Analysis Of Distortion Parameters Of Eight Node .

Analysis of Distortion Parameters of Eight node Serendipity Elementon the Elements PerformanceVishal Jagota & A. P. S. SethiDepartment of Mechanical Engineering , Shoolini University, Solan (HP), IndiaDepartment of Mechanical Engineering, B. B. S. B. Engineering College, Fatehgarh Sahib (Pb), IndiaE-mail : j4jagota@yahoo.com & aps.sethi@bbsbec.ac.inAbstract - Often the result produced using FEM technique on the distorted shaped element gives poor result. Moreover if the resultsare poor then the design will fail. Eight noded serendipity elements is the most widely used element in 2-D analysis of structures.But despite its benefits it remains distortion sensitive. In the majority of computer programs, automatic mess generation is an integralpart of the program. For complex shapes, the automatic grid generation will result in distorted quadrilateral shapes. Thus the solutionobtained by using these meshes will produce erroneous results so it becomes necessary to incorporate the distortion measures in tothese automatic mesh generations to limit the errors. Here in this paper distortion parameters are defined in terms of the coefficientof the element’s shape polynomials and are tested for a range of distortion.Keywords-FEM; serendipity element; distortion parametersI.opening a wide field of modeling of curved boundaries[5].INTRODUCTIONThe present day FEM stands on three legs:mathematical models, matrix formulation of the discreteequation and computing tools to do the numerical work.The third is the one that has undergone the mostdramatic changes. In early days of FEM, Hermikoff andMcHenry introduced the idea of modeling thecontinuum by replacing it with structural elements [1].Later Courant in 1943 established a variation solution ofthe Poisonous equation using what we know today aslinear triangular elements. He calculated the torsionalstiffness and observed the convergence by refining themesh [2].Melosh classified the errors in FEM as firstly due toidealization by modeling of curved surfaces as the flatsurfaces, secondly discretization errors due to replacingthe continuous structure by a finite number of smallpieces which can be minimized on refining andmanipulation errors due to round off and truncation [6].Eight noded serendipity element is the most widelyused element in 2-D analysis of structures. The elementwas first defined by Zhu & Zienckiewicz in 1968. Thiselement was an extension of isoparametric family ofelements proposed by Taig. Irons have proposed thenumerical integration technique for the isoparametricelements thus making it possible to evaluate the elementstiffness [5]. The eight noded quadrilateral was differentfrom the bilinear elements in number of ways, thebilinear is contained to have only straight sides. Theeight nodded quadrilateral have curved edges also. Thiselement also satisfies the two necessary criteria forconformability i.e. the element is able to represent theconstant strain exactly & the displacement remainscontinuous across the element even when the edges arecurved.Almost a decade later, Levy in 1953 recommendedthe derivation of stiffness matrix separately andassembling them for better accuracy [3]. Argyris in aseries entitled Energy theorem and structural analysishas given different procedures for stress anddisplacement analysis [4]. Taig in 1961 introduced theisoparametric concept through the quadrilateral element.His ideas systematically interpreted by Irons resulted ina major breakthrough in the formulation ofisoparametric elements in FEM. He later proposed thatisoparametric elements can also have curved edges thusInternational Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 2231 –6477, Vol-2, Issue-1, 2012110

Analysis of Distortion Parameters of Eight node Serendipity Element on the Elements PerformanceHowever this element despite of its so manybenefits remains distortion sensitive. Bathe proposedsome tests for isoparametric element and showed that itperforms well when distorted slightly [7]. The problemstill undefined is that how much distortion is allowablein eight nodded quadrilateral and how the parametershould incorporate the distortion measure. So here inthis paper we will make an attempt to define theparameters in terms of the coefficient of the element’sshape polynomials and test the element for a range ofdistortion in each of them.Where Ni, Ni’ are shape functions in terms ofintrinsic coordinates. The element is isoparametric if m n and Ni Ni’ and the same nodal points are used todefine the both element geometry and elementdisplacement. The element is sub parametric if m n, andsuper parametric if m n. The isoparametric element cancorrectly display rigid body and constant strainmodes.For plane stress problems the three unknownfields are displacement field U(x y 0), stress field σ(σxxσyy σxy), strain field ε(εxx εyy εxy). The in-planecomponents of all the above fields are assumed to beuniform throughout the thickness of the plate.Consequently the dependence of z disappears and allsuch component becomes the function of x and y only.II. FINITE ELEMENTS BASED ONDISPLACEMENT FIELDSThe most widely used elements in structuralmechanics are based on assumed displacement fields.Thus the x, y, z displacement components u, v, w of anarbitrary points within an element are interpolated fromdisplacements as(u v w)T N IV. DISTORTION PARAMETERA flat quadrilateral has four shape parameters,aspect ratio, skew angle and two tapers. The parameterswere based on a step by step drawing procedure forconstructing a flat quadrilateral. In this paper we hadtaken a new look at the definition parameters, that theycan be expressed in terms of coefficients in simplepolynomials which express the shape of thequadrilateral. These polynomial coefficients have asimple physical meaning and are functions of the cornerpoint coordinates of the flat or projected quadrilateral. Asimple representation of the shape parameters for a fournoded quadrilateral is given first and than extend to theeight node quadrilateral.(1)Matrix N is called the shape function matrix. Itcontains the interpolation polynomial. is the nodaldisplacement vector. Compatibility is satisfied withinelements because the polynomial displacement field iscontinuous. Compatibility is enforced at the node. At theinterface, however, elements may or may not becompatible depending upon their assumed displacementfields. Equilibrium prevails at the nodes. At elementboundaries, equilibrium is rarely satisfied. A stepchange is seen as we move from one element to anotherin general.yη34III. ISOPARAMETRIC FAMILIES OFELEMENTSξIsoparametric elements were introduced in 1966 byIrons. They are able to have curved sides and they usean intrinsic coordinate system for formulating theelement stiffness matrix. The same interpolation schemeis used to define both the geometry and the displacementfield of an element.xFig. 3.2 shows a flat quadrilateral in general localxy-system. The shape of the quadrilateral can be writtenin the form of interpolation function as restraint, so thatPlane stress is a condition that prevails in a flatplate which lies in a x-y plane and is loaded only in itsown plane and without z direction restraint, so that4x mu N uii21(2) N (ξ,η) xii(4)i 1i 14y mx N' uii(3) N (ξ,η) yii(5)i 1i 1WhereInternational Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 2231 –6477, Vol-2, Issue-1, 2012111

Analysis of Distortion Parameters of Eight node Serendipity Element on the Elements PerformanceN1 N2 N3 N4 (1-ξ)(1-η)4(1 ξ)(1-η)4(1 ξ)(1-η)(6)4(1-ξ)(1 η)x e4ξη4And ξ and η are non dimensional coordinates withlimits -1 to 1. Above equations shows that the eightparameters are needed to define a quadrilateral. In thiscase, the corner point’s coordinates are (x1, x2, x3, x4) (y1,y2, y3, y4).Fig. 2 : Interpretation of coefficient e1f1 to e4f4There are eight parameters (e1 to e4 and f1 to f4). Thephysical significance of the e and f coefficients is known.It is clear that e1 and f1 define an origin (translational ofaxis). e2 and f3 define the size of the rectangle (aspectratio), e3 and f2 gives two rotations (skew and rotation ofaxis) and e4 and f4 give tow tapers.A. Interpretation of coefficient e1f1 to e4f4The use of interpolation functions is very elegant butthe mathematical elegance can hide the practicalsignificance of various quantities. An alternative form ofthe shape representation is to use the simple and basicpolynomials.x e1 e2ξ e3η e4ξηy f1 f2ξ f3η f4ξηB. Interpretation of coefficient e5f5 to e8f8Similarly for eight node quadrilateral the shapefunction can be expressed in the simple polynomial. x e1 e2ξ e3η e4ξη e5ξ2 e6η2 e7ξ2η e8ξη2 (2)(7)y f1 f2ξ f3η f4ξη f5ξ2 f6η2 f7ξ2η f8ξη2(8)(3)For eight node quadrilateral, additional distortionparameter emerge which are associated with the offset ofthe boundary nodes. For a particular boundary, this offsetmay be due to the curvature or when the side is straight,because the boundary node is not in the centre.The procedure used for the four node quadrilateralfor obtaining the distortion parameters from thecoefficients in the simple shape polynomial is extendedhere to the eight node quadrilateral with curvedboundaries.where e and f coefficient are related to the nodalcoordinates x1 to x4 and y1 to y4 by linear relationx e1, y f1y f4ξηIn case of eight node quadrilateral the eight moreparameter are there. These parameters are radius andangle for each boundary.x e2ξ, y f3ηx e5ξ2x e3η, y f2ξy f5ξ2International Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 2231 –6477, Vol-2, Issue-1, 2012112

Analysis of Distortion Parameters of Eight node Serendipity Element on the Elements Performancex e6 η2y f 6η 2Fig. 4 : Undistorted configuration of 8 node elementx e 7ξ 2 ηThese distortions are illustrated in Fig. 5 and Fig. 6.Any element distortion can generally be considered tobe composed of some or all of the above types ofdistortions. We may also distinguish between distortionsthat are present in the undeformed finite element modeland distortions that are created by large deformations.The latter type occurs only in non-linear analysis withlarge strain effects.y f7ξ2ηFig. 5 : Aspact-ratio and Taper distortion2x e8ξη2y f8ξηFigure 3 : Interpretation of coefficient e5f5 to e8f8V. TESETING OF THE ELEMENTThe objective of this paper is to present asystematic study with numerical and analytical resultson the performance of the element when it is used indistorted shapes and summarize the findings that havepractical consequences.Fig. 6 : Unevenly-spaced-nodes and Curved-edgedistortionA. Classification of Element DistortionThe attention is focused on the types of elementdistortions that occur most frequently in practice andthat affect the performance of the elements. Theclassification of element distortions based on the shapeof the element and the location of its nodes. Consideringa square element with evenly spaced nodes to be theundistorted element, we may identify the followingbasic types of distortions: B. Aspect-ratio Sensitivity TestAspect-ratio is given by a/b, it is the ratio betweenlength and breadth. The test conducted on the cantileverbeam, problem is shown belowBoundary conditionsu 0u v 0Aspect-ratio distortionTaper distortionUnevenly-spaced-nodes distortionCurved-edge distortionE 200x109 N/m2, υ 0.3, P 60 NInternational Journal of Mechanical and Industrial Engineering (IJMIE), ISSN No. 2231 –6477, Vol-2, Issue-1, 2012113