考試大禮包有限元版finite elements

1、Two dimensional elements 二維單元Gauss-Legendre quadrature 高斯勒讓德求積法modal identification 模式識別element types in Ansys Ansys中的單元Lecture 5In this lecture We look further at the finite element method by using the example of two-dimensional elements with in-plane deformation plane elements or membrane elements

2、.平面單元或膜單元 This gives further insight into the process of finite element calculations. In particular,考察有限元的運(yùn)算過程o how to deal with multiple degrees of freedom 多自由度問題o how to deal with the integrations in the evaluation of element matrices in a systematic numerical way.系統(tǒng)化的數(shù)值方法估算單元矩陣中的積分 In doing this

3、we learn about the existence of a good numerical integration technique called Gauss-Legendre quadrature 高斯勒讓德求積法 We cast an eye over the different types of element there are. 單元一覽 I slip in a quick note about identifying modes.模態(tài)識別簡介This completes our account of constructing FE formulations. After t

4、hat we will look atsolution methods a bit more and do more practical work with ANSYS.本章將完成構(gòu)建有限元格式的表述，此后只有不斷的實(shí)踐。Element type 1Element type 2Element type 6Element type 5Energies 連續(xù)體的能量The kinetic energy of any continuum is the sum of the kinetic energy of the infinitesimal particles that make it up.連續(xù)

5、體的動能為無限小的質(zhì)點(diǎn)的動能之和where u and v are the displacements in x and y directionsThe strain energy is calculated as integral of 1/2 of stress times strain. The stress is calculated from strain using the stress-strain matrix 連續(xù)體的應(yīng)變能whereShear modulusPoissons ratioModulus of elasticityWe notice in these expre

6、ssions that the highest derivative appearing in the expressions for energy is the first - so we can use linear shape functions.上述表達(dá)式中出現(xiàn)的最高階導(dǎo)數(shù)為一階，因此，線性形函數(shù)可用。We take u and v to be our nodal degrees of freedom. These must be continuous i.e. displacements of one element must match those of the neighbori

7、ng elementat the nodes (and by proper choice of shape functions, all along the edges).節(jié)點(diǎn)和單元之間的位移必需連續(xù)Lets look at a rectangular element with linear shape functions. It therefore has 4nodes.4節(jié)點(diǎn)，矩形單元，Local coordinates 局部坐標(biāo)The displacement functions can be written in the combined formDisplacements are a

8、pproximated over elements using the shape function method在單元內(nèi)部，利用形函數(shù)得到位移場的近似表達(dá)Shape functions must be such thatNow we can use this expression for the displacements in the energy expressions for an element 將位移表達(dá)式用于能量表達(dá)式wherewithSo, to fill inB, we justdifferentiate NjChanging to the local coordinate

9、system (, ) givesThis only works if we arewith a rectangular elementalong the x and y directions.It does not work for ageneral quadrilateral.此式只適用于矩形單元Notice that the integration is carriedout on a local coordinate square.在局部坐標(biāo)系統(tǒng)中進(jìn)行積分These two expressions are evaluated using numerical integration (q

10、uadrature).In this case the integrands are quadratic（二次型） in and .Numerical integration 數(shù)值積分We are already familiar with numerical integration by the trapezium rule and Simpsons rule. These just amount to a sum of function evaluations times weighting factors. If we can choose where we evaluate the f

11、unction we can construct a more efficient scheme than constant width strips.Gauss-Legendre Quadrature where Hj are weights and j are abscissae（橫坐標(biāo)） or sampling points.（樣點(diǎn)）With n sample points this method can integrate a polynomial of order (2n - 1) exactly.n個(gè)積分樣點(diǎn)能對(2n - 1)階多項(xiàng)式精確積分。 For example with

12、only 3 sample points, the integration accuracy corresponds to that of the best fit 5th order polynomial approximation to the function on the element.Example weights and abscissae for n = 2, 3, 4 (note that they are symmetrical)Abramowicz and Stegun Handbook of Mathematical Functions lists rules up t

13、o n = 96 to 21 decimal places.n=4n=3 n=2For two dimensionswhere the weights and the abscissae are the same as for the one-dimensional case.This allows us to integrate the functions over an element that is from 1 to +1 inboth and , the local coordinates on the generic element. 母體單元Mapping onto the ge

14、neric element 實(shí)際單元向母體單元映射To do the integration over a general (distorted) quadrilateral element we must mapthe element onto this generic element.對一般的四邊形單元積分需將實(shí)際單元向母體單元映射Now the factor formapping is not constantwith positionThe determinant of the matrix |J| is called the Jacobian of the mapping ortra

15、nsformation. |J| 稱為映射的可比行列式，變換矩陣The integration of a function f (x, y) over the element esApproximation of geometry on the elementNow we need to choose the mapping functionsThey must satisfy 4 requirements1. Smooth and invertible （可逆）within each element.2. Generate mesh with no gaps or overlaps 無縫隙，

16、無重疊3. Easy to construct from geometrical data (nodal co-ordinates) 便于由節(jié)點(diǎn)坐標(biāo)構(gòu)造4. x(, ) and y(, ) should be easy to manipulate mathematically. 便于數(shù)學(xué)運(yùn)算Here, we choose the same interpolation scheme for the geometry that we used for the displacements. 利用與位移相同的內(nèi)插方案Different element typesWe have followed the

17、 example of: quadrilateral 四邊形 linear shape function 線性形函數(shù) isoparametric element 等參數(shù)單元Isoparametric means we have used the same order shape function interpolation for the displacements as for the geometry approximation.If we used a lower order approximation for the geometry then the element would be

18、 sub-parametric.（亞參元）An element type is defined by(單元的類型由以下因素決定) the shape and number of nodes 形狀和節(jié)點(diǎn)數(shù) the interpolation for the displacements 位移的插值函數(shù) the interpolation of the geometry 幾何形狀的插值函數(shù) the numerical integration 數(shù)值積分 the theory implemented - Euler beam, Timoshenko beam, thin plate, thick pla

19、te 理論依據(jù)，歐拉梁，鐵木辛克梁，薄板，中厚板Element types in ANSYSModal identification 模態(tài)識別How do we describe the modes that we have found for structures?如何描述結(jié)構(gòu)的模態(tài)beams/rods - We say we have modes of different order.梁/桿有不同階的模態(tài)Generally, we need to be able to describe or classify modes.我們需要能夠?qū)Y(jié)構(gòu)的模態(tài)進(jìn)行描述或分類 to discuss the

20、m 對模態(tài)進(jìn)行討論 to relate them to measurements 將模態(tài)與實(shí)測相聯(lián)系 to understand how each is affected by different parameters to describethe physical behavior of our structure is the first step to understandinghow to change its behavior in some useful way.不同的參數(shù)如何對各階模態(tài)產(chǎn)生影響，描述結(jié)構(gòu)的物理行為是改變結(jié)構(gòu)動態(tài)性能的基礎(chǔ)。These are the mode sh

21、apes for a clamped-simply supported beam that we had from the lecture on the Rayleigh-Ritz method.Can you put these into order without knowing their corresponding frequencies before hand?More wiggles meanshigher strain energymeaning highereigen-frequencies.We count the number of nodes and antinodes.

22、 (Remember, this is a different meaning of the word node from the FE one.)There are different types of mode in the same beam; the lateral bending modes (possibly in two separate, uncoupled lateral directions) and the axial modes. These are uncoupled from the bending modes as well. 關(guān)于耦合Uncoupled mean

23、s they have no component of displacement in common and socan exist independently. (In the case of a beam we can write different equationsfor bending and extending.)In two-dimensional structures things might get slightly more complicated 二維結(jié)構(gòu)的模態(tài)Consider a rectangular plate we classify modes by the no

24、dal lines .用節(jié)線來給模態(tài)分類this is a n = 1, m = 3, where n describes the order in the width direction and m in the length direction.This illustrates how we often characterize the modes as families of modes. A mode family is a sequence of modes that is enumerated by their order.通常，以模態(tài)族的方式來描述結(jié)構(gòu)的模態(tài)特征，按階數(shù)排列In

25、structures that are not simple shapes: it is still as valid to want to describe the behavior of different modes 描述實(shí)際結(jié)構(gòu)的模態(tài)行為有意義 modes can often follow families of similar simple structures but:o modes are more likely to be coupled, 結(jié)構(gòu)的模態(tài)可能耦合o still shows what modes you expect to find 仍能給出期望的模態(tài)o still

26、 provides insight on how to change modal family behavior 有助于改變模態(tài)族的行為(The + and signs already show that we are thinking about the consequences of the order of modes at different frequencies in terms of acoustic cancelling.)從消聲的角度考慮不同頻率的模態(tài) “How do we alter the structure to change the frequencies of th

27、e modes in thisfamily?”如何修改結(jié)構(gòu)來改變模態(tài)族的頻率 “Does our model correctly predict one family of modes but not another?” Thismay tell us what is wrong with the model and what is right.In FE analyses modal identification has an important role in getting something useful out of your model. e.g. 用有限元分析進(jìn)行模態(tài)識別，能從模型中得到一些有用的信息。 “What family of modes is responsible for sound radiation?” - not the in-plane ones, not the ones which acoustically short circuit except at edges, corners (plates). Simpl