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Application Courseware of Material Science in Computer

Review the application of computer in materials science.

Examination questions:

Indefinite item selection: 20 points;

Fill in the blanks: 20 points;

Noun explanation: 12 points;

Short answer: 30 points;

Calculation: 18 (1)

Examination time and place:

On the afternoon of July 5th (Tuesday of 20th week) 14: 00— 16: 00, Jiang 'an Comprehensive C504.

Review content:

Optional, name:

1, material classification:

According to its composition and structure, it can be divided into metal materials, inorganic nonmetal materials, organic polymer materials and composite materials.

According to its performance characteristics and functions, it can be divided into structural materials and functional materials.

It is divided into building materials, energy materials, electronic materials, refractory materials, medical materials and corrosion-resistant materials according to their uses.

2, curve fitting and least square method:

Least square method: a method to realize the best fitting of experimental data in the sense of variance.

Curve fitting: according to a set of data, that is, several points, it is required to determine a function, that is, a curve, so that these points are as close as possible to the curve as a whole. Objective: According to the experimental data, to establish an effective empirical function relationship between dependent variables and independent variables, and to provide clues for further research. )

3, the basic steps of establishing a mathematical model:

1) modeling preparation (collect relevant information and data, and make clear the background and purpose)

2) Modeling assumptions (purpose, simplicity, authenticity and comprehensiveness)

3) Build a model (distinguish parameters, select appropriate tools and construction methods)

4) model solving (designing or selecting mathematical methods and algorithms for solving models)

5) Model analysis (stability analysis, sensitivity analysis and error analysis)

6) Model test (whether objective or not)

7) Model application (the purpose of modeling, the most objective and fair test of the model)

4. Basic principle and essence of finite difference method (FDM):

Basic principle: Finite difference method (FDM) divides the solution domain into difference grids, and replaces the continuous solution domain with finite grid nodes. FDM uses Taylor series expansion and other methods to discretize the derivative in the control equation by replacing the difference quotient of function values on grid nodes, thus establishing the algebraic equation of unknown values on grid nodes.

Essence: The process of using finite check instead of infinite differential, differential-algebraic equations instead of differential equation, and numerical calculation instead of mathematical derivation, thus discretizing continuous function and replacing continuous function distribution with finite and discrete values.

5. The foundation and basic idea of finite element method and grid division method:

The finite element method is based on variational principle and weighted residual method. The basic idea is to discretize a continuous geometric structure into finite elements, and set a finite number of nodes in each element, so that the continuum can be regarded as a set of elements connected only at nodes, and the node values of field functions are selected as basic unknowns, and an approximate interpolation function is assumed in each element to represent the distribution law of field functions in the elements. Then, the finite element equation for solving unknown nodes is established, so that the infinite degree of freedom problem in the continuous domain is transformed into the finite degree of freedom problem in the finite domain. After solving the node value, the field function on the element and even the whole aggregate can be determined by the set interpolation function.

The foundation of finite element method is to replace the original continuum with a group of finite elements. Therefore, it is necessary to simplify the elastic body first, and then divide the elastic body into discrete bodies composed of finite units. These units are connected by unit nodes. A collection of cells, nodes and node lines is called a grid.

Usually, three-dimensional entities are divided into four-sided or six-sided grids, and plane problems are divided into triangular or quadrilateral grids.

6. Noun explanation: nodes and units

Nodes: Nodes are used to determine cell shapes, express cell characteristics and connect adjacent cells. Nodes are the smallest elements in a finite element model. Multiple units can use a node, which plays the role of connecting units and realizing data transmission.

Element: Each small block in the finite element model is called an element. According to different shapes, the elements can be divided into the following types: line element, triangle element, quadrilateral element, tetrahedron element and hexahedron element.

The difference between 7.7. FDM and finite element method;

1) finite element method does not need the step of establishing differential equations, and its physical problems always have clear physical significance in the whole discretization process. The finite difference method is not. There are great differences between the two methods in solving problems.

2) There are obvious differences between finite difference method and finite element method in discrete areas. The finite element method is arbitrary in the configuration of triangle division area, which approaches the boundary and interface well and has good calculation accuracy. The calculation format is complex, but it can be computerized, and the program is easy to standardize, which does not affect its practical application.

3) The finite element method lists the calculation formats of nodes in the region and boundary nodes from a unified point of view. In this way, the calculation accuracy of each node is generally coordinated. However, the accuracy of each node of the finite difference method is generally inconsistent.

4) The finite element method needs to input a lot of computer memory and data, which is one of its shortcomings. In fact, the finite difference method is more widely used than the finite element method, and there are many physical problems that the finite element method can't handle at present, but the finite difference method can always handle. Especially when the boundary shape is regular, the finite difference method is the most suitable.

8. Monte Carlo random number generation method, two basic principles of pseudo-random number test:

Physical method: the basic principle of generating random numbers by physical method is to use some physical phenomena and add some special equipment to the computer to directly generate random numbers on the computer. These special devices are called random number generators. There are two main physical sources used as random number generators: one is based on the radioactivity of radioactive substances, and the other is using the inherent noise of computers.

Mathematical method: The most practical and commonly used method for generating random numbers on computers is mathematical method, that is, generating random number sequences through recursive formulas. For given initial values ξ 1, ξ2…, ξk, determine ξn+k, n= 1, 2, …. Commonly used K= 1. For a given initial value ξ 1, determine ξn+ 1, n= 1, 2…

Because there are two problems with random numbers generated by mathematical methods, they are often called pseudo-random numbers. Pseudo-random numbers generated by mathematical methods can be easily obtained on computers and can be recalculated, which is not limited by computer models. Therefore, although this method has some problems, it is still widely used in computers, and it is the main method to generate pseudo-random numbers on computers.

At present, the most popular and commonly used congruence generator is congruence generator, which generates a series of numbers through the following linear congruence relationship.

Where x0 is called the seed. A, c, x0 and m are integers greater than zero, which are called multipliers, increments, initial values and modulo respectively. It is necessary to carefully select the modulus m and multiplier a to make the cycle period of the generated pseudo-random number as long as possible. C0 can reach the maximum period, but the obtained pseudo-random number has poor characteristics. Generally, x0 is any non-negative integer, and the values of multiplier a and increment c are as follows: a=4q+ 1, c = 2p+ 1p, and q is a positive integer. The values of p, q, x0 and m are generally selected through qualitative analysis and computer experiments, so that the obtained pseudo-random number sequence has a long enough period, and its independence and uniformity can pass a series of tests.

The characteristics of pseudo-random numbers are determined by various statistical tests, including uniformity test, independence test, combination law test, irrelevance test, parameter test and so on. The most basic thing is the test of unity and independence.

9, the potential function in molecular dynamics and its basic limitations:

Potential function: For potential (two-body potential), it is considered that the interaction between atoms is between two, regardless of the position of other atoms. It has achieved great success in the simulation calculation of molecular crystals, ionic compounds and some metals. For example, Lennard-Jones potential (below) is often used to describe the force between gas molecules or water molecules; Morse potential and Johnson potential are often used to describe metals. But for transition metals, there are some valence bonds in metal bonds, so it is more difficult.

Like the stochastic simulation method, MD method faces two basic limitations: one is the limitation of limited observation time; The second is the limited system size limit.

10, Fourier heat conduction equation:

The French mathematician Fourier summed up the law of heat conduction as Fourier's law through the inductive study of heat conduction data and practical experience, that is, the heat flow through the isothermal surface per unit time is proportional to the temperature gradient and heat transfer area:

DQ is the heat flow (w) through the isothermal surface per unit time; K is the thermal conductivity of the material (w/m.k); N is the boundary normal; S is the isothermal surface area (m2); T is the temperature (k).

1 1, stress field and stress-strain relationship;

1) stress

Under the action of external force, the size and geometry of a material will change, and at the same time, there will be "additional internal force" between the parts of the material, which is called "internal force" for short. The stress at a certain point on the cross section, that is, the concentration degree of internal force distributed at that point, reflects the magnitude and direction of internal force at that point on the cross section. The stress at a point can be regarded as a function of the position coordinates of the point and the direction of the taken section.

In order to describe the stress state of point P in elastic materials, a micro-cell with sides of dx, dy and dz is taken out around point P. Since dx, dy and dz tend to be infinitesimal, this cell can be equivalent to point P to be investigated, so the stress on each section of the research cell is equivalent to the stress state of point P, as shown in the following figure:

Elasticity proves that the six shear stress components have the following relationship:

Therefore, if the six stress components x, y, z, xy, zy and zx of any point p of the material are known, the normal stress and shear stress of any section passing through this point can be obtained. In other words, these six stress components are independent of each other and can uniquely determine the stress state of any point in the material.

2) Strain

The mechanical quantity describing the relative displacement of an object after deformation is called strain. Strain is divided into positive strain and shear strain, which are expressed by six strain components, namely X, Y, Z, xy, yz and zx. Normal strain refers to the relative expansion and contraction of unit length of each side of parallelepiped; Shear strain refers to the change of right angles between the sides of a parallelepiped, expressed in radians. For positive strain, it is positive when it is extended and negative when it is shortened; For shear strain, the right angle formed by two line segments along the positive direction of the coordinate axis is positive when it decreases and negative when it increases.

3) Physical equation (stress-strain relation equation)

The stress-strain relationship of elastic body can be described by Hooke's law. In the three-dimensional case, there are six independent stress components at any point in the elastic body, and the stress-strain relationship can be expressed by generalized Hooke's law as follows:

Where e is elastic modulus, v is Poisson's ratio,

12, diffusion law in metals:

Fick's first law:

In heterogeneous system, the process of independent molecular groups migrating from high concentration region to low concentration region is called diffusion. Under the condition of steady diffusion, the diffusion flux (Ji) of the diffused substance vertically passing through the ith unit cross section is directly related to the concentration gradient (ci/ x) and its diffusion coefficient (Di) passing through the diffusion equation:

This is the one-dimensional form of Fick's first diffusion law, and the negative sign indicates that the flux is the direction of decreasing concentration. The gradient is mainly caused by uneven concentration distribution.

Fick's second law:

In fact, the most important diffusion is unstable. In the process of diffusion, the concentration of diffused substances changes with time. In order to study this situation, Fick's second law is derived on the basis of Fick's first law according to the mass balance of diffused substances, namely:

If Di is a constant, we get:

In the three-dimensional case, the diffusion coefficients in the x, y, z, y and z directions are dx, dy and d z respectively, and the following results are obtained:

When it is isotropic, that is, Dx=Dy=Dz=D, we get:

13, database composition and characteristics:

Database system refers to a man-machine system composed of database, database management system, application program, database administrator and user. Modern database system includes at least the following three parts: i) database, a structured collection of related data, including the relationship between data itself and data, which exists independently of application programs and is the core and management object of database system; Ii) Physical storage, hardware media for storing data, such as magnetic disks, optical disks and other mass storage; Iii) Database software, responsible for database management and maintenance. It has the functions of defining, describing, operating and maintaining data, accepting and completing different requests from user programs and terminal commands to the database, and is responsible for protecting data from all kinds of interference and destruction.

Main features: Compared with file management, the data managed by computer database system has the following features:

A) data * * *

B) data independence

C) reduce data redundancy

D) data structuring

E) unified data protection function

14, composition of expert system:

Expert system consists of knowledge base, comprehensive database, inference engine, knowledge acquisition mechanism, interpretation mechanism and man-machine interface.

Knowledge base is a collection of domain knowledge needed to solve problems, including basic facts, rules and other related information.

The comprehensive database is mainly composed of the initial data of the problem and the intermediate information generated in the process of system solution.

Inference machine is the core execution mechanism to solve problems. It is actually a program to interpret knowledge. According to the semantics of knowledge, it interprets and executes the knowledge discovered through certain strategies, and records the results in the appropriate space of the dynamic library.

The knowledge acquisition mechanism is responsible for the establishment, modification and expansion of the knowledge base, mainly to realize the self-learning of the expert system, automatically acquire knowledge during the use of the system, and constantly improve and expand the functions of the existing system.

The explanation mechanism is to explain the solution process and answer the user's questions. The two basic questions are "why" and "how".

The main function of the man-machine interface is to realize the two-way information conversion between the system and the user, that is, the system translates the user's input information into a familiar information expression.

The working process of expert system is that the system takes the comprehensive database as the starting point according to the goals put forward by users, and under the guidance of control strategy, the inference engine uses the relevant knowledge in the knowledge base and realizes the goal of solving through continuous exploration and reasoning.

15, the concept of material design and its three levels:

Definition: Using high-performance computers and powerful materials professional software, the basic elements of materials science and engineering and their relationships are quantitatively or semi-quantitatively characterized, and the composition and technology of materials are designed on computers to predict their structures and properties, which is called material design and simulation, also known as computational materials science.

At present, there is no unified and strict division of the research level of material design. Generally speaking, according to the spatial scale of the research object, it can be divided into three levels: micro-design level, and the spatial scale is about1nm; Continuous model hierarchy, the scale is about1m; ; The level and scale of engineering design correspond to macro-materials, involving the processing and performance of large materials.

16, the concept of first principles:

The so-called first principle means that the total energy, microstructure and state of the system can be calculated very accurately only by five basic physical constants (electron mass me, electron quantity e, Planck constant h, vacuum light speed c and Boltzmann constant kB) and the arrangement of atoms in space (that is, crystal structure), without other empirical parameters.

Second, short answer

1, five applications of computers in materials science and engineering: (2-5 pages of textbooks, summary by yourself)

1) is used for the design of new materials and new alloys;

2) Simulation for materials science research:

3) Used for optimization and automatic control of material process;

4) Used to characterize the composition and microstructure of materials:

5) Used for data and image processing and others:

2. The meaning and classification of mathematical model:

Mathematical model definition:

Mathematical concepts, formulas and theories abstracted from the corresponding objective prototypes are called mathematical models. Or those mathematical symbol systems that reflect a specific problem or a specific thing system are called mathematical models. Its purpose is to solve practical problems.

Mathematical model classification:

According to the mathematical method of establishing the model, it can be divided into graph theory model, differential equation model, stochastic model and simulation model.

According to the characteristics of the model, it can be divided into discrete model, continuous model, linear model and nonlinear model.

3. The solution steps of 3.FDM and finite element method:

FDM problem solving steps:

1) to establish a differential equation.

According to the nature of the problem, the calculation area is selected, the differential equation is established, and the initial conditions and boundary conditions are written.

2) constructing a difference scheme

Firstly, the solution area is discretized, the calculation nodes are determined, and the grid layout, difference form and step size are selected. Then the infinite differential is replaced by finite difference, the WeChat service is replaced by difference quotient, and the differential equation and boundary conditions are replaced by difference equation.

3) Solving the difference equation

Difference equations are usually a large number of linear algebraic equations, and their solutions mainly include accurate method and approximate method. Accurate method is also called direct method, which mainly includes matrix method, gauss elimination method and principal component elimination method. Approximation method, also known as indirect method, is mainly iterative method, including direct iterative method, indirect iterative method and over-relaxation iterative method.

4) Accuracy analysis and inspection

The accuracy and convergence of the obtained values are analyzed and tested.

FEM problem solving steps:

The calculation steps of finite element method can be summarized as the following three basic steps: grid division, element analysis and overall analysis.

1) grid division

The foundation of finite element method is to replace the original continuum with a group of finite elements. Therefore, it is necessary to simplify the elastic body first, and then divide the elastic body into discrete bodies composed of finite units. These units are connected by unit nodes. A collection of cells, nodes and node lines is called a grid.

Usually, three-dimensional entities are divided into four-sided or six-sided grids, and plane problems are divided into triangular or quadrilateral grids.

2) Unit analysis

For elastic mechanics, element analysis is to establish the relationship between node displacement and node force of each element.

Because the node displacement of the element is regarded as the basic variable, it is necessary to determine an approximate expression of the internal displacement of the element, then calculate the strain and stress of the element, and then establish the relationship between the node force and the node displacement in the element.

3) Overall analysis

The whole of each element is analyzed, the relationship between the external load and the node displacement is established, and the node displacement is solved. This process is a holistic analysis. Taking the plane problem of elasticity as an example, as shown in the right figure, the boundary node I is subjected to concentrated force, and the node I is the joint point of three elements, so the nodal forces of these three elements at the same node should be gathered together to establish the equilibrium equation.

4. The classification of expert system:

According to the different nature of solving engineering problems, expert systems are divided into the following categories:

1) interpretation expert system: through the analysis and interpretation of known information and data, determine its meaning, such as image analysis, chemical structure analysis, signal interpretation, etc.

2) Forecast expert system: Through the analysis of past and present known situations, the possible future situations, such as weather forecast, population forecast, economic forecast and military forecast, can be inferred.

3) Diagnosis expert system: Infer the reason of an object's failure (i.e. failure) according to the observed situation, such as medical diagnosis, software fault diagnosis, material fault diagnosis, etc.

4) Design expert system: according to the requirements of tool design, find out the target configuration that meets the constraints of design problems, such as circuit design, civil engineering design, computer structure design, mechanical product design, production process design, etc.

5) Planning expert system: find out the action sequence or steps that can achieve the given goal, such as robot planning, transportation scheduling, engineering project demonstration, communication and military command, crop fertilization scheme, etc.

6) Monitoring expert system: continuously observe the behaviors of systems, objects or processes, and compare the observed behaviors with their proper behaviors, so as to find abnormal situations and give an alarm, such as safety monitoring of nuclear power plants.

7) Control expert system: adaptively manage the overall behavior of a controlled object to meet the expected requirements, such as air traffic control, business management, combat management, autonomous robot control, production process control, etc.

Third, calculation:

Application of finite difference method in heat conduction;

Examples of solving FDM problems

1. problem

There is a furnace wall with a thickness of, the inner wall temperature T0=900C and the outer wall temperature Tm= 100 C, and the temperature distribution along the thickness direction of the furnace wall is obtained.

analyse

This is a one-dimensional steady-state heat conduction problem, with boundary conditions T0=900C and Tm= 100 C. The temperature values of several nodes along the thickness direction of the furnace wall can be obtained by finite difference method.

Mathematical basis of FDM;

In numerical calculation, functions are considered as two tabular forms. One column is the (discrete) value xi of the independent variable, and the other column is the corresponding function value of xi, which is expressed as fi or f(xi).

From the perspective of operators, three kinds of operators are defined:

(Forward Difference Operator): fi fi+ 1 fi.

(to difference operator): fi fi1.

(Center difference operator): fi fi+ 1/2 fi 1/2.

Where, fi 1 = f (Xihe), fi 1/2 = f (Xihe /2), Xi+ 1xi = h, which is the same for all I.

The above difference corresponding to the first derivative is called the first difference, and the difference corresponding to the second derivative is called the second difference:

2fi =(fi+ 1 fi)= fi+22fi+ 1+fi

2fi =(fi fi 1)= fi2fi 1+fi2

2fi =fi+ 12fi+fi 1

Three operators are related: 2=. Other higher-order differences can be analogized in turn.

The ratio of the difference between the function and the independent variable is called the difference quotient between the function and the independent variable. Take the second order as an example, its three forms are:

Forward difference quotient:

Backward difference quotient:

Center difference quotient:

The difference and difference quotient of multivariate functions can also be obtained in a similar way.

The essence of finite difference method is to use difference instead of differential, and the geometric significance of using difference quotient instead of WeChat service is to use the average change rate of function in a certain area instead of the real change rate of function. For the first-order WeChat service, there are three typical forms of price difference:

Forward difference quotient:

Backward difference quotient:

Center difference quotient:

According to Taylor series, the errors of the above three difference forms can be calculated, namely:

From these three formulas, we can see that the difference quotient defined by different methods is different to replace the WeChat service. When WeChat service is replaced by forward difference quotient or backward difference quotient, its phase error is O(x), which is the order of magnitude of the first power of X; Using central difference quotient to replace WeChat service, the truncation error is O(x)2, which is the order of X, that is, the error of using central difference quotient to replace WeChat service is one order of magnitude smaller than that of using forward difference quotient or reverse difference quotient to replace WeChat service.

Therefore, when using FDM to calculate, we must pay attention to the form of difference equation, the establishment method and the resulting errors.

Note: 1, 4-5 nodes should be selected;

2. Answer the questions in strict accordance with the problem-solving steps, especially don't leave out the final accuracy analysis and inspection steps.