Model and simulation. Presentation on the topic “models and simulation” Classification of models according to the presence of impacts on the system

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What is a model? A model is an object that has some properties of another object (the original) and is used instead of it. Originals and models The first Russian battleship "Goto Predestination"

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What can be modeled? Models of objects: reduced copies of buildings, ships, airplanes, ... models of the atomic nucleus, crystal lattices, drawings... Models of processes: changes in the environmental situation, economic models, historical models... Models of phenomena: earthquake, solar eclipse, tsunami...

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Modeling Modeling is the creation and use of models to study originals. When modeling is used: the original does not exist ancient Egypt the consequences of a nuclear war (N.N. Moiseev, 1966) researching the original is life-threatening or expensive: controlling a nuclear reactor (Chernobyl, 1986) testing a new spacesuit for astronauts developing a new aircraft or ship the original is difficult to research directly: Solar system, galaxy (large dimensions) atom, neutron (small dimensions) processes in the internal combustion engine (very fast) geological phenomena (very slow) only some properties of the original are of interest checking the paint for the fuselage of the aircraft

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The goals of modeling are researching the original studying the essence of an object or phenomenon “Science is the satisfaction of one’s own curiosity at public expense” (L.A. Artsimovich) analysis (“what will happen if ...”) learning to predict the consequences of various influences on the original synthesis (“how to make it so that ...") learn to manage the original, influencing it optimization (“how to make it better”) choosing the best solution under given conditions

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One original - one model? a material point can correspond to several different models and vice versa! !

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Why do you need many models? studying the structure of the body trying on clothes studying heredity training rescuers recording the citizens of the country The type of model is determined by the goals of the modeling! !

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The nature of models is material (physical, subject) models: information models represent information about the properties and state of an object, process, phenomenon, and its relationship with the outside world: verbal - verbal or mental symbolic - graphic expressed using a formal language (drawings, diagrams, maps, ...) tabular mathematical (formulas) logical (various options for choosing actions based on an analysis of conditions) special (notes, chemical formulas)

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Models by area of application training (including simulators) experimental - when creating new technical means scientific and technical wind tunnel testing in an experimental pool solar radiation simulator vacuum chamber at the Space Research Institute vibration stand NPO Energia

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Models based on the time factor, static – describe the original at a given moment in time forces acting on the body at rest results of a doctor’s examination photograph dynamic model of body movement natural phenomena (lightning, earthquake, tsunami) medical history video recording of an event

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Models by the nature of the connections, deterministic connections between input and output quantities are rigidly specified with the same input data, the same results are obtained each time Examples: body movement without taking into account the wind, calculations using known formulas probabilistic (stochastic) take into account the randomness of events in the real world with the same input data, each time the results are slightly different different results Examples of body movement taking into account the wind Brownian motion of particles model of ship movement in waves models of human behavior

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4 What can be modeled? Models of objects: reduced copies of buildings, ships, airplanes, ... models of the atomic nucleus, crystal lattices, drawings... Models of processes: changes in the environmental situation, economic models, historical models... Models of phenomena: earthquake, solar eclipse, tsunami...

5 Modeling Modeling is the creation and use of models to study originals. When modeling is used: the original does not exist - ancient Egypt - the consequences of a nuclear war (N.N. Moiseev, 1966) research of the original is life-threatening or expensive: - control of a nuclear reactor (Chernobyl, 1986) - testing a new spacesuit for astronauts - development of a new aircraft or a ship, the original is difficult to study directly: -Solar system, galaxy (large sizes) -atom, neutron (small sizes) -processes in the internal combustion engine (very fast) -geological phenomena (very slow) only some properties of the original are of interest -checking paint for aircraft fuselage

6 Goals of modeling research of the original study of the essence of an object or phenomenon “Science is the satisfaction of one’s own curiosity at public expense” (L.A. Artsimovich) analysis (“what will happen if ...”) learning to predict the consequences of various influences on the original synthesis (“how to do, in order to...") learn to manage the original, influencing it optimization ("how to make it better") selection of the best solution under given conditions

8 The nature of models material (physical, subject) models: information models represent information about the properties and state of an object, process, phenomenon, and its relationship with the outside world: verbal - verbal or mental symbolic - graphic expressed using formal language (drawings, diagrams , maps, ...) tabular mathematical (formulas) logical (various options for choosing actions based on an analysis of conditions) special (notes, chemical formulas)

9 Models by area of application training (including simulators) experimental - when creating new technical means scientific and technical wind tunnel testing in an experimental pool solar radiation simulator vacuum chamber at the Space Research Institute vibration stand NPO Energia

10 Special types of game models - taking into account the actions of the enemy; models of economic situations; models of military operations; sports games; personnel training; simulation - it is impossible to calculate or predict the behavior of the system in advance; - you can imitate its reaction to external influences; - maximum consideration of all factors; - only numerical results; - selection of the best solution by trial and error during repeated experiments Examples: drug testing on mice, monkeys, ... mathematical modeling of biological systems business and management models learning process models

11 Models based on the nature of the connections; deterministic connections between input and output quantities are rigidly specified; with the same input data, the same results are obtained each time; Examples: movement of a body thrown at an angle to the horizon; calculations using known formulas; model of the normal operation of the mechanism; probabilistic (stochastic); take into account the randomness of events in the real world with the same input data, slightly different results are obtained each time Examples: body motion taking into account wind Brownian motion of particles influence of waves on a ship modeling of human actions

12 Models based on the time factor static - describe the original at a given moment in time forces acting on the body at rest results of a doctor’s examination photograph dynamic model of body movement natural phenomena (lightning, earthquake, tsunami) medical history video recording of an event

13 Models by structure tabular models (correspondence pairs) hierarchical (multi-level) models network models (graphs) Director Chief engineer VasyaPetya Chief accountant MashaDashaGlasha start finish

15 I. Statement of the problem research of the original study of the essence of an object or phenomenon analysis (“what will happen if ...”) learning to predict the consequences of various influences on the original synthesis (“how to do so that ...”) learning to manage the original by influencing it optimization ( “how to do it better”) choosing the best solution under given conditions Errors in setting the problem lead to the most severe consequences! ! !

16 I. Statement of the problem A well-posed problem: all connections between the initial data and the result are described; all initial data are known; the solution exists; the problem has a unique solution. Examples of poorly posed problems: Winnie the Pooh and Piglet built a trap for a heffalump. Will they be able to catch him? The Kid and Carlson decided to share two nuts like brothers - a big one and a small one. How to do it? Find the maximum value of the function y = x 2 (no solutions). Find a function that passes through the points (0,1) and (1,0) (non-unique solution).

17 II. Model development select the type of model determine the essential properties of the original that need to be included in the model, discard those that are not essential (for a given task) build a formal model - this is a model written in a formal language (mathematics, logic, ...) and reflecting only the essential properties of the original develop an algorithm for the model an algorithm is a clearly defined order of actions that must be performed to solve a problem

18 III. Model testing Testing is a test of a model on simple initial data with a known result. Examples: a device for adding multi-digit numbers - checking the ship's motion model on single-digit numbers - if the rudder is level, the course should not change; if the rudder is turned to the left, the ship should go to the right. model of saving money in a bank - at a rate of 0%, the amount should not change The model has been tested. Does this guarantee its correctness? ? ?

19 IV. Experiment An experiment is a study of a model under conditions of interest to us. Examples: a device for adding numbers - working with multi-digit numbers, a model of ship motion - research in rough seas, a model of saving money in a bank - calculations with a non-zero rate. Can you trust the results 100%? ? ?

22 I. Statement of the problem Assumptions: we consider the coconut and the banana to be material points, the distance to the palm tree is known, the height of the monkey is known, the height at which the banana hangs is known, the monkey is known to throw the banana with a known initial speed, we do not take into account air resistance. Under these conditions, it is necessary to find the initial angle at which throw a nut. Is there always a solution? ? ? 24 24 III. Testing the model at zero speed the coconut falls vertically down at t=0 the coordinates are equal to (0, h) when thrown vertically upward (=90 o) the x coordinate does not change at some t the y coordinate begins to decrease (downward branches of the parabola) Mathematical model No contradictions were found ! ! !

25 IV. Experiment Method I. Change the angle. For the selected angle we construct the flight path of the nut. If it passes above the banana, we reduce the angle, if below, we increase it. Method II. From the first equality we express the flight time: Change the angle. For the selected angle, we calculate t, and then the y value at this t. If it is greater than H, we reduce the angle; if it is less, we increase it. there is no need to build the entire trajectory for each

26 V. Analysis of the results 1.Can a monkey always knock down a banana? 2.What will change if the monkey can throw the coconut with different forces (with different initial speeds)? 3.What will change if coconut and bananas are not considered as material points? 4.What changes if you need to take into account air resistance? 5.What will change if the tree sways?

"Models of gliders" - Precision landing. Vocabulary work. Extra glue doesn't make your craft any prettier. Glider, keel, wing, airplane, plane, window. What is the name of the sport where athletes fly gliders and hang gliders? Safety rules for working with scissors and glue. Fuselage. Trace the template. What parts does the glider consist of?

“Fashion and Model” - And only at the age of 42 does he achieve success. Christian Dior. Sleep disturbance. Women suffer more from the disorder, but men also suffer from anorexia. Distorted ideas about the norm of one’s own weight.” The work of a model is to be beautiful and slim. Mini project "Fashion Now". And finally... Gabrielle Chanel.

“Airplane models” - Goals and objectives. Project. Yak-3 USSR 1944 Wing. French pilots of the Normandie-Niemen regiment fought on Yak-3 fighters. Keel. 4,797 aircraft were produced. Fuselage. Stabilizer. Aviation museum. Armament: 2 12.7 mm machine guns, 1 20 mm cannon. Cabin. Cook. Magazine "Modelist-Constructor" 1972-1974. Project implementation.

“Types of models” - Non-scale: doll; children's drawing. The model may also be NOT ADEQUATE. 9. Types of models by branches of knowledge. 7. Types of models depending on time. 6. Types of models depending on the form of presentation. Models modeling. 2. The need to create models. Insert clip!!! Modeling is the process of creating and using models.

“Object model” - Formalization. Representation of the object model. Answer questions on the topic. Know the definitions of modeling, formalization, the concept of visualization of models. Homework. The material model is a) a globe; b) world map; c) drawing; d) schedule. Modeling as a method of cognition. Information models play a very important role in human life.

“Model representation” - The behavior of the system can be represented as a function of time. It is recommended to use an equivalent circuit for representing a linear element. Model of the environment - description of the environment at the input and output. In connection with the above, parentheses acquire a very important, additional role. The property of linearity is also called the principle of superposition.

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Models and Simulation

A model is an object that has some properties of another object (the original) and is used instead of it. Originals and models

What we can model Models of objects: small copies of buildings, ships, airplanes, ... models of the atomic nucleus, crystal lattices, drawings ... Models of processes: changes in the environmental situation, economic models, historical models ... Models of phenomena: earthquake, solar eclipse, tsunami

What is modeling Modeling is the creation and use of models to study originals. When modeling is used: the original does not exist ancient Egypt the consequences of a nuclear war (N.N. Moiseev, 1966) researching the original is life-threatening or expensive: controlling a nuclear reactor (Chernobyl, 1986) testing a new spacesuit for astronauts developing a new aircraft or ship the original is difficult to research directly: Solar system, galaxy (large dimensions) atom, neutron (small dimensions) processes in the internal combustion engine (very fast) geological phenomena (very slow) only some properties of the original are of interest checking the paint for the fuselage of the aircraft

Types of models: material (physical, subject) models: information models represent information about the properties and state of an object, process, phenomenon, and its relationship with the outside world: verbal - verbal or mental symbolic - graphic expressed using formal language (drawings, diagrams, maps, ...) tabular mathematical (formulas) logical (various options for choosing actions based on an analysis of conditions) special (notes, chemical formulas) educational (including simulators) experimental - when creating new technical means scientific and technical

Classification of models 1. According to the time factor static - describe the original at a given moment in time forces acting on the body at rest results of a doctor's examination photograph dynamic model of body movement natural phenomena (lightning, earthquake, tsunami) medical history video recording of an event

By the nature of the connections, deterministic connections between input and output quantities are rigidly specified with the same input data, the same results are obtained each time; probabilistic (stochastic) take into account the randomness of events in the real world, with the same input data, slightly different results are obtained each time

By structure: tabular models (matching pairs), hierarchical (multi-level) models, network models (graphs)

Main stages of modeling Stage I Problem formulation Stage II Model development Stage III Computer experiment Stage IV Analysis of results The result corresponds to the goal The result does not correspond to the goal

9 The nature of models material (physical, subject) models: information models represent information about the properties and state of an object, process, phenomenon, and its relationship with the outside world: verbal - verbal or mental symbolic - graphic expressed using formal language (drawings, diagrams , maps, ...) tabular mathematical (formulas) logical (various options for choosing actions based on an analysis of conditions) special (notes, chemical formulas)

10 Models by area of application training (including simulators) experimental - when creating new technical means scientific and technical wind tunnel testing in an experimental pool solar radiation simulator vacuum chamber at the Space Research Institute vibration stand NPO Energia

11 Models based on the time factor static - describe the original at a given moment in time forces acting on the body at rest results of a doctor’s examination photograph dynamic model of body movement natural phenomena (lightning, earthquake, tsunami) medical history video recording of an event

12 Models by the nature of the connections, deterministic connections between input and output quantities are rigidly specified, with the same input data, the same results are obtained each time Examples: body movement without taking into account the wind, calculations using known formulas, probabilistic (stochastic) take into account the randomness of events in the real world, with the same input data, each time the results are obtained slightly different results Examples of body motion taking into account the wind Brownian motion of particles model of ship motion in waves models of human behavior

14 Special types of simulation models - it is impossible to calculate or predict the behavior of the system in advance, but you can simulate its reaction to external influences; -maximum consideration of all factors; -only numerical results; Examples: drug testing on mice, monkeys, ... mathematical modeling of biological systems business and management models learning process models The task is to find the best solution by trial and error (multiple experiments)! ! !

16 Adequacy of the model Adequacy is the coincidence of the essential properties of the model and the original: the modeling results are consistent with the conclusions of the theory (conservation laws, etc.) ... confirmed by experiment. The adequacy of the model can only be proven by experiment! ! ! A model is always different from the original. Any model is adequate only under certain conditions! ! !

17 Systematic approach A system is a group of objects and connections between them, isolated from the environment and considered as one whole. Examples: family ecological system computer technical system society A A B B C C D D environment The system has (due to connections!) special properties that no individual object has! ! !

19 System approach A graph is a set of vertices and edges connecting them vertex edge edge weight (weighted graph) Rurik Igor Svyatoslav Vladimir Yaropolk Oleg directed graph (digraph) - edges have a direction

22 I. Statement of the problem research of the original study of the essence of an object or phenomenon analysis (“what will happen if ...”) learning to predict the consequences of various influences on the original synthesis (“how to do so that ...”) learning to manage the original by influencing it optimization ( “how to do it better”) choosing the best solution under given conditions Errors in setting the problem lead to the most severe consequences! ! !

23 I. Statement of the problem A well-posed problem: all connections between the initial data and the result are described; all initial data are known; the solution exists; the problem has a unique solution. Examples of poorly posed problems: Winnie the Pooh and Piglet built a trap for a heffalump. Will they be able to catch him? The Kid and Carlson decided to share two nuts like brothers - a big one and a small one. How to do it? Find the maximum value of the function y = x 2 (no solutions). Find a function that passes through the points (0,1) and (1,0) (non-unique solution).

24 II. Model development select the type of model determine the essential properties of the original that need to be included in the model, discard those that are not essential (for a given task) build a formal model - this is a model written in a formal language (mathematics, logic, ...) and reflecting only the essential properties of the original develop an algorithm for the model an algorithm is a clearly defined order of actions that must be performed to solve a problem

25 III. Model testing Testing is a test of a model on simple initial data with a known result. Examples: a device for adding multi-digit numbers - checking the ship's motion model on single-digit numbers - if the rudder is level, the course should not change; if the rudder is turned to the left, the ship should go to the right. model of saving money in a bank - at a rate of 0%, the amount should not change The model has been tested. Does this guarantee its correctness? ? ?

26 IV. Experiment with a model An experiment is a study of a model under conditions of interest to us. Examples: a device for adding numbers - working with multi-digit numbers, a model of ship motion - research in rough seas, a model of saving money in a bank - calculations with a non-zero rate. Can you trust the results 100%? ? ?

27 V. Testing by practice, analysis of results Possible conclusions: the problem is solved, the model is adequate, it is necessary to change the algorithm or modeling conditions, it is necessary to change the model (for example, take into account additional properties), it is necessary to change the formulation of the problem

29 I. Statement of the problem Assumptions: we consider a coconut and a banana to be material points, the distance to the palm tree is known, the height of the monkey is known, the height at which the banana hangs is known, the monkey is known to throw a coconut with a known initial speed, we do not take into account air resistance. Under these conditions, it is necessary to find the initial angle at which toss in the coconut. Is there always a solution? ? ?

31 III. Testing the model at zero speed the coconut falls vertically down at t=0 the coordinates are equal to (0, h) when thrown vertically upward (=90 o) the x coordinate does not change at some t the y coordinate begins to decrease (downward branches of the parabola) Mathematical model No contradictions were found ! ! !

32 IV. Experiment Method I. Change the angle. For the selected angle we construct the flight path of the nut. If it passes above the banana, we reduce the angle, if below, we increase it. Method II. From the first equality we express the flight time: Change the angle. For the selected angle, we calculate t, and then the y value at this t. If it is greater than H, we reduce the angle; if it is less, we increase it. there is no need to build the entire trajectory for each

33 V. Analysis of the results 1.Can a monkey always knock down a banana? 2.What will change if the monkey can throw the coconut with different forces (with different initial speeds)? 3.What will change if coconut and bananas are not considered as material points? 4.What changes if you need to take into account air resistance? 5.What will change if the tree sways?

Models and modeling © K.Yu. Polyakov, Topic 3. Models of biological systems (based on the textbook by A.G. Gein et al., Computer Science and ICT, grade 10, M.: Prosveshchenie, 2008)

37 Model of limited growth (P. Verhulst) L – maximum number of animals Ideas: 1) the growth rate K L depends on the number N 2) with N = 0 there should be K L = K (initial value) 3) with N = L there should be K L = 0 (limit reached) The model is adequate if the error

Models and modeling © K.Yu. Polyakov, Topic 4. Modeling of random processes (based on the textbook by A.G. Gein et al., Computer Science and ICT, grade 10, M.: Prosveshchenie, 2008)

45 Random numbers on a computer Electronic generator requires a special device cannot reproduce the results short period (the sequence is repeated after 10 6 numbers) Mid-square method (J. von Neumann) squared Pseudo-random numbers - have the properties of random numbers, but each next number is calculated according to a given formula .

46 Random numbers on a computer Linear congruent method a, c, m - integers prime number period m What is the period? ? ? remainder of the Mersenne Vortex division: period

48 Distribution of random numbers Features: distribution is a characteristic of the entire sequence, not just one number, uniform distribution one, computer sensors of (pseudo) random numbers give a uniform distribution of uneven - many any uneven can be obtained using uniform a b a b uniform distribution

49 Calculating the area (Monte Carlo method) 1. We fit a complex figure into another figure for which it is easy to calculate the area (rectangle, circle, ...). 2.Evenly N points with random coordinates inside the rectangle. 3. Count the number of points that fall on the figure: M. 4. Calculate the area: Total N points There are M points on the figure 1. Approximate method. 2.The distribution must be uniform. 3.The more points, the more accurate. 4.Accuracy limited by random number sensor. !

51 Brownian motion Random step: Random direction (in rad): alpha:= 2*pi*random; h:= hMax*random; Program: for i:=1 to N do begin (find random direction and step) x:= x + h*cos(alpha); y:= y + h*sin(alpha); end; for i:=1 to N do begin (find random direction and step) x:= x + h*cos(alpha); y:= y + h*sin(alpha); end;

52 Queuing systems Examples: 1) calls at a telephone exchange 2) ambulance calls 3) customer service in a bank, how many teams? how many lines? how many operators? Features: 1) clients (requests for service) arrive constantly, but at random intervals 2) the time spent servicing each client is a random variable. You need to know the characteristics (distributions) of “randomness”! ! !

Q*K then count:= count + 1; end; writeln(count/L:0:2); c" title="56 Clients in the bank (program) count:= 0; (counter of “bad” minutes) for i:=1 to L do begin in:= (random number of incoming ones) out:= (random number of served ) N:= N + in – out; if N > Q*K then count:= count + 1; end; writeln(count/L:0:2); c" class="link_thumb"> 56 !} 56 Clients in the bank (program) count:= 0; (counter of “bad” minutes) for i:=1 to L do begin in:= (random number of incoming) out:= (random number of served) N:= N + in – out; if N > Q*K then count:= count + 1; end; writeln(count/L:0:2); count:= 0; (counter of “bad” minutes) for i:=1 to L do begin in:= (random number of incoming) out:= (random number of served) N:= N + in – out; if N > Q*K then count:= count + 1; end; writeln(count/L:0:2); What is output? ? ? simulation period L minutes Q*K then count:= count + 1; end; writeln(count/L:0:2); c"> Q*K then count:= count + 1; end; writeln(count/L:0:2); count:= 0; (counter of “bad” minutes) for i:=1 to L do begin in: = ( random number of incoming ) out:= ( random number of served ) N:= N + in – out; if N > Q*K then count:= count + 1; end; writeln(count/L:0:2); What is output? ? simulation period L minutes"> Q*K then count:= count + 1; end; writeln(count/L:0:2); c" title="56 Clients in the bank (program) count:= 0; (counter of “bad” minutes) for i:=1 to L do begin in:= (random number of incoming ones) out:= (random number of served ) N:= N + in – out; if N > Q*K then count:= count + 1; end; writeln(count/L:0:2); c"> title="56 Clients in the bank (program) count:= 0; (counter of “bad” minutes) for i:=1 to L do begin in:= (random number of incoming) out:= (random number of served) N:= N + in – out; if N > Q*K then count:= count + 1; end; writeln(count/L:0:2); c"> !}

Model and simulation. Presentation on the topic “models and simulation” Classification of models according to the presence of impacts on the system

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