Modeling the research process and its algorithmization. Construction of modeling algorithms: formalization and algorithmization of processes. Description of the mathematical model

Sovetov B.Ya., Yakovlev S.A.

Systems modeling. 4th ed. – M.: Higher School, 2005. – P. 84-106.

The second stage of modeling is the stage of model algorithmization and its machine implementation. This stage is a stage aimed at implementing ideas and mathematical schemes in the form of a machine model M systems functioning process S.

System operation process S can be considered as a sequential change of its states in k-dimensional space. The task of modeling the process of functioning of the system under study S is the construction of functions z, on the basis of which it is possible to calculate the characteristics of interest in the process of system functioning. This requires relationships connecting the functions z with variables, parameters and time, as well as initial conditions at the moment of time t=t 0 .

There are two types of system states:

• 1) special, inherent in the process of functioning of the system only at certain points in time;
• 2) non-singular, in which the process remains the rest of the time. In this case the state function z i (t) can change abruptly, and between special ones - smoothly.

Modeling algorithms can be built according to the “principle of special states”. Let us denote the jump-like (relay) change of state z How z, and the “principle of special states” - as principle z.

« Principle z" makes it possible for a number of systems to significantly reduce the cost of computer time for the implementation of modeling algorithms. mathematical modeling model statistical

A convenient form of representing the logical structure of models of processes of functioning of systems and computer programs is a diagram. At various stages of modeling, the following schemes of modeling algorithms and programs are compiled:

Generalized (enlarged) diagram of the modeling algorithm specifies the general procedure for modeling a system without any further details.

Detailed diagram of the modeling algorithm contains clarifications that are missing in the generalized scheme.

Logical diagram of the modeling algorithm represents the logical structure of the system functioning process model S.

Program outline displays the order of software implementation of the modeling algorithm using specific mathematical software. A program diagram is an interpretation of the logical diagram of a modeling algorithm by a program developer based on a specific algorithmic language.

Stages of algorithmization of the model and its machine implementation:

• 1. Construction of a logical diagram of the model.
• 2. Obtaining mathematical relationships.
• 3. Checking the reliability of the system model.
• 4. Selection of tools for modeling.
• 5. Drawing up a plan for performing programming work.
• 6. Specification and construction of the program diagram.
• 7. Verification and verification of the reliability of the program scheme.
• 8. Carrying out model programming.
• 9. Checking the reliability of the program.
• 10. Drawing up technical documentation for the second stage.

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Introduction

1. Analytical review existing methods and means of solving the problem

1.1 Concept and types of modeling

1.2 Numerical calculation methods

1.3 General concept of the finite element method

2. Algorithmic analysis of the problem

2.1 Problem statement

2.2 Description of the mathematical model

2.3 Graphic diagram algorithm

3. Software implementation of the task

3.1 Deviations and tolerances of cylindrical pipe threads

3.2 Implementation of deviations and tolerances of cylindrical pipe threads in the Compass software

3.3 Implementation of the task in the C# programming language

3.4 Implementation of a structural model in ANSYS package

3.5 Study of the obtained results

Conclusion

List of used literature

Introduction

IN modern world Increasingly, there is a need to predict the behavior of physical, chemical, biological and other systems. One of the ways to solve the problem is to use a fairly new and relevant scientific direction - computer modeling, a characteristic feature of which is high visualization of the stages of calculations.

This work is devoted to the study of computer modeling in solving applied problems. Such models are used to obtain new information about the modeled object for an approximate assessment of the behavior of systems. In practice, such models are actively used in various fields of science and production: physics, chemistry, astrophysics, mechanics, biology, economics, meteorology, sociology, other sciences, as well as in applied and technical problems in various fields of radio electronics, mechanical engineering, automotive industry and others. The reasons for this are obvious: and this is the opportunity to quickly create a model and quickly make changes to the source data, enter and adjust Extra options models. Examples include studying the behavior of buildings, parts and structures under mechanical load, predicting the strength of structures and mechanisms, modeling transport systems, designing materials and its behavior, designing Vehicle, weather forecasting, work emulation electronic devices, simulated crash tests, testing the strength and adequacy of pipelines, thermal and hydraulic systems.

Purpose course work is the study of computer modeling algorithms, such as the finite element method, the boundary difference method, the finite difference method with further application in practice for calculation threaded connections for strength; Development of an algorithm for solving a given problem with subsequent implementation in the form software product; ensure the required calculation accuracy and evaluate the adequacy of the model using different software products.

1 . Analytical review of existing methods and means for solving the problem

1.1 Concept and types of modelsAndroving

Research problems solved by modeling various physical systems can be divided into four groups:

1) Direct problems, in the solution of which the system under study is specified by the parameters of its elements and the parameters of the initial mode, structure or equations. It is required to determine the response of the system to the forces (disturbances) acting on it.

2) Inverse problems, in which, based on a known reaction of a system, it is required to find the forces (perturbations) that caused this reaction and force the system under consideration to arrive at a given state.

3) Inverse problems that require determining the parameters of the system based on the known course of the process, described by differential equations and the values ​​of forces and reactions to these forces (disturbances).

4) Inductive problems, the solution of which is aimed at drawing up or clarifying equations that describe processes occurring in a system whose properties (disturbances and reactions to them) are known.

Depending on the nature of the processes being studied in the system, all types of modeling can be divided into the following groups:

Deterministic;

Stochastic.

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

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

Depending on the behavior of the object over time, modeling is classified into one of two types:

Static;

Dynamic.

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

Depending on the form of representation of the object (system), we can distinguish

Physical modeling;

Math modeling.

Physical modeling differs from observation of a real system (full-scale experiment) in that research is carried out on models that preserve the nature of phenomena and have a physical similarity. An example is a model of an aircraft being studied in a wind tunnel. In the process of physical modeling, some characteristics of the external environment are specified and the behavior of the model under given external influences is studied. Physical modeling can take place on real and unreal time scales.

Mathematical modeling is understood as the process of establishing a correspondence between a given real object and a certain mathematical object, called a mathematical model, and the study of this model on a computer in order to obtain the characteristics of the real object in question.

Mathematical models are built on the basis of laws identified by fundamental sciences: physics, chemistry, economics, biology, etc. Ultimately, one or another mathematical model is chosen on the basis of practice criteria, understood in a broad sense. After the model is formed, it is necessary to study its behavior.

Any mathematical model, like any other, describes a real object only with a certain degree of approximation to reality. Therefore, in the modeling process it is necessary to solve the problem of correspondence (adequacy) of the mathematical model and the system, i.e. conduct additional research into the consistency of simulation results with the real situation.

Mathematical modeling can be divided into the following groups:

Analytical;

Imitation;

Combined.

Using analytical modeling, the study of an object (system) can be carried out if explicit analytical dependencies are known that connect the desired characteristics with the initial conditions, parameters and variables of the system.

However, such dependencies can only be obtained for relatively simple systems. As systems become more complex, studying them using analytical methods encounters significant difficulties that are often insurmountable.

In simulation modeling, the algorithm that implements the model reproduces the process of system functioning over time, and the elementary phenomena that make up the process are simulated while preserving the logical structure, which allows, from the source data, to obtain information about the states of the process at certain points in time in each link of the system.

The main advantage of simulation modeling compared to analytical modeling is the ability to solve more complex problems. Simulation models make it possible to quite simply take into account factors such as the presence of discrete and continuous elements, nonlinear characteristics of system elements, numerous random influences, etc.

Currently, simulation modeling is often the only practically available method for obtaining information about the behavior of a system, especially at the design stage.

Combined (analytical-simulation) modeling allows you to combine the advantages of analytical and simulation modeling.

When building combined models, a preliminary decomposition of the object’s functioning process into its constituent subprocesses is carried out, and for those of them, where possible, analytical models are used, and simulation models are built for the remaining subprocesses.

From the point of view of describing an object and depending on its nature, mathematical models can be divided into models:

analog (continuous);

digital (discrete);

analog-to-digital.

By analogue model we mean a similar model that is described by equations relating continuous quantities. A digital model is understood as a model that is described by equations relating discrete quantities presented in digital form. By analog-to-digital we mean a model that can be described by equations connecting continuous and discrete quantities.

1.2 Numerical methodsWithcouple

Solving a problem for a mathematical model means specifying an algorithm to obtain the required result from the original data.

Solution algorithms are conventionally divided into:

precise algorithms that allow you to obtain the final result in a finite number of actions;

approximate methods - allow, due to certain assumptions, to reduce the solution to a problem with an exact result;

numerical methods - involve the development of an algorithm that provides a solution with a given controlled error.

Solving problems of structural mechanics is associated with great mathematical difficulties, which are overcome with the help of numerical methods, which make it possible, using a computer, to obtain approximate solutions that satisfy practical purposes.

The numerical solution is obtained by discretization and algebraization of the boundary value problem. Discretization is the replacement of a continuous set with a discrete set of points. These points are called grid nodes, and only at them are the function values ​​searched. In this case, the function is replaced by a finite set of its values ​​at the grid nodes. Using the values ​​at the grid nodes, partial derivatives can be approximately expressed. As a result, the partial differential equation is transformed into algebraic equations (algebraization of the boundary value problem).

Depending on the way discretization and algebraization are performed, various methods are distinguished.

The first method for solving boundary value problems that has become widespread is the finite difference method (FDM). IN this method discretization consists of covering the solution area with a grid and replacing a continuous set of points with a discrete set. A grid with constant step sizes (regular grid) is often used.

The MKR algorithm consists of three stages:

1. Construction of a grid in a given area. Approximate values ​​of the function (nodal values) are determined at the grid nodes. A set of node values ​​is a grid function.

2. Partial derivatives are replaced by difference expressions. In this case, the continuous function is approximated by a grid function. The result is a system of algebraic equations.

3. Solution of the resulting system of algebraic equations.

Another numerical method is the boundary element method (BEM). It is based on considering a system of equations that includes only the values ​​of variables at the boundaries of the region. The discretization scheme requires only the surface to be partitioned. The boundary of the region is divided into a number of elements and it is believed that it is necessary to find an approximate solution that approximates the original boundary value problem. These elements are called boundary elements. Discretizing only the boundary leads to a smaller system of problem equations than discretizing the whole body. BEM reduces the dimension of the original problem by one.

When designing various technical objects, the finite element method (FEM) is widely used. The emergence of the finite element method is associated with the solution of space research problems in the 1950s. Currently, the scope of application of the finite element method is very extensive and covers all physical problems that can be described by differential equations. The most important advantages of the finite element method are the following:

1. The material properties of adjacent elements do not have to be the same. This allows the method to be applied to bodies composed of several materials.

2. A curved region can be approximated using straight-line elements or described exactly using curved elements.

3. Item sizes may be variable. This allows you to enlarge or refine the network of dividing the area into elements, if necessary.

4. Using the finite element method, it is easy to consider boundary conditions with a discontinuous surface load, as well as mixed boundary conditions.

Solving problems using FEM contains the following steps:

1.Partition of a given area into finite elements. Numbering of nodes and elements.

2.Construction of finite element stiffness matrices.

3. Reduction of loads and impacts applied to finite elements to nodal forces.

4.Formation common system equations; taking into account boundary conditions. Solution of the resulting system of equations.

5. Determination of stresses and strains in finite elements.

The main disadvantage of FEM is the need to discretize the entire body, which leads to a large number of finite elements and, therefore, unknown problems. In addition, FEM sometimes leads to discontinuities in the values ​​of the quantities under study, since the method procedure imposes continuity conditions only at the nodes.

To solve the problem, the finite element method was chosen, since it is the most optimal for calculating a structure with a complex geometric shape.

1.3 General concept of the finite element method

The finite element method consists of breaking down a mathematical model of a structure into some elements, called finite elements. Elements are one-dimensional, two-dimensional and multi-dimensional. An example of finite elements is provided in Figure 1. The type of element depends on the initial conditions. The set of elements into which a structure is divided is called a finite element mesh.

The finite element method generally consists of the following steps:

1. Partitioning the area into finite elements. The division of an area into elements usually begins from its boundary, in order to most accurately approximate the shape of the boundary. Then the internal areas are divided. Often, the division of an area into elements is carried out in several stages. First, they are divided into large parts, the boundaries between which pass where the properties of the materials, geometry, and applied load change. Each subarea is then broken down into elements. After dividing the area into finite elements, the nodes are numbered. Numbering would be a trivial task if it did not affect the efficiency of subsequent calculations. If we consider the resulting system of linear equations, we can see that some non-zero elements in the coefficient matrix are between the two lines; this distance is called the bandwidth of the matrix. It is the numbering of nodes that affects the width of the stripe, which means that the wider the stripe, the more iterations are needed to obtain the desired answer.

modeling algorithm software ansys

Figure 1 - Some finite elements

2. Determination of the approximating function for each element. At this stage, the required continuous function is replaced by a piecewise continuous function defined on a set of finite elements. This procedure can be performed once for a typical area element and then the resulting function can be used for other area elements of the same type.

3. Combination of finite elements. At this stage, the equations relating to individual elements are combined, that is, into a system of algebraic equations. The resulting system is a model of the desired continuous function. We get the stiffness matrix.

4. Solution of the resulting system of algebraic equations. The real structure is approximated by many hundreds of finite elements, and systems of equations with many hundreds and thousands of unknowns arise.

Solving such systems of equations is the main problem in implementing the finite element method. The solution methods depend on the size of the resolving system of equations. In this regard, special methods for storing the stiffness matrix have been developed to reduce the volume required for this. random access memory. Stiffness matrices are used in each strength analysis method using a finite element mesh.

To solve systems of equations, various numerical methods are used, which depend on the resulting matrix; this is clearly visible in the case when the matrix is ​​not symmetrical; in this case, methods such as the conjugate gradient method cannot be used.

Instead of constitutive equations, a variational approach is often used. Sometimes a condition is set to ensure a small difference between the approximate and true solutions. Since the number of unknowns in the final system of equations is large, matrix notation is used. Currently, there are a sufficient number of numerical methods for solving a system of equations, which makes it easier to obtain the result.

2. Algorithmic analysis of the problem

2 .1 Problem statement

It is required to develop an application that simulates the stress-strain state of a flat structure and carry out a similar calculation in the Ansys system.

To solve the problem, it is necessary to: divide the area into finite elements, number the nodes and elements, set the characteristics of the material and boundary conditions.

The initial data for the project are a diagram of a flat structure with an applied distributed load and fastening (Appendix A), values ​​of material characteristics (elastic modulus -2*10^5 Pa, Poisson's ratio -0.3), load 5000H.

The result of the course work is obtaining the movements of the part in each node.

2.2 Description of the mathematical model

To solve the problem, the finite element method described above is used. The part is divided into triangular finite elements with nodes i, j, k (Figure 2).

Figure 2 - Finite element representation of a body.

The displacements of each node have two components, formula (2.1):

six components of displacements of element nodes form a displacement vector (d):

The displacement of any point inside the finite element is determined by relations (2.3) and (2.4):

When combining (2.3) and (2.4) into one equation, the following relation is obtained:

Deformations and displacements are related to each other as follows:

When substituting (2.5) into (2.6), we obtain relation (2.7):

Relationship (2.7) can be represented as:

where [B] is a gradient matrix of the form (2.9):

The shape functions linearly depend on the x, y coordinates, and therefore the gradient matrix does not depend on the coordinates of the point inside the finite element, and the deformations and stresses inside the finite element are constant in this case.

In a plane deformed state in an isotropic material, the matrix of elastic constants [D] is determined by formula (2.10):

where E is the elastic modulus, and is Poisson's ratio.

The finite element stiffness matrix has the form:

where h e is the thickness, A e is the area of ​​the element.

The equilibrium equation of the i-th node has the form:

To take into account the fastening conditions, there is the following method. Let there be some system N of equations (2.13):

In the case when one of the supports is motionless, i.e. U i =0, use the following procedure. Let U 2 =0, then:

that is, the corresponding row and column are set to zero, and the diagonal element is set to one. Accordingly, F 2 is also equal to zero.

To solve the resulting system, we choose the Gaussian method. The solution algorithm using the Gauss method is divided into two stages:

1. direct stroke: by elementary transformations above the lines, the system is reduced to a stepped or triangular shape, or it is established that the system is incompatible. The kth resolving row is selected, where k = 0…n - 1, and for each subsequent row the elements are converted

for i = k+1, k+2 … n-1; j = k+1,k+2 … n.

2. reverse: the values ​​of the unknowns are determined. From the last equation of the transformed system the value of the variable x n is calculated, after which from the penultimate equation it becomes possible to determine the variable x n -1 and so on.

2. 3 Graphic diagram of the algorithm

The presented graphical diagram of the algorithm shows the main sequence of actions performed when modeling a structural part. In block 1, the initial data is entered. Based on the entered data, the next step is the construction of a finite element mesh. Next, in blocks 3 and 4, local and global stiffness matrices are constructed, respectively. In block 5, the resulting system is solved by the Gaussian method. Based on the solution in block 6, the required movements in the nodes are determined, and the results are displayed. A brief graphical diagram of the algorithm is presented in Figure 7.

Figure 7 - Graphic diagram of the algorithm

3 . Aboutgrammaticallysuccessful implementation of the task

3.1 Deviations and tolerances of cylindrical pipe threads

Pipe cylindrical thread (GOST 6357-73) has a triangular profile with rounded peaks and valleys. This thread is mainly used for connecting pipes, pipeline fittings and fittings.

To achieve the proper joint density, special sealing materials (linen threads, red lead yarn, etc.) are placed in the gaps formed by the arrangement of the tolerance fields between the bolt cavities and the nut protrusions.

The maximum deviations of cylindrical pipe thread elements for diameter “1” of external and internal threads are given in tables 1 and 2, respectively.

Table 1 - deviations of external cylindrical pipe threads (according to GOST 6357 - 73)

Table 2 - deviations of pipe internal cylindrical threads (according to GOST 6357 - 73)

Limit deviations of the external thread of the minimum outer diameter, formula (3.1):

dmin=dн + ei (3.1)

where dн is the nominal size of the outer diameter.

The maximum deviations of the external thread of the maximum outer diameter are calculated using formula (3.2):

dmax=dн + es (3.2)

Limit deviations of external threads of minimum average diameter, formula (3.3):

d2min=d2 + ei (3.3)

where d2 is the nominal size of the average diameter.

Limit deviations of external threads of maximum average diameter are calculated using formula (3.4):

d2max=d2 + es (3.4)

Limit deviations of the external thread of the minimum internal diameter, formula (3.5):

d1min=d1 + ei (3.5)

where d1 is the nominal size of the internal diameter.

The maximum deviations of the external thread of the maximum internal diameter are calculated using formula (3.6):

d1max=d1 + es (3.6)

Limit deviations of the internal thread of the minimum outer diameter, formula (3.7):

Dmin=Dн + EI, (3.7)

where Dн is the nominal size of the outer diameter.

The maximum deviations of the internal thread of the maximum outer diameter are calculated using formula (3.8):

Dmax=Dн + ES (3.8)

Limit deviations of internal threads of minimum average diameter, formula (3.9):

D2min=D2 + EI (3.9)

where D2 is the nominal size of the average diameter.

Limit deviations of internal threads of maximum average diameter are calculated using formula (3.10):

D2max=D2 + ES (3.10)

Limit deviations of the internal thread of the minimum internal diameter, formula (3.11):

D1min=D1 + EI (3.11)

where D1 is the nominal size of the internal diameter.

The maximum deviations of the internal thread of the maximum internal diameter are calculated using formula (3.12):

D1max=D1 + ES (3.12)

A fragment of the thread sketch can be seen in Figure 6 of Chapter 3.2.

3.2 Implementation of deviations and tolerances of cylindrical pipe threads inSoftware "Compass"

Figure 6 - Pipe cylindrical thread with tolerances.

The coordinates of the points are shown in Table 1 of Appendix D

Copying a constructed thread:

Select the thread > Editor > copy;

Thread insertion:

We place the cursor at the place we need>editor>paste.

The result of the constructed thread can be seen in Appendix D

3.3 Implementation of the taskchi in the C# programming language

To implement the strength calculation algorithm, the MS Visual Studio 2010 development environment was selected using the language C# from the package . NETFramework 4.0. Using the object-oriented programming approach, we will create classes containing the necessary data:

Table 3 - Element class structure