Computer experiment Computer experiment To give life to new design developments, to introduce new technical solutions into production. Computer experiment
Computer experiment Computer experiment To give life to new design developments, to introduce new technical solutions into production or to test new ideas, an experiment is needed. In the recent past, such an experiment could be carried out either in laboratory conditions on installations specially created for it, or in situ, i.e. on a real sample of the product, subjecting it to all kinds of tests. This requires large material costs and time. Computer studies of models came to the rescue. When conducting a computer experiment, the correctness of the models is checked. The behavior of the model is studied under various object parameters. Each experiment is accompanied by an understanding of the results. If the results of a computer experiment contradict the meaning of the problem being solved, then the error must be looked for in an incorrectly chosen model or in the algorithm and method for solving it. After identifying and eliminating errors, the computer experiment is repeated. To give life to new design developments, introduce new technical solutions into production, or test new ideas, an experiment is needed. In the recent past, such an experiment could be carried out either in laboratory conditions on installations specially created for it, or in situ, i.e. on a real sample of the product, subjecting it to all kinds of tests. This requires large material costs and time. Computer studies of models came to the rescue. When conducting a computer experiment, the correctness of the models is checked. The behavior of the model is studied under various object parameters. Each experiment is accompanied by an understanding of the results. If the results of a computer experiment contradict the meaning of the problem being solved, then the error must be looked for in an incorrectly chosen model or in the algorithm and method for solving it. After identifying and eliminating errors, the computer experiment is repeated.
A mathematical model is understood as a system of mathematical relationships of formulas, inequalities, etc., reflecting the essential properties of an object or process. A mathematical model is understood as a system of mathematical relationships of formulas, inequalities, etc., reflecting the essential properties of an object or process.
Modeling problems from various subject areas Modeling problems from various subject areas Economics Economics Economics Astronomy Astronomy Astronomy Physics Physics Physics Ecology Ecology Ecology Biology Biology Biology Geography Geography Geography
The machine-building plant, selling products at negotiated prices, received a certain revenue, having spent a certain amount of money on production. Determine the ratio of net profit to invested funds. The machine-building plant, selling products at negotiated prices, received a certain revenue, having spent a certain amount of money on production. Determine the ratio of net profit to invested funds. Statement of the problem Statement of the problem The purpose of the simulation is to study the process of production and sales of products in order to obtain the greatest net profit. Using economic formulas, find the ratio of net profit to invested funds. The purpose of modeling is to explore the process of production and sales of products in order to obtain the greatest net profit. Using economic formulas, find the ratio of net profit to invested funds.
The main parameters of the modeling object are: revenue, cost, profit, profitability, profit tax. The main parameters of the modeling object are: revenue, cost, profit, profitability, profit tax. Input data: Input data: revenue B; revenue B; costs (cost) S. costs (cost) S. We will find other parameters using the basic economic dependencies. The profit value is defined as the difference between revenue and cost P=B-S. We will find other parameters using the basic economic dependencies. The profit value is defined as the difference between revenue and cost P=B-S. Profitability r is calculated using the formula:. Profitability r is calculated using the formula:. The profit corresponding to the marginal level of profitability of 50% is 50% of the cost of production S, i.e. S*50/100=S/2, therefore the profit tax N is determined as follows: Profit corresponding to the marginal level of profitability of 50% is 50% of the cost of production S, i.e. S*50/100=S/2, so the profit tax N is determined as follows: if r
Analysis of results Analysis of results The resulting model allows, depending on profitability, to determine the profit tax, automatically recalculate the amount of net profit, and find the ratio of net profit to invested funds. The resulting model allows, depending on profitability, to determine the profit tax, automatically recalculate the amount of net profit, and find the ratio of net profit to invested funds. A computer experiment shows that the ratio of net profit to invested funds increases with increasing revenue and decreases with increasing production costs. A computer experiment shows that the ratio of net profit to invested funds increases with increasing revenue and decreases with increasing production costs.
Task. Task. Determine the speed of the planets in orbit. To do this, create a computer model of the solar system. Statement of the problem The purpose of the simulation is to determine the speed of the planets in orbit. Modeling object: Solar system, the elements of which are planets. The internal structure of the planets is not taken into account. We will consider planets as elements with the following characteristics: name; R - distance from the Sun (in astronomical units; astronomical units. average distance from the Earth to the Sun); t is the period of revolution around the Sun (in years); V is the orbital speed (astro units/year), assuming that the planets move around the Sun in circles at a constant speed.
Analysis of results Analysis of results 1. Analyze the calculation results. Is it possible to say that planets located closer to the Sun have a higher orbital speed? 1. Analyze the calculation results. Is it possible to say that planets located closer to the Sun have a higher orbital speed? 2. The presented model of the Solar system is static. When constructing this model, we neglected changes in the distance from the planets to the Sun during their orbital motion. To know which planet is further away and what the approximate relationships between the distances are, this information is quite enough. If we want to determine the distance between Earth and Mars, then we cannot neglect temporary changes, and here we will have to use a dynamic model. 2. The presented model of the Solar system is static. When constructing this model, we neglected changes in the distance from the planets to the Sun during their orbital motion. To know which planet is further away and what the approximate relationships between the distances are, this information is quite enough. If we want to determine the distance between Earth and Mars, then we cannot neglect temporary changes, and here we will have to use a dynamic model.
Computer experiment Enter the initial data into the computer model. (For example: =0.5; =12) Find the friction coefficient at which the car will go down the mountain (at a given angle). Find the angle at which the car will stand on the mountain (for a given friction coefficient). What will be the result if the friction force is neglected? Analysis of the results This computer model allows you to conduct a computational experiment instead of a physical one. By changing the values of the source data, you can see all the changes occurring in the system. It is interesting to note that in the constructed model the result does not depend either on the mass of the car or on the acceleration of gravity.
Task. Task. Imagine that there will be only one source of fresh water left on Earth, Lake Baikal. For how many years will Baikal provide the population of the whole world with water? Imagine that there will be only one source of fresh water left on Earth, Lake Baikal. For how many years will Baikal provide the population of the whole world with water?
Model development Model development To build a mathematical model, we determine the initial data. We denote: To build a mathematical model, we define the initial data. Let us denote: V - volume of Lake Baikal km3; V is the volume of Lake Baikal km3; N - Earth population 6 billion people; N - Earth population 6 billion people; p - water consumption per day per person (on average) 300 l. p - water consumption per day per person (on average) 300 l. Since 1l. = 1 dm3 of water, it is necessary to convert V of the lake water from km3 to dm3. V (km3) = V * 109 (m3) = V * 1012 (dm3) Since 1l. = 1 dm3 of water, it is necessary to convert V of the lake water from km3 to dm3. V (km3) = V * 109 (m3) = V * 1012 (dm3) The result is the number of years during which the population of the Earth uses the waters of Lake Baikal, let us denote it as g. So, g=(V*)/(N*p*365) The result is the number of years during which the Earth's population uses the waters of Lake Baikal, let's denote it as g. So, g=(V*)/(N*p*365) This is what the spreadsheet looks like in formula display mode: This is what the spreadsheet looks like in formula display mode:
Task. Task. To produce the vaccine, it is planned to grow a bacterial culture at the plant. It is known that if the mass of bacteria is x g, then after a day it will increase by (a-bx)x g, where coefficients a and b depend on the type of bacteria. The plant will daily collect m bacteria for vaccine production. To draw up a plan, it is important to know how the mass of bacteria changes after 1, 2, 3,..., 30 days. To produce the vaccine, it is planned to grow a bacterial culture at the plant. It is known that if the mass of bacteria is x g, then after a day it will increase by (a-bx)x g, where coefficients a and b depend on the type of bacteria. The plant will daily collect m bacteria for vaccine production. To draw up a plan, it is important to know how the mass of bacteria changes after 1, 2, 3,..., 30 days..
Statement of the problem Statement of the problem The object of modeling is the process of population change depending on time. This process is influenced by many factors: the environment, the state of medical care, the economic situation in the country, the international situation and much more. Having summarized the demographic data, scientists derived a function expressing the dependence of the population on time: The object of modeling is the process of changing the population depending on time. This process is influenced by many factors: the environment, the state of medical care, the economic situation in the country, the international situation and much more. Having generalized the demographic data, scientists derived a function expressing the dependence of the population on time: f(t)=where the coefficients a and b are different for each state, f(t)=where the coefficients a and b are different for each state, e is the base of the natural logarithm. e is the base of the natural logarithm. This formula only approximately reflects reality. To find the values of coefficients a and b, you can use a statistical reference book. Taking the values f(t) (population size at time t) from the reference book, you can approximately select a and b so that the theoretical values of f(t) calculated using the formula do not differ much from the actual data in the reference book. This formula only approximately reflects reality. To find the values of coefficients a and b, you can use a statistical reference book. Taking the values f(t) (population size at time t) from the reference book, you can approximately select a and b so that the theoretical values of f(t) calculated using the formula do not differ much from the actual data in the reference book.
The use of a computer as a tool for educational activities makes it possible to rethink traditional approaches to the study of many issues in natural sciences, strengthen the experimental activities of students, and bring the learning process closer to the real process of cognition based on modeling technology. The use of a computer as a tool for educational activities makes it possible to rethink traditional approaches to the study of many issues in natural sciences, strengthen the experimental activities of students, and bring the learning process closer to the real process of cognition based on modeling technology. Solving problems from various areas of human activity on a computer is based not only on students’ knowledge of modeling technology, but, naturally, also on knowledge of a given subject area. In this regard, it is more expedient to conduct the proposed lessons on modeling after students have studied the material in a general education subject; a computer science teacher needs to collaborate with teachers from different educational fields. There is known experience in conducting binary lessons, i.e. lessons taught by a computer science teacher together with a subject teacher. Solving problems from various areas of human activity on a computer is based not only on students’ knowledge of modeling technology, but, naturally, also on knowledge of a given subject area. In this regard, it is more expedient to conduct the proposed lessons on modeling after students have studied the material in a general education subject; a computer science teacher needs to collaborate with teachers from different educational fields. There is known experience in conducting binary lessons, i.e. lessons taught by a computer science teacher together with a subject teacher.
At the end of the chapter, we will consider the question: where to classify a computer experiment and computer modeling ( computer simulations) !
Initially, computer modeling appeared in meteorology and nuclear physics, but today the range of its applications in science and technology is extremely wide. A very indicative example in this regard is “global modeling”, where the world is considered as a set of subsystems interacting with each other: population, society, economy, food production, innovation complex, Natural resources, habitats, countries and regions of the world (the first example is the report to the Club of Rome “The Limits to Growth” published in 1972). The development and interaction of these subsystems determine global dynamics.
It is obvious that we are dealing here with a super complex system with a mass of nonlinear interactions, for which It is not possible to build a VIO-type model. Therefore, here they proceed as follows. A multidisciplinary group is assembled, consisting of specialists belonging to various subsystems. This group, based on the knowledge of its members, draws up a flowchart from a large variety of elements and connections. This block diagram is converted into a mathematical computer model that represents the system being modeled. After which numerical experiments are carried out with a computer model, i.e. computer experiments that resemble a real complex experiment in terms of creating models of objects and processes, debugging and execution.
There is a certain analogy between thought experiments and computer experiments. In the case of a computer experiment, the computer model developed during it is an analogue of the VIO model in the thought experiment. In both cases, experimental research is an element of the search for an adequate theoretical model. During this search, in the first case, PIOs and interactions between them (and their magnitude) are selected, and in the second case, elements and connections (and their magnitude) are selected. From this comparison it is obvious that the result of such experimental activity in both cases is the possible emergence of new knowledge. That is, computer models correspond to theoretical VIO models of the phenomenon, and a computer experiment is a means for constructing them. In this case, experimentation occurs with a model, and not with a phenomenon (according to the work, the same is indicated in the works).
In physics and other natural sciences, in the case of “laboratory” phenomena, a real experiment can change something in the phenomenon itself (“asking it a question”). If this turns out to be enough to create a VIO-model, and the only question remains about clarifying its parameters, then in this case the computer model has a more trivial application than described above - solving complex equations that describe the physical or technical system, and selection of parameters for systems for which the VIO-model has already been specified. This case is often called a "numerical experiment".
However, physics also deals with phenomena that need to be qualitatively studied before placing them in the laboratory, for example, the release of nuclear energy or the birth of elementary particles. A similar situation may arise: 1) in the cases of economic or technical complexity of a real experiment listed for a thought experiment, 2) in the absence of a VIO model, i.e. lack of a theory of the phenomenon (as in the case of turbulent flows). In nuclear and particle physics we have the first, if not both, cases. Here we have a situation similar to "global modeling" and begin to experiment with theoretical models through computer simulations. It is therefore not surprising that computer modeling appeared very early in nuclear physics.
So, a computer experiment and computer models in a non-trivial case, as in the example with “global modeling”, correspond, respectively, to a mental VIO experiment and theoretical VIO models of the phenomenon.
An experiment is a form of communication between two sides - a phenomenon and a theoretical model. In principle, this implies the possibility of manipulation with two sides. In the case of a real experiment, experimentation occurs with a phenomenon, and in the case of a mental and computer experiment, which can be considered as an analogue of a mental experiment, with a model. But in both cases, the goal is to obtain new knowledge in the form of an adequate theoretical model.
- This includes the remark of E. Winsberg: “It is not true that a real experiment always manipulates only the object of interest. In fact, in both a real experiment and a simulation, there is a complex relationship between what is manipulated in the study, on the one hand, and the systems of the real world, which are the goal of the study, on the other hand... Mendel, for example, manipulated peas, and was interested in studying the phenomenon of general heredity."
L. V. Pigalitsyn,
, www.levpi.narod.ru, Municipal educational institution secondary school No. 2, Dzerzhinsk, Nizhny Novgorod region.
Computer physical experiment
4. Computational computer experiment
Computational experiment turns
into an independent field of science.
R.G. Efremov, Doctor of Physical and Mathematical Sciences
A computational computer experiment is in many ways similar to a conventional (full-scale) one. This includes planning experiments, creating an experimental setup, performing control tests, conducting a series of experiments, processing experimental data, interpreting them, etc. However, it is carried out not on a real object, but on its mathematical model; the role of the experimental setup is played by a computer equipped with a special program.
Computational experimentation is becoming more and more popular. It is practiced in many institutes and universities, for example, at Moscow State University. M.V. Lomonosov, MPGU, Institute of Cytology and Genetics SB RAS, Institute of Molecular Biology RAS, etc. Scientists can already obtain important scientific results without a real, “wet” experiment. For this, there is not only computer power, but also the necessary algorithms, and most importantly, understanding. If previously they divided - in vivo, in vitro, – then now more has been added in silico. In fact, computational experiment is becoming an independent field of science.
The advantages of such an experiment are obvious. It is, as a rule, cheaper than natural. It can be easily and safely interfered with. It can be repeated and interrupted at any time. This experiment can simulate conditions that cannot be created in the laboratory. However, it is important to remember that a computational experiment cannot completely replace a full-scale one, and the future lies in their reasonable combination. A computational computer experiment serves as a bridge between natural experiment and theoretical models. The starting point of numerical modeling is the development of an idealized model of the physical system under consideration.
Let's consider several examples of computational physical experiments.
Moment of inertia. In “Open Physics” (2.6, part 1) there is an interesting computational experiment on finding the moment of inertia of a rigid body using the example of a system consisting of four balls strung on one knitting needle. You can change the position of these balls on the knitting needle, and also select the position of the axis of rotation, drawing it both through the center of the knitting needle and through its ends. For each arrangement of balls, students calculate the value of the moment of inertia using Steiner's theorem on parallel translation of the axis of rotation. The data for calculations is provided by the teacher. After calculating the moment of inertia, the data is entered into the program and the results obtained by the students are checked.
"Black box". To implement the computational experiment, my students and I created several programs to study the contents of an electrical “black box”. It may contain resistors, incandescent light bulbs, diodes, capacitors, coils, etc.
It turns out that in some cases it is possible, without opening the “black box,” to find out its contents by connecting various devices to the input and output. Of course, at the school level this can be done for a simple three- or four-terminal network. Such tasks develop students' imagination, spatial thinking and creativity, not to mention the fact that solving them requires deep and solid knowledge. Therefore, it is no coincidence that at many All-Union and international Olympiads in physics, the study of “black boxes” in mechanics, heat, electricity and optics is proposed as experimental problems.
In my special course classes, I conduct three real laboratory works in a “black box”:
– resistors only;
– resistors, incandescent lamps and diodes;
– resistors, capacitors, coils, transformers and oscillatory circuits.
Structurally, “black boxes” are designed in empty matchboxes. Inside the box is placed electrical diagram, and the box itself is sealed with tape. Research is carried out using instruments - avometers, generators, oscilloscopes, etc. - because To do this, you have to build the current-voltage characteristic and frequency response. Students enter instrument readings into a computer, which processes the results and plots the current-voltage characteristic and frequency response. This allows students to figure out what parts are in the black box and determine their parameters.
When performing frontal laboratory work With “black boxes,” difficulties arise due to the lack of instruments and laboratory equipment. Indeed, to conduct research it is necessary to have, say, 15 oscilloscopes, 15 sound generators, etc., i.e. 15 sets of expensive equipment that most schools do not have. And this is where virtual “black boxes” come to the rescue - the corresponding computer programs.
The advantage of these programs is that research can be carried out simultaneously by the whole class. As an example, consider a program that uses a random number generator to implement “black boxes” containing only resistors. There is a “black box” on the left side of the desktop. It contains an electrical circuit consisting only of resistors that can be located between the points A, B, C And D.
The student has three devices at his disposal: a power source (its internal resistance is taken equal to zero to simplify calculations, and the emf is randomly generated by the program); voltmeter (internal resistance is infinity); ammeter (internal resistance is zero).
When the program is launched, an electrical circuit containing from 1 to 4 resistors is randomly generated inside the “black box”. The student can make four attempts. After pressing any key, he is asked to connect any of the proposed devices in any sequence to the terminals of the “black box”. For example, he connected to the terminals AB current source with EMF = 3 V (the EMF value was generated randomly by the program, in this case it turned out to be 3 V). To terminals CD I connected a voltmeter, and its readings turned out to be 2.5 V. From this it should be concluded that the “black box” has at least a voltage divider. To continue the experiment, instead of a voltmeter, you can connect an ammeter and take readings. This data is clearly not enough to solve the mystery. Therefore, two more experiments can be carried out: the current source is connected to the terminals CD, and the voltmeter and ammeter - to the terminals AB. The data obtained in this case will be quite enough to unravel the contents of the “black box”. The student draws a diagram on paper, calculates the parameters of the resistors and shows the results to the teacher.
The teacher, having checked the work, enters the appropriate code into the program, and the circuit located inside this “black box” and the parameters of the resistors appear on the desktop.
The program was written by my students in BASIC. To run it in Windows XP or in Windows Vista you can use an emulator program DOS, For example, DosBox. You can download it from my website www.physics-computer.by.ru.
If there are nonlinear elements inside the “black box” (incandescent lamps, diodes, etc.), then in addition to direct measurements, the current-voltage characteristic will have to be taken. For this purpose, it is necessary to have a current source, voltage, at the outputs of which the voltage can be changed from 0 to a certain value.
To study inductances and capacitances, it is necessary to remove the frequency response using a virtual sound generator and an oscilloscope.
Speed selector. Let's consider another program from “Open Physics” (2.6, part 2), which allows you to conduct a computational experiment with a speed selector in a mass spectrometer. To determine the mass of a particle using a mass spectrometer, it is necessary to perform a preliminary selection of charged particles by velocities. This purpose is served by the so-called speed selectors.
In the simplest speed selector, charged particles move in crossed homogeneous electric and magnetic fields. An electric field is created between the plates of a flat capacitor, and a magnetic field is created in the gap of the electromagnet. starting speed υ charged particles is directed perpendicular to the vectors E And IN .
A charged particle is acted upon by two forces: the electric force q E and Lorentz magnetic force q υ × B . Under certain conditions, these forces can exactly balance each other. In this case, the charged particle will move uniformly and rectilinearly. After flying through the capacitor, the particle will pass through a small hole in the screen.
The condition of a rectilinear trajectory of a particle does not depend on the charge and mass of the particle, but depends only on its speed: qE = qυB→υ = E/B.
In the computer model, you can change the values of the electric field strength E, induction magnetic field B and initial particle speed υ . Velocity selection experiments can be performed for electrons, protons, alpha particles, and fully ionized atoms of uranium-235 and uranium-238. The computational experiment in this computer model is carried out as follows: students are informed about which charged particle flies into the speed selector, the electric field strength and the initial speed of the particle. Students calculate the magnetic field induction using the above formulas. After this, the data is entered into the program and the flight of the particle is observed. If the particle flies horizontally inside the velocity selector, then the calculations are done correctly.
More complex computational experiments can be carried out using the free package "MODEL VISION for WINDOWS". Plastic bag ModelVisionStudium (MVS) is an integrated graphical shell for quickly creating interactive visual models of complex dynamic systems and conducting computational experiments with them. The package was developed by the Experimental Object Technologies research group at the Department of Distributed Computing and Computer Networks, Faculty of Technical Cybernetics, St. Petersburg State Technical University. Freely redistributable free version package MVS 3.0 is available on the website www.exponenta.ru. Environment Simulation Technology MVS is based on the concept of a virtual laboratory bench. The user places virtual blocks of the simulated system on the stand. Virtual blocks for the model are either selected from the library or created again by the user. Plastic bag MVS is designed to automate the main stages of a computational experiment: constructing a mathematical model of the object under study, generating a software implementation of the model, studying the properties of the model and presenting the results in a form convenient for analysis. The object under study may belong to the class of continuous, discrete or hybrid systems. The package is best suited for the study of complex physical and technical systems.
As an example Let's consider a fairly popular problem. Let a material point be thrown at a certain angle to a horizontal plane and collide absolutely elastically with this plane. This model has become almost mandatory in the demo set of modeling packages. Indeed, this is a typical hybrid system with continuous behavior (flight in a gravitational field) and discrete events (bounces). This example also illustrates the object-oriented approach to modeling: a ball flying in the atmosphere is a descendant of a ball flying in airless space, and automatically inherits all the common features, while adding its own characteristics.
The last, final, from the user's point of view, stage of modeling is the stage of describing the form of presentation of the results of a computational experiment. These can be tables, graphs, surfaces, and even animations that illustrate the results in real time. Thus, the user actually observes the dynamics of the system. Points in phase space, user-drawn design elements can move, the color scheme can change, and the user can monitor, for example, heating or cooling processes on the screen. In the created packages for the software implementation of the model, it is possible to provide special windows that allow you to change the values of parameters during the course of a computational experiment and immediately see the consequences of the changes.
A lot of work on visual modeling of physical processes in MVS held at Moscow State Pedagogical University. There, a number of virtual works have been developed for the course of general physics, which can be associated with real experimental installations, which allows you to simultaneously observe on the display in real time changes in the parameters of both the real physical process and the parameters of its model, clearly demonstrating its adequacy. As an example, I cite seven laboratory works on mechanics from a laboratory workshop on the Internet portal of open education, corresponding to existing state educational standards for the specialty “Physics Teacher”: the study of rectilinear motion using the Atwood machine; measuring the speed of a bullet; addition of harmonic vibrations; measurement of the moment of inertia of a bicycle wheel; study of the rotational motion of a rigid body; determining the acceleration of free fall using a physical pendulum; study of free oscillations of a physical pendulum.
The first six are virtual and are simulated on a PC in ModelVisionStudiumFree, and the latter has both a virtual version and two real ones. In one, intended for distance learning, the student must independently make a pendulum from a large paper clip and an eraser and, hanging it under the shaft of a computer mouse without a ball, obtain a pendulum, the angle of deflection of which is read by a special program and must be used by the student when processing the results of the experiment. This approach allows some of the skills necessary for experimental work to be practiced only on a PC, and the rest - when working with available real devices and with remote access to equipment. In another option, intended for home preparation of full-time students to perform laboratory work in the workshop of the Department of General and Experimental Physics, Faculty of Physics, Moscow State Pedagogical University, the student practices skills in working with an experimental setup on a virtual model, and in the laboratory conducts an experiment simultaneously on a specific real setup and with its virtual model. At the same time, he uses both traditional measuring instruments in the form of an optical scale and a stopwatch, as well as more accurate and fast-acting means - a displacement sensor based on an optical mouse and a computer timer. Simultaneous comparison of all three representations (traditional, refined with the help of electronic sensors associated with a computer, and model) of the same phenomenon allows us to draw a conclusion about the limits of adequacy of the model when computer modeling data begin after some time to differ more and more from the readings, filmed on a real installation.
The above does not exhaust the possibilities of using a computer in a physical computing experiment. So for a creative teacher and his students there will always be untapped opportunities in the field of virtual and real physical experiments.
If you have any comments or suggestions on various types of physical computer experiments, please write to me at:
U modern computer many uses. Among them, as you know, the capabilities of the computer as a means of automating information processes are of particular importance. But no less significant are its capabilities as tool carrying out experimental work and analyzing its results.
Computational experiment has long been known in science. Remember the discovery of the planet Neptune “at the tip of a pen.” Often, the results of scientific research are considered reliable only if they can be presented in the form of mathematical models and confirmed by mathematical calculations. Moreover, this applies not only to physics
or technical design, but also to sociology, linguistics, marketing - traditionally humanitarian disciplines far from mathematics.
The computational experiment is theoretical method knowledge. A development of this method is numerical modeling- a relatively new scientific method that has become widespread thanks to the advent of computers.
Numerical modeling is widely used both in practice and in scientific research.
Example. Without constructing mathematical models and carrying out a variety of calculations on constantly changing data coming from measuring instruments, the operation of automatic production lines, autopilots, tracking stations, and automatic diagnostic systems is impossible. Moreover, to ensure the reliability of systems, calculations must be carried out in real time, and their errors can amount to millionths of a percent.
Example. A modern astronomer can often be seen not at the eyepiece of a telescope, but in front of a computer display. And not only a theorist, but also an observer. Astronomy is an unusual science. She, as a rule, cannot directly experiment with research objects. Different kinds radiation (electromagnetic, gravitational, neutrino or cosmic ray fluxes) astronomers only “spy” and “eavesdrop.” This means that you need to learn to extract as much information as possible from observations and reproduce them in calculations to test hypotheses describing these observations. The applications of computers in astronomy, as in other sciences, are extremely diverse. This includes automation of observations and processing of their results (astronomers see images not in an eyepiece, but on a monitor connected to special instruments). Computers are also needed to work with large catalogs (stars, spectral analyses, chemical compounds, etc.).
Example. Everyone knows the expression “a storm in a teacup.” To study in detail such a complex hydrodynamic process as a storm, it is necessary to use sophisticated numerical modeling methods. Therefore, in large hydrometeorological centers there are powerful computers: “the storm is played out” in the computer processor crystal.
Even if you are carrying out not very complex calculations, but you need to repeat them a million times, it is better to write the program once, and the computer will repeat it as many times as necessary (the limitation, of course, will be the speed of the computer).
Numerical simulation can be independent method research when only the values of some indicators are of interest (for example, the cost of production or the integral spectrum of the galaxy), but more often it acts as one of the means of constructing computer models in the broader sense of the term.
Historically, the first works on computer modeling were associated with physics, where a whole class of problems in hydraulics, filtration, heat transfer and heat exchange, and mechanics were solved using numerical modeling solid etc. Modeling, basically, was the solution of complex nonlinear problems of mathematical physics and, in essence, was, of course, mathematical modeling. The successes of mathematical modeling in physics contributed to its extension to problems in chemistry, electrical power engineering, and biology, and the modeling schemes were not too different from each other. The complexity of the problems solved on the basis of modeling was limited only by the power of available computers. This type of modeling is still widespread today. Moreover, during the development of numerical modeling, entire libraries of subroutines and functions have been accumulated that facilitate application and expand modeling capabilities. And yet, at present, the concept of “computer modeling” is usually associated not with fundamental natural science disciplines, but primarily with system analysis complex systems from the perspective of cybernetics (that is, from the perspective of management, self-government, self-organization). And now computer modeling is widely used in biology, macroeconomics, in the creation of automated control systems, etc.
Example. Remember Piaget's experiment described in the previous paragraph. It, of course, could be carried out not with real objects, but with an animated image on the display screen. But the movement of the toys could be filmed on regular film and shown on TV. Is it appropriate to call the use of a computer in this case computer simulation?
Example. A model of the flight of a body thrown vertically upward or at an angle to the horizon is, for example, a graph of the height of the body as a function of time. You can build it
a) on a sheet of paper, dotted;
b) in a graphic editor at the same points;
c) using a business graphics program, for example, in
spreadsheets;
d) by writing a program that not only displays
wounds flight path, but also allows you to set different
ny initial data (angle of inclination, initial speed
growth).
Why do you not want to call option b) a computer model, but options c) and d) fully correspond to this name?
Under computer model often refers to a program (or a program plus a special device) that provides an imitation of the characteristics and behavior of a specific object. The result of this program is also called a computer model.
In the specialized literature, the term “computer model” is more strictly defined as follows:
A conventional image of an object or some system of objects (processes, phenomena), described using interconnected computer tables, flowcharts, diagrams, graphs, drawings, animation fragments, hypertexts, and so on, and displaying the structure (elements and relationships between them) of the object. Computer models of this type are called structural and functional;
A separate program or a set of programs that allows, using a sequence of calculations and graphical display of their results, to reproduce (simulate) the processes of the functioning of an object, subject to the influence of various, usually random, factors on it. Such models are called imitation.
Computer models can be simple or complex. Simple models you created repeatedly when you were learning programming or building your database. In three-dimensional graphics systems, expert systems, and automated control systems, very complex computer models are built and used.
Example. The idea of constructing a model of human activity using a computer is not new, and it is difficult to find an area of activity in which it has not been attempted. Expert systems are computer programs that simulate the actions of a human expert when solving problems in any subject area on the basis of accumulated knowledge that makes up the knowledge base. ES solve the problem of modeling mental activity. Due to the complexity of the models, the development of ES usually takes several years.
Modern expert systems, in addition to a knowledge base, also have a precedent base - for example, the results of a survey of real people and information about the subsequent success/failure of their activities. For example, the New York Police expert system precedent base is 786 000 people, Hobby Center (personnel policy at the enterprise) - 512 000 people, and according to the specialists of this center, the ES they developed began working with the expected accuracy only when the base exceeded 200 000 man, it took 6 years to create.
Example. Progress in the creation of computer graphics has advanced from wireframe images of three-dimensional models with simple halftone images to modern realistic pictures that are examples of art. This resulted from success in more precisely defining the modeling environment. Transparency, reflection, shadows, lighting patterns and surface properties are some of the areas where research teams are hard at work, constantly coming up with new algorithms for creating ever more realistic artificial images. Today, these methods are also used to create high-quality animation.
Practical needs V computer modeling pose challenges for hardware developers funds computer. That is, the method actively influences not only the emergence of new and new programs But And on development technical means.
Example. People first started talking about computer holography in the 80s. So, in computer-aided design systems, in geographic information systems, it would be nice to be able to not only look at an object of interest in three-dimensional form, but to present it in the form of a hologram that can be rotated, tilted, and looked inside it. To create a holographic image useful in real applications, you need
holographic
Pictures 
displays with a gigantic number of pixels - up to a billion. This work is now actively underway. Simultaneously with the development of the holographic display, work is in full swing to create a three-dimensional workstation based on the principle called “reality substitution.” Behind this term is the idea of the widespread use of all those natural and intuitive methods that a person uses when interacting with natural (material-energy) models, but at the same time the emphasis is on their comprehensive improvement and development using the unique capabilities of digital systems. It is expected, for example, that it will be possible to manipulate and interact with computer holograms in real time using gestures and touches.
Computer modeling has the following advantages:
Provides visibility;
Available to use.
The main advantage of computer modeling is that it allows not only to observe, but also to predict the result of an experiment under some special conditions. Thanks to this opportunity, this method has found application in biology, chemistry, sociology, ecology, physics, economics and many other fields of knowledge.
Computer simulation is widely used in teaching. Using special programs, you can view models of such phenomena as the phenomena of the microworld and the world with astronomical dimensions, the phenomena of nuclear and quantum physics, the development of plants and the transformation of substances in chemical reactions.
The training of specialists in many professions, especially such as air traffic controllers, pilots, nuclear and power plant dispatchers, is carried out using computer-controlled simulators that simulate real situations, including emergency ones.
Laboratory work can be carried out on a computer if the necessary real devices and instruments are not available or if solving the problem requires the use of complex mathematical methods and labor-intensive calculations.
Computer modeling makes it possible to “revive” the physical, chemical, biological, and social laws being studied, and to conduct a number of experiments with the model. But we should not forget that all these experiments are of a very conditional nature and their educational value is also very conditional.
Example. Before the practical use of the nuclear decay reaction, nuclear physicists simply did not know about the dangers of radiation, but the first mass use of “achievements” (Hiroshima and Nagasaki) clearly showed how radiation
c is dangerous for humans. Physicists start with nuclear electro-
stations, humanity would not have learned about the dangers of radiation for a long time. The achievement of chemists at the beginning of the last century - the most powerful pesticide DDT - was considered absolutely safe for humans for quite a long time -
In the context of the use of powerful modern technologies, widespread replication and thoughtless use of erroneous software products, seemingly highly specialized issues such as the adequacy of a computer model of reality can acquire significant universal significance.
Computer experiments- it is a tool for studying patterns rather than natural or social phenomena.
Therefore, simultaneously with a computer experiment, a full-scale experiment should always be carried out so that the researcher, by comparing their results, can evaluate the quality of the corresponding model, the depth of our understanding of the essence of the phenomena of the phenomenon.
childbirth. Do not forget that physics, biology, astronomy, computer science are sciences about the real world, and not about virtual reality.
IN scientific research, both fundamental and practically oriented (applied), the computer often acts as necessary tool experimental work.
A computer experiment is most often associated with:
With complex mathematical calculations (number
linear modeling);
With the construction and study of visual and/or dynamic
mic models (computer modelling).
Under computer model is understood as a program (or a program in combination with a special device) that provides simulation of the characteristics and behavior of a certain object, as well as the result of the execution of this program in the form of graphic images (fixed or dynamic), numerical values, tables, etc.
There are structural-functional and simulation computer models.
Structural-functional a computer model is a conventional image of an object or some system of objects (processes, phenomena), described using interconnected computer tables, flowcharts, diagrams, graphs, drawings, animation fragments, hypertexts, and so on, and displaying the structure of the object or its behavior.
A computer simulation model is a separate program or software package that allows, using a sequence of calculations and graphical display of their results, to reproduce (simulate) the functioning processes of an object, subject to the influence of various random factors on it.
Computer modeling is a method for solving the problem of analyzing or synthesizing a system (most often a complex system) based on the use of its computer model.
Advantages of computer modeling are that it:
Allows you not only to observe, but also to predict the result of an experiment under some special conditions;
Allows you to simulate and study phenomena predicted by any theories;
It is environmentally friendly and does not pose a danger to nature and humans;
Provides visibility;
Available to use.
The computer modeling method has found application in biology, chemistry, sociology, ecology, physics, economics, linguistics, law and many other fields of knowledge.
Computer modeling is widely used in education, training and retraining of specialists:
For a visual representation of models of phenomena of the microcosm and the world with astronomical dimensions;
To simulate processes occurring in the world of living and inanimate nature
To simulate real situations of managing complex systems, including emergency situations;
To carry out laboratory work when the necessary devices and instruments are not available;
To solve problems, if this requires the use of complex mathematical methods and labor-intensive calculations.
It is important to remember that it is not objective reality that is modeled on a computer, but our theoretical ideas about it. The object of computer modeling is mathematical and other scientific models, and not real objects, processes, and phenomena.
Computer experiments- it is a tool for studying patterns rather than natural or social phenomena.
The criterion for the correctness of any of the results of computer modeling was and remains a full-scale (physical, chemical, social) experiment. In scientific and practical research, a computer experiment can only accompany a natural one, so that the researcher can compare
By studying their results, I could evaluate the quality of the model and the depth of our understanding of the essence of natural phenomena.
It is important to remember that physics, biology, astronomy, economics, computer science are sciences about the real world, not about 
virtual reality.
Exercise 1
Hardly anyone would call a letter written in a word processor and sent by email a computer model.
Text editors often allow you to create not only ordinary documents (letters, articles, reports), but also document templates in which there is permanent information that the user cannot change, there are data fields that are filled in by the user, and there are fields in which the calculations based on the entered data. Can such a pattern be considered a computer model? If so, what is the object of modeling in this case and what is the purpose of creating such a model?
Task 2
You know that before you can create a database, you first need to build a data model. You also know that an algorithm is a model of activity.
Both data models and algorithms are most often developed with computer implementation in mind. Is it fair to say that at some point they become a computer model, and if so, when does this happen?
Note. Check your answer against the definition of “computer model.”
Task 3
Describe the stages of building a computer model using the example of developing a program that simulates some physical phenomenon.
Task 4
Give examples when computer modeling brought real benefits and when it led to undesirable consequences. Prepare a report on this topic.
| Planning lessons for the school year | Main stages of modeling
Lesson 2
Main stages of modeling
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After studying this topic, you will learn:
What is modeling;
- what can serve as a prototype for modeling;
- what place does modeling occupy in human activity;
- what are the main stages of modeling;
- what is a computer model;
- What is a computer experiment?
Computer experiment
To give life to new design developments, introduce new technical solutions into production, or test new ideas, an experiment is needed. An experiment is an experience that is performed with an object or model. It consists of performing certain actions and determining how the experimental sample reacts to these actions.
At school you conduct experiments in biology, chemistry, physics, and geography lessons.
Experiments are carried out when testing new product samples at enterprises. Usually, a specially created installation is used for this, which allows an experiment to be carried out in laboratory conditions, or the real product itself is subjected to all kinds of tests (full-scale experiment). To study, for example, the operational properties of any unit or component, it is placed in a thermostat, frozen in special chambers, tested on vibration stands, dropped, etc. It’s good if it’s a new watch or vacuum cleaner - the loss due to destruction is not great. What if it’s an airplane or a rocket?
Laboratory and field experiments require large material costs and time, but their significance is nevertheless very great.
With development computer equipment A new unique research method has appeared - a computer experiment. In many cases, computer studies of models have come to help, and sometimes even replace experimental samples and test benches. The stage of conducting a computer experiment includes two stages: drawing up an experiment plan and conducting research.
Experimental plan
The experimental plan must clearly reflect the sequence of work with the model. The first point of such a plan is always testing the model.
Testing is the process of checking the correctness of the constructed model.
A test is a set of initial data that allows one to determine the correctness of the construction of the model.
To be sure of the correctness of the obtained modeling results, you need to: ♦ check the developed algorithm for constructing the model; ♦ make sure that the constructed model correctly reflects the properties of the original that were taken into account during the modeling.
To check the correctness of the model construction algorithm, a test set of initial data is used, for which the final result is known in advance or predetermined in other ways.
For example, if you use in modeling calculation formulas, then you need to select several options for the initial data and calculate them “manually”. These are test tasks. Once the model is built, you test with the same variations of the input data and compare the simulation results with the conclusions obtained by calculation. If the results coincide, then the algorithm is developed correctly; if not, we need to look for and eliminate the reason for their discrepancy. Test data may not reflect the real situation at all and may not carry any semantic content. However, the results obtained during the testing process may lead you to think about changing the original information or symbolic model, primarily in the part where the semantic content is embedded.
To make sure that the constructed model reflects the properties of the original that were taken into account during the modeling, it is necessary to select a test example with real source data.
Conducting research
After testing, when you have confidence in the correctness of the constructed model, you can proceed directly to conducting research.
The plan must include an experiment or series of experiments that satisfy the modeling objectives. Each experiment must be accompanied by an understanding of the results, which serves as the basis for analyzing the modeling results and making decisions.
The scheme for preparing and conducting a computer experiment is shown in Figure 11.7.
Rice. 11.7. Computer experiment scheme
Analysis of simulation results
The ultimate goal of modeling is making a decision, which should be made on the basis of a comprehensive analysis of the modeling results. This stage is decisive - either you continue the research or finish it. Figure 11.2 shows that the results analysis stage cannot exist independently. The findings often contribute to conducting an additional series of experiments, and sometimes to changing the problem.
The basis for developing a solution is the results of testing and experiments. If the results do not correspond to the goals of the task, it means that mistakes were made at the previous stages. This may be either an incorrect formulation of the problem, or an overly simplified construction of an information model, or an unsuccessful choice of a modeling method or environment, or a violation of technological techniques when building a model. If such errors are identified, then the model needs to be adjusted, that is, a return to one of the previous stages. The process is repeated until the experimental results meet the modeling goals.
The main thing is to always remember: an identified error is also a result. As it says folk wisdom, learn from mistakes. The great Russian poet A. S. Pushkin also wrote about this:
Oh, how many wonderful discoveries we have
Prepare the spirit of enlightenment
And experience, the son of difficult mistakes,
And genius, friend of paradoxes,
And chance, God the inventor...
Test questions and assignments
1. Name the two main types of modeling problems.
2. In the famous “Problem Book” by G. Oster there is the following problem:
The evil witch, working tirelessly, turns 30 princesses a day into caterpillars. How many days will it take her to turn 810 princesses into caterpillars? How many princesses will have to be turned into caterpillars per day to complete the job in 15 days?
Which question can be classified as “what will happen if...” type, and which question can be classified as “how to do so that...”?
3. List the most well-known purposes of modeling.
4. Formalize the humorous problem from G. Oster’s “Problem Book”:
From two booths located at a distance of 27 km from one another, two pugnacious dogs jumped out towards each other at the same time. The first one runs at a speed of 4 km/h, and the second one runs at 5 km/h.
How long will it take for the fight to start?
5.
Name as many characteristics of the object “pair of shoes” as possible. Create an information model of an object for different purposes:
■ choosing shoes for a hiking trip;
■ selection of a suitable shoe box;
■ purchase of shoe care cream.
6. What characteristics of a teenager are important for recommendations on choosing a profession?
7. For what reasons is the computer widely used in modeling?
8. Name the computer modeling tools you know.
9. What is a computer experiment? Give an example.
10. What is model testing?
11. What errors occur during the modeling process? What should you do when an error is discovered?
12. What is the analysis of simulation results? What conclusions are usually drawn?
