材料數(shù)值模擬基礎(chǔ)（朱鳴芳）chapter8finiteelementmethod

1、chapter 8 finite element method8.1 basics of the finite element method&11 introductionvirtually every phenomenon in nature, whether biological, geological, or mechanical, can be described with the aid of the laws of physics, in terms of algebraic, differential, or integral equations relating var

2、ious quantities of interest. determining the stress distribution in a pressure vessel with oddly shaped holes and numerous stiffeners and subjected to mechanical, thermal, and/or aerodynamic loads, finding the concentration of pollutants in seawater or in the atmosphere, and simulating weather in an

3、 attempt to understand and predict the mechanics of formation of tornadoes and thunderstorms are a few examples of many important practical problems.most engineers and scientist studying physical phenomena are involved with two major tasks:1. mathematical formulation of the physical process2. numeri

4、cal analysis of the mathematical modelwhile the derivation of the governing equations for most problems is not unduly difficult, their solution by exact methods of analysis is a formidable task. in such cases, approximate methods of analysis provide alternative means of finding solutions. among thes

5、e, the finite difference method and the finite element method are most frequently used in the literature. the finite element method is derived by the variational methods such as the rayleigh-ritz and galerkin methods.in the finite difference approximation of a differential equation, the derivatives

6、in the latter are replaced by difference quotients (or the function is expanded in a taylor series) that involve the values of the solution at discrete mesh points of the domain. the resulting algebraic equations are solved, after imposing the boundary conditions, for the values of the solution at t

7、he mesh points.in the solution of a differential equation by a variational method, the equation is put into an equivalent weighted-integral form and then the approximate solution over the domain is assumed to be a linear combination 心) of appropriately chosen approximation functions j and undetermin

8、ed coefficients, cj. the coefficients cf are determined such that the integral statement equivalent to the original differential equation is satisfied. various variational methods, e.g, the rayleigh-ritz, galerkin, and least-squares methods, differ from each other in the choice of the integral form,

9、 weight functions, and/or approximation functions. they suffer from the disadvantage that the approximation functions for problems with arbitrary domains are difficult to constructthe finite element method overcomes the disadvantage of the traditional variational methods by providing a systematic pr

10、ocedure for the derivation of the approximation functions over subregions of the domain. the method is endowed with three basic features that account for itssuperiority over other competing method. first, a geometrically complex domain of the problem is represented as a collection of geometrically s

11、imple subdomains, called finite elements. second, over each finite element, the approximate functions are derived using the basic idea that any continuous function can be represented by a linear combination of algebraic polynomials. third, algebraic relations among the undermined coefficients (i.e.?

12、 nodal values) are obtained by satisfying the governing equations, often in a weighted-integral sense, over each element thus, the finite element method can be viewed, in particular, as an element-wise application of the rayleigh-ritz or weighted-residual methods. in it, the approximation functions

13、are often taken to be algebraic polynomials, and the undetermined parameters represent the values of the solution at a finite number of preselected points, called nodes, on the boundary and in the interior of the element. the approximation functions are derived using concepts from interpolation theo

14、ry, and are therefore called interpolation function. one finds that the degree of the interpolation functions depends on the number of nodes in the element and the order the differential equation being solved.the term "finite element55 was first used by clough in 1960. since its inception, the

15、literature on finite element application has grown exponentially, and today there are numerous journals that are primarily devoted to the theoiy and application of the methodin the present book, the finite-element method is introduced as a variationally based technique of solving differential equati

16、ons. a continuous problem described by a differential equation is put into an equivalent variational form, and the approximate solution is assumed to be a linear combination,工cj& of approximation functions 卜 the parameters cj are detennined using the associated variational form. the finiteelemen

17、t method provides a systematic technique for deriving the approximation function for simple subregions by which a geometrically complex region can be represented. in the finite-element method, the approximation functions are piece-wise polynomials (i.e., polynomials that are defined only on a subreg

18、ion, called an element).8.1.2 the basic concept of the finite element methodthe most distinctive feature of the finite-element method that separates it from others is the division of a given domain into a set of simple subdomains, called finite elements. any geometric shape that allows computation o

19、f the solution or its approximation, or provides necessary relations among the values of the solution at selected points, called nodes, of the subdomain, qualified as a finite element. other features of the method include seeking continuous, often polynomial, approximations of the solution over each

20、 element in terms of nodal values, and assembly of element equations by imposing the interelement continuity of the solution and balance of interelement forces. here the basic ideas underlying the finite element method are introduced via a simple example: determination of the circumference of a circ

21、le using a finite number of line segments.consider the problem of determining the perimeter of a circle of radius r (see fig. 8.1a). ancient mathematicians estimated the value of the circumference by approximating it by line segments, whose lengths they were able to measure the approximate value of

22、the circumference is obtained by summing the lengths of the line segments used to represent it. although this is a triviallength hefigure 8.1approximation of the circumference of a circle by line elements: (a) circle of radius r; (b) uniform and nonuniform meshes used to represent the circumference

23、of the circle; (c) a typical element.1. finite element discretization first, the domain (i.e., the circumference of the circle) is represented as a collection of a finite number n of subdomains, namely, line segments. this is called discretization of the domain. each subdomain (i.e., line segment) i

24、s called an element. the collection of elements is called the finite element mesh. the elements are connected to each other at points called nodes. in the present case, we discretize the circumference into a mesh of five (n=5) line segments. the line segments can be of different lengths. when all el

25、ements (i.e., line segment) are of the same length, the mesh is said to be uniform; otherwise, it is called a nonuniform mesh (see fig. 8.1b).2. element equations. a typical element (ie, line segment, qe) is isolated and its required properties, i.e., length, are computed by some appropriate means.

26、let he be the length of element qe in the mesh. for a typical element q； he is given by (see fig. &lc)心= 27?sin*0(8.1.1)where r is the radius of the circle and 產(chǎn)兀 is the angle subtended by the line segment. the above equations are called element equations. ancient mathematicians most likely made

27、 measurements, rather than using (8 1 1), to find he.3. assembly of element equations and solution. the approximate value of the circumference (or perimeter) of the circle is obtained by putting together the element properties in a meaningful way;this process is called the assembly of the element eq

28、uations. it is based, in the present case, on the simple idea that the total perimeter of the polygon (assembled elements) is equal to the sum of the lengths of individual elements:n幾(8.1.2) e=then pn represents an approximation to the actual perimeter, p. if the mesh is uniform, or he is the same f

29、or each of the elements in the mesh, then 0e= 2兀/« and we have4. convergence and error estimate. for this simple problem, we know the exact solution: p=2ttr. we can estimate the error in the approximation and show that the approximation solution pn converges to the exact p in the limit as moo.

30、consider the typical element qe. the error in the approximation is equal to the difference between the length of the sector and that of the line segment (see fig. 8.1c):ee= sehe(8.1.4)where se = r3e is the length of the sect0匚 thus, the error estimate for an element in the mesh is given by27t7te =7?

31、(-2sin-)' nnthe total error (called global error) is given by multiplying ee by n:(8.1.6)ee=2r(7i-n sin -) = itir pcnwe now show that e goes to zero as letting x = /n, we havern = 2rn sin = 2r sin nxand(k、a r sin%兀(cos tlx )2r-limmri x丿xto1丿lim p = lim幵 t8xt0=2兀rhence, en goes to zero as moo. th

32、is completes the proof of convergence.in summary, it is shown that the circumference of a circle can be approximated as closely as we wish by a finite number of piecewise-linear functions. as the number of elements is increased, the approximation improves, i.c., the error in the approximation decrea

33、ses.8.1.3 solution of differential equationconsider the temperature variation in a composite cylinder consisting of two coaxial layers in perfect thermal contact (see fig. 8.2). heat dissipation from a wire (with two insulations) carrying an electric current and heat flow across a thick-walled compo

34、site circular cylindrical tube are typical examples. the temperature t is a function of the radial coordinate r. the variation of t with r is, in general, nonunifbrm. we wish to determine an approximation te(r) to t(r) over the thicknesses of the cylinde匚 the exact solution is determined by solving

35、the differential equation1 dr drrkdt)dr)subject to approximate boundary condition, for example, insulated at r = and subjected to a temperature to at r = ro:(8.1.96)at r =; t(r)=to at r = ro人（廠）=丈珈;（廠）where k is the thermal conductivity, which varies from layer to layer, 7?/ and ro are the inner and

36、 outer radii of the cylinder, and q is the rate of energy generation in the medium. note that the temperature is independent of the circumferential coordinate (because of the axisymmetric geometry, boundary conditions, and loading), and it has the same variation along any radial line. when it is dif

37、ficult to obtain an exact solution of the problem (8.1.9), either because of complex geometry and material properties or because q(r) is a complicated function that does not allow exact evaluation of its integral, we seek an approximate one. in the finite element method, the domain (/?/, ro) is divi

38、ded into n subintervals, and the approximate solution is sought in the form(r. <r< ri + h、; first interval)n3(廠)=t；屮;(廠)(r. + h<r< rt + hy + /72; second interval)(&1.10)./=1v(r) = xt；呎(廠)(r + /?1 + + hn_x <r<ro; nth interval)7=1where he denotes the length of the eth interval, i

39、s the value of the temperature te(r) at the 丿thgeometric point of the eth interval, and 0； are polynomials on the eth interval. the continuous function t(r) is approximated in each interval by a desired degree of polynomial, and the polynomial is expressed in terms of the values of the function at a

40、 selected number of points in the interval. the number of points is equal to the number of parameters in the polynomial. for example, a linear polynomial approximation of the temperature over the interval requires two values, and hence two points are identified in the interval. the endpoints of the

41、interval are selected for this purpose because the two points also define the length of the interval (see fig. 83a). forhigher-order polynomial approximation, additional points are identified interior to the interval (see fig. 8.3b) the intervals are called finite element, the points used to express

42、 the polynomial approximation of the function are called nodes, t； are called nodal values, and 叭 are called finite element approximation functions. the nodal values t； are determined such that te(r)satisfies the differential equation (8.1.9a) and boundary conditions (8.1.9b) in some sense. usually,

43、 the differential equation is satisfied in a weighted-integral sense, and boundary conditions on the function itself are satisfied exactlyelement(a)(b)figure 8.2 (a) coaxial (composite) cylinder made of two different materials.(b) finite element representation of a radial line of the cylinder山 九 爲ii

44、i1 oo0 32(a)(b)figure 8.3 (a) linear approximation of a function t(r). (b) quadratic approximation of a functiont(dthe piecewise (i.e., element-wise) approximation of the solution allows us to include any discontinuous data, such as the material properties, and to use meshes of many lower-order elem

45、ents or a mesh of few higher-order elements to represent large gradients of the solution. polynomial approximation of the form (8.1.10) can be derived systematically for any assumed degree of variation.the satisfaction of the differential equation in a weighted-integral sense leads, for steady-state

46、 problems, to algebraic relations among nodal temperature tj and heats 0； of the element. the algebraic equations of all elements are assembled (i.e., related to each other) such that the temperature is continuous and the heats are balanced at nodes common to elements.8.1.4 some remakesin summary, i

47、n the finite element method, a given domain is divided into subdomains, called finite element, and an approximation solution to the problem is developed over each of these. the subdivision of a whole into parts has two advantages:1. it allows accurate representation of complex geometries and inclusi

48、on of dissimilar materials.2. it enables accurate representation of the solution within each element, to bring out local effects (e.g., large gradients of the solution).the three fundamental steps of the finite element method that are illustrated via the examples are:1. divide the whole into parts (

49、both to represent the geometry and solution of the problem).2. over each part, seek an approximation to the solution as a linear combination of nodal values and approximation functions.3. derive the algebraic relations among the nodal values of the solution over each part, and assemble the parts to

50、obtain the solution to the whole.8.1.5 summaryin a numerical simulation of a physical process, we employ a numerical method and computer to evaluate a mathematical model of the process. the finite element method is a powerful numerical technique devised to evaluate complex physical process. the meth

51、od is characterized by three features:1. the domain of the problem is represented by a collection of simple subdomains, called finite elements. the collection of finite element is called the finite element mesh.2. over each finite element, the physical process is approximated by functions of desired

52、 type (polynomials or otherwise), and algebraic equation relating physical quantities at selective points, called nodes, of the element are developed based on the variational methods.3. the element equations are assembled using continuity and/or “balance" of physical quantities.8.2 variational

53、methods8.2.1 functionalsan integral expression of the formi(u) = f(x,u.u)dx, u = w(x)5 u -dudx(&2.1)where the integrand f(x, u.u is a given function of the arguments x, u. and , is called a dxfunctional. the value i(u) of the integral depends on u; hence the notation i(u) is appropriate. however

54、, for a given u, i(u) represents a scalar value. we shall use the term functioned to describe functions defined by integrals whose arguments themselves are functions. loosely speaking, a functional is a "function of functions/' mathematically, a functional is an operator i mapping u into a

55、scalar i(u).functional is a function whose domain is a set of functions and whose range is a set of functions or a set of numbers.簡單的說，泛函就是定義域是一個函數(shù)集，而值域是實數(shù)集或者實數(shù)集的一個子集，推 廣開來，泛函就是從任意的向量空間到標量的映射。也就是說，它是從函數(shù)空間到數(shù)域的映 射。泛函也是一種“函數(shù)”，它的獨立變量一般不是通常函數(shù)的“自變量”，而是通常函數(shù)木 身。泛函是函數(shù)的函數(shù)。由于函數(shù)的值是由自變量的選取而確定的，而泛函的值是由自變量 函數(shù)確定的，故

56、也可以將其理解為函數(shù)的函數(shù)。8.2.2 the variational symbolconsider the functional f = f(x,u.u). for an arbitrary fixed value of the independent variable x, f depends on u and u . the change av in u, where a is a constant and y is a function, is called the variation of u and is denoted by 6usu = av(8.2.2)the operato

57、r 8 is called the variational symbol. the variation of a function 8u represents an admissible change in the function w(x) fixed value of the independent variable x. if ” is specified at a point (usually on the boundary), the variation of u is zero there because the specified value cannot be varied,

58、thus the variation of a function u should satisfy the homogeneous form of the boundary conditions for u. the variation du in u is a virtual change. associated with this change in u (i.e., u going to u+av), there is a change in f. in analogy with the total differential of a function of two variables, the first variation of f at u is defined by(8.2.3)(&2.4)8f = 3u8udu dunote the analogy between the first variation, (8.2.3), and the total differential of f.df = dx du + dudu du dusince x is not varied during the variation of u to u+su, dx=0 and the analogy betwe