Shape-3-1
- 格式：pdf
- 大小：106.11 KB
- 文档页数：16
3. SHAPE FUNCTIONS AND DISCRETIZATIONLinear linear quadratic cubic Rectangle quadrilateral quadrilateral quadrilateraliii) Three-dimensional Elementsz z z znodeeach per d.o.f. 5 : ,,,,nodeeach per d.o.f. 3 : ,,node each per d.o.f. 2 : ,node each per d.o.f. 3 : ,,yy xy xx v u p v u v u yx σσσφφφ∂∂∂∂Sometimes, one may want to introduce somewhat special element type for which each node may have different d.o.f. (e.g. 2 d.o.f. for vertex node, 1 d.o.f. for mid-side node).3.2 Shape Functions, Interpolationi i e N φφ=i : element nodal numberinge φ : approximation of φ over an elementi φ : nodal value of φi N : shape function()i i y x , : nodal coordinates3.2.1 Polynomial Approximation1-Dimensional case : n n x x x ααααφ++++=L 2210 : complete n -th orderpolynomial in 1-Dor∑==ni i i n x x P 0)(α1)1(+=n T n :number of terms for completen -th order polynomial in 1-D2-Dimensional case :L L ++++++++=j i k y x y xy x y x αααααααφ26524321or n j i y x y x P n T k j i k n ≤+=∑= ,),()2(1α:complete n -th orderpolynomial in 2-D2)2)(1()2(++=n n T n : number of terms for completen -th order polynomial in 2-DPascal Triangleorder name )2(n T1linear 3quadratic 6 cubic 10 quartic 15 quintic 21 hexatic 28 septic 36i α, involved in the complete n )2(n T ), is the3-Dimensional case : or n k j i z y x z y x P k T l j i l n n ≤++=∑= ,),,()3(1α:complete n -th orderpolynomial in 2-D2)3)(2)(1()3(+++=n n n T n : number of terms for completen -th order polynomial in 3-DGeneralized Coordinate i α is called “generalized coordinate”, which has no physical interpretation.In the subsequent section, it will be explained that the shape function isbetter to be used than this generalized coordinate for the interpolation over an element.At this point, one should think about how one selects the element type for a certain problem, in particular, with respect to the order of polynomial in interpolation, in other words, with respect to the number of nodes within each element. The higher the order is, more accurate finite element analysis result is expected, but at the cost of computation time. Certainly, there is a trade-off between the accuracy and the computational cost. We will discuss this matter in more detail later.Geometric IsotropyField variable representation should remain unchanged under a coordinate transformation. This fact is called “Geometric Isotropy” This concept is applied to, for instance, temperature, displacement, strain field, etc. The guideline to ensure the geometry isotropy is as follows: 1. complete polynomial 2. incomplete polynomial with appropriate terms to preserve “symmetry” e.g.2928265243213103726524321),(),(xyy x y xy x y x y x P y x y xy x y x y x P αααααααααααααααα+++++++=+++++++=3.2.2 Relationship between Generalized Coordinate and Shape FunctionField variable was represented in terms of polynomial series with “Generalized Coordinate” having no physical meaning. From this representation, one can derive a representation of a field variable in terms of interpolation functions and physical degree of freedom.∑∑===≤+=)2()2(11),( ,n n T k ii T k j i k y x N nj i y x φαφ∑∑+=+==≤≤=22)1(1)1(1),( , ,n k iin k j i ky x N n j n i y x φαφ⎣⎦⎣⎦⎟⎟⎠⎞⎜⎜⎝⎛=+++=xy y x P xy y x 1 e.g.4321ααααφi φ at ()i i y x , is the nodal value at node i :{}[]{}αφG =[]⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=4444111111 e.g.y x y x y x y x G M M M M {}α can be determined by the inverse:{}[]{}φα1−=G∴ ⎣⎦[]{}φφ1−=G Pi.e. ⎣⎦{})( i i N N φφφ== with ⎣⎦⎣⎦[]1−=G P Nwhere ⎣⎦N is called “Shape Function” or “Interpolation function”There is a more systematic procedure to obtain ⎣⎦N without having to find []1−G .We will cover this method extensively later. In any case, there are important characteristics of shape functions based on which one can generate formulae for various shape functions.3.2.3 Characteristics of Shape Functions()i j i i j i jjy x N N φφφφφ===),( , Q),(1y x N ),(2y x N(for C 0 continuity, piecewise continuous element)For a constant field variable φ, i.e. φ = constant, .any for constant k k =φ(constant ),( ====∴∑∑k ii k ii i N N y x φφφφ)3) Note thatφ along one side of an element should depend only on the nodalvalues associated with the side due to the continuity and compatibility.In general, nodes along a side of an element should be shared with a side of an adjacent element. However, for some reasons, one may introduce incompatible element which, for instance, looks like the following figure. [Hughes P.159]This kind of mesh is somehow useful with a special care for transient region from a coarse one to a finer one.We will discuss a special coordinate system, namely natural coordinates, which is more suitable for representing shape functions than a global coordinate system, before explaining the method of finding shape functions in detail.3.2.4 Natural CoordinatesNote that ),(y x N i in our current form is represented in terms of the nodal coordinates ()i i y x , and a global coordinate ()y x ,. One can have a better form in terms of so-called “Natural Coordinate ”, in particular for triangular type of elements (or “Normalized Coordinate ” for a quadrilateral type of elements )1) one-dimensional case22112211 x L x L x l l x l l x p +=+=→ Naturalcoordinate L 1 , L 2 “Length coordinate ” 121=+L Llike a linear interpolation)2)AA L i i = : “Area coordinate ”())( ,,321x,y L L L ↔⎩⎨⎧==ii ii y L y x L x The relationship between the natural coordinate and global coordinate system for 2-dimensional case is as follows:321332211332211 1L L L y L y L y L y x L x L x L x ++=++=++= → ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡1111321321321y x L L L y y y x x x solve for L i :⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡∆=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧121333222111321y x a c b a c b a c b L L L where a i , b i , c i ’s have already been defined when i i N φφ= for a triangular element in plane elasticity.3) three-dimensional caseVVL i i =: “Volume coordinate ”∑=====411i ii i ii i i L z L z y L y x L xNormalized coordinate for a rectangular element:orA)B)A)n-th order polynomial needs (n+1) points.x∑==nk k k x L x 0)()(φφOne should find )(x L k such that⎩⎨⎧=≠=ki ki x L i k when 1 when 0)( , satisfying the characteristics of shape function.It is easily proved that the formula below satisfies the requirement described by the equation above.)())(())(()())(())(()(111011100n k k k k k k k n k k nk m m m k m k x x x x x x x x x x x x x x x x x x x x x x x x x L −−−−−−−−−−=−−=+−+−≠=∏L L L L (3.1)It may be noted that )(x L k is an n -th order polynomial.Question : Is 1)(0=∑=nk k x L ? i.e. 1=∑ii NAnswer: yes.∑=nk kx L)( is an n-th order polynomial and is equal to unity at (n +1) points.The only n-th order polynomial being equal to unity at (n +1) points is justone over the whole domain of x.The interpolation via Lagrange polynomials is demonstrated below:⎣⎦{}⎣⎦{}φφφφφN L x L x x i i i ===≈∑=3)()(~)()()( x L x N i i =∴L i (x ) look like the following figures.It may be noted that, instead of global coordinate x , one can use normalized (local)coordinate ξ in the Lagrange interpolation function.The Lagrange interpolation for 1-dimensional case can be easily extended to 2-dimensional case:)()(ηξJ I IJ i L L N N ==3-dimensional extension:)()()(ςηξK J I IJK i L L L N N ==B) Hermitian InterpolationBasic type of interpolation used to represent a function in terms of values of both the function itself and derivatives of the function.One method uses end values at only (2 nodes) (e.g. beam bending analysis). General form for two end nodes and derivatives up to order m -1:)1(22232221)1(1131211 −+++−++′′+′++++′′+′+=m m m m m m m N N N N N N N N φφφφφφφφφL L (3.2)which is good for C m -1-interelement continuity. Note that this interpolation function isξ(2m -1)-th order polynomials.Alternatively, one can introduce m points and make use of function and its first order derivatives at each node:m m m m N N N N N N φφφφφφφ′+++′++′+=−21224231211L (3.3)which is good for C 1-interelement continuity.More generally, one can further introduce n points and make use of function and its derivatives up to (m -1)-th order at each node as depicted below:In this case, the interpolation can be expressed by)1(2)1(1)1()1(222221)1(11211 −+−+−−++−++′+++++′++++′+=m n nm nm n n m n m m m m m m N N N N N N N N N φφφφφφφφφφL L L L L (3.4)In this case, there are total degree of freedom of nm and there are nm number of shape functions. Each shape function should satisfy nm requirements 1 for each degree of freedom. In this regard, each shape function is described in terms of (nm -1)-th order polynomial.Because of the variety of strategies as described above, no fixed formulae for Hermitian interpolation are available. Instead, just use generalized parameters approach as described below.12321−++++=nm nm i x a x a x a a N Lwhich can be determined by characteristics of shape functions. [Huebner & Thornton Sect. 5.6.2]1Note that the shape function corresponding to a degree of freedom should provide itself at the node when the corresponding derivative is taken for the expression of φ. Other order of derivatives at the node should be zero and any order of derivatives are zero at other nodes. As a result, total nm requirementsExample : Beam bending analysis (m =2, C 1 –continuity)ξd d l dx de 1=2211212211 ,θθφφ===′=′dx dydx dy 24231211)(φφφφξ′++′+=N N N N y3)(42)(3)(2)(1)(ξξξξi i i i i a a a a N +++=Requirements:fori =1 : 0)1(1 ,0)1( ,0)0(1 ,1)0(1111====ξξd dN l N d dN l N e efor i =2 : 0)1(1 ,0)1( ,1)0(1,0)0(2222====ξξd dN l N d dN l N e efori =3 : 0)1(1 ,1)1( ,0)0(1 ,0)0(3333====ξξd dN l N d dN l N e efor i =4 : 1)1(1 ,0)1( ,0)0(1 ,0)0(4444====ξξd dN l N d dN l N e e∴())1()32(121)1(224232221−=+−=−=⎟⎠⎞⎜⎝⎛+−=ξξξξξξξξe e l N N l N N* Check if this interpolation function can describe the rigid body rotation.。
![英语单词详解系列[高中外研必修3单元1]第六十二篇](https://imgs-1438308264.cos.ap-hongkong.myqcloud.com/fd864bce856a561252d36fcc.webp)