有限元分析离散方法（Finite element analysis of discrete methods）

1、有限元分析离散方法（Finite element analysis of discrete methods）Discretization method: finite difference method, finite element method, finite volume methodThe so-called region discretization,In essence, a finite set of discrete points is used to replace the original continuous space. The implementation proce

2、ss is; Divide the calculated area into a number of non-overlapping sub-regions, determine the node location of each sub-region and the control volume represented by the node. Node: the geometrical position of unknown physical quantities that need to be solved; Control volume: minimum geometric unit

3、of applied control equation or conservation law. A node is generally considered to be a representative of volume control. The control volume and subregions are not always coincident. At the beginning of the discretization process, the small region of a series of lines or curves that correspond to th

4、e axes of the axes or curves is called subregions. Grid is a discrete basis, and grid nodes are the storage locations of discrete physical quantities.As you know, the commonly used discretization methods are finite difference method, finite element method, finite volume method.1. Finite difference m

5、ethod is the most classical method in numerical solution. It is to solve the division for difference grid, using a finite number of grid nodes instead of continuous solving domain, then the partial differential equation (control equation) derivative difference quotient instead of containing the disc

6、rete point of a finite number of unknowns is deduced differential equations. This method has been developed earlier and more mature, which is used to solve the hyperbolic and parabolic problems. Using it to solve the complicated boundary conditions, especially the elliptical problem is less convenie

7、nt than finite element method or finite volume method.2. The finite element method (fem) is a continuous domain arbitrary into appropriate shapes of many small units, and construct the interpolation function in each small unit divided, and then according to the principle of extremum (variational or

8、weighted residual method), the problem of control equations into all elements on the finite element equations, the overall extremum extremum as each unit, the sum of the local unit total synthesis, is embedded in the specified boundary conditions of algebraic equations, solve the equations of each n

9、ode and get function value. There is a better adaptability to the elliptical problem. The finite element method and finite volume method are not widely used in commercial CFD software. Current commercial CFD software, FIDAP USES finite element method.3. Also known as the control volume method, finit

10、e volume method is divided into a grid computing area, and make each grid point surrounded by a repetitive control volume, each will stay for each control volume integral solution of the differential equation, a set of discrete equation is obtained. The dependent variables on the unknown dozens of g

11、rid nodes. The subdomain method is discrete, which is the basic method of finite volume method. In the case of discrete methods, finite volume method can be regarded as the intermediate product of finite element method and finite difference method.4. Finite analysis method: as with the finite differ

12、ence method, the region is separated by a series of grid lines. The difference is that each node is composed of eight neighboring points. In computing unit the control equation of the nonlinear item of local linear, and the change of unknown functions on the cell type line make assumptions, the sele

13、cted line shape coefficient and the constant term in the expression with unit boundary node of the unknown variable values, so the cell is problem is converted into the first kind boundary condition of a definite solution problem, can find the analytical solution;Then, the algebraic equation of the

14、unknown value between the middle and the boundary of the unit is obtained by using this analytical solution, which is the discrete equation of the point in the unit. Two methods of discrete method: the node in the four corners of the subdomain, the node location is determined first, and the correspo

15、nding interfacial node method is calculated: the node is in the subdomain center, and the subdomain overlaps with the control volume. Calculate the interface and then calculate the node position.5. The boundary element method (Boudarv element method, the BEM) the above four methods must be made to t

16、he whole area discretization processing, with distribution function of a finite number of nodes in the whole area of approximation instead of a continuous solution of the problem. Application of greens function formula in the boundary element method, and by selecting the appropriate weighting functi

17、on to solve the space domain Bi partial differential equations into the boundary integral equation, it is area point in solving variables (e.g., temperature) and boundary conditions. The algebraic equation of unknown value on the boundary node is derived from the integral equation by discretization.

18、 By solving the unknown values on the boundary, the boundary integral equation can be used to obtain the value of the internal function. The maximum advantage of the boundary element method is that the space dimension of the solution can be reduced by one order, so that the computational workload an

19、d the computer capacity are greatly reduced. One of the biggest limitations of the application of boundary element method is the basic solution of the green function of the partial differential equation. Although the basic solutions of many partial differential equations have been found, the fundame

20、ntal solutions to nonlinear partial differential equations such as navier-stoles equations have not been found. One way to do this is to treat the nonlinear term in the navier-stokes equation as the source term for the diffusion equation and to solve it iteratively, but generally only the solution o

21、f the lower case is obtained. In the recent literature, the boundary element method of high order vortex-first-rate function equation has been used to stabilize Re - up to 10000.This article from: to get oil BBS Detailed source reference The finite element methodFinite element methodFinite element m

22、ethod (fem) is a kind of highly efficient, commonly used calculation method. The finite element method developed in the early based on variational principle, so it is widely used in Laplaces equation and poisson equation described by various physical fields (such as field and closely related to the

23、functional extremum problems). Since 1969, some scholars in the fluid mechanics in the application of weighted remainder method and liao method (Galerkin) or least square method and so on also obtained the finite element equations, so the finite element method can be applied to any differential equa

24、tion described by various physical fields, and no longer require such physical fields and functional extremum problems.Basic idea: to solve the problem of the extremum of the functional by solving the given poisson equation.Basic steps of the method:Step 1: dissection:Will stay solutions for region

25、segmentation, divided into the set of finite element. The shape of the elements (unit) in principle is arbitrary. The two-dimensional problem generally adopts the triangular or rectangular units, three dimensional space can use tetrahedron or polyhedron. Each unit of vertices are called node (or nod

26、es).Step 2: unit analysis:A linear interpolation function is established by using the function values of the shape functions and discrete grid points in the partition unitStep 3: solve the approximate variation equationA finite element method is used to solve the problems of mechanical and physical

27、problems by means of a finite element. Finite element method is a finite element method of finite element method. The unit of a continuum is a unit of various shapes such as triangles, quadrangle, hexahedron, etc. The field function of each unit is a simple field function that contains only a finite

28、 number of undetermined nodes. The collection of these unit field functions can approximate the field function of the entire continuum. Based on the energy equation or weighted residual equation, the algebraic equations of finite number of undetermined parameters can be established, and the numerica

29、l solution of finite element method is obtained by solving the discrete equations. The finite element method has been used to solve linear and nonlinear problems, and various finite element models have been established, such as coordination, incoordination, mixing, hybridization and fitting. Finite

30、element method is very effective, versatile and widely used, and many large or special program systems have been used for engineering design. The finite element method is also used in computer aided manufacturing.Finite element method can be traced back to the 1940s. For the first time, Courant USES

31、 the split continuous function and the minimum energy principle of the triangle region to solve the problem of st. Venant torsion. Modern finite element method is the first successful attempt in 1956, Turner, Clough and others on the analysis of the plane structure, the steel displacement method app

32、lied in mechanics of elasticity plane problems, presents a triangle unit of plane stress problem is obtained using the correct answer. In 1960, Clough further dealt with the problem of plane elasticity, and first proposed the finite element method to make people realize its efficacy. The famous mech

33、anic of China, academician xu zhilun (professor of hohai university), first introduced the finite element method to our country, and played a very important role in promoting its application.The finite volume method (FVM) is also called the control volume method.The basic idea is to divide the compu

34、tation area into a series of unrepeatable control volumes and to have a control volume around each grid point. A set of discrete equations is obtained by integrating the differential equations of the solution to each of the control volume integrals. The unknown is the number of dependent variables o

35、n the grid points. In order to calculate the integral of the control volume, it is necessary to assume the variation rule of the value between the grid point, i.e. the distribution profile of the distribution of the assumed value.From the selection method of the integral region, the finite volume me

36、thod is the subregion method of the weighted residual method. From the approximate method of unknown solution, the finite volume method is a discrete method with local approximation. In short, the subregional method is the basic method of finite volume.The basic idea of finite volume method is easy

37、to understand and can draw a direct physical explanation. Physical meaning of the discrete equation is the dependent variable in the limited size of the control volume conservation principle, as a dependent variable differential equation is said in the control volume of the infinitesimal conservatio

38、n principle. The discrete equation obtained by finite volume method,The integral conservation of the dependent variable is satisfied for any set of control volume, and the whole calculation region is also satisfied. This is the attractive advantage of finite volume method. There are some discrete me

39、thods, such as finite difference method, which can satisfy the integral conservation only when the grid is extremely fine. The finite volume method, even in the case of coarse grid, shows the exact conservation of integrals.In the case of discrete methods, finite volume method can be regarded as int

40、ermediate between finite element method and finite difference method. The finite element method must assume the change rule (interpolation function) between grid points and use it as an approximate solution. The finite difference method only considers the values of the grid points without considerin

41、g how the values change between grid points. The finite volume method only seeks the node value, which is similar to the finite difference method; However, when the finite volume method seeks to control the integral of volume, it must assume the distribution of the value between the grid points, whi

42、ch is similar to the finite element method. In the finite volume method, the interpolation function is only used to calculate the integral of the control volume. After the discrete equation is obtained, the interpolation function can be forgotten. If you need to, you can use different interpolation

43、functions for different items in the differential equation.Finite difference methodFinite difference methodThe numerical solution of differential equations and integral differential equations. The basic idea is to replace the continuous solution area with a grid of finite discrete points. These disc

44、rete points are called nodes of the grid. The function of continuous variables in the continuous solution region is approximated by the discrete variable functions defined on the grid. The original equation and the definite condition of micro commercial difference quotient to approximate integral wi

45、th integral and approximation, then the original differential equation and the definite condition approximately to algebraic equations, the finite difference equations, the solution of the equations can get the approximate solution of the original problem in discrete points. Then the interpolation m

46、ethod can be used to obtain the approximate solution of the fixed solution problem in the whole region.The main content of finite-difference method includes: how to make grid dissection in the region according to the characteristics of the problem; How to transform the original differential equation

47、 into differential equations and how to solve the algebraic equations. In addition to ensure the calculation process is feasible and correct results of the calculation of need analysis in theory of differential equations, including, the existence and the uniqueness of solutions of difference format

48、compatibility, the convergence and stability. For a differential equation established by a differential equation, for practical purposes, a basic requirement is that they can arbitrarily approximate the differential equation, which is the compatibility requirement. In addition, if a difference schem

49、e is useful, it is important to see whether the exact solution of the difference equation can approximate any solution to the differential equation, which is the concept of convergence. In addition, there is an important concept that must be considered, namely, the stability of the difference format

50、. Because the calculation process of the difference scheme is advanced by layer, the approximation of the n-th layer is used to calculate the approximate value of the NTH + 1 layer until it is related to the initial value. If in front of each layer has a rounding error, will inevitably affect the va

51、lue of each layer, if the influence of the error is bigger and bigger, so that the difference scheme of the appearance of exact solution is completely cover, this format is not stable, on the contrary, if the spread of error can be controlled,The format is considered stable. Only in this case, the a

52、pproximate solution of the difference scheme can approximate the exact solution of the difference equation. There are three ways to construct a difference format. The most commonly used method is numerical differentiation, such as replacing wechat business with the difference quotient. The other met

53、hod is called integral interpolation method, because the differential equations obtained in practical problems often reflect some kind of conservation principle in physics, which can be expressed in terms of integral form. In addition, the method of undetermined coefficients can be used to construct

54、 some high - precision difference schemes.From .Finite difference method (FDM), finite difference method (FDM) and finite volume method (FDM) are the earliest methods used in numerical simulation of computer and are still widely used. This method divides the solution domain into the difference grid

55、and replaces the continuous solution domain with a finite grid node. Finite difference method with Taylor series expansion method, the control equation of derivative with the grid node function value difference quotient instead of discrete, to establish a grid nodes on the value of the algebraic equ

56、ations for the unknown. This method is an approximate numerical solution to solve the problem of differential problems directly into algebraic problems. The mathematical concept is intuitive and simple, and is a numerical method of early and mature development. For a finite difference format, the fo

57、rmat is divided into a first-order format, a second order format, and a higher-order format. From the space form of difference, it can be divided into center format and upwind format. Considering the influence of time factor, the difference scheme can be divided into explicit format, implicit format

58、, and explicit alternate format. At present, the common difference scheme is mainly composed of the combination of the above forms, and different combinations constitute different difference formats. The difference method is mainly applied to the structure grid. The step length of the grid is usuall

59、y determined according to the actual terrain and the stable conditions of the coren. The method of constructing difference has many forms, and the main method is Taylor series expansion. The basic difference expression mainly has three forms: first-order differential forward, backward one order difference, first central difference and second order central difference, etc., the top two formats for a first order ca