Exam Questions:
Indeterminate choice: 20 points;
Fill in the blanks: 20 points;
Nouns explanation: 12 points;
Short answer: 30 points;
Calculation: 18 points (1)
Exam time and place:
July 5 (20 weeks Tuesday) 14:00-16:00 pm Jiang'an synthesis C504
Review content:
Optional, famous solution:
1, the classification of materials:
Based on its composition and structure, is divided into metal materials, inorganic non-metallic materials, organic polymer materials and composite materials, etc.;
Based on its performance characteristics and role, divided into structural materials and functional materials, etc.
Based on the use, divided into construction materials, energy materials, electronic materials, refractory materials, medical materials and corrosion-resistant materials.
2, curve fitting and least squares:
Least squares: the method to achieve the best fit to the experimental data under the significance of variance.
Curve fitting: based on a set of data, i.e., a number of points, it is required to determine a function, i.e., a curve, so that these points are as close as possible to the curve in general. (Purpose: To establish a valid empirical functional relationship between the dependent and independent variables based on experimentally obtained data to provide clues for further in-depth research.)
3. The basic steps of building a mathematical model:
1) Modeling preparation (collecting relevant information and data, clarifying the background and purpose)
2) Modeling assumptions (purposefulness, simplicity, truthfulness, comprehensiveness)
3) Constructing the model (distinguishing between covariates, choosing the appropriate tools and construction methods)
4) Model solving (designing or select the mathematical methods and algorithms for solving the model)
5) Model analysis (stability analysis, sensitivity analysis, error analysis)
6) Model testing (whether it meets the objective)
7) Model application (the purpose of modeling, the most objective, fair test of the model)
4, finite difference method (FDM) basic principles, the essence of:
Basic principles:
Basic principles:
4. p>
The basic principle: Finite Difference Method (FDM) will solve the domain divided into difference grid, with a finite number of grid nodes instead of a continuous solution domain.FDM through the Taylor series expansion and other methods, the derivative in the control equation with the grid nodes on the function value of the difference quotient instead of discretization, so as to establish the value of grid nodes as the unknown set of algebraic equations.
Substance: the process of replacing infinite differentiation with finite checking, differential algebraic equations with difference algebraic equations, and mathematical derivation with numerical computation, thus discretizing continuous functions and replacing continuous function distributions with finite, discrete values.
5. Basis of Finite Element Method (FEM), Basic Idea, Mesh Division Method:
The basis of Finite Element Method (FEM) is the principle of variational and weighted residual method, and the basic idea is to discretize the continuous geometric structure into a finite number of units, and to set a finite number of nodes in each unit, so that the continuum can be regarded as a set of units that are connected at the nodes only, and the node value of the selected field function as the basic unknown quantity, and at the same time, the node value of the chosen field function as the basic unknown quantity. The node value of the field function as the basic unknown quantity, and in each unit assumes an approximate interpolation function to represent the distribution law of the field function in the unit, and then set up a finite element equation system for solving the unknown quantity of the node, so that a continuous domain of infinite degrees of freedom into a finite degree of freedom in the finite domain, the node value can be obtained after the solution of the interpolation function can be set to determine the field function of the unit on the unit to the entire ensemble on the The field function on the cell and on the whole ensemble can be determined by a set interpolation function.
The finite element method is based on replacing the original continuum with a finite set of cells. Therefore, the elastomer is first simplified as necessary, and then divided into a discrete body of finite cells. The units are connected to each other by unit nodes. The set of cells, nodes, and node links is called a mesh.
Usually, three-dimensional solids are divided into meshes of 4-plane or 6-plane cells, and planar problems are divided into meshes of triangular or quadrilateral cells.
6, terminology: nodes, units
nodes: used to determine the shape of the unit, the expression of unit characteristics and connect neighboring units of the point is called nodes. Nodes are the smallest constituent elements in the finite element model. Multiple units can be **** with a node, nodes connecting units and realize the role of data transfer.
Cell: Each small block in a finite element model is called a cell. According to its shape, the unit can be divided into the following types: line segment unit, triangular unit, quadrilateral unit, tetrahedral unit and hexahedral unit and so on.
7. Differences between FDM and FEM:
1) The finite element method deals with physical problems without the step of setting up differential equations, and its physical problems always have a clear physical meaning in the whole process of discretization. This is not the case with the finite difference method. There is a large difference in the mathematical approach to the problem between the two methods.
2) The finite difference method and the finite element method also have obvious differences in the discretization method of the region. Finite element method of triangular division of the region configuration is more arbitrary, and its approximation of the boundary and interface is good, there is a good calculation accuracy. The calculation format is complex, but it can be computerized, the program is also easy to standardize, so it does not affect its practical application.
3) Finite element method with a unified view of the nodes in the region and the boundary nodes listed in the calculation format. So that the calculation of the nodes of the accuracy of the overall coordination. And the finite difference method of each node accuracy in general is not consistent enough.
4) The finite element method requires a larger amount of computer memory and a larger amount of data to be prepared for input, which is one of its disadvantages. In fact, the finite difference method is more widely used than the finite element method, there are a lot of physical problems can not be dealt with by the finite element method, but can always be dealt with by the finite difference method. Especially when the shape of the boundary is relatively regular, the finite difference method is the most appropriate.
8, Monte Carlo method of random number generation, pseudo-random number test of the two most basic principles:
Physical methods: physical methods to generate random numbers of the basic principle is: the use of certain physical phenomena, in the computer to add some special equipment, you can generate random numbers directly on the computer. These special devices are called random number generator. There are two main types of physical sources used as random number generators: one is based on the radioactivity of radioactive substances, and the other is to use the inherent noise of the computer.
Mathematical methods: the most practical and common method of generating random numbers on a computer is the mathematical method, where a recursive formula is used to generate a sequence of random numbers. For a given initial value ξ1,ξ2..., ξk, determine ξn+k, n=1, 2, .... Often used is the case k=1, for a given initial value ξ1, determine ξn+1, n=1,2...
Random numbers generated by mathematical methods are often referred to as pseudo-random because of two problems with them. Pseudo-random numbers generated by mathematical methods are easily obtained on computers, can be recalculated, and are not limited by computer model. Therefore, this method, although there are some problems, but still widely used on computers, is the main method of generating pseudo-random numbers on computers.
Today the more popular and most used is the congruence generator, which generates a sequence of numbers by means of the following linear congruence relation equation
where x0 is called the seed. a,c,x0,m are integers greater than zero, called the multiplier, the increment, the primitive, and the modulus, respectively. The modulus m and multiplier a need to be carefully selected so that the cycle period of the pseudorandom number generated is as long as possible. c0 can achieve the maximum period, but the pseudorandom number obtained is not well characterized. Usually x0 is chosen as an arbitrary non-negative integer, and the multiplier a and the increment c are taken as:a=4q+1, c=2p+1 p,q are positive integers. p, q, x0, m values are generally chosen by qualitative analysis and computer experiments, so that the pseudo-random number series obtained has a sufficiently long period, and the independence and uniformity can be passed a series of tests.
The pseudo-random number characteristics are determined by a variety of statistical tests, these tests include uniformity test, independence test, combinatorial law test, no coherence test, parametric test, etc. The most basic is the uniformity and independence test, the combination of the law test, no coherence test, parametric test and so on. The most basic is the uniformity and independence of the good and bad test.
9, molecular dynamics in the potential function and its basic limitations:
Potential function: the potential for the potential (two-body potential) that the interaction between atoms is the interaction between two and two, independent of the position of the other atoms in the molecular crystals, ionic compounds, and some of the simulation of the metal in the calculation of the relatively large success. For example, the Lennard-Jones potential (below) is often used to describe the forces between gas molecules or water molecules; the Morse and Johnson potentials are often used to describe metals. However, for transition metals, difficulties are encountered because the metallic bonds also contain some *** valence bonds.
The MD method, like the stochastic simulation method, faces two basic limitations: first, the limit of finite observation time; and second, the limit of finite system size.
10, Fourier's thermal conductivity equation:
French mathematician Fourier through the inductive study of thermal conductivity data and practical experience, the law of thermal conductivity is summarized as Fourier's law, that is, the heat flux through the isothermal surface per unit of time is directly proportional to the temperature gradient and the area of heat transfer:
dQ is the heat flux through the isothermal surface per unit of time (W); k is the material thermal conductivity (W/m.K); n is the boundary normal; s is the isothermal surface area (m2); T is the temperature (K).
11, stress field and stress-strain relationship:
1) Stress
Materials in the external force, its size and geometry will change, in the generation of "deformation" at the same time, the material will produce internal parts of the material between the "additional internal force". "Additional internal forces", or "internal forces" for short. The stress at a point on the cross-section, that is, the set of distributed internal forces at this point, reflects the size and direction of the internal forces at this point on the cross-section. The stress at a point can be regarded as a function of the coordinates of the position of the point and the direction of the section taken.
In order to describe the stress state at a point P in an elastic material, a micro-unit with prism lengths dx, dy, dz is taken out around the point P. Since dx, dy, dz tends to be infinitesimal, this unit can be equated to the point P to be examined, so the study of stresses on the various cross-sections of the unit is also equivalent to the study of the state of stresses at the point P. The stress state at the point P is also equated to the study of the unit. The following figure:
Elastic mechanics proves that the six tangential stress components have the following relationship:
Thus, if the material is known at any point P x, y, z, xy , zy , zx the six stress components, you can find out through the point of any cross-section of the positive stresses and tangential stresses. In other words, these six stress components are independent of each other, and can uniquely determine the stress state at any point in the material.
2) Strain
The mechanical quantity that describes the relative displacement of an object deformed by a force is called strain. Strain is divided into positive strain and tangential strain, represented by six strain components, respectively x, y, z, xy, yz, zx. Positive strain refers to the relative expansion and contraction per unit length of each side of a parallel hexahedron; tangential strain refers to the change in the right angle between the sides of a parallel hexahedron, expressed in radians. For positive strain, it is positive when elongated and negative when shortened; for tangential strain, it is positive when the right angle formed by two line segments along the positive direction of the coordinate axis becomes small and negative when it becomes large.
3) Physical equation (stress-strain relationship equation)
The stress-strain relationship of elastomer can be described by Hooke's law. In the three-dimensional case, there are six independent stress components at any point in the elastomer, and its stress-strain relationship can be expressed by the generalized Hooke's law:
Where: E is the modulus of elasticity, v is the Poisson's ratio,
12. Diffusion laws in metals:
Fick's first law:
In inhomogeneous systems, groups of molecules, each of them independent, migrate from the The process of migration of separate groups of molecules in an inhomogeneous system from a region of high concentration to a region of low concentration is called diffusion. Under steady-state diffusion conditions, the diffusive flux of a diffusing substance vertically across the ith unit cross-section (Ji) is directly related to the concentration gradient (ci/ x) across the diffusion equation and to its diffusion coefficient (Di):
This is the one-dimensional form of Fick's first law of diffusion, with the negative sign indicating that the flux is in the direction of decreasing concentration. The cause of the gradient is primarily an uneven distribution of concentration.
Fick's second law:
In fact, most important diffusions are non-stationary, and the concentration of the diffusing substance varies with time during diffusion. In order to study this situation, Fick's second law is derived from Fick's first law based on the mass balance of the diffusing substance, i.e.
If Di is a constant, it is obtained:
If it is a three-dimensional case, the diffusion coefficients in the x,y,z directions are Dx,Dy,Dz, respectively, and it is obtained:
When isotropic, i.e., Dx=Dy=Dz=D, get:
13, database composition and characteristics:
Database system refers to the human-computer system consisting of a database, database management system, application programs, database administrators, users and so on. Modern database system includes at least the following three parts: i) database, a structured collection of related data, including the data itself and the link between the data, which exists independently of the application program, is the core of the database system and the management object; ii) physical memory, the hardware media to save the data, such as disks, CD-ROMs and other high-capacity memory; iii) database software, is responsible for the database management The database software is responsible for the management and maintenance of the database. It has the functions of defining, describing, operating and maintaining the data, accepting and completing different requests to the database from user programs and terminal commands, and is responsible for protecting the data from all kinds of interference and damage.
Key features: Compared with file management, computer database systems manage data with the following features:
a) Data **** enjoyment
b) Data independence
c) Reduction of data redundancy
d) Data structuring
e) Unified data protection
14, Components of an expert system:
The expert system consists of six parts: a knowledge base, a comprehensive database, a reasoning machine, a knowledge acquisition mechanism, an interpretation mechanism, and a human-machine interface.
The knowledge base is a collection of domain knowledge needed for problem solving, including basic facts, rules and other relevant information.
The comprehensive database consists mainly of the relevant initial data of the problem and the intermediate information generated during the system solution.
The reasoner is the core actuator that implements the problem solving, it is actually the program that interprets the knowledge and executes the interpretation of the knowledge found according to a certain strategy based on the semantics of the knowledge and records the results in the appropriate space of the dynamic library.
The knowledge acquisition mechanism is responsible for establishing, modifying and expanding the knowledge base, mainly for realizing the self-learning of the expert system, which can automatically acquire knowledge during the use of the system and continuously improve and expand the existing system functions.
The explanation mechanism is to explain the solution process and answer the user's questions. The two most basic questions are "Why" and "How".
The main function of the human-computer interface is to realize two-way information conversion between the system and the user, i.e., the system translates the user's input information into information expressions familiar to the system.
The working process of the expert system is that the system, according to the goal proposed by the user, takes the comprehensive database as the starting point, and under the guidance of the control strategy, the reasoning machine applies the relevant knowledge in the knowledge base, and realizes the goal of the solution through continuous exploratory reasoning.
15, the concept of materials design and its three levels:
Definition: the use of high-performance computers and powerful materials-specific software to quantitatively or semi-quantitatively characterize the basic elements of the materials science and engineering disciplines and the relationship between the elements, the composition and process design of materials on the computer, and to predict the structure and properties of the material, which is known as materials design and simulation,. This is called materials design and simulation, also known as computational materials science.
The research level of materials design, there is no uniform and strict division. Generally speaking, according to the spatial scale of the research object can be divided into three levels: microscopic design level, the spatial scale of about 1nm; continuous modeling level, the scale of about 1m; engineering design level, the scale corresponds to the macro-materials, related to the processing of large pieces of material and the use of performance.
16, the concept of first principles:
The so-called first principles, means that only five basic physical constants (electron mass me, electron power e, Planck's constant h, the speed of light in the vacuum, c and Boltzmann's constant kB), as well as the type of atoms and atomic positional arrangement of the atoms in space (i.e., the crystal structure), do not need other empirical parameters, you can calculate the total energy of the system, the microscopic energy, the microcosmic energy and the microscopic energy of the system can be calculated with great accuracy. The total energy, microstructure and state of the system can be calculated without other empirical parameters.
Two, short answer
1, the five major applications of computers in materials science and engineering: (textbook page 2-5, their own summary of the summary)
1) for the design of new materials and new alloys:
2) for the simulation of materials science research:
3) for the optimization of materials process and Automatic control:
4) For the characterization of material composition and microstructure:
5) For data and image processing and others:
2. Meaning and classification of mathematical models:
Definition of mathematical model:
Mathematical concepts, mathematical formulas, mathematical theories, etc., which are abstracted in the context of the corresponding objective prototype are called mathematical models. Mathematical model. Or the system of mathematical symbols that reflect a particular problem or a particular system of things is called a mathematical model. Its purpose is to solve practical problems.
Classification of mathematical models:
Classification according to the mathematical method of modeling: graphical models, differential equation models, stochastic models, simulation models, etc..
Classified according to the characteristics of the model: discrete model, continuity model, linear model and nonlinear model, etc.
3, FDM and FEM solution steps:
FDM solution steps:
1) Establish the differential equation
Select the computational region according to the nature of the problem, establish the differential equation formula, and write the initial conditions and boundary conditions.
2)Construct the differential format
First, discretize the solution region, determine the computational nodes, and choose the grid layout, differential form and step size; then replace the infinite differential with finite difference, the differential quotient with differential quotient, and the differential equation with differential equation and boundary conditions.
3)Solving Difference Equations
Difference equations are usually a set of linear algebraic equations with a large number, and their solution methods mainly include: exact method and approximate method. The exact method is also called the direct method, including the matrix method, Gauss elimination method and the main element elimination method, etc. The approximate method is also called the indirect method, which is mainly based on the iterative method, including the direct iteration method, the indirect iteration method, and the super-relaxation iteration method.
4) Accuracy analysis and test
The accuracy and convergence of the obtained values are analyzed and tested.
FEM solution steps:
The computational steps of the finite element method (FEM) are summarized in the following three basic steps: mesh division, cell analysis, and overall analysis.
1) Mesh delineation
The basis of the finite element method is to replace the original continuum with a finite set of cells. Therefore, first of all, the necessary simplification of the elastomer, and then the elastomer will be divided into a finite number of cells composed of discrete body. The units are connected to each other by unit nodes. The set of cells, nodes, and node links is called a mesh.
It is common to divide three-dimensional solids into meshes of 4-plane or 6-plane cells, and planar problems into meshes of triangular or quadrilateral cells.
2) Unit Analysis
For elastic mechanics problems, unit analysis, is to establish the relationship between the node displacement and node force of each unit.
Since the node displacement of the unit as the basic variable, unit analysis is carried out first to determine an approximate expression for the displacement within the unit, and then calculate the unit's strain, stress, and then establish the relationship between the node force and node displacement in the unit.
3) Integral analysis
The analysis of the whole composed of each unit, establish the relationship between the node external load and the displacement of the node, in order to solve the node displacement, this process is the overall analysis. Then take the plane problem of elastic mechanics as an example, as shown in the figure on the right, a concentrated force is applied at the boundary node i. Node i is the union of three units, so it is necessary to bring together the nodal forces of these three units at the same node to establish the equilibrium equation.
4. Classification of expert systems:
According to the different nature of the problem to be solved in engineering, expert systems are classified into the following categories:
1) Interpretation of expert systems: by analyzing and interpreting the known information and data, to determine their meanings, such as image analysis, chemical structure analysis and signal interpretation.
2) Forecasting expert system: inferring what is likely to happen in the future by analyzing known conditions in the past and present, such as weather forecasting, population forecasting, economic forecasting, and military forecasting.
3) Diagnostic expert system: based on the observed situation to infer the cause of an object malfunction (i.e., failure), such as medical diagnosis, software failure diagnosis, material failure diagnosis.
4) design expert system: tool design requirements, to find the target configuration to meet the constraints of the design problem, such as circuit design, civil engineering design, computer structure design, mechanical product design and production process design.
5) planning expert system: to find out the sequence of actions or steps to achieve a given goal, such as robotics planning, transportation scheduling, engineering project justification, communications and military command, as well as crop fertilization programs.
6) Surveillance expert system: the behavior of the system, object or process is constantly observed, and the observed behavior is compared with the behavior it should have, in order to find anomalies, issue an alarm, such as nuclear power plant safety monitoring.
7) control expert system: adaptively manage the overall behavior of a controlled object so that it meets expected requirements, such as air traffic control, business management, combat management, autonomous robot control, production process control.
III. Calculation:
Application of Finite Difference Method (FDM) to heat transfer:
Example of FDM solution
1. Problems
Furnished with a furnace wall of thickness , the temperature of the inner wall of the furnace wall T0 = 900 C, the temperature of the outer wall of the temperature of the wall Tm = 100 C, to find the temperature distribution of the furnace wall along the thickness of the direction of the temperature distribution.
2. Analysis
This is a one-dimensional steady state heat transfer problem with boundary conditions T0=900C, Tm=100 C. The finite difference method can be used to find the temperature values of a number of nodes along the thickness direction of the furnace wall.
Mathematical foundations of FDM:
In numerical computation, a function (function) is considered in the form of a two-column table. One column is the (discrete) value xi of the independent variable, and the other column is the corresponding function value at xi, denoted as fi or f(xi).
Taking the operator view, three operators are defined:
(Forward difference operator): fi fi+1 fi
(Backward difference operator): fi fi fi1
(Center difference operator): fi fi+1/2 fi1/2
Where fi1 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), fi1/2 = f(xih), and fi1/2 = f(xih). fi1/2 = f(xih/2), xi+1xi=h,same for all i.
The above difference corresponding to the first-order derivative is called the first-order difference, and the corresponding difference corresponding to the second-order derivative is called the second-order difference:
2fi = ( fi+1 fi)=fi+22fi+1+fi
2fi = ( fi fi1)=fi2fi1+fi2
2fi =fi+12fi+fi1< /p>
The three operators are related: 2= . The rest of the higher order differentials can be followed in turn.
The ratio of the difference of a function to the difference of the independent variable is called the difference quotient of the function with respect to the independent variable. In second order, for example, its three forms are:
Forward Difference Quotient:
Backward Difference Quotient:
Center Difference Quotient:
Difference and Difference Quotients of multivariate functions can be obtained in a similar way.
The essence of the finite difference method is to replace the differential with the difference, and the geometrical meaning of replacing the differential with the quotient is to replace the true rate of change of the function with the average rate of change of the function in a region. For the first-order differential quotient, there exist the following three typical forms of difference:
Forward differential quotient:
Backward differential quotient:
Center differential quotient:
Based on the Taylor level, the errors of the above three forms of difference can be calculated as:
From these three equations, it can be seen that the errors caused by replacing the differential quotient by the difference quotient defined by different methods are different. Replacing the differential quotient with the forward differential quotient or the backward differential quotient has a stage error of O(x), which is of the order of magnitude of the primary of x. Replacing the differential quotient with the center differential quotient has a truncation error of O(x)2, which is of the order of magnitude of the quadratic of x, i.e., replacing the differential quotient with the center differential quotient is one order of magnitude smaller than replacing the differential quotient with the forward differential quotient or the backward differential quotient.
Therefore, when applying the FDM to a calculation, attention must be paid to the form of the difference equation, the method of building it, and the resulting error.
Note: 1, the number of selected nodes should be appropriate 4-5 appropriate;
2, in strict accordance with the solution steps to answer the question, in particular, do not omit the final accuracy of the analysis and testing steps.