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基于adams的小车式起落架着陆及全机滑跑动态仿真

Key Words: track-like landing gear dynamic simulation landing full aircraft taxing ADAMS
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基于 ADAMS 的小车式起落架着陆及全机滑跑动态仿真
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图 2.1 多体系统动力学建模与求解一般过程 .....................................................8 图 2.2 ADAMS 软件求解方法及过程....................................................................... 11 图 2.3 飞机数字功能样机组成 ...........................................................................14 图 3.1 小车式起落架的结构 ...............................................................................16 图 3.2 小车式起落架结构模型 ...........................................................................18 图 3.3 外筒受力图 ...............................................................................................18 图 3.4 内筒受力图 ...............................................................................................19 图 3.5 车架受力图 ...............................................................................................20 图 3.6 后轮受力图 ...............................................................................................20 图 3.7 前轮受力模型 ...........................................................................................21 图 3.8 缓冲器结构模型 .......................................................................................22 图 3.9 缓冲器受力图 ...........................................................................................24 图 3.10 圆角方形截面结构 .................................................................................25 图 4.1 起落架 CATIA 三维建模..........................................................................29 图 4.2 设计过程 Step 函数结果曲线 ..................................................................31 图 4.3 IMPACT 函数示意图.................................................................................32 图 4.4 ADAMS 仿真模型 .....................................................................................32 图 4.5 缓冲器随行程变化曲线 ...........................................................................35 图 4.6 空气弹簧力随行程变化曲线 ...................................................................35 图 4.7 油液阻尼力随行程变化曲线 ...................................................................35 图 4.8 缓冲器的功量图 .......................................................................................36 图 4.9 轮胎作用力 ...............................................................................................36 图 4.10 后轮冲击载荷 .........................................................................................37 图 4.11 前轮冲击载荷..........................................................................................37 图 4.12 缓冲器行程变化曲线 .............................................................................38 图 4.13 不同重量下的缓冲器行程 .....................................................................39 图 4.14 不同重量下起落架对机身的冲击载荷 .................................................39 图 4.15 空气弹簧力随行程变化的比较曲线 .....................................................40 图 4.16 缓冲器的功量图 .....................................................................................41 图 5.1 理想的变油孔面积曲线形式 ...................................................................46

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Mesh-free analysis of cracks in isotropic functionallygraded materialsB.N.Rao,S.Rahman *College of Engineering,The University of Iowa,2140Seamans Center,Iowa City,IA 52242,USAReceived 21August 2001;received in revised form 22February 2002;accepted 17March 2002AbstractThis paper presents a Galerkin-based meshless method for calculating stress-intensity factors (SIFs)for a stationary crack in two-dimensional functionally graded materials of arbitrary geometry.The method involves an element-free Galerkin method (EFGM),where the material properties are smooth functions of spatial coordinates and two newly developed interaction integrals for mixed-mode fracture analysis.These integrals can also be implemented in con-junction with other numerical methods,such as the ﬁnite element method (FEM).Five numerical examples including both mode-I and mixed-mode problems are presented to evaluate the accuracy of SIFs calculated by the proposed parisons have been made between the SIFs predicted by EFGM and available reference solutions in the literature,generated either analytically or by FEM using various other fracture integrals or analyses.Agood agreement is obtained between the results of the proposed meshless method and the reference solutions.Ó2002Elsevier Science Ltd.All rights reserved.Keywords:Crack;Functionally graded materials;Element-free Galerkin method;Stress-intensity factor;J -integral;Interaction integral1.IntroductionIn recent years,functionally graded materials (FGMs)have been introduced and applied in the devel-opment of structural components subject to non-uniform service requirements.FGMs,which possess continuously varying microstructure and mechanical and/or thermal properties,are essentially two-phase particulate composites,such as ceramic and metal alloy phases,synthesized such that the composition of each constituent changes continuously in one direction,to yield a predetermined composition proﬁle [1].Even though the initial developmental emphasis of FGMs was to synthesize thermal barrier coating for space applications [2],later investigations uncovered a wide variety of potential applications,including nuclear fast breeder reactors [3],piezoelectric and thermoelectric devices [4–6],graded refractive index *Corresponding author.Tel.:+1-319-335-5679;fax:+1-319-335-5669.E-mail address:rahman@ (S.Rahman).URL:/$rahman.0013-7944/03/$-see front matter Ó2002Elsevier Science Ltd.All rights reserved.PII:S 0013-7944(02)00038-3materials in audio–video disks[7],thermionic converters[8],dental and medical implants[9],and others [10].The absence of sharp interfaces in FGM largely reduces material property mismatch,which has been found to improve resistance to interfacial delamination and fatigue crack propagation[11].However,the microstructure of FGM is generally heterogeneous,and the dominant type of failure in FGM is crack initiation and growth from inclusions.The extent to which constituent material properties and micro-structure can be tailored to guard against potential fracture and failure patterns is relatively unknown.Such issues have motivated much of the current research into the numerical computation of crack-driving forces and the simulation of crack growth in FGMs.Analytical work on FGMs begins as early as1960when soil was modeled as a non-homogeneous ma-terial by Gibson[12].Due to the complexity,plane elasticity problems involving cracks in FGM are solved assuming a functional form of the material property variation,usually a linear or exponential function. Assuming an exponential spatial variation of the elastic modulus,Atkinson and List[13],Dhaliwal and Singh[14],and Delale and Erdogan[15]solved crack problems for non-homogeneous materials subjected to mechanical loads.Delale and Erdogan[15]showed that the asymptotic crack-tip stressﬁeld in FGMs possesses the same square root singularity as in homogeneous materials.Eischen[16]studied mixed-mode conditions in non-homogeneous materials using theﬁnite element method(FEM).He also veriﬁed that the leading term of the asymptotic expansion for stresses was square-root singular.This result was reconﬁrmed by Jin and Noda[17]for materials with piecewise diﬀerentiable property variations.By further assuming the exponential variation of thermal properties of the material,Jin and Noda[18]and Erdogan and Wu[19] computed thermal stress-intensity factor(SIF)for non-homogeneous solids.Yang and Shih[20]considered a semi-inﬁnite crack in an interlayer between two dissimilar materials,and they obtained an approximate solution from a known bimaterial solution.Gu and Asaro[21]considered a semi-inﬁnite crack in a strip of FGM under edge loading and obtained SIF relations for many commonly used fracture specimen con-ﬁgurations.Erdogan[11]reviewed the elementary concepts of fracture mechanics of FGM and identiﬁed a number of typical problems relating to FGM fracture.Crack deﬂection in FGM has been considered by Gu and Asaro[22]who reported the strong inﬂuence of the material gradient on the crack kink angle when the crack is in the middle of the gradient zone.Tohgo et al.[23]carried out a numerical analysis of particulate FGM,and studied the inﬂuence of the material gradient on the size of a singularﬁeld by comparing the FGM results with those obtained for a homogeneous medium.Gu et al.[24]presented a simpliﬁed method for calculating the crack-tipﬁeld of FGMs using the equivalent domain integral technique.Anlas et al.[25] evaluated SIFs in FGMs by the FEM where the material property variation was discretized by assigning diﬀerent homogeneous elastic properties to each element.Both Gu et al.[24]and Anlas et al.[25]considered a mode-I crack where the crack is parallel to the material gradation,and used commercial FEM software in their analyses.Marur and Tippur[26]considered a crack normal to the elastic gradient and performed FEM analysis in conjunction with their experiments.Bao and Wang[27]studied multi-cracking in an FGM coating.Bao and Cai[28]studied delamination cracking in a functionally graded ceramic/metal substrate. Lee and Erdogan[29]evaluated residual thermal stresses in FGMs.Recently,Kim and Paulino[30] evaluated the mixed-mode fracture parameters in FGMs using FEM analysis with three diﬀerent ap-proaches:the path-independent JÃk -integral method,the modiﬁed crack-closure integral method,and thedisplacement correlation technique.Zou et al.[31]proposed a multiple isoparametric FEM to evaluate the SIFs of cracks in FGMs.Thus,most of the analytical studies on FGM reviewed above have used FEM as the numerical tool.FEM may present some limitations in solving solid mechanics problems characterized by a continuous change in geometry of the domain under analysis.Crack propagation is a prime example in which the use of FEM requires a large number of remeshings of theﬁnite element model to represent arbitrary and complex paths.The underlying structures of FEM and similar methods,which rely on a mesh,is quite cumbersome in treating cracks that are not coincident with the original mesh geometry. Consequently,the only viable option for dealing with moving cracks using FEM is to remesh during each discrete step of model evolution so that the mesh lines remain coincident with the cracks throughout the 2 B.N.Rao,S.Rahman/Engineering Fracture Mechanics70(2003)1–27B.N.Rao,S.Rahman/Engineering Fracture Mechanics70(2003)1–273 analysis.This creates numerical diﬃculties,often leading to degradation of solution accuracy,complexity in computer programming,and a computationally intensive environment.In recent years,various Galerkin-based meshless or mesh-free methods have been developed or inves-tigated to solve fracture-mechanics problems without the use of a structured grid[32–38].These meshless methods employ moving least-squares(MLS)approximation of a function that permits the resultant shape functions to be constructed entirely in terms of arbitrarily placed nodes.Since no element connectivity data is required,the burdensome meshing or remeshing characteristic of FEM is avoided.Since the mesh generation of complex cracked structures can be a far more time-consuming and costly eﬀort than the solution of a discrete set of linear equations,the meshless method provides an attractive alternative to FEM.However,to date most developments in meshless methods have focused on the fracture of homo-geneous materials.Fracture analysis of cracks in FGMs using meshless methods has not been widespread and is only currently gaining attention.As a result,there is considerable interest in developing meshless methods for the evaluation of crack-driving force in FGMs.This paper presents a meshless method for calculating the fracture parameters of a stationary crack in FGM with arbitrary geometry.This method involves an element-free Galerkin method(EFGM),where the material properties are smooth functions of spatial coordinates and two newly developed interaction in-tegrals for mixed-mode fracture analysis.In conjunction with the proposed method,both mode-I and mixed-mode two-dimensional problems have been solved.Five numerical examples are presented to evaluate the accuracy of SIFs calculated by the proposed parisons have been made between the SIFs predicted by the proposed method and the existing results available in the current literature. 2.Crack-tipﬁelds in FGMConsider a two-dimensional structure with a rectilinear crack of length2a,subjected to external loads S1;S2;...;S M,as shown in Fig.1.It is assumed that the material properties,such as the modulus of elas-ticity E and the Poisson’s ratio m,vary accordingtoE¼Eðx1;x2Þ¼EðxÞ;ð1Þm¼mðx1;x2Þ¼mðxÞ;ð2Þwhere x¼f x1;x2g T2R2,EðxÞP0andÀ16mðxÞ61=2are continuous,bounded,and at least piecewise diﬀerentiable functions on domain X,and the x1–x2coordinate system is deﬁned in Fig.1.In reality,FGMs are multi-phase materials with generally,locally discontinuous material properties.Hence,EðxÞand mðxÞin Eqs.(1)and(2)should be viewed as smoothly varying‘‘eﬀective’’material properties of FGMs.In this case, FGMs can be modeled as non-homogeneous materials,for which the elastic constitutive equation ise ij¼1þmÃðxÞEÃðxÞr ijþmÃðxÞEÃðxÞr kk d ij;ð3Þwhere e ij and r ij are the strain and stress components,respectively,and d ij is the Kronecker delta.In Eq.(3), EÃðxÞand mÃðxÞare given by EðxÞand mðxÞunder plane stress condition and by EðxÞ=½1ÀmðxÞ2 and mðxÞ=½1ÀmðxÞ under plane strain condition,respectively.For non-homogeneous materials undergoing plane stress or plane strain linear-elastic deformation,in the absence of body forces the Airy stress function Fðx1;x2Þsatisﬁes[16]r2r2FEÃðxÞÀo2o x221þmÃðxÞEÃðxÞo2Fo x21Ào2o x211þmÃðxÞEÃðxÞo2Fo x22þ2o2o x1o x21þmÃðxÞEÃðxÞo2Fo x1o x2¼0;ð4Þwhere r2¼o2=o x21þo2=o x22is the two-dimensional Laplacian operator.Eischen[16]and later Jin andNoda[17]showed that for piecewise diﬀerentiable material property variations,the elastic stress and dis-placementﬁelds in FGM can be derived using the stress function in variable separable form,identical to the homogeneous case.Hence,the linear-elastic singular stressﬁeld near the crack tip can be obtained as[16]r11¼1ﬃﬃﬃﬃﬃﬃﬃ2p rp½K I f I11ðhÞþK II f II11ðhÞ ;ð5Þr22¼1ﬃﬃﬃﬃﬃﬃﬃ2p rp½K I f I22ðhÞþK II f II22ðhÞ ;ð6Þr12¼1ﬃﬃﬃﬃﬃﬃﬃ2p rp½K I f I12ðhÞþK II f II12ðhÞ ;ð7Þwhere K I and K II are the mode-I and mode-II SIFs,respectively,and f Iij ðhÞand f IIijðhÞ(i,j¼1,2)are thestandard angular functions for a crack in a homogeneous elastic medium.Similarly,the near tip dis-placementﬁeld u¼f u1;u2g T can be obtained as[16]u1¼1ltipﬃﬃﬃﬃﬃﬃr2pr½K I g I1ðhÞþK II g II1ðhÞ ð8Þandu2¼1ltipﬃﬃﬃﬃﬃﬃr2pr½K I g I2ðhÞþK II g II2ðhÞ ;ð9Þwhere l tip¼E tip=½2ð1þm tipÞ is the shear modulus,E tip is the elastic modulus,and m tip is the Poisson’s ratio,all evaluated at the crack tip,and g Ii ðhÞand g IIiðhÞ,i¼1,2are standard angular functions for a crack in ahomogeneous elastic medium[39].Even though the material gradient does not inﬂuence the square-root singularity or the singular stress distribution,the material gradient does aﬀect the SIFs.Hence,the fracture parameters are functions of the material gradients,external loading,and geometry.4 B.N.Rao,S.Rahman/Engineering Fracture Mechanics70(2003)1–273.The interaction integral methodThe interaction integral method is an eﬀective tool for calculating mixed-mode fracture parameters in homogeneous materials[40,41].In this section the interaction integral method for homogeneous materials isﬁrst brieﬂy summarized,then extended for cracks in FGM.In fact,the study of FGM would enhance the understanding of a fracture in a generic material,since upon shrinking the gradient layer in FGM is ex-pected to behave like a sharp interface,and upon expansion,the fracture behavior would be analogous to that of a homogeneous material.3.1.Homogeneous materialsThe path independent J-integral for a homogeneous cracked body is given by[42]J¼ZCW d1jÀr ijo u io x1n j d C;ð10Þwhere W¼Rr ij d e ij is the strain energy density and n j is the j th component of the outward unit vectornormal to an arbitrary contour C enclosing the crack tip.For linear elastic material models it can shown that W¼r ij e ij=2¼e ij D ijkl e kl=2,where D ijkl is a component of constitutive tensor.Applying the divergence theorem,the contour integral in Eq.(10)can be converted into an equivalent domain form,given by[43]J¼ZAr ijo u io x1ÀW d1jo qo x jd AþZAoo x jr ijo u io x1ÀW d1jq d A;ð11Þwhere A is the area inside the contour and q is a weight function chosen such that it has a value of unity at the crack tip,zero along the boundary of the domain,and arbitrary elsewhere.By expanding the second integrand,Eq.(11)reduces toJ¼ZAr ijo u io x1ÀW d1jo qo x jd AþZAo r ijo x jo u io x1þr ijo2u io x j o x1Àr ijo e ijo x1À12e ijo D ijklo x1e klq d A:ð12ÞUsing equilibrium(o r ij=o x j¼0)and compatibility(e ij¼o u i=o x j)conditions and noting that o D ijkl=o x1¼0 in homogeneous materials,the second integrand of Eq.(12)vanishes,yieldingJ¼ZAr ijo u io x1ÀW d1jo qo x jd A;ð13Þwhich is the classical domain form of the J-integral in homogeneous materials.Consider two independent equilibrium states of the cracked body.Let state1correspond to the actual state for the given boundary conditions,and let state2correspond to an auxiliary state,which can be either mode-I or mode-II near tip displacement and stressﬁelds.Superposition of these two states leads to another equilibrium state(state S)for which the domain form of the J-integral isJðSÞ¼ZAðrð1Þij"þrð2ÞijÞoðuð1Þiþuð2ÞiÞo x1ÀWðSÞd1j#o qo x jd A;ð14Þwhere superscript i¼1,2,and S indicateﬁelds and quantities associated with state i and WðSÞ¼1ðrð1Þijþrð2ÞijÞðeð1Þijþeð2ÞijÞ:ð15ÞBy expanding Eq.(14),JðSÞ¼Jð1ÞþJð2ÞþMð1;2Þ;ð16ÞB.N.Rao,S.Rahman/Engineering Fracture Mechanics70(2003)1–275whereJð1Þ¼ZArð1Þijo uð1Þio x1"ÀWð1Þd1j#o qo x jd Að17ÞandJð2Þ¼ZArð2Þijo uð2Þio x1"ÀWð2Þd1j#o qo x jd Að18Þare the J-integrals for states1and2,respectively,andMð1;2Þ¼ZArð1Þijo uð2Þio x1"þrð2Þijo uð1Þio x1ÀWð1;2Þd1j#o qo x jd Að19Þis an interaction integral.In Eqs.(17)–(19),Wð1Þ¼1rð1Þij eð1Þij,Wð2Þ¼1rð2Þij eð2Þij,and Wð1;2Þ¼1ðrð1Þij eð2Þijþrð2Þij eð1ÞijÞrepresent various strain energy densities,which satisfyWðSÞ¼Wð1ÞþWð2ÞþWð1;2Þ:ð20ÞFor linear-elastic solids under mixed-mode loading conditions,the J-integral is also equal to the energy release rate and hence,the J-integral can be written asJ¼1EÃðK2IþK2IIÞ:ð21ÞApplying Eq.(21)to states1,2,and the superimposed state S givesJð1Þ¼1EÃðKð1Þ2IþKð1Þ2IIÞ;ð22ÞJð2Þ¼1EÃðKð2Þ2IþKð2Þ2IIÞð23ÞandJðSÞ¼1EÃðKð1ÞIhþKð2ÞIÞ2þðKð1ÞIIþKð2ÞIIÞ2i¼1EÃðKð1Þ2IhþKð1Þ2IIÞþðKð2Þ2IþKð2Þ2IIÞþ2ðKð1ÞIKð2ÞIþKð1ÞIIKð2ÞIIÞi¼Jð1ÞþJð2Þþ2EÃKð1ÞIKð2ÞIþKð1ÞIIKð2ÞII:ð24ÞComparing Eqs.(16)and(24),Mð1;2Þ¼2EÃðKð1ÞIKð2ÞIhþKð1ÞIIKð2ÞIIÞi:ð25ÞThe individual SIFs for the actual state can obtained by judiciously choosing the auxiliary state(state2). For example,if state2is chosen to be state I,i.e.,the mode-I near tip displacement and stressﬁeld is chosenas the auxiliary state,then Kð2ÞI ¼1and Kð2ÞII¼0.Hence,Eq.(25)can be reduced toMð1;IÞ¼2Kð1ÞIEÃ;ð26Þ6 B.N.Rao,S.Rahman/Engineering Fracture Mechanics70(2003)1–27from whichK ð1ÞI ¼M ð1;I ÞE Ã2:ð27ÞSimilarly,if state 2is chosen to be state II,i.e.,the mode-II near tip displacement and stress ﬁeld is chosen as the auxiliary state,then K ð2ÞI ¼0and K ð2ÞII ¼1.Following similar considerations,K ð1ÞII ¼M ð1;II ÞE Ã2:ð28ÞThe interaction integrals M ð1;I Þand M ð1;II Þcan be evaluated from Eq.(19).Eqs.(27)and (28)have been successfully used for calculating SIFs under various mixed-mode loading conditions [32–38].3.2.Functionally graded materialsFor non-homogeneous materials,even though the equilibrium and compatibility conditions are satisﬁed,the material gradient term of the second integrand of Eq.(12)does not vanish.So Eq.(12)reduces to a more general integral,henceforth referred to as the e J -integral [24],which is e J ¼Z A r ij o u i o x 1 ÀW d 1j o q o x j d A ÀZ A12e ij o D ijkl o x 1e kl q d A :ð29ÞBy comparing Eq.(29)to the classical J -integral (see Eq.(13)),the presence of material non-homogeneity results in the addition of the second domain integral.Although this integral is negligible for a path very close to the crack tip,it must be accounted for with relatively large integral domains,so that the e J -integral can be accurately calculated.The e J -integral in Eq.(29)is actually the ﬁrst component of the J Ã¼f J Ã1;J Ã2g T vector integral (i.e.,J Ã1)proposed by Eischen [16].Hence,e J also represents the energy release rate of an elastic body.It is ele-mentary to show that the e J -integral becomes zero for any closed contour in an uncracked homogeneous,as well as in non-homogeneous bodies,and therefore remains path independent when used in conjunction with cracks in FGM [16,44].In order to derive interaction integral for FGMs,consider again actual (state 1),auxiliary (state 2),and superimposed (state S )equilibrium states.For the actual state,Eq.(29)can be directly invoked to represent the e J -integral.However,a more general form,such as Eq.(11),must be used for auxiliary and superim-posed states.For example,the e J -integral for the superimposed state S can written as e J ðS Þ¼Z A ðr ð1Þij þr ð2Þij Þo ðu ð1Þi þu ð2Þi Þo x 1ÀW ðS Þd 1j !o q o x j d A þZ Ao o x j ðr ð1Þij þr ð2Þij Þo ðu ð1Þi þu ð2Þi Þo x 1ÀW ðS Þd 1j !q d A :ð30ÞClearly,the evaluations of e JðS Þand the resulting interaction integral depend on how the auxiliary ﬁeld is deﬁned.There are several options in choosing the auxiliary ﬁeld.Two methods,developed in this study,are described in the following.3.2.1.Method I:homogeneous a uxilia ry ﬁeldThe method I involves selecting the auxiliary stress and displacement ﬁelds given by Eqs.(5)–(9)and calculating the auxiliary strain ﬁeld from the symmetric gradient of the auxiliary displacement ﬁeld.In this approach,the auxiliary stress and strain ﬁelds are related through a constant constitutive tensor evaluated at the crack tip.Hence,both equilibrium (o r ð2Þij =o x j ¼0)and compatibility (e ð2Þij ¼o u ð2Þi =o x j )conditions are satisﬁed in the auxiliary state.However,the non-homogeneous constitutive relation of FGM is not strictly satisﬁed in the auxiliary state,which would introduce gradients of stress ﬁelds as extra terms in the in-teraction integral.B.N.Rao,S.Rahman /Engineering Fracture Mechanics 70(2003)1–277Using Eq.(20)and invoking both equilibrium and compatibility conditions,Eq.(30)can be further simpliﬁed to e J ðS Þ¼Z Aðr ð1Þij þr ð2Þij Þo ðu ð1Þi þu ð2Þi Þo x 1ÀðW ð1ÞþW ð2ÞþW ð1;2ÞÞd 1j !o q o x j d A þZ A 12"Àe ð1Þij o D ijkl o x 1e ð1Þkl þr ð1Þij o e ð2Þij o x 1Ào r ð2Þij o x 1e ð1Þij þr 2ij o e ð1Þij o x 1Ào r ð1Þij o x 1e ð2Þij #q d A :ð31ÞBy expanding Eq.(31),e J ðS Þ¼e J ð1Þþe J ð2Þþe Mð1;2Þ;ð32Þwheree J ð1Þ¼Z A r ð1Þij o u ð1Þi o x 1"ÀW ð1Þd 1j #o q o x j d A ÀZ A 12e ð1Þij o D ijkl o x 1e ð1Þkl q d A ;ð33Þe J ð2Þ¼Z A r ð2Þij o u ð2Þi o x 1"ÀW ð2Þd 1j #o q o x j d A ð34Þare the e J -integrals for states 1and 2,respectively,and e M ð1;2Þ¼Z A r ð1Þij o u ð2Þi 1"þr ð2Þij o u ð1Þi 1ÀW ð1;2Þd 1j #o q j d A þZ A 1r ð1Þij o e ð2Þij 1"Ào r ð2Þij 1e ð1Þij þr ð2Þij o e ð1Þij 1Ào r ð1Þij 1e ð2Þij #q d A ð35Þis the modiﬁed interaction integral for non-homogeneous materials.3.2.2.Method II:non-homogeneous a uxilia ry ﬁeldThe method II entails selecting the auxiliary stress and displacement ﬁelds given by Eqs.(5)–(9)and calculating the auxiliary strain ﬁeld using the same spatially varying constitutive tensor of FGM.In thisapproach,the auxiliary stress ﬁeld satisﬁes equilibrium (o r ð2Þij =o x j ¼0);however,the auxiliary strain ﬁeld isnot compatible with the auxiliary displacement ﬁeld (e ð2Þij ¼o u ð2Þi =o x j ).If the auxiliary ﬁelds are not com-patible,extra terms that will arise due to lack of compatibility should be taken into account while evalu-ating the interaction integral,even though they may not be suﬃciently singular in the asymptotic limit to contribute to the value of the integral [45–47].Hence,this method also introduces additional terms to the resulting interaction integral.Following similar considerations,but using only equilibrium condition in the auxiliary state,Eq.(30)can also be simpliﬁed toe J ðS Þ¼Z Aðr ð1Þij þr ð2Þij Þo ðu ð1Þi þu ð2Þi Þo x 1ÀðW ð1ÞþW ð2ÞþW ð1;2ÞÞd 1j !o q o x j d A þZ A ðr ð1Þij þr ð2Þij Þo 2u ð2Þi o x j o x 1 Ào e ð2Þij o x 1!À12ðe ð1Þij þe ð2Þij Þo D ijkl o x 1ðe ð1Þkl þe ð2Þkl Þ!q d A :ð36Þ8 B.N.Rao,S.Rahman /Engineering Fracture Mechanics 70(2003)1–27Comparing Eqs.(36)and(32),e Jð1Þ¼ZArð1Þijo uð1Þio x1"ÀWð1Þd1j#o qo x jd AÀZA12eð1Þijo D ijklo x1eð1Þklq d A;ð37Þe Jð2Þ¼ZArð2Þijo uð2Þio x1"ÀWð2Þd1j#o qo x jd AþZArð2Þijo2uð2Þio x j o x1"Ào eð2Þijo x1!À12eð2Þijo D ijklo x1eð2Þkl#q d Að38Þare the e J-integrals for states1and2,respectively,ande Mð1;2Þ¼ZArð1Þijo uð2Þio x1"þrð2Þijo uð1Þio x1ÀWð1;2Þd1j#o qo x jd AþZArð1Þijo2uð2Þio x j o x1"Ào eð2Þijo x1!Àeð1Þijo D ijklo x1eð2Þkl#q d Að39Þis another modiﬁed interaction integral for non-homogeneous materials.Recently,Dolbow and Gosz[48] have also derived a path independent interaction integral which is the same as the one given by Eq.(39). Note,for homogeneous materials,o D ijkl=o x1¼0,eð2Þij¼o uð2Þi=o x j,rð1Þij o eð2Þij=o x1¼o rð2Þij=o x1eð1Þij and rð2Þij o eð1Þij=o x1¼o rð1Þij=o x1eð2Þij,regardless of how the auxiliaryﬁeld is deﬁned.As a result,the e Jð1Þ,e Jð2Þ,and e Mð1;2Þintegrals in methods I and II degenerate to their corresponding homogeneous solutions,as expected.3.2.3.Stress-intensity factorsFor linear-elastic solids,the e J-integral also represents the energy release rate and,hence,e J¼1Etip ðK2IþK2IIÞ;ð40Þwhere EÃtip is evaluated at the crack tip.Regardless of how the auxiliaryﬁelds are deﬁned,Eq.(40)applied tostates1,2,and S yieldse Jð1Þ¼1Etip ðKð1Þ2IþKð1Þ2IIÞ;ð41Þe Jð2Þ¼1Etip ðKð2Þ2IþKð2Þ2IIÞð42Þande JðSÞ¼e Jð1Þþe Jð2Þþ2Etip ðKð1ÞIKð2ÞIþKð1ÞIIKð2ÞIIÞ:ð43ÞComparing Eq.(32)with Eq.(43),e Mð1;2Þ¼2EtipKð1ÞIKð2ÞIhþKð1ÞIIKð2ÞIIi:ð44ÞFollowing a similar procedure and judiciously choosing the intensity of the auxiliary state as described earlier,the SIFs for non-homogeneous materials can also be derived asKð1ÞI ¼e Mð1;IÞEÃtip2ð45ÞB.N.Rao,S.Rahman/Engineering Fracture Mechanics70(2003)1–279。

减反结构综述

highlights
• Colloidal self-assembly enables selfcleaning broadband anti-reﬂection (AR) coatings.
• Bioinspired moth-eye AR coatings show much better AR performance than quarter-wavelength AR coatings.
1.1. Quarter-wavelength AR coatings
The antireﬂection performance of traditional quarterwavelength AR coatings depends on both the coating thickness and the refractive index of the material. Carefully controlling both
1.1.1. Vacuum deposition of AR coatings Some of the most popular vacuum-based technologies for
generating AR coatings include plasma-enhanced chemical vapor deposition (PECVD) [13–19], electron-beam (E-beam) evaporation [20–22], and sputtering [23,24]. For crystalline silicon substrates, which have been widely used in fabricating solar cells, the typical AR coatings are quarter-wavelength titanium dioxide (TiO2) and silicon nitride (SiNx) ﬁlms [25]. Silicon has a high refractive index of ∼3.5 [26], therefore when light travels from air to a silicon substrate, the amount of light reﬂected will be about 30% from each air/silicon interface. This could signiﬁcantly impact the conversion efﬁciencies of crystalline silicon solar cells as an ideal solar cell should adsorb all useful photons. Therefore it is essential to utilize AR coatings to maximize the amount of light adsorbed by a solar cell [27].

基于SPH方法的钛合金切削仿真分析

-机械研究与应用•2020年第4期（第33卷，总第168期）研究与试验doi：10.16576/ki.1007-4414.2020.04.001基于SPH方法的钛合金切削仿真分析朱留宪1-2-3，孙勇,张永盛“，武友德"，冯颖珊3(1.四川省高温合金切削工艺技术工程实验室，四川德阳618000；2.西南交通大学机械工程学院，四川成都610031；3.四川工程职业技术学院机电工程系，四川德阳618000)摘要:运用SolidWorks、hypermesh和Is-prepost软件建立了切削仿真模型，基于光滑流体动力学方法对Ti6A14V钛合金切削进行了仿真,对切屑形成过程以及切削力进行了分析。

仿真分析结果表明:SPH方法解决了切削仿真过程中的网格畸变造成的无法求解问题，同时，切削仿真锯齿形切屑形貌与试验相一致，验证了SPH方法切削仿真的有效性。

从而为进一步研究钛合金切削机理以及优化切削参数提供了理论支持。

关键词:SPH；钛合金;仿真;锯齿形切屑中图分类号:TH161文献标志码:A文章编号：1007-4414(2020)04-0001-03Simulation Analysis of Titanium Alloy Cutting Based on the SPH Method ZHU Liu-xian1，2,3，SUN Yong1，3，ZHANG Yong-sheng1，3，WU You-de1，3，FENG Ying-shan3(1.Sichuan Provincial Engineering Laboratory qf Super Alloy Cutting Technology，Deyang Sichuan618000，China；2.School of Mechanical Engineering，Southwest Jiaotong University，Chengdu Sichuan610031，China；3.Department of Mechanical Electrical Engineering，Sichuan Engineering Technical College，Deyang Sichuan618000，China)Abstract:A model of cutting simulation was established with the Solidworks，hypermesh and ls-prepost software in this paper.The process of Ti6Al4V Titanium alloy cutting was simulated based on the smoothed particle hydrodynamics method；the process of chip formation and cutting force were analyzed.The results show that the SPH method avoid the element distortion problem which always occurs in the process of cutting simulation，and the shape of serrated chip in cutting simulation also is consistent with the test，thus the validity of the cutting simulation is verified by means of the SPH method.Therefore，this research provides a theoretical support for the further study on the cutting mechanism of Titanium alloy and optimization of the cutting parameters.Key words:SPH；Titanium alloy；simulation；serrated chip0引言钛合金因其具有高比强度、高比刚度以及良好的耐腐蚀性等特点,在航空、航天、国防、医疗、电子等领域应用广泛。

基于生成式对抗网络的画作图像合成方法

收稿日期：２０２００３１４；修回日期：２０２００５０６基金项目：国家自然科学基金资助项目（９１７４６１０７）作者简介：赵宇欣（１９９５），女，山西晋中人，硕士研究生，主要研究方向为机器学习、深度学习、计算机视觉（ｚｈａｏｙｕｘｉｎ＿ａｌｉｃｅ＠ｔｊｕ．ｅｄｕ．ｃｎ）；王冠（１９９２），女，内蒙古呼伦贝尔人，博士研究生，主要研究方向为深度学习、数学物理反问题．基于生成式对抗网络的画作图像合成方法赵宇欣，王　冠（天津大学数学学院，天津３００３５４）摘　要：画作图像合成旨在将两个不同来源的图像分别作为前景和背景融合在一起，这通常需要局部风格迁移。

现有算法过程繁琐且耗时，不能做到实时的图像合成。

针对这一缺点，提出了基于生成式对抗网络（ｇｅｎｅｒａｔｉｖｅａｄｖｅｒｓａｒｉａｌｎｅｔ，ＧＡＮ）的前向生成模型（ＰａｉｎｔｅｒＧＡＮ）。

ＰａｉｎｔｅｒＧＡＮ的自注意力机制和ＵＮｅｔ结构控制合成过程中前景的语义内容不变。

同时，对抗学习保证逼真的风格迁移。

在实验中，使用预训练模型作为ＰａｉｎｔｅｒＧＡＮ的生成器，极大地节省了计算时间和成本。

实验结果表明，比起已有方法，ＰａｉｎｔｅｒＧＡＮ生成了质量相近甚至更好的图像，生成速度也提升了４００倍，在解决局部风格迁移问题上是高质量、高效率的。

关键词：图像风格迁移；生成对抗网络；图像合成；自注意力机制中图分类号：ＴＰ３９１４１文献标志码：Ａ文章编号：１００１３６９５（２０２１）０４０４７１２０８０４ｄｏｉ：１０．１９７３４／ｊ．ｉｓｓｎ．１００１３６９５．２０２０．０３．００８２ＰａｉｎｔｅｒｌｙｉｍａｇｅｃｏｍｐｏｓｉｔｉｏｎｂａｓｅｄｏｎｇｅｎｅｒａｔｉｖｅａｄｖｅｒｓａｒｉａｌｎｅｔＺｈａｏＹｕｘｉｎ，ＷａｎｇＧｕａｎ（ＳｃｈｏｏｌｏｆＭａｔｈｅｍａｔｉｃｓ，ＴｉａｎｊｉｎＵｎｉｖｅｒｓｉｔｙ，Ｔｉａｎｊｉｎ３００３５４，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｐａｉｎｔｅｒｌｙｉｍａｇｅｃｏｍｐｏｓｉｔｉｎｇａｉｍｓｔｏｈａｒｍｏｎｉｚｅａｆｏｒｅｇｒｏｕｎｄｉｍａｇｅｉｎｓｅｒｔｅｄｉｎｔｏａｂａｃｋｇｒｏｕｎｄｐａｉｎｔｉｎｇ，ｗｈｉｃｈｉｓｄｏｎｅｂｙｌｏｃａｌｓｔｙｌｅｔｒａｎｓｆｅｒ．Ｔｈｅｃｈｉｅｆｄｒａｗｂａｃｋｏｆｔｈｅｅｘｉｓｔｉｎｇｍｅｔｈｏｄｓｉｓｔｈｅｈｉｇｈｃｏｍｐｕｔａｔｉｏｎａｌｃｏｓｔ，ｗｈｉｃｈｍａｋｅｓｒｅａｌｔｉｍｅｏｐｅｒａｔｉｏｎｄｉｆｆｉｃｕｌｔ．Ｔｏｏｖｅｒｃｏｍｅｔｈｉｓｄｒａｗｂａｃｋ，ｔｈｉｓｐａｐｅｒｐｒｏｐｏｓｅｄａｆｅｅｄｆｏｒｗａｒｄｍｏｄｅｌｂａｓｅｄｏｎｇｅｎｅｒａｔｉｖｅａｄｖｅｒｓａｒｉａｌｎｅｔｗｏｒｋ（ＧＡＮ），ｃａｌｌｅｄＰａｉｎｔｅｒＧＡＮ．ＰａｉｎｔｅｒＧＡＮｉｎｔｒｏｄｕｃｅｄａｓｅｌｆａｔｔｅｎｔｉｏｎｎｅｔｗｏｒｋａｎｄａＵＮｅｔｔｏｃｏｎｔｒｏｌｔｈｅｓｅｍａｎｔｉｃｃｏｎｔｅｎｔｉｎｔｈｅｇｅｎｅｒａｔｅｄｉｍａｇｅ．Ｍｅａｎｗｈｉｌｅ，ａｄｖｅｒｓａｒｉａｌｌｅａｒｎｉｎｇｇｕａｒａｎｔｅｅｄａｆａｉｔｈｆｕｌｔｒａｎｓｆｅｒｏｆｓｔｙｌｅ．ＰａｉｎｔｅｒＧＡＮａｌｓｏｉｎｔｒｏｄｕｃｅｄａｐｒｅｔｒａｉｎｅｄｎｅｔｗｏｒｋｗｉｔｈｉｎｔｈｅｇｅｎｅｒａｔｏｒｔｏｅｘｔｒａｃｔｆｅａｔｕｒｅｓ．ＴｈｉｓａｌｌｏｗｅｄＰａｉｎｔｅｒＧＡＮｔｏｄｒａｍａｔｉｃａｌｌｙｒｅｄｕｃｅｔｒａｉｎｉｎｇｔｉｍｅａｎｄｓｔｏｒａｇｅ．Ｅｘｐｅｒｉｍｅｎｔｓｓｈｏｗｔｈａｔ，ｃｏｍｐａｒｅｄｔｏｓｔａｔｅｏｆａｒｔｍｅｔｈｏｄｓ，ＰａｉｎｔｅｒＧＡＮｇｅｎｅｒａｔｅｄｉｍａｇｅｓｈｕｎｄｒｅｄｓｏｆｔｉｍｅｓｆａｓｔｅｒｗｉｔｈｃｏｍｐａｒａｂｌｅｏｒｓｕｐｅｒｉｏｒｑｕａｌｉｔｙ．Ｔｈｅｒｅｆｏｒｅ，ｉｔｉｓｅｆｆｅｃｔｉｖｅａｎｄｅｆｆｉｃｉｅｎｔｆｏｒｌｏｃａｌｓｔｙｌｅｔｒａｎｓｆｅｒ．Ｋｅｙｗｏｒｄｓ：ｉｍａｇｅｓｔｙｌｅｔｒａｎｓｆｅｒ；ＧＡＮ；ｉｍａｇｅｃｏｍｐｏｓｉｔｉｎｇ；ｓｅｌｆａｔｔｅｎｔｉｏｎ０　引言图像合成属于图像变换问题，目的是通过模型将一个简单的粘贴合成图像转变成一个融合为一体的图像。

基于SPH_方法的钢筋混凝土切削模拟研究

基于SPH 方法的钢筋混凝土切削模拟研究*谭松成，石恒超，王伟雄，方小红，段隆臣（中国地质大学(武汉) 工程学院, 武汉 430074）摘要　钢筋混凝土材料目前已得到广泛应用，但其加工切削过程十分复杂。

为解决常规有限元模拟中网格变形尺寸有限的局限性，采用光滑粒子流体动力学（SPH ）算法模拟研究金刚石磨粒对钢筋混凝土的切削破碎过程。

在数值模拟中，根据金刚石出刃状态将其简化成方形、圆形2种磨粒，并以0.45 mm/ms 的速度和0.1 mm 的切削深度对钢筋、混凝土和不同组合的钢筋混凝土材料进行表面切削，分析不同情况下的基体碎屑形态特征、切削后基体材料表面形态变化、内部裂纹延伸变化以及磨粒切削面上的应力变化情况。

模拟结果表明：SPH 方法能够较好地模拟钢筋混凝土在切削过程中的裂纹扩展、碎屑形成和分离；磨粒在切削过程中受到间断性冲击后，会优先破坏混凝土和钢筋材料之间的连接强度，使2种材料逐渐分离，且磨粒以面切削方式进行加工时，相较于点切削方式，能够形成更大的破碎区域。

关键词　钢筋混凝土；数值模拟；SPH 算法；金刚石；切削过程中图分类号　TG58; TU528.7　文献标志码　A 文章编号　1006-852X(2023)02-0161-09DOI 码　10.13394/ki.jgszz.2022.3006收稿日期　2022-07-03　修回日期　2022-08-20金刚石切削工具具有切割精度高、速度快、振动幅度小的特点，以锯片、串珠和钻头等形式已广泛应用于道路、桥梁、大坝、房屋等钢筋混凝土结构的修复或拆除工作中，并表现出优异的性能[1-5]。

为进一步提高金刚石工具的工作效率，需要不断调整工具结构特征、胎体成分和金刚石参数[6-7]，并建立合理的切削模型对切削过程进行预测[8]，从而选择更优的加工参数，提高工具的综合使用性能。

通过合理的物理试验方法能够提高金刚石工具的使用性能，但传统试验方法成本高、周期长，难以及时改变金刚石工具的各项参数进行试验，因此数值模拟已成为一种非常重要的研究方法。

基于kriging代理模型的优化设计方法及其在注塑成型中的应用共3篇

基于kriging代理模型的优化设计方法及其在注塑成型中的应用共3篇基于kriging代理模型的优化设计方法及其在注塑成型中的应用1随着工程设计的深入发展，更为高效、快速、精确的优化设计方法在注塑成型领域成为了追求的目标。

在这方面，经典的kriging代理模型逐渐成为注塑成型中实验设计和优化设计的一种重要方法。

本文将简要介绍基于kriging代理模型的优化设计方法以及该方法在注塑成型中的应用。

kriging代理模型，也称为高斯过程、克里金插值或克里格代理模型，是一种基于统计学方法的回归分析工具，通常用于建立输入与输出之间的关系，能有效地预测因数之间的关联性，并支持衡量不确定性和进行假设检验。

kriging代理模型的核心思想是插值，即通过对已知数据进行插值来预测可能的输出结果。

其中，典型的kriging代理模型主要有普通kriging、简单kriging、泛指kriging和克里金样条等。

在注塑成型中，kriging代理模型通常被用于实验设计和优化设计。

实验设计主要是为了了解注塑成型的各项因素对于最终产品性能的影响，以集体调节各个因素并找出最优的组合，从而使注塑成型更为高效、经济、精确。

优化设计则是在实验设计的基础上，利用kriging代理模型进行预测和优化，以实现优化的目标。

在这方面，kriging代理模型由于其高度灵活和精确，被广泛应用于应对不同的注塑成型挑战。

在应用kriging代理模型的优化设计方法中，首先需要通过design of experiment (DOE)方法进行实验设计，以获取样本数据。

其次，根据样本数据建立kriging代理模型，通过该模型预测变量间的相互作用，并得出最优的组合方案。

最后，根据kriging代理模型预测结果，进行优化设计，并进行相关性检验，以校准模型。

总的来说，kriging代理模型作为一种优秀的分析工具，有良好的精度和稳定性，并可以适应各种优化目标。

在注塑成型中的应用，不仅有助于提高注塑成型的效率和效益，同时也能够为行业的发展注入新的活力总之，kriging代理模型在注塑成型中的应用具有重要意义。

基于共形几何代数的并联机器人逆运动学分析方法

基于共形几何代数的并联机器人逆运动学分析方法柴馨雪;李翔毅;汤陈昕;李秦川;徐灵敏【期刊名称】《农业机械学报》【年(卷),期】2024(55)3【摘要】运动学分析是并联机器人运动学性能评估和结构尺寸优化的基础。

现有并联机器人运动学分析方法存在几何建模与几何计算相分离的问题,本文利用共形几何代数(Conformal geometric algebra, CGA)集几何表示和计算为一体的优势,提出一种并联机器人逆运动学分析方法。

根据动平台位姿参数给出动平台刚体运动算子,通过共形几何代数框架下的几何积实现动平台上任意点的刚体变换,得到任意点在运动过程中的共形几何表达式;结合机构中尺寸、几何约束,利用内积运算,建立机构运动学方程;根据运动学方程,进行运动学反解计算和速度分析。

以3自由度的3-RPS并联机器人和6自由度6-UPS并联机器人为例,对所提方法进行验证,并将逆运动学推导结果与仿真软件所得结果进行了对比,验证了本文提出方法的正确性。

该方法将空间向量和旋转表示等几何对象与矩阵乘法、矢量外积等计算方式相结合,使得并联机器人空间几何问题统一在一个代数系统中进行处理,因此分析过程几何直观性较强,简化了运动学逆解分析计算过程。

【总页数】10页(P421-430)【作者】柴馨雪;李翔毅;汤陈昕;李秦川;徐灵敏【作者单位】浙江理工大学机械工程学院;上海交通大学机械系统与振动国家重点实验室【正文语种】中文【中图分类】TH112【相关文献】1.利用共形几何代数的串联机器人位置逆解求解方法2.6自由度关节机器人运动学反解的共形几何代数方法3.空间并联机构运动学分析的共形几何代数方法4.一种6自由度关节机器人逆运动学共形几何代数法5.基于共形几何代数的工业机器人运动学分析因版权原因，仅展示原文概要，查看原文内容请购买。

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KiMPA:AKinematics-BasedMethodforPolygonApproximation

EdizS¸aykol,G¨urcanG¨ule¸sir,U˘gurG¨ud¨ukbayand¨Ozg¨urUlusoyDepartmentofComputerEngineering,BilkentUniversity06533Bilkent,Ankara,Turkey{ediz,gulesir,gudukbay,oulusoy}@cs.bilkent.edu.tr

Abstract.Indiﬀerenttypesofinformationsystems,suchasmultimediainformationsystemsandgeographicinformationsystems,object-basedinformationisrepresentedviapolygonscorrespondingtotheboundariesofobjectregions.Inmanyapplications,thepolygonshavelargenumberofverticesandedges,thusawayofrepresentingthepolygonswithlessnumberofverticesandedgesisdeveloped.Thisapproach,calledpoly-gonapproximation,orpolygonsimpliﬁcation,isbasicallymotivatedwiththediﬃcultiesfacedinprocessingpolygonswithlargenumberofvertices.Besides,largememoryusageanddiskrequirements,andthepossibilityofhavingrelativelymorenoisecanalsobeconsideredasthereasonsforpolygonsimpliﬁcation.Inthispaper,akinematics-basedmethodforpolygonapproximationisproposed.Theverticesofpolygonsaresim-pliﬁedaccordingtothevelocitiesandaccelerationsoftheverticeswithrespecttothecentroidofthepolygon.Anotherpropertyoftheproposedmethodisthattheusermaysetthenumberofverticestobeintheapproximatedpolygon,andmayhierarchicallysimplifytheoutput.Theapproximationmethodisdemonstratedthroughtheexperimentsbasedonasetofpolygonalobjects.

1IntroductionInmostoftheinformationsystemsemployingmethodsforrepresentingorretrievalofobject-basedinformation,polygonsareusedtorepresentobjectsorobjectregions,correspondingtotheboundariesoftheob-jectregions.Forexample,inmultimediainformationsystems,polygonsareusedinpatternrecognition,object-basedsimilarity[1,14],andimageprocessing[12].Anotherexampleapplicationcanbetherepresentationofgeographicinformationbythehelpofpolygonsingeographicinfor-mationsystems[6].Inmanycases,thepolygonshavelargenumberofvertices,andmanagingthesepolygonsisnotaneasytask.Thus,polygonapproximationisrequiredtofacilitateinformationprocessing.Obviously,theoutputoftheapproximationmustbeapolygonpreservingallofthecriticalshapeinformationintheoriginalpolygon.Besidesthediﬃcul-tiesinmanagingpolygonswithlargenumberofvertices,storingsuchapolygonmayrequirerelativelylargediskspaceandmemoryusageduringprocessing.However,thesimpliﬁedformrequireslessmemoryanddiskspacethantheoriginalone.Anotherreasonforusingpolygonapproxi-mationcanbethefactthatthesimpliﬁedformoftheoriginalpolygontendstohaveless‘noise’asfarasthesmalldetailsontheobjectregionboundariesareconcerned.In[13],atoolforobjectextractioninvideoframes,calledObjectEx-tractor,ispresentedinwhichtheboundariesoftheextractedobjectre-gionsarerepresentedaspolygons.Basically,theextractedpolygonshaveatleast360verticesforeachanglewithrespecttothecentroidoftheobject.Sincesuchasizeisquitelargeaswemighthavealargenumberofobjectsinasinglevideoframe,anapproximationforthepolygonsisinevitable.Sincedealingwithpolygonshavingappropriatesizeisoneofthemainmotivationsofpolygonapproximation,areasonablesizefortheapproximatedpolygonsisaround30vertices.Forexample,TurningAn-gleMethod[1],afamouspolygonalshapecomparisonmethod,maynotbefeasibleforpolygonshavingmorethan30vertices.Inthismethod,a(polygonal)objectisrepresentedbyasetofverticesandshapecompar-isonbetweenanytwoobjectsisperformedwithrespecttotheirturninganglerepresentations.Aturninganglerepresentationofanobjectisthelistofturninganglescorrespondingtoeachvertexonthepolygon.Inthispaper,Kinematics-BasedMethodforPolygonApproximation(KiMPA)isproposed.Themainmotivationofusingkinematicsistoﬁgureoutthemost‘important’verticestoappearintheapproximatedformviaassociatingalevelofimportanceforeachofthevertices.Thus,KiMPAeliminateslessimportantverticesuntilthepolygonreducestoanappropriatesize.Therestofthepaperisorganizedasfollows:Sec-tion2deﬁnesthepolygonapproximationproblemformallyanddiscussessomeoftheexistingmethodsintheliterature.TheKiMPAapproxima-tionprocessisdiscussedinSection3.InSection4,theexperimentstodemonstrateapproximationmethodarepresentedforasetofpolygonalobjects.Finally,Section5concludesthepaper.

2RelatedWorkInparalleltothewideuseofpolygonsininformationsystems,polygonapproximationmethodsarerequiredinvariousapplicationareas,suchaspatternrecognition[10],processingspatialinformation[6],andshapecod-ing[8].Besides,therearealsomethodsapplicabletomanyapplications.