Axisymmetric solid-of-revolution finite elements with rotational degrees of freedom.pdf
基于APDL语言的机床滑枕结构优化设计说明书
收藏
资源目录
压缩包内文档预览：
编号：91866774
类型：共享资源
大小：1.80MB
格式：ZIP
上传时间：2020-08-10
上传人：QQ14****9609
认证信息
个人认证
郭**（实名认证）
陕西
IP属地：陕西
15
积分
- 关 键 词：
-
基于
APDL
语言
机床
结构
优化
设计
说明书
- 资源描述：
-
基于APDL语言的机床滑枕结构优化设计说明书,基于,APDL,语言,机床,结构,优化,设计,说明书
- 内容简介：
-
Finite Elements in Analysis and Design 45 (2009) 121-131Contents lists available at ScienceDirectFiniteElementsinAnalysisandDesignjournal homepage: /locate/finelAxisymmetricsolid-of-revolutionfiniteelementswithrotationaldegreesoffreedomCraig S. Longa, Philip W. Lovedaya, Albert A. GroenwoldbaSensor Science and Technology, CSIR Material Science and Manufacturing, Box 395, Pretoria 0001, South AfricabDepartment of Mechanical and Mechatronic Engineering, University of Stellenbosch, Matieland 7602, South AfricaA R T I C L EI N F OA B S T R A C TArticle history:Received 30 October 2007Received in revised form 31 July 2008Accepted 5 August 2008Available online 23 September 2008Keywords:AxisymmetricFinite elementRotational degrees of freedomTwo new solid-of-revolution axisymmetric finite elements, which account for hoop fibre rotations, areintroduced. The first is based on an irreducible formulation, with only displacement and rotation fieldsassumed independently. The second element, based on a HellingerReissner like formulation, possessesan additional assumed stress field. Furthermore, an element correction, often employed in membraneelements with drilling degrees of freedom to alleviate membrane-bending locking, is adapted to theaxisymmetric case. The supplemental nodal rotations introduced herein enhance modelling capability,facilitating for instance the connection between axisymmetric shell and solid models. The new elementsare shown to be accurate and stable on a number of popular benchmark problems when compared withpreviously proposed elements. In fact, for cylinders under internal pressure analysed with a regular mesh,our mixed elements predict displacement exactly, a phenomenon known as superconvergence. The newelements are also shown to be robust and accurate on a number of bending dominated problems. 2008 Elsevier B.V. All rights reserved.1. IntroductionIn recent times, flat shell finite elements with in-plane rotational(drilling) degrees of freedom have become quite popular. Apart fromenrichment of the displacement field, resulting in increased elementaccuracy, drilling degrees of freedom allow for the modelling of, forinstance, folded plates and beamslab intersections.The motivation for developing axisymmetric elements with rota-tional degrees of freedom is similar. Hoop fibre rotational degrees offreedom are desirable in solid-of-revolution axisymmetric elements,to accommodate their connection to axisymmetric shell elements.Furthermore, the accuracy benefits attributed to the enhanced dis-placement interpolations are shown herein to be similar to the pla-nar case.Early obstacles in the development of finite elements with drillingdegrees of freedom were largely overcome by Hughes and Brezzi 1,who presented a rigorous mathematical framework in which to for-mulate such elements. They proposed a modified variational princi-ple, based on the work of Reissner 2, but with improved stabilityproperties in the discrete form. Early finite element implementa-tions(inCartesiancoordinates)employingtheformulationofHughesand Brezzi were presented by Hughes et al. 3 and Ibrahimbegovicet al. 4,5.Corresponding author. Tel.: +27128412498; fax: +27128413895.E-mail address: clongcsir.co.za (C.S. Long).0168-874X/$-see front matter2008 Elsevier B.V. All rights reserved.doi:10.1016/j.finel.2008.08.001Developments in mixed/hybrid membrane finite elements havebeen equally important during recent years. Since the assumed stressfinite element presented by Pian 6, numerous formulations havebeen proposed. A compilation is presented by Pian 7. Eventually,assumed stress formulations were applied to elements with drillingdegrees of freedom in a single element formulation, e.g. see 812.Compared to finite element development in the Cartesian coor-dinate frame, advances in axisymmetric finite elements residing ina cylindrical coordinate system have been relatively slow. Besidesthe restrictions these elements place on the geometry and boundaryconditions that can be modelled, the slower pace of developmentis presumably due to difficulties associated with directly extendingdevelopments in planar elements to axisymmetric elements. For ex-ample, reciprocal coordinate terms as well as stress and displace-ment terms appearing directly in equations of equilibrium and thestraindisplacement operator, respectively, add complexities.For the purposes of this discussion, axisymmetric finite elementdevelopments may be grouped based on the origins of their varia-tional formulations. For example, several authors have made use ofHu-Washizu like functionals to derive accurate elements. Of these,some of the most notable are Bachrach and Belytschko 13, whoused a projection method employing GrammSchmidt orthogonal-ization, to ensure the resultant stiffness matrix requires only block-diagonalinversion,thereby reducingcomputationaleffort.Wanji andCheung 14, after earlier efforts 15, proposed a non-conformingand a refined hybrid quadrilateral axisymmetric element. Theydeveloped a general approach for constituting non-conformingdisplacement functions. More recently, Kasper and Taylor 16122C.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-131presented a mixed-enhanced formulation for axisymmetric prob-lems extending their previous works in Cartesian coordinates 17.OtherauthorsdrawonformulationsbasedonmodifiedHellingerReissner functionals. Examples include Tian and Pian 18who extended the rational approach for assumed stress finite ele-ment development, proposed by Pian and Sumihara 19, to applyto axisymmetric problems. Unfortunately, as pointed out in 14,these elements do not pass the patch test. Sze and Chow 20 intro-duced an incompatible element, incorporating an inversion crime(making use of a simplified Jacobian in the straindisplacementoperator). This element was then modified using an extendedHellingerReissner principle, and finally simplified using the con-cept of free formulation 21.Finally,variousresearchershaveusedmoreconventionalHellingerReissner principles to formulate axisymmetric elements.Zongshu 22 developed a series of 8-node axisymmetric solid ele-ments using the traditional HellingerReissner model. The assumedstress fields were chosen in global coordinates, so as to satisfy equi-librium and such that the element has proper rank and is invariantwith respect to a co-coordinate system shift in the direction of theaxis of radial symmetry.Weissman and Taylor 23 introduced two elements based on theHellingerReissner functional. Their elements employ the popularPian and Sumihara interpolation, modified to obtain correct rank forthe axisymmetric case. The?rr,?zzand?rzterms are interpolatedin the local coordinate system while the hoop stress,?, is inter-polated in terms of global coordinates.Renganathan et al. 24 proposed a hybrid stress element withthe minimum seven?-parameters and stress interpolations in globalcoordinates which satisfy equilibrium as well as compatibility con-ditions. Most recently, Jog and Annabattula 25 presented a proce-dure for the development of mixed axisymmetric elements basedon the HellingerReissner variational principle in which stress inter-polations in local coordinates are formulated based on zero-energymodes resulting from reduced integration.The aim of this paper is to extend the existing work on planarmembrane elements with drilling degrees of freedom to the torsion-lessaxisymmetric case. Furthermore, inordertoensurethat accuracybenefits are maintained in the limit of near incompressibility, we in-troduce a mixed (assumed stress) element accounting for rotationaldegrees of freedom, based on a HellingerReissner like functional.2. Problem statementWe proceed, as in 26, by defining?as a closed and boundeddomain occupied by a body in three-dimensional space. The interiorpart of?is denoted by?and its boundary by?,?=?.The measure of?is V and the measure of?is S.Vis the vectorspace associated with the Euclidean point space andLthe space ofall linear applications ofVintoV, which possesses inner productAB=tr(AtB), A,B Land Atthe transpose of A (see 26). Referencewill also be made to subsets ofL, namelySandWwhich contain,respectively, symmetric and skew-symmetric tensors inL.The boundary?is split into two parts,?uand?t, such that?u?t=?and?u?t= . On?udisplacements u areprescribed, while on?tthe tractiont is prescribed. The discussionis limited to linear elastic problems and boundary terms are omitted.Boundary conditions may, however, be incorporated in the standardmanner, e.g. see 4,2729.In the most general case, the stress tensor,rL(which is nota priori assumed to be symmetric), the displacement vector field u,the skew-symmetric infinitesimal rotational tensor,wW, and thestrain tensor?Sare taken as independent variables. The varia-tionalformulationrequires thatthe rotationsw,strains?and stressesr, together with the displacement generalised derivativesu,belong to the space of square-integrable functions over the region?. The Euclidean decomposition of a second-rank tensor is used, i.e.,r= symmr+ skewr,(1)wheresymmr=12(r+rt)(2)andskewr=12(rrt).(3)The problem under consideration is now constructed as follows:Given f, the body force vector, find u,w,rand?such that:divr+ f = 0,(4)skewr= 0,(5)w= skewu,(6)?= symmu,(7)symmr= c?,(8)for all x ?, and where c is the elasticity tensor. Eqs. (4)(8) are,respectively, the linear and angular momentum balance equations,the definition of rotation in terms of displacement gradients, thecompatibility condition for strain in terms of displacement gradientand the constitutive equations.Now, assuming a cylindrical coordinate system, the strain tensorcontains the terms?rr=?ur?r,?r?=12?1r?ur?+?u?ru?r?,(9)?=1r?u?+urr,?rz=12?uz?r+?ur?z?,(10)?zz=?uz?z,?z=12?u?z+1r?uz?,(11)and the components of rotations are defined from continuummechanics considerations as?r=12r?(ru?)?r?ur?,?rz=12?ur?z?uz?r?and?z?=12?1r?uz?u?z?.(12)However, assuming torsionless axisymmetry, these relations sim-plify to?rr=?ur?r,?r?= 0,(13)?=urr,?rz=12?uz?r+?ur?z?,(14)?zz=?uz?z,?z= 0,(15)and?r= 0,?rz=12?ur?z?uz?r?and?z?= 0.(16)In the sections to follow, this problem will be embodied in an ir-reducible finite element formulation, which requires only indepen-dent displacement and rotation fields. A second element, based onan assumed stress HellingerReissner like formulation, will also bepresented.C.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-1311233. Irreducible element with rotational degrees of freedomIn this section, an irreducible element with rotational degreesof freedom is introduced. Here, the term irreducible refers to aformulation with the minimum number of independent assumedfields, i.e. ur, uzand?rzwhich adequately embodies Eqs. (4)(8).1The resulting elements have three degrees of freedom per node,namely two displacements, urand uz, and a rotational degree offreedom,?rz, representing a fibre hoop rotation tangent to the fibre.Details of the variational formulation of elements with rotationaldegrees of freedom are not repeated in this paper, since a morethorough discussion can be found in the references presented inSection 1. Instead, only the functionals from which the elements arederived, are presented here.3.1. Variational formulationThe irreducible functional with only kinematic independent vari-ables can be written as?(u,w) =12?csymmu symmudV +?2?|skewu w|2dV?f udV + boundary terms.(17)The variational equations arising from?can be shown to be0 =?=?csymmu symm?udV+?(skewu w) (skew?u ?w)dV?f ?udV + boundary terms.(18)3.2. Finite element implementationThe finite element interpolations employed in the elements aris-ing from the variational formulations highlighted in the foregoingare presented in this section. It is required that only the independenttranslation and hoop rotation fields are interpolated.Consider a 4-node quadrilateral element with degrees of free-dom as depicted in Fig. 1. The reference surface of the element isdefined byx =4?I=1NI(?,?)xI,(19)where x represents coordinates (r,z) and NI(?,?) are the isopara-metric shape functions 30. The independent rotation field isinterpolated as a standard bilinear field over each element?rz=4?I=1NI(?,?)?rzI.(20)The rz (in-plane) elemental displacement approximation is takenas an Allman-type interpolation, similar to the development ofmembrane elements 5, i.e.?uruz?= u =4?I=1NI(?,?)uI+8?I=5NSI(?,?)lJK8(?rzK?rzJ)nJK,(21)1Technically speaking, only urand uzare required to describe this problem ifskewris a priori assumed zero, since?rzcan be derived from the displacements.We nevertheless refer to these elements as irreducible to distinguish them fromthe mixed elements to follow.where lJKand nJKare the length and the outward unit normal vectoron the element side associated with the corner nodes J and K, andNSIare the serendipity shape functions. Employing matrix notationwe definesymmu = B1IuI+ B2I?rzI,(22)where uIand?rzIare nodal values of the displacement and therotation fields, respectively. The B1Imatrix in (22) has the standardformB1I=NI,r0NIr00NI,zNI,zNI,r,I = 1,2,3,4,(23)where, for example, NI,r=?NI/?r. The part of the displacement in-terpolation associated with the rotation field is defined asB2I=18(lIJcos?IJNSL,r lIKcos?IKNSM,r)0(lIJsin?IJNSL,z lIKsin?IKNSM,z)lIJcos?IJNSL,z lIKcos?IKNSM,z+lIJsin?IJNSL,r lIKsin?IKNSM,r.(24)Now considering terms associated with the skew-symmetric part ofthe displacement gradient, we begin by denotingskewu ?rz= b1IuI+ b2I?rzI,(25)whereb1I=?12NI,z12NI,r?,I = 1,2,3,4,(26)andb2I= 116(lIJcos?IJNSL,zlIKcos?IKNSM,z)+116(lIJsin?IJNSL,rlIKsin?IKNSM,r) NI,I = 1,2,3,4.(27)The indices J, K, L, M in (21), (24) and (27) are defined in, for example5,31. The foregoing definitions may now be used to construct theelement stiffness matrix for the irreducible element.3.3. A4R finite element based on functional?The stiffness matrix for the element derived from (18) may bedirectly written asKA4Ra = fwhere a =?uwrz?(28)and withKA4R= K + p.(29)The first term in the stiffness matrix can be written explicitly asK =?B1B2TCB1B2d?.(30)Employing the interpolations for displacement (21) and rotations(20) and combining with (25) leads top =?b1b2?b1b2d?,(31)representing the second term in (18). The matrix p is integratedby a single point Gaussian quadrature. By integrating K using a full9-point scheme and combining with p, spurious zero energy modesare prevented.22This is also true if a 5-point modified reduced integration scheme (e.g. see32,33) is employed.124C.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-131Fig. 1. Quadrilateral axisymmetric element with rotational degrees of freedom.This element depends on a penalty parameter?. Appropriate val-ues for?have been the topic of a number of studies 1,4,12,34,35.A value of?=?, with?the shear modulus, is used throughout, eventhough this value may not necessarily be optimal.Note that an element in which the skew-symmetric part of thestress tensor is retained has also been implemented, and results forthis element are available from the first author upon request. Inthe interests of brevity, however, we present only the irreducibleelement considered above.4. Mixed element with rotational degrees of freedomWe now present a mixed assumed stress element, accounting fornodal hoop rotations. Once again, the full variational formulation isnot given, since it may be found in for instance 1,12.4.1. Variational formulationThe HellingerReissner like functional, corresponding to the?functional, can be shown to be given by?(u,w,symmr) = 12?ssymmr symmrdV+?symmu symmrdV+?2?|skewu w|2dV ?f udV+ boundary terms,(32)where s = c1. Setting the first variation of?to zero, results in?= 0 = ?ssymmr symm?rdV +?symmu symm?rdV+?symmr symm?udV+?(skewu w) (skew?u ?w)dV?f ?udV + boundary terms.(33)4.2. Finite element implementationThe element derived from?requires interpolations for notonly displacements and rotations, but an additional (independently)assumed symmetric stress field. Since the interpolations and opera-tors presented in Section 3.2 remain applicable here, only the stressinterpolation is considered in more detail in this section.Practically all issues surrounding the selection of stress interpo-lations for mixed low-order planar elements have been resolved,and community consensus on the appropriate interpolation has es-sentially been reached. The situation with axisymmetric elements isvery different, with a wide variety of approaches and interpolationscontinually being proposed.Some authors propose interpolation in the global coordinatesystem 22,24, however the selection of stress fields in cylindricalcoordinates is far more complex than in the case of planar or solidelements in Cartesian coordinates. Finding an interpolation that hasthe minimum number of stress modes, that is coordinate frameinvariant and that does not result in element rank deficiencies, isalready a challenge.Additional complications are associated with equilibrium equa-tions in cylindrical coordinates. The first is the necessity for recip-rocal terms in the stress interpolation, which becomes problematicas r 0. Furthermore, as pointed out by Sze and Chow 20, radialand hoop stresses appear directly in the equilibrium equations. Asa result, elements which a priori or a posterior satisfy equilibrium,demonstrate a false shear phenomenon due to coupling betweenconstant and higher order stress components. For these reasons weopt to interpolate in a local coordinate system, as was done in forexample 15,16,18,23,25.In matrix form, the stress interpolation can be expressed in termsof element parametersb, assymmr= symmrc+ symmrh= Icbc+ TPhbh,(34)wheresymmrT= ?rr?zz?rz.(35)Icis a 4 4 identity matrix and there are four corresponding?-parameters inbc, allowing for the accommodation of constant stressstates required to pass the patch test. The transformation matrix isgiven byT =(J11)20(J21)22J11J210100(J12)20(J22)22J12J22J11J120J21J22J11J22+ J12J21,(36)C.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-131125with terms defined using the Jacobian, relating the?and the rzcoordinate systems, i.e.J(?,?) =?J11J12J21J22?.(37)Kasper and Taylor 16 proposed the use of an average Jacobianin computing the transformation matrix T, but we have found thetransformation based on the expression in (36) to be adequate. Al-ternatively, some numerical cost may be spared by using a constantT, evaluated at the element centroid.There are now various possibilities for constructing the higher-order (non-constant) stress interpolations Ph. We restrict ourselvesto a single option however. Essentially, we adopt the procedure sug-gested by Jog and Annabattula 25, who proposed the selection of in-terpolation functions such that zero-energy modes (associated withreduced integrations schemes) are captured.In their paper Sze and Ghali 11 proposed such an interpolationscheme for planar elements employing Allman shape functions. Sim-ilar to Jog and Annabattula 25, we append the interpolation of Szeand Ghali with a term to capture the mode associated with the pla-nar rigid body rotation (a mode which contributes to strain in theaxisymmetric case). Specifically, we choosePhbh=?000?2000000zg0?00?2000?00?5?6?7?8?9?10T,(38)wherezg= J12?+ J22?.(39)Once again, J12and J22could be replaced by their values at the originof the local coordinate system. Considering zgfurther, although Jogand Annabattula 25 suggest setting zg= J12?+ J22?, they reportthis to be similar to the interpolation used in Weissman and Taylorsdegenerate stress field (DSF) element 23, in which this term isinterpolated in global coordinates, i.e. zg= b1?+ b3?+ b2?, whereb1,b2and b3represent Jacobian entries computed at?=?= 0. Forthe proposed element, we have found little difference between thetwo methods. The results presented in Section 6 therefore use zgasdefined in (39) throughout.There are naturally many other interpolations which could beevaluated, but in the interests of brevity, we will evaluate only theone presented here.It is now possible to construct the stiffness matrix for the mixedelement with hoop rotations using the assumed stress field proposedin the foregoing.4.3. A4R?element based on functional?Developing a discrete form of (33), we get:?pGTGH?ab?=?f0?.(40)The matrix forms of G and H take on their standard forms, namelyG =?PTB1B2d?(41)andH =?PTSPd?.(42)In our current implementation, a full 9-point Gauss quadrature isused to evaluate both G and H, even though a modified reduced5-point integration scheme is sufficient to suppress spurious zeroenergy modes. In order to write the problem in the standardforcedisplacement form, i.e.KA4R?a = f,(43)simplifications using static condensation on the element level arerequired. Specifically, the unknown?-parameters are condensed out,resulting inb= H1Ga(44)and we can defineK?= GTH1G.(45)Finally, the stiffness matrix becomesKA4R?= K?+ p,(46)where K?is given by (45) and p, integrated using a one-point rule,is given by (31).5. Element strain correctionOften, membrane elements with in-plane rotations defined usingAllman interpolations, employ a so-called membrane-bending lock-ing correction 12,31,36. This is especially true when the membraneelement is used as the in-plane component of a flat shell element.This correction is reported to alleviate undesirable interactions be-tween membrane and (plate) bending actions 36.The correction, introduced by Jetteur and Frey 37 and furtherdeveloped by Taylor 36 modifies the strain definition in (22) asfollows:?= B1IuI+ B2I?rzI+ ?0,(47)where ?0is found by augmenting the potential energy functionalwith the statement?rT(B2I?rzI+ ?0)d?,(48)with both rand ?0being constant over the element, and therebyenforcing the statementB2I?rzI+ ?0= 0,(49)in a weak sense. Thus, ?0may be shown to be?0= 1?B2Id?rzI,(50)which can in turn be substituted back into (47).This correction is reminiscent of a correction to ensure satisfac-tion of the patch test, popular in incompatible elements, see for ex-ample 38. The net result of the modified strain definition in (47) isthat, at least in a weak sense, the contribution of higher-order com-ponents of the Allman interpolations are eliminated. As a result, topass the patch test, although higher-order displacement interpola-tions are employed the consistent nodal loads are computed usingonly bilinear interpolations, thereby simplifying pre-processing.If however this correction is not employed, the patch test canstill be passed. However, the full Allman interpolations are requiredto compute the consistent nodal loads, generally resulting in asso-ciated nodal moments. Complications may however be experiencedat boundaries where essential boundary conditions are specified.The numerical results to follow report on elements both with andwithout the correction presented in (47).126C.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-1316. Numerical resultsThe proposed elements are now evaluated on a number of stan-dard test problems. Their performance is judged based on a compar-ative study with several existing elements, whose results are sourcedfrom data collated by Wanji and Cheung 14. Elements consideredin the comparison include the following:A4: the standard four-node isoparametric axisymmetric ele-ment (see for example 27,30),LA1: the quadrilateral non-conforming element proposed bySze and Chow 20,AQ4: the quadrilateral non-conforming element proposed byWu and Cheung 39,SQ4: the generalised hybrid element proposed by Wanji andCheung 15,HA1/FA1: the hybrid stress elements proposed by Sze and Chow20,FSF/DSF: the mixed element proposed by Weissman and Taylor23,NAQ6: the non-conforming element proposed by Wanji andCheung 14, andRHAQ6: the refined non-conforming hybrid element presented byWanji and Cheung 14.Furthermore, the elements proposed in this paper are denoted asfollows:A4R: the irreducible 4-node axisymmetric element with rotationaldegrees of freedom based on functional?, and with stiffnessmatrix given by (29), andA4R?: the 4-node axisymmetric element with rotational degrees offreedom based on functional?, with an assumed stress field,with stiffness matrix as in (46).6.1. Eigenvalue analysisFirstly, an eigenvalue analysis of a regular, undistorted element iscarried out to confirm that the new elements possess the appropri-ate number of non-zero eigenvalues. Indeed, the proposed elementsfulfil the requirement of displaying only a single zero eigenvalue,associated with a rigid body translation along the axis of radial sym-metry.6.2. Patch testNext, each element is subjected to a standard patch test. Theproblem under consideration is similar to the patch test proposedby Wanji and Cheung 14, in which the boundary displacement isgiven by (in the radial direction) ur=2r, and (in the direction of theaxis of radial symmetry) uz= 1 + 4?, with?representing Poissonsratio and r the radial coordinate.Fig. 2 depicts the arbitrary mesh used for the patch test, slightlyoffset from the axis of symmetry. Our new elements pass this pre-scribed displacement, as well as an equivalent prescribed force, patchtest. The patch test is passed for any value of?0, with or withoutthe correction highlighted in Section 5. However, as pointed out inSection 5, the appropriate consistent nodal loads should be used.6.3. Node numbering invarianceThis single element problem, proposed by Sze and Chow 20 anddepicted in Fig. 3, is used to assess the invariance (or lack thereof)of an element to node numbering. All four possible node numberingFig. 2. Patch test geometry and material properties.Fig. 3. Mesh used to test the sensitivity to node numbering.sequences are used to compute the displacement of point A, and theresults compared. The newly proposed elements are found to be, tomachine precision, invariant with respect to node numbering.6.4. Thick walled cylinder under internal pressureAn infinitely long thick walled cylinder under internal pressure,analysed by considering a slice of unit thickness, is depicted inFig. 4. Fig. 4(a) depicts a cylinder with an inner radius of 5 and awall thickness of 5, analysed using a regular mesh, while Fig. 4(b)and (c) represent the MacNealHarder test 40, employing a regu-lar and a distorted mesh, respectively. The corresponding results arepresented in Tables 13.As expected, our irreducible element does not perform as wellas the assumed stress element, especially in the near incompress-ibility limit. In fact, apart from the added modelling capability, theirreducible A4R element does not offer significant benefits over thestandard isoparametric A4 element. This is not surprising since therotational degrees of freedom are not activated in these tests, andare in fact prescribed to be zero.On the other hand, the assumed stress A4R?element performsvery well when judged on either displacement or on stress predic-tion, and is almost completely insensitive to incompressibility ef-fects. For both problems analysed with a regular mesh, displacementis predicted exactly, a phenomenon known as superconvergence25 which has only previously been observed with elements devel-oped by Jog and Annabattula 25 and the DSF element of Weissmanand Taylor 23.6.5. Uniformly loaded simply supported circular plateThe next test problem is depicted in Fig. 5 and represents a simplysupported, uniformly loaded, circular plate of thickness t. The plateis discretized using two different meshes as depicted in Figs. 5(a)and (b), respectively. Results for the mesh depicted in Fig. 5(a) withvarying distortion e, and plate thickness t are presented in Tables46, while results for the highly distorted mesh shown in Fig. 5(b)are presented in Table 7.C.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-131127Fig. 4. Thick walled cylinder(s) under internal pressure.Table 1Normalised results for a thick-walled cylinder with regular mesh and variable Poissons ratio (see Fig. 4(a).Element?= 0.49?= 0.499?= 0.4999ura?rA?A?zAura?rA?A?zAura?rA?A?zAA40.9060.8170.9631.1290.4940.0260.7781.7040.0890.8560.5972.275LA10.9921.2820.8510.3560.9921.0900.5845.8210.99230.73114.93767.698AQ60.9920.9961.0111.0290.9920.9961.0111.0280.9920.9961.0111.028NAQ60.9920.9961.0111.0290.9920.9961.0111.0280.9920.9961.0111.028SQ40.9920.9961.0111.0290.9920.9961.0111.0280.9920.9961.0111.028HA1/FA10.9920.9961.0111.0290.9920.9961.0111.0280.9920.9961.0111.028RHAQ60.9920.9961.0111.0290.9920.9961.0111.0280.9920.9961.0111.028A4Rb0.9060.8170.9631.1300.4940.0280.7781.7060.0890.8570.5972.277A4R?b1.0001.0111.0061.0001.0001.0111.0060.9991.0001.0111.0061.001Exact31.7802.4504.5701.04031.8302.4504.5701.06031.8302.4504.5701.060Exact results in 102.aRadial displacement measured at r = 5.bSame results with and without element strain correction.Table 2MacNealHarder test, regular mesh (see Fig. 4(b).Element?= 0.0?= 0.3?= 0.49?= 0.499?= 0.4999?= 0.499999A40.9940.9880.8470.3590.0535.61 104FSF0.9940.9900.9860.9860.986aDSF1.0001.0001.0001.0001.000aAQ60.9940.9900.9860.9850.9860.985NAQ60.9940.9900.9330.9860.9860.986RHAQ60.9940.9900.9860.9860.9860.986A4Rb0.9940.9880.8460.3580.0535.59 104A4R?b1.0001.0001.0001.0001.0001.000Normalised displacement at point A.aNo results reported.bSame results with and without element strain correction.From the results of this bending dominated problem presentedin Tables 4 and 5, and for values of Poissons ratio sufficientlylower than 0.5, it is apparent that the irreducible A4R elementTable 3MacNealHarder test, distorted mesh (see Fig. 4(c).Element?= 0.0?= 0.3?= 0.49?= 0.499?= 0.4999?= 0.499999A40.9890.9820.8160.3150.0444.43 104FSF0.9850.9810.9760.9760.976aDSF0.9970.9970.9970.9970.997aAQ60.9910.9850.9380.7180.4720.410NAQ60.9890.9850.9390.7130.4450.372RHAQ60.9890.9870.9830.9830.9830.983A4Rb0.9880.9820.8150.3130.0444.55 104A4R?b0.9940.9930.9840.9820.9820.982Normalised displacement at point A.aNo results reported.bSame results with and without element strain correction.performs significantly better than the standard isoparametric A4element on displacement accuracy. Stress accuracy is, however,not compared since displacement based elements are incapable of128C.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-131Fig. 5. Simply supported uniformly loaded circular plate.Table 4Normalised results for uniformly loaded circular plate with varying Poissons ratio, modelled using a 1 4 regular mesh.ElementuzA?rA?= 0.25?= 0.49?= 0.4999?= 0.499999?= 0.25?= 0.49?= 0.4999?= 0.499999A40.6960.0790.0160.015aaaaLA11.0341.0110.7850.761aaaaAQ61.0341.0110.7850.7610.9990.9370.5970.567NAQ61.0341.0110.7850.7610.9990.9360.5960.566SQ41.0341.0110.7850.7610.9990.9370.5970.567HA1/FA11.0371.0431.0431.0431.0061.0071.0071.007RHAQ61.0371.0431.0431.0431.0061.0081.0081.008A4R0.8680.0800.0160.015aaaaA4Rb0.8490.0770.0150.015aaaaA4R?1.0141.0111.0101.0100.9520.9390.9360.936A4R?b1.0041.0091.0091.0091.0001.0001.0001.000Exact738.280524.980515.720515.630121.880130.880131.250131.250See Fig. 5(a), t = 1, e = 0.aNot possible to compute?rAdue to singularity at r = 0.bElement without element strain correction.Table 5Normalised results for uniformly loaded circular plate with varying Poissons ratios, modelled using a 1 4 slightly distorted mesh.ElementuzA?rA?= 0.25?= 0.49?= 0.4999?= 0.499999?= 0.25?= 0.49?= 0.4999?= 0.499999A40.6940.0790.0160.015aaaaLA11.0271.0060.7770.753aaaaAQ61.0301.0080.7810.7561.0050.9330.5850.555NAQ61.0301.0080.7810.7570.9800.8791.3441.329SQ41.0301.0080.7810.7561.0110.9480.5990.568HA11.0301.0401.0401.0430.9880.9300.9240.924FA11.0301.0401.0401.0400.9970.9380.9330.933RHAQ61.0301.0411.0411.0410.9940.9980.9980.998A4R0.8650.0800.0160.015aaaaA4Rb0.8490.0770.0150.015aaaaA4R?1.0091.0081.0071.0070.9710.9560.9530.953A4R?b1.0041.0091.0091.0091.0031.0031.0031.003Exact738.280524.980515.720515.630121.880130.880131.250131.250See Fig. 5(a), t = 1, e = 0.025.aNot possible to compute?rAdue to singularity at r = 0.bElement without element strain puting stress directly at the axis of symmetry due to the re-ciprocal term in hoop strain, from which the radial stress is partlyderived.Again, the assumed stress element (with or without the elementstrain correction) performs very well indeed, and exhibits littleaccuracysensitivitytoPoissonsratio,eveninthelimitofC.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-131129Table 6Normalised uzAt3values for various element aspect ratios (=2.5/t) with?= 0.25, see Fig. 5(a).Computed quantityuzAt3for 1 4 regular meshuzAt3for 1 4 distorted meshElement aspect ratio2.51005002.5100500A40.6962.22 1038.89 1050.6948.87 1043.84 105LA11.0341.0251.0251.0270.5490.546AQ61.0341.0251.0251.0300.4930.491NAQ61.0341.0251.0251.0300.4930.491SQ41.0341.0251.0251.0300.4930.491HA11.0371.0281.0281.0300.5590.556FA11.0371.0281.0281.0300.5590.556RHAQ61.0371.0281.0281.0300.7380.738A4R0.8680.4210.1900.8655.37 1037.05 105A4Ra0.8490.8390.8380.8490.5151.42103A4R?1.0140.4720.2121.0090.0156.13103A4R?a1.0040.9940.9951.0040.9410.897Exact738.28Distorted mesh with e = 0.025.aElement without element strain correction.Table 7Normalised results for a uniformly loaded circular plate with varying Poissons ratio, modelled using a 1 4 highly distorted mesh.ElementuzA?rA?= 0.25?= 0.49?= 0.4999?= 0.499999?= 0.25?= 0.49?= 0.4999?= 0.499999A40.6350.0880.0170.019aaaaLA10.9090.9030.5900.563aaaaAQ60.9070.9060.5930.5661.3891.1320.3700.328NAQ60.9120.9060.5920.5651.1540.8003.2543.590SQ40.9060.9040.5930.5661.3411.1640.4690.430HA10.9180.9760.9790.9791.2261.0431.0301.030FA10.9150.9710.9740.9741.3521.1671.1541.153RHAQ60.9280.9770.9790.9790.9741.0291.0321.032A4R0.7730.1130.0180.016aaaaA4Rb0.8800.0910.0150.015aaaaA4R?0.8640.9110.9110.9111.4871.3981.3881.388A4R?b0.9981.0051.0051.0051.0531.0391.0391.039Exact738.280524.980515.720515.630121.880130.880131.250131.250See Fig. 5(b), t = 1.aNot possible to compute?rAdue to singularity at r = 0.bElement without element strain correction.incompressibility. The A4R?element also achieves favourable dis-placement and stress predictions when compared to results frompreviously published elements.Still considering the problem depicted in Fig. 5(a), Table 6 com-pares the element displacement accuracy for various values of ele-ment aspect ratio and distortion, while Poissons ratio is fixed (?=0.25). These results indicate that the strain correction presented inSection 5 significantly stiffens the A4R and the A4R?elements whenelement aspect ratio is high. However, without this (optional3) cor-rection, the elements are shown to be extremely robust. In particular,the A4R element is apparently only susceptible to element aspect ra-tio when the elements are distorted and the aspect ratio is extreme.The A4R?element is shown to be extremely robust, demonstratingsuperior performance when compared to results for previously pub-lished elements.It should however be emphasised that the differences between el-ements with and without the element strain correction only becomenotable when element aspect ratios are extreme. The correctiontherefore nevertheless has merit since it simplifies pre-processing ofthe element consistent nodal loads.3Recall that both elements pass the patch test and have the proper rank withor without the correction.Fig. 6. Thin sphere subjected to unit internal pressure.130C.S. Long et al. / Finite Elements in Analysis and Design 45 (2009) 121-131Table 8Normalised displacements for a thin sphere under internal pressure, see Fig. 6.ElementuzAurB?= 0?= 0.3?= 0.49?= 0.499?= 0.4999?= 0?= 0.3?= 0.49?= 0.499?= 0.4999A41.0241.0270.8960.4240.0680.9970.9920.8760.4190.067FSF1.0611.0761.0851.0851.0850.9940.9900.9840.9840.984DSF1.0591.0741.0851.0861.0860.9950.9920.9880.9880.988A4R1.0561.0630.9100.4270.0680.9960.9900.8730.4180.067A4Ra0.9990.9950.8830.4210.0680.9980.9930.8780.4200.069A4R?1.0521.0651.0721.0721.0720.9970.9940.9900.9900.990A4R?a0.9940.9910.9870.9860.9860.9990.9970.9940.9940.994Exact4.0813.1272.5232.4942.4914.0813.1272.5232.4942.491Exact solution in 102.aElement without element strain correction.Finally, the results for the highly distorted mesh depicted inFig. 5(b) are presented in Table 7. Once again, the assumed stressA4R?element performs well when compared to previously proposedelements on both displacement and stress accuracy.6.6. Sphere under internal pressureThe final test of element accuracy is depicted in Fig. 6, and rep-resents a thin sphere with inside radius, ri, and outside radius, ro,subjected to a unit internal pressure. Only the top hemisphere ismodelled using 10 evenly spaced elements, thereby exploiting theproblem symmetry. An analytical solution to this problem may befound in, for example, 41. The material properties are as shown inthe figure.The normalised displacements at points A and B are summarisedin Table 8 for various values of Poissons ratio?. For values of Pois-sons ratio sufficiently lower than 0.5, the irreducible A4R elementagain performs very well. Once again, compared to previously pub-lished elements, our assumed stress A4R?element achieves excel-lent results.7. ConclusionsIn this paper, two new axisymmetric solid-of-revolution elementswith hoop fibre rotational degrees of freedom were introduced. Theprimary objective for developing these elements was to enhance ex-isting modelling capability, the additional rotational degree of free-dom facilitating, for instance, the connection between axisymmetricshell and axisymmetric solid models. However, the new elementswere shown to also demonstrate improved accuracy and robustnesson a number of popular benchmark problems when compared toexisting elements.The first element, denoted A4R, possesses two displacement de-grees of freedom per node, standard in axisymmetric elements, aswell a single rotational degree of freedom accounting for hoop fi-bre rotations. A second element, based on a mixed assumed stressformulation, possessing the aforementioned rotational nodal degreeof freedom was also proposed. This element, denoted A4R?, furtherimproved element performance, especially in the near incompress-ibility limit.The elements were derived from a variational framework, shownto be stable in the discrete form. Rotations are based on the contin-uum mechanics definition of rotation, and the stress tensor is nota priori assumed to be symmetric. Stress interpolations, possessingthe minimum number of stress parameters, for the mixed elementswere proposed.Furthermore, an element strain correction often used in mem-brane elements to alleviate membrane-bending locking in flat shellelements, was adapted to the axisymmetric case. This correctioneffectively removes strain contributions from higher-order compo-nents of the Allman interpolation in a weak sense. The patch testwas accordingly passed using the consistent nodal loads of the stan-dard 4-node isoparametric axisymmetric element, thereby simplify-ing pre-processing. The correction may, however, hamper elementperformancewhentheelementaspectratiobecomesverylarge.Nev-ertheless, the correction is optional since the patch test is passedeven if it is omitted. The proposed elements were evaluated withand without the correction.The performance of the irreducible (i.e. without assumed stress)A4R element was shown to be superior to standard displacementelements, especially when used to analyse bending dominated prob-lems. However, since the element is essentially displacement-based,it tends to lock in the near incompressibility limit. Furthermore, it isincapable of directly predicting stress at the axis of symmetry dueto reciprocal coordinate terms in the strain evaluation.On the other hand, the mixed assumed stress A4R?element wasfound to be extremely accurate and robust when compared to anumber of previously proposed elements on a variety of test prob-lems. This is especially true when the aforementioned element cor-rection was omitted. For undistorted meshes, the element predictsthe exact displacement for a cylinder under internal pressure, a phe-nomenon known as superconvergence. The element was also foundto perform excellently on a number of other benchmark problemswhen compared to previously proposed elements.References1 T.J.R. Hughes, F. Brezzi, On drilling degrees of freedom, Comput. Methods Appl.Mech. Eng. 72 (1989) 105121.2 E. Reissner, A note on variational principles in elasticity, Int. J. Solids Struct. 1(1965) 9395.3 T.J.R. Hughes, F. Brezzi, A. Masud, I. Harari, Finite element with drilling degreeof freedom: theory and numerical evaluation, in: R. Gruber, J. Periaux, R.P.Shaw (Eds.), Proceedings of the Fifth International Symposium on NumericalMethods in Engineering, vol. 1, Springer, Berlin, 1989, pp. 317.4 A. Ibrahimbegovic, R.L. Taylor, E.L. Wilson, A robust quadrilateral membranefinite element with drilling degrees of freedom, Int. J. Numer. Methods Eng.30 (1990) 445457.5 A. Ibrahimbegovic, E.L. Wilson, A unified formulation for triangular andquadrilateral flat shell finite elements with six nodal degrees of freedom,Commun. Appl. Numer. Methods 7 (1991) 19.6 T.H.H. Pian, Derivation of element stiffness matrices by assumed stressdistributions, AIAA J. 2 (1964) 13331336.7 T.H.H. Pian, State-of-the-art development of the hybrid/mixed finite elementmethod, Finite Element Anal. Des. 21 (1995) 520.8 M.A. Aminpour, An assumed-stress hybrid 4-node shell element with drillingdegrees of freedom, Int. J. Numer. Methods Eng. 33 (1992) 1938.9 G. Rengarajan, M.A. Aminpour, N.F. Knight, Improved assumed-stress hybridshell element with drilling degrees of freedom for linear stress, buckling andfree vibration analyses, Int. J. Numer. Methods Eng. 38 (1995) 19171943.C.S. Long et al. / Finite Element
- 温馨提示:
1: 本站所有资源如无特殊说明，都需要本地电脑安装OFFICE2007和PDF阅读器。图纸软件为CAD,CAXA,PROE,UG,SolidWorks等.压缩文件请下载最新的WinRAR软件解压。
2: 本站的文档不包含任何第三方提供的附件图纸等，如果需要附件，请联系上传者。文件的所有权益归上传用户所有。
3.本站RAR压缩包中若带图纸，网页内容里面会有图纸预览，若没有图纸预览就没有图纸。
4. 未经权益所有人同意不得将文件中的内容挪作商业或盈利用途。
5. 人人文库网仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对用户上传分享的文档内容本身不做任何修改或编辑，并不能对任何下载内容负责。
6. 下载文件中如有侵权或不适当内容，请与我们联系，我们立即纠正。
7. 本站不保证下载资源的准确性、安全性和完整性, 同时也不承担用户因使用这些下载资源对自己和他人造成任何形式的伤害或损失。
人人文库网所有资源均是用户自行上传分享，仅供网友学习交流，未经上传用户书面授权，请勿作他用。
|
2:不支持迅雷下载,请使用浏览器下载
3:不支持QQ浏览器下载,请用其他浏览器
4:下载后的文档和图纸-无水印
5:文档经过压缩，下载后原文更清晰
|
川公网安备: 51019002004831号