of a harmonic oscillator.

Mathematical pendulum. Programming notes.

Damped oscillations. Linear response to external force.

Principles of superposition. Oscillations in external circuits.

Wave phenomena.

Introduction. Related oscillators.

Fourier analysis.

Wave motion.

Interference and diffraction.

Polarization.

Geometric optics.

4. Examples of problems solved by students.

Previously, we have already integrated individual tasks from the computer modeling course in physics into elective classes in computer science.

The results we obtained inspired us to organize a separate elective course. Participants who solved problems on modeling physical processes better mastered new material and easily solved problems related to the topics for which they created physical models.

Example. Modeling harmonic vibrations.

An example of a program created by one of the students is shown in Figure No. 1

Figure 1.

At the same time, 11th grade students wrote a test paper on the topic “Mechanical vibrations, waves, sound”

The results were as follows

The average score for the test work of students participating in the course is 4.5

The average score for the test work of all 11th grade students of Municipal Educational Institution Lyceum No. 38 is 3.9

In addition, students' performance in computer science also increased.

So we see that the quality of knowledge on the topic of harmonic vibrations of the students participating in the course was average. This confirms the effectiveness of this course.

The model created by the students can also be used by the teacher as a demonstration experiment in physics lessons in the topic “Mechanical vibrations, waves, sound.”

4. Conclusions.

Currently, the quality of students’ knowledge in basic and necessary subjects like air in a modern world filled with innovations is falling. (Physics, computer science, mathematics) There are many ways to combat this.

However, the course of elective classes that we developed not only stimulates students’ interest in physics, but also strengthens the theoretical and practical knowledge base in this subject, while simultaneously improving students’ practical skills in computer science and mathematics. At the same time, the teacher’s toolkit is expanding, which he can use for demonstration experiments in physics lessons.

Thanks to all these features, we achieve high results in the quality of knowledge in several subjects at once.

Literature:

D.Heerman. Methods of computer experiment in statistical physics. Translation from English, "Science", Moscow, 1990.

K. Binder, D. Heerman. Monte Carlo simulation in statistical physics. Translation from English, "Science", Moscow, 1995.

Monte Carlo methods in statistical physics. Ed. K. Binder, Moscow, Mir, 1982.

H. Gould, J. Tobochnik. Computer modeling in physics. In 2 volumes, Moscow, Mir, 1990.

M. P. Allen, D. J. Tildesley. Computer simulation of liquids.Clarendon Press, Oxford, 1987.

K. Binder (editor), Applications of the Monte Carlo method in statistical physics, Springer-Verlag, 1987.

M. P. Allen, D. J. Tildesley (eds.). Computer simulation in Chemical Physics.Kluwer Academic Publishers, 1993.

Monte Carlo and Molecular Dynamics Simulations in Polymer Science.K. Binder (ed.), Oxford University Press, 1995.

Monte Carlo and Molecular Dynamics of Condensed Matter Physics, edited by K. Binder and G. Ciccotti, (proceedings of the conference in Como, Italy), 1996.

D.Frenkel, B.Smit, Understanding molecular simulation: from algorithms to applications. Academic Press, 1996.

Kobelnitsky Vladislav

Computer simulation. Simulation of physical and mathematical processes on a computer.

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Research work

"COMPUTER SIMULATION"

COMPLETED:

KOBELNITSKY VLADISLAV

9TH CLASS STUDENT

MKOU secondary school No. 17

Supervisor:

mathematics and computer science teacher

Tvorozova E.S.

KANSK, 2013

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

INTRODUCTION

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

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

Topic of my research:"Computer Modeling".

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

The purpose of the work is study the possibilities of computer modeling and its use in various subject areas.

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

– give theoretical information about modeling;

– describe the stages of modeling;

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

Draw a general conclusion about computer modeling in subject areas.

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

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

COMPUTER SIMULATION

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

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

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

Modeling – the process of creating and using a model.

Modeling Goals

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

Classification by presentation form

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

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

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

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

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

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

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

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

1. 3.3. Quite formalized(mathematical) models.

Model properties

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

It should also be noted:

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

The process of building a model is called modeling.

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

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

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

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

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

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

PRACTICAL PART

Modeling of various processes and phenomena

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

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

First stage

Second stage

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

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

Determination of the specific heat capacity of a substance

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

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

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

Work order:

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

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

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

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

From the heat balance equation

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

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

m 1 – mass of the aluminum vessel;

c 1 – specific heat capacity of aluminum;

m 2 - mass of water;

from 2 - specific heat capacity of water;

t 1 - initial water temperature

m - cylinder mass;

t 2 - initial temperature of the cylinder;

T - general temperature

Work 2 “Study of oscillations of a spring pendulum”

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

First stage . A mathematical model is compiled.

Second stage . Working with the compiled model.

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

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

Description of work in the laboratory workshop:

A load suspended on a steel spring and disturbed from a state of equilibrium undergoes harmonic oscillations under the influence of gravity and the elasticity of the spring. The natural frequency of oscillation of such a spring pendulum is determined by the expression

where k – spring stiffness; m – body weight.

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

Getting the job done.

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

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

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

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

where n – number of oscillations.

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

Work 3 “Law of conservation of mechanical energy”

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

First stage . Drawing up a mathematical model.

Second stage . Working with the compiled model.

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

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

Description of work in laboratory workshop

CHECKING THE LAW OF CONSERVATION OF MECHANICAL ENERGY.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Conclusion

I have identified the advantages of using Excel:

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

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

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

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

REFERENCES

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

Let's start with the definition of the word modeling.

Modeling is the process of constructing and using a model. A model is understood as a material or abstract object that, in the process of study, replaces the original object, preserving its properties that are important for this study.

Computer modeling as a method of cognition is based on mathematical modeling. A mathematical model is a system of mathematical relationships (formulas, equations, inequalities and signed logical expressions) that reflect the essential properties of the object or phenomenon being studied.

It is very rarely possible to use a mathematical model for specific calculations without the use of computer technology, which inevitably requires the creation of some kind of computer model.

Let's look at the computer modeling process in more detail.

2.2. Introduction to Computer Modeling

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 conduct computational experiments in cases where real experiments are difficult due to financial or physical obstacles or may give unpredictable results. The logic 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.

Computer modeling as a new method scientific research is based on:

1. Construction of mathematical models to describe the processes being studied;

2. Using the latest high-speed computers (millions of operations per second) and capable of conducting a dialogue with a person.

Distinguish analytical And imitation 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 that reproduces the functioning of the system under study by sequentially performing a large number of elementary operations.

2.3. Building a computer model

The construction of a computer model is based on abstraction from the specific nature of phenomena or the original object being studied and consists of two stages - first creating 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.

So, The main stages of computer modeling include:

1. Statement of the problem, definition of the modeling object:

At this stage, information is collected, a question is formulated, goals are defined, forms for presenting results, and data is described.

2. System analysis and research:

system analysis, meaningful description of the object, development of an information model, analysis of hardware and software, development of data structures, development of a mathematical model.

3. Formalization, that is, the transition to a mathematical model, the creation of an algorithm:

choosing a method for designing an algorithm, choosing a form for writing an algorithm, choosing a testing method, designing an algorithm.

4. Programming:

choosing a programming language or application environment for modeling, clarifying ways to organize data, writing an algorithm in the selected programming language (or in an application environment).

5. Conducting a series of computational experiments:

debugging of syntax, semantics and logical structure, test calculations and analysis of test results, program modification.

6. Analysis and interpretation of results:

modification of the program or model if necessary.

There are many software packages and environments that allow you to build and study models:

Graphics environments

Text editors

Programming environments

Spreadsheets

Math packages

HTML editors

2.4. Computational experiment

An experiment is an experience that is performed with an object or model. It consists of performing certain actions to determine how the experimental sample reacts to these actions. A computational experiment involves carrying out calculations using a formalized model.

Using a computer model that implements a mathematical one is similar to conducting experiments with a real object, only instead of a real experiment with an object, a computational experiment is carried out with its model. By specifying a specific set of values ​​for the initial parameters of the model, as a result of a computational experiment, a specific set of values ​​for the required parameters is obtained, the properties of objects or processes are studied, their optimal parameters and operating modes are found, and the model is refined. For example, having an equation that describes the course of a particular process, you can, by changing its coefficients, initial and boundary conditions, study how the object will behave. Moreover, it is possible to predict the behavior of an object under various conditions. To study the behavior of an object with a new set of initial data, it is necessary to conduct a new computational experiment.

To check the adequacy of the mathematical model and the real object, process or system, the results of computer research are compared with the results of an experiment on a prototype full-scale model. The test results are used to adjust the mathematical model or the question of the applicability of the constructed mathematical model to the design or study of specified objects, processes or systems is resolved.

A computational experiment allows you to replace an expensive full-scale experiment with computer calculations. It allows, in a short time and without significant material costs, to study a large number of options for a designed object or process for various modes of its operation, which significantly reduces the time required for the development of complex systems and their implementation in production.

2.5. Simulation in various environments

2.5.1. Simulation in a programming environment

Modeling in a programming environment includes the main stages of computer modeling. At the stage of building an information model and algorithm, it is necessary to determine which quantities are input parameters and which are results, and also determine the type of these quantities. If necessary, an algorithm is drawn up in the form of a block diagram, which is written in the selected programming language. After this, a computational experiment is carried out. To do this, you need to load the program into the computer's RAM and run it. A computer experiment necessarily includes an analysis of the results obtained, on the basis of which all stages of solving the problem (mathematical model, algorithm, program) can be adjusted. One of the most important stages is testing the algorithm and program.

Debugging a program (the English term debugging means “catching bugs” appeared in 1945, when a moth got into the electrical circuits of one of the first Mark-1 computers and blocked one of thousands of relays) is the process of finding and eliminating errors in the program , are produced based on the results of a computational experiment. Debugging involves localizing and eliminating syntax errors and obvious coding errors.

In modern software systems debugging is carried out using special software tools called debuggers.

Testing is checking the correct operation of the program as a whole or its components. The testing process checks the functionality of the program and does not contain obvious errors.

No matter how carefully the program is debugged, the decisive stage that establishes its suitability for work is monitoring the program based on the results of its execution on the test system. A program can be considered correct if, for the selected system of test input data, correct results are obtained in all cases.

2.5.2. Modeling in Spreadsheets

Modeling in spreadsheets covers a very wide class of problems in different subject areas. Spreadsheets are a universal tool that allows you to quickly perform labor-intensive work on calculating and recalculating the quantitative characteristics of an object. When modeling using spreadsheets, the algorithm for solving the problem is somewhat transformed, hiding behind the need to develop a computing interface. The debugging stage is retained, including the elimination of data errors in connections between cells and in computational formulas. Additional tasks also arise: work on the convenience of presentation on the screen and, if it is necessary to output the received data on paper, on their placement on sheets.

The modeling process in spreadsheets follows a general pattern: goals are defined, characteristics and relationships are identified, and a mathematical model is compiled. The characteristics of the model are necessarily determined by purpose: initial (affecting the behavior of the model), intermediate, and what is required to be obtained as a result. Sometimes the representation of an object is supplemented with diagrams and drawings.