An Adaptive Meshless Method for Modeling Large Mechanical Deformation and Contacts

Transcription

1 2007 IEEE Iteratoal Coferece o Robotcs ad Automato Roma, Italy, 0-4 Aprl 2007 WeD7. A Adaptve Meshless Method for Modelg Large Mechacal Deformato ad Cotacts Qag L ad Kok-Meg Lee*, Fellow, IEEE Abstract- Growg ew robotc applcatos agrculture, food-processg, asssted surgery ad haptcs, whch requres hadlg of hghly deformable objects, preset a umber of desg challeges; amog these are methods to aalyze deformable cotacts. Recetly, meshless methods (MLM), whch hert may advatages of fte elemet method (FEM) ad yet eed o explct mesh structure to dscretze geometry, have bee proposed as a attractve alteratve to FEM for solvg egeerg problems where automatc re-meshg s eeded. Ths paper offers a adaptve MLM (automatcally sertg odes to large error regos) for solvg cotact problems. We employ the sldg le algorthm wth the pealty method to hadle cotact costrats; t does ot rely o small dsplacemet assumptos ad thus, t ca solve o-lear cotact problems wth large deformato. Alog wth three practcal applcatos, we valdate the method agast results computed usg commercal FEM software ad aalytcal solutos. I. INTRODUCTION Numercal methods have bee creasgly used o-tradtoal robotc applcatos volvg mechacal deformable cotact; for example, complat graspg [, 2], robotc assembly of sap-fts [] ad asssted surgery [4]. Mechacal cotacts are commo robotc problems agrculture, food-processg ad surgcal robotc systems. Solutos of cotact wth large deflecto ad/or volvg deformable objects are essetal to help optmze desgs ad mprove performace of these systems. Wth few exceptos, solutos to these hghly olear problems are solved umercally. Amog the methods, FEM has bee most popular because t s a geeral method ad ca hadle complcated geometry. After decades of developmet, commercal FEM software has bee wdely avalable to solve may egeerg cotact problems [5, 6]. Ulke lumped-parameter methods, methods usg dstrbuted models such as FEM eed a loger computatoal tme but ca provde more detaled ad accurate solutos. Recetly, FEM has bee appled to aalyze robotc systems volvg deformable bodes. Durez etc. [7] developed a cotact model based o a lear FEM for haptc smulato. Alterovtz et al. [8] used FEM to study the effect of frcto o surgcal eedle serto. Cocarle et al. [9] appled FEM to study the grasp qualty of deformable fgers. Rapd mprovemet computer speeds has made FEM more acceptable eve for some real-tme applcatos. Pcboo et al. [0] developed a lear FEM model for real tme force feedback of a haptc devce, ad exteded ther Ths work was supported part by the Georga Agrculture Techology Research Program ad the U.S. Poultry ad Eggs Assocato. The authors are wth the Woodruff School of Mechacal Egeerg at the Georga Isttute of Techology. * Correspodg author: Phoe: +(404) ; fax: +(404) ; e-mal: kokmeg.lee@ me.gatech.edu. method to olear soft tssue models []. Cot et al. [2] preseted a pre-processg techque to allow real-tme computato of deformatos/forces for surgery smulato. Although FEM meshes provde the geeralty to hadle complcated geometres, approprate mesh structures are ofte dffcult to be created or modfed especally for applcatos where meshes must be recostructed automatcally durg the computatoal process. For example, [4] cosderable research effort must be devoted to developg a adaptve mesh geerato algorthm order to smulate the process of eedle serto. The accuracy of FEM depeds sgfcatly o the qualty of ts mesh. Addtoally, the mesh desty must be mataed at a suffcetly hgh level aroud the cotact rego to obta reasoably accurate results. However, addtoal elemets o-cotact regos do ot geerally help mprove the overall accuracy. Thus, the mesh desty should ot be uformly hgh as they would smply demad more computatoal tme; clearly, a approprately desged mesh s very mportat for FEM aalyss to obta accurate results effcetly. Ths s especally true for solvg a cotact problem where a large umber of teratos are ofte eeded for the soluto to the hghly olear problem to coverge. Exstg mesh geerato programs for FEM, geeral, have dffcultes to meet the demads of both accuracy ad computatoal effcecy smultaeously due to the strget shape requremets of FEM elemets; addtoal maual modfcatos of the meshes are ofte ecessary. For cotact problems volvg large deformato, t s very dffcult to costruct a good mesh eve wth the help from a expereced FEM aalyst because cotact regos such problems ca ot be located accurately before computg. Thus, t s desred to have a method that automatcally detfes regos of large errors, ad systematcally crease ther odal desty to mprove the overall accuracy wth o huma volvemet. ML methods (bult o the theoretcal framework of FEM) have bee gag atteto [-5]. The costructo of a bass fucto for MLM, however, does ot rely o the mesh structure. Ths sgfcatly reduces the dffcultes of developg a algorthm to adaptvely crease ode destes, ad makes MLM a very attractve alteratve to FEM for solvg problems where automatc re-meshg s eeded. Recetly, a adaptve MLM [6] wth automatc error estmato ad ode serto has bee developed for solvg lear electromagetc problems effectvely reducg tme eeded to desg tal meshes or re-mesh computato Ths paper exteds the adaptve MLM for solvg o-lear problems ad offers the followgs: /07/$ IEEE. 207

2 WeD7.. We preset a formulato for solvg deformable cotact problems usg MLM. Ths formulato, whch apples the sldg le algorthm [7] alog wth the pealty method [6] for hadlg the cotact costrats, does ot assume small (or lear) dsplacemets. Ulke [6] where magetc problems are formulated terms of dsplacemets, we derve the error estmato, whch detfes regos of large computatoal errors for automatc ode serto, based o mechacal stresses sce dsplacemet of a rgd body moto does ot ecessarly result mechacal stresses. 2. Three examples are gve to llustrate the automatc sertg procedure of the adaptve MLM ad ts applcatos deformable cotact. Ulke FEM where excessvely large deformato could cause severe elemet dstorto ad cosequetly breakdow the smulato, the adaptve MLM algorthm s able to costruct bass fuctos wthout usg mesh structure.. The adaptve MLM algorthm for solvg cotact problems has bee valdated by comparg computed results agast aalytcal solutos wheever possble, ad those smulated usg ANSYS (a commercal FEM package). I. FORMULATION OF DEFORMABLE CONTACT Mechacal cotact volvg large deformato s formulated usg MLM weak form. Cotact s modeled as a costrat mposed oto the weak form formulato. A. Formulato of Mechacal Cotact Cosder two bodes, Ω A ad Ω B, boud by boudares, Γ A ad Γ B, respectvely as show Fg. a, where X s the orgal u-deformed coordate of a partcle; ad x A (X, t) ad x B (X, t) represet the deformed coordates of a arbtrary partcle o Ω A ad Ω B at tme t respectvely. Γ A Γ B Ω A Ω B x A x B g (X,t) x m Slave x s a) Cotact betwee two bodes b) Dscretzed represetato Fg. cotact costrat Cotact s posed as a dsplacemet costrat o dscretzed odes. I the dscretzed doma, the two cotact bodes are referred as the slave ad master. The dscrete odes coordates are defed Fg. b, where x s ad x c are the slave ode ad cotact pot o the master segmet respectvely; x m ad x m2 are two adjacet master odes; ad x c0 s the cotact pot of the last computatoal step. I Fg. b, ad t are respectvely the ut ormal ad tagetal vectors at x c. The vector t s computed from the master odes: t = ( xm2 xm ) / where = xm2 xm () ad ca the be obtaed from the orthogoalty = ez t where e z s a ut vector alog the z axs. The dstace from the slave ode to the master segmet s x c0 x c Master t g x m2 defed as the ormal gap fucto g as follows: g = ( xs xm) (2) The ormal cotact force τc s proportoal to g : 0 g 0 whe two pots are ot cotact τc = 0 g = 0 whe two pots are at cotact () kg g < 0 peetrato occurs where the pealty proportoalty k s a very large umber. Ths approxmato approaches deal cotact as k. The tagetal compoet of the cotact force (or the frcto force) τct ca be obtaed by the Coulomb frcto law: Stck occurs f 0 < τ ct µτ c (4) Slp occurs f τ ct = µτc where µ s the frcto coeffcet. To determe the curret state of cotact ( stck or slp ), aother gap fucto g t (or the dstace from x co to x c ) s defed to depct the dstace that the cotact pot slps for two adjacet tme steps: g t = ( xc xc0 ) t (5) Wth g t, τct ca be approxmately obtaed as follows: kg t t µτ c kg t t (stck) τ ct = (6) µτ c sg( gt ) µτ c < kt gt (slp) where k t s the tagetal pealty parameter. As k t, (6) approaches the deal Coulomb law. B. Formulato of Large Deformato Mechacs For quas-statc problems volvg large deformato, the three goverg equatos gve by [8] are Pj + ρ0b0 = 0 ( =, 2,) (7) j= j where ρ 0 ad b 0 are the desty ad body force of the orgal u-deformed state; ad P j s the elemet of the st Pola- Krchhorff (PK) stress tesor P. To solve (7) for the dsplacemet fucto u=x-x as a depedet varable, the asymmetrc stress tesor P s trasformed to the symmetrc 2 d PK stress tesor S by xr Pj = Sr where Sr = Crkl ( εkl + εkl ) (8a, b) r = j k= l= C rkl s a elemet of the materal complat tesor C (a materal property); ad εkl ad ε kl are the terms the Gree stra tesor gve by uk u l um um ε kl = + ad εkl = 2 l (9a, b) k k l m= For lear small dsplacemet problems, the hgher order terms the Gree stra tesor ca be gored or εkl 0 ; the Gree stra tesor reduces to Cauchy stra ε kl. The two types of BC s for a cotuum body (Drchlet ad Neuma) are the dsplacemet u ad tracto t (or force/area) BC s: u = u ( =,2,) o Γ u (0) 208

3 WeD7. P j j = t ( =,2,) o Γ t () j= where s the ormal vector of boudary. C. Weak form Formulato of Cotact Mechacs The MLM approxmates a ukow dsplacemet fucto u(x) usg (2): ux ( ) = x X= Ψ ( Xu ) (2) = where u s the odal cotrol value assocated wth the th ode; ad Ψ(X) s a ML bass fucto; see for example, reproducg kerel (RPK) method [9]. If the ML bass fucto at the th ode s a terpolatg fucto, u s the dsplacemet at ths ode u =u(x ). Otherwse, u u(x ). The goverg equato weak form ca be formulated usg the varatoal method alog wth eergy coservato. The varatos of the vrtual teral ad exteral works wthout cotact are gve respectvely by ad Ψ δw P d I = jδx I Ω0 () Ω0 j δw = Ψ ρ bδx dω + Ψ tδx dγ (4) e I 0 I 0 I I 0 Ω0 Γ0 Icorporatg () ad (6) or the assumptos the pealty method, the varato of the vrtual work cotrbuted by the cotact force ( τ c ad τ ct ) ca be wrtte as From (2) ad (5), δg = τ δg + τ δg dγ P c ct t Γc T δg =, ( α), α δ w x ; ad T t = ( / o) t, / ( ) t, / t xw T δ x w = δ x s, δ x m, δ x m2 ; α = ( ) s m / δg g α g α δ (5) where x x t ; ad ad o are the curret ad prevous dstaces defed (). The weak form equatos ca thus be formed usg the eergy coservato, δw = δwe + δgp. As show (8a, b) ad (9a, b), the st PK stress tesor P s a olear fucto of dsplacemet u. Thus, for the case of large deformato, the dscretzed weak form equatos are a set of olear equatos whch ca be solved usg Newto method. The atural (or Neuma) BC s are appled whe the goverg equatos are coverted to weak forms. Oce the weak-form equatos are learzed, the essetal (or Drchlet) BC s are appled before solvg the learzed set of equatos. II. ADAPTIVE MLM FOR COMPUTATION MECHANICS A smple way to mprove the accuracy of the umercal approxmato s to uformly crease the odal desty the whole computatoal doma, whch s effcet f large errors oly occur certa regos. A more effectve way s to estmate the error dstrbuto ad sert addtoal odes accordgly, or more specfcally, to the large error regos. A. Error Estmato The umercal error ca be estmated by,d,d, 2d, 2d = = e( x ) = Ψ ( x ) Φ Ψ ( x ) Φ (6) where e( x ) s estmated error; Ψ,d ad Ψ,2d deote the bass fuctos at the th ode wth a support sze d ad 2d respectvely; Φ,d s the soluto solved the prevous computato step; ad Φ,2d s the ftted result usg the bass fucto wth a support sze of 2d. The ratoale for (6) ca be explaed by comparg two dfferet support szes of a RPK bass fucto [9] as show Fg. 2. I geeral, the larger the support sze the smoother s the bass fucto, ad more dffcult to approxmate a fucto wth a abrupt chage the soluto. Thus, regos of large errors ca be characterzed by comparg the approxmato solutos solved usg the two dfferet bass fuctos. Fg. 2 RKP bass fucto wth two dfferet support szes B. Adaptve Node Iserto Oce the error s estmated from (6), locatos of large errors are detfed as follows: x a : e( xa ) > e (7) p where x a s the test locato; ad e p s a specfed error threshold. Addtoal odes ca be serted to the computatoal doma usg the Voroo plot [20] techque that costructs oe Voroo cell for each ode. As show Fg., a Voroo cell s a polygo cotag all the pots closest to the ode that t surrouds. The error at the vertexes of each Voroo cell s computed from (6) ad f the error satsfes crtero (7), a ew ode s created at that pot as llustrated Fg.. The support sze of the serted ode s calculated usg (8): r = ap max( xj x ) (8) where r s the support radus for th ode; a p s a costat coeffcet ormally take a value betwee to ; x ad x j are the coordates of th ad j th odes respectvely. The Voroo cell of the j th ode s adjacet to the Voroo cell of the th ode. For the ewly serted ode, the choce of the support radus of the bass fucto s a trade-off betwee two cosderatos: It must be suffcetly large to cover eough odes for costructg the ML bass fucto but kept small to localze the effect of the ewly serted odes. C. Partto Uty Itegrato Most of the bass fuctos (cludg the RKP method [9]) used MLM have the partto uty property: Ψ ( x ) = (9) = wth whch the tegrato for a arbtrary fucto f(x) the 209

4 WeD7. computatoal doma ca be computed usg as follows: Ω Ω = = Ω f ( x)dx= f ( x) Ψ ( x)dx = f ( x) Ψ ( x )dx (20) where Ω s the computatoal doma. To exclude pots outsde the computatoal doma, (20) s wrtte such that the tegrato s wth the support doma S of th bass fucto: f ( x) Ψ ( x)dx = f( x)p( x) Ψ( x )dx (2) = Ω = S whe x Ω where P( x ) = 0 whe x Ω The global tegrato for the whole computatoal doma s dvded to sub-tegrato domas ad performed upo the support doma of bass fuctos. Because the support doma of the bass fuctos, geeral, has a regular shape, the covetoal umercal tegrato scheme such as Gaussa quadrature ca be appled easly. It s worth otg that usg the partto uty tegrato, a ew tegrato cell s automatcally created oce a ew ode s serted as llustrated Fg. 4 ad thus, ths umercal tegrato scheme s very sutable for adaptve computato. 5. Compute T for the orgal ad the ew results usg (22). 6. Estmate the error as the dfferece betwee two stress magtudes. III. SIMULATION OF DEFORMABLE CONTACT We llustrate three examples. The st example valdates the adaptve MLM ad cotact algorthm agast aalytcal solutos ad results obtaed from ANSYS. The 2d example vestgates the effect of frcto for a sap-ft mechasm. The rd example shows the potetal of MLM medcal surgery applcatos. Example : Cotact betwee rgd ad elastc objects Fgure 5 shows a classc two-body cotact problem, where a small rgd object (whch may be a rgd puch or robotc fger) s drve ormally to a elastc body. Both objects are fte the z-axs. The structure s symmetrc about the y-axs; thus oly half of the geometry o the +x-axs s solved. Closed form aalytcal soluto descrbg the y-dsplacemet for the frctoless case ca be foud [2]. 0 whe x a 2 2 uy( x) = δ y 2( ν ) P x x (2) l + whe x > a 2 π E a a where u y s the dsplacemet the y-drecto; P s the force appled o rgd puch; a s half wdth of rgd puch; ad δ s dstace that the rgd object moves to the elastc body. y Rgd object P y x Fg. Voroo plot Fg. 4 Itegrato cells D. Nodal serto for computato mechacs For solvg mechacal problems, the stress or stra s a more approprate varable for error estmato sce dsplacemet of a rgd body moto does ot ecessarly result mechacal stresses. The mechacal stress s a 9-compoet tesor σ j (or S j the case of large deformato) ad ca be represeted as a x symmetrc matrx. The three prcpal stress compoets (commoly used as crtera to determe materal falure) are the egevalues of the stress matrx. They are coordate depedet, ad ca be utlzed to locate the rego of hgh stresses. The overall magtude of the stress T ca be wrtte as T = λ = σ + 2 κj ( σj σσjj) (22) = = = j= where λ s the egevalue of stress matrx; ad f j κj =. 0f = j The error estmato for sertg addtoal odes solvg mechacal problems ca be executed as follows:. Determe a approprate support sze for the ML bass fucto. 2. Compute the dsplacemet feld u(x) wth the orgal bass fucto.. Ft the dsplacemet usg the bass fucto but a larger support sze. 4. Compute the stress feld σ(x) from the lear or olear stra (9a, b) usg the orgal ad the ew dsplacemets. Elastc body L Fg. 5 Rgd puch cotacts wth elastc foudato Table Geometry parameters of example 2 L (m) H (m) a (m) δ y To demostrate the effectveess of the adaptve method, o specal ode refemet s made aroud the cotact rego. The computato starts wth a uform dstrbuto of odes. After three successve computatos, the total umber of odes creases from ts tal 2 odes to 94. Fgures 6(a) ad 6(b) show the Voroo dagrams of the tal ad 2 d ode dstrbutos. The fal ode dstrbuto s show Fg. 6(c). The MLM ad FEM results are compared agast the aalytcal solutos Fg. 7, where FEM uses a total of 544 odes wth specal refemet aroud cotact area. As llustrated Fg. 6, the adaptve MLM automatcally serts addtoal odes aroud the cotact rego. Fg. 7 shows that the fal MLM result s greatly mproved after three computatos. FEM ad MLM agree very well results but both are slghtly hgher tha the aalytcal soluto. The dscrepacy s somewhat expected because the aalytcal soluto assumes the body s ftely log the x-drecto H δ y a Elarged cotact rego 20

5 WeD7. whle umercal solutos base o a fte dmeso. a) Ital Voroo dagram ode vertexes of a Voroo cell b) 2 d Voroo dagram c) Fal ode dstrbuto Fg. 6 adaptve odes serto commo procedures employed moder clcal practce. Applcatos of these procedures clude the bopsy of prostate brachytherapy ad eurosurgcal probe serto, whch are usually wthout vsual feedback from below the sk s surface. Maxmum force ad stresses geerally occur at cotact before peetrato. The adaptve MLM ca provde computatoally effcet detaled formato at the cotact rego betwee the surgery tool ad tssues for medcal surgery smulato applcatos. Specfcally, we smulate a eedle cotactg a ellptcal elastc body. The materal propertes of the deformable body ad the tal geometry ad ode dstrbuto are Fg. 0. No specal refemet has bee made aroud the cotact rego for the tal ode dstrbuto. The eedle moves vertcally dowward from ts tal posto. The cotact s computed from the tp of the eedle at four locatos startg from the locato at 9.99mm ad the creasg at a terval of 0.25mm. Fg. 7 Comparso betwee MLM, FEM ad aalytcal result Example 2: Cotact of a sap-ft We demostrate here the use of MLM for aalyze cotact forces of a sap-ft, ad compare the results agast those computed by ANSYS. A typcal sap-ft geometry s show Fg. 8, where the reteto block (assumed u-deformable) moves horzotally from rght to left ad cotacts wth the catlever-hook. The geometry ad materal parameters of the catlever-hook alog wth the optos used for ANSYS ad MLM are gve Table 2. Fg. 8 Geometry of a sap-ft mechasm Fgure 9 compares the cotact forces computed usg MLM agast those wth ANSYS for both frctoless (µ=0) ad frctoal (µ=0.2) cotacts. The MLM ad FEM agree closely wth each other up to the locato where the edge of reteto block passes the jaw tp, beyod whch ANSYS computato breaks dow due to the large dstorto of elemets. Ulke FEM, MLM s free from mesh dstorto, ad predcts the cotact forces throughout the sap fttg process. Example : Cotact smulato of eedle serto Subcutaeous serto of eedles s oe of the most a) x drecto force b) y drecto force Fg. 9 Cotact forces Table 2 Smulato parameters of sap-ft mechasm Parameters Values Numercal 2D model Youg s modulus 2.62 GPa Plae stress (thckess; 0mm) Posso s rato 0.4 Thckess w (mm).2 l f (mm) 57 l h (mm) 76 Radus r (mm) 50 x a, y a (mm) 49.9, -4.0 y c (mm) 2.6 ANSYS wth 282 odes Elemet type: PLANE2, CONTACT75, TARGET69 MLM Number of odes: 69 (tal) 80~200 (after two adaptve computatos) At the st cotact posto, four adaptve computatos are performed. The coverged results for the tal ode dstrbuto ad cotact force are show Fg.. Fgure (a) shows that the computato wth a small umber of tal odes caot reflect the detal deformato at the cotact locato. Wth more odes aroud the cotact rego, shape chages ca be see at that locato Fg. (b). The cotact forces of the four computatos at the tal posto are show Fg. (c). The covergece ca be observed from the fact that the cotact force dfferece betwee two cosecutve computatos becomes smaller as the adaptve procedure proceeds. Ihertg the odes added from the st posto, three addtoal computatos are performed at the 2 d posto. No sgfcat mprovemet was observed betwee these computatos, dcatg that the ode desty s suffcetly large. The deformed geometry ad ode dstrbutos at the other three postos are plotted Fgs. 2(a)-2(c). The fal cotact force results are lsted Table. The cotact force creases as the eedle moves dowward. 2

6 WeD7. Three examples have bee llustrated demostratg the potetals of the adaptve MLM for robotc applcatos (such as surgcal smulato ad food processg) where hadlg of hghly deformable objects s essetal. The adaptve MLM (curretly programmed Matlab) wll be evaluated agast commercal FEM packages terms of computatoal tme. Fg. 0 tal geometry ad ode dstrbuto (Deformable body: Youg s modulus E= E6 Pa; Posso s rato µ=0.4) (a) Ital (b) After st computato (e) Cotact force Fg. Result after each adaptve computato at the st posto (a) 2 d posto (b) rd posto (c) 4 th posto Fg. 2 Results of MLM smulato Table : Cotact Force Locato of eedle tp(mm) Cotact force (N) Fgure shows the equvalet stress dstrbuto aroud the cotact for the st ad 4 th eedle postos. As expected, the magtude of stress creases as the eedle moves from posto to 4 ad ts maxmum occurs at the cotact locato. The stress formato, whch serves as the crtero for materal falure the theory of fracture mechacs, provdes a meas to judge whe the peetrato happes. (a) st posto (b) 4th posto Fg. Equvalet stress dstrbuto (N/m 2 ) IV. CONCLUSIONS A adaptve MLM method for solvg deformable cotact problems has bee preseted. Ths method utlzg the sldg le ad pealty methods to hadle cotact costrats has bee valdated agast aalytcal ad umercal solutos. The results show that the adaptve MLM effectvely detfy regos of large computatoal errors ad progressvely add odes accordgly. As demostrated wth termedate results, the overall error rapdly reduces as the adaptve procedure proceeds. Apart from elmatg the eed to maually re-mesh as ofte requred FEM, the adaptve algorthm drastcally reduces the computatoal tme of the MLM. REFERENCES [] K. M. Lee, J. Jo, ad X. Y, "Complat graspg force modelg for hadlg of lve objects," ICRA200, pp , Seoul, South Korea. [2] C.-H. Xog, M. Y. Wag, Y. Tag, ad Y.-L. Xog, "Complat graspg wth passve forces," Joural of Robotc Systems, vol. 22, pp , [] C.-C. La ad K.-M. Lee, "A aalytcal method for desg of complat grppers wth macro/mcro mapulato ad assembly applcatos," AIM2005, pp , Moterey, CA, Uted States. [4] S. P. DMao ad S. E. Salcudea, "Needle serto modelg ad smulato," IEEE Tras. o Robotcs ad Automato, vol. 9, pp , 200. [5] J. C. Smo ad T. A. Laurse, "A augmeted Lagraga treatmet of cotact problems volvg frcto," Computers ad Structures, vol. 42, pp. 97, 992. [6] P. Wrggers, T. Vu Va, ad E. Ste, "Fte elemet formulato of large deformato mpact-cotact problems wth frcto," Computers ad Structures, vol. 7, pp. 9-, 990. [7] C. Durez, C. Adrot, ad A. Kheddar, "Sgor's cotact model for deformable objects haptc smulatos," IROS2004, pp. 22-7, Seda, Japa. [8] R. Alterovtz, K. Goldberg, J. Poulot, R. Taschereau, ad I. C. Hsu, "Needle serto ad radoactve seed mplatato huma tssues: Smulato ad sestvty aalyss," ICRA200, pp. 79, Tape. [9] M. Cocarle, A. Mller, ad P. Alle, "Grasp aalyss usg deformable fgers," IROS 2005, pp , Edmoto, Alta., Caada. [0] G. Pcboo, J.-C. Lombardo, H. Delgette, ad N. Ayache, "Asotropc elastcty ad force extrapolato to mprove realsm of surgery smulato," ICRA2000, pp. 596, Sa Fracsco, CA, USA. [] G. Pcboo, H. Delgette ad N. Ayache, "No-lear ad asotropc elastc soft tssue models for medcal smulato," ICRA200, pp. 70, Seoul. [2] S. Cot ad H. Delgette, "Real-tme surgery smulato wth haptc feedback usg fte elemets," ICRA 998, pp. 79, Leuve, Belgum. [] T. Belytschko, Y. Krogauz, D. Orga, M. Flemg, ad P. Krysl, "Meshless methods: a overvew ad recet developmets," Computer Methods Appled Mechacs ad Egeerg, vol. 9, pp., 996. [4] I. Babuska ad J. M. Melek, "The Partto of Uty Method," Iteratoal Joural for Numercal Methods Egeerg, vol. 40, pp , 997. [5] C. A. Duarte ad J. T. Ode, "h-p adaptve method usg clouds," Computer Methods Appled Mechacs ad Egeerg, vol. 9, pp. 27, 996. [6] Q. L ad K. M. Lee, "A adaptve meshless method for magetc feld computato," IEEE Tras. o Magetcs, Vol. 42, No. 8, pp , [7] J. O. Hallqust, "A umercal treatmet of sldg terfaces ad mpact," Computatoal Techques for Iterface Problems 978, pp. 7-, Sa Fracsco, CA, USA. [8] L. E. Malver, Itroducto to the mechacs of a cotuous medum Eglewood Clffs, N.J., Pretce-hall, 969. [9] W. K. Lu, S. Ju, ad Y. F. Zhag, "Reproducg kerel partcle methods," Iteratoal Joural for Numercal Methods Fluds, vol. 20, pp. 08, 995. [20] C. B. Barber, D. P. Dobk, ad H. Huhdapaa, "The Quckhull algorthm for covex hulls," ACM Tras. o Mathematcal Software, vol. 22, pp. 469, 996. [2] K. L. Johso, Cotact mechacs. Cambrdge; New York: Cambrdge Uversty Press,