Surface and Volume Discretization of Functionally Based Heterogeneous Objects

Transcription

1 Surfae and Volume Dsretzaton of Funtonally Based Heterogeneous Objets Elena Kartasheva Insttute for Mathematal Modelng Russan Aademy of Sene Mosow, Russa Oleg Fryaznov Insttute for Mathematal Modelng Russan Aademy of Sene Mosow, Russa Valery Adzhev he Natonal Centre for Computer Anmaton, Bournemouth Unversty Poole, BH1 5BB UK Alexander Pasko Hose Unversty okyo, Japan Vladmr Gaslov Insttute for Mathematal Modelng Russan Aademy of Sene Mosow, Russa ABSRAC he presented approah to dsretzaton of funtonally defned heterogeneous objets s orented towards applatons assoated wth numeral smulaton proedures, for example, fnte element analyss (FEA). Suh applatons mpose spef onstrants upon the resultng surfae and volume meshes n terms of ther topology and metr haratersts, exatness of the geometry approxmaton, and onformty wth ntal attrbutes. he funton representaton of the ntal objet s onverted nto the resultng ellular representaton desrbed by a smplal omplex. We onsder n detal all phases of the dsretzaton algorthm from ntal surfae polygonzaton to fnal tetrahedral mesh generaton and ts adaptaton to speal FEA needs. he ntal objet attrbutes are used at all steps both for ontrollng geometry and topology of the resultng objet and for alulatng new attrbutes for the resultng ellular representaton. Categores and Subjet Desrptors I.3.5 [Computer Graphs]: Computatonal Geometry and Objet Modelng Curve, surfae, sold, and objet modelng, Physally based modelng; I.3.6 [Computer Graphs]: Methodology and ehnques; I.3.8 [Computer Graphs]: Applatons. General erms Algorthms, Desgn. Keywords Heterogeneous objets, attrbutes, funton representaton, ellular representaton, volume modelng, onstrutve hypervolume, fnte element analyss, mesh. 1. INRODUCION In ths paper, we deal wth generaton of dsrete models for heterogeneous objets defned usng real-valued funtons. Generally, heterogeneous objets have an nternal struture wth non-unform dstrbuton of materal and other attrbutes of an arbtrary nature (photometr, physal, statstal, et.), and elements of dfferent dmenson. Reently we an observe a steady nterest n funtonally based geometr models suh as mplt surfaes, funtonally defned solds and heterogeneous objets (see [1-3] for detals). hese models provde ompat and ntutve mathematal representaton for omplex heterogeneous objets, support set-theoret and other operatons suh as offsettng, blendng, and sweepng. Rapd development of hardware, omputatonal algorthms, and spealzed software allow for manpulaton of suh models at nteratve rates. Pratal applatons of funtonally based models n CAD/CAM/CAE and fnte element analyss (FEA) requre some key proedures to be also applable to these models. Numeral FEA methods use dsrete models (surfae and volume meshes) of geometr objets, although meshfree analyss and smulaton methods are also emergng [4]. Algorthms for fnte element mesh generaton are well developed for boundary and spatal enumeraton representatons [5]. Meshes are also atvely used now n vsualzaton, anmaton, omputatonal geometry, mage proessng, and other areas. However, requrements for dsrete models n FEA are strter than n other areas. Suh requrements are formulated n terms of dsrete model topology and metr haratersts, exatness of the geometry approxmaton, and onformty wth ntal attrbutes. hs results from the use of meshes n FEA for approxmaton of systems of equatons of mathematal physs. he sze and shape of mesh elements and mesh struture serously nfluene the stablty of numeral smulaton proedures and auray of obtaned solutons. hat s why meshes used for vsualzaton usually do not satsfy FEA requrements and speal refnement of them s needed. he above s the motvaton for work on dsrete model generaton for funtonally based surfaes, solds, and heterogeneous objets. In ths paper, we deal wth the dsretzaton problem wthn the hybrd ellular-funtonal model [6] of heterogeneous objets. he funton representaton of the ntal 3D heterogeneous objet s onverted nto the resultng ellular representaton desrbed by a smplal omplex. We onsder n detal the followng phases of the dsretzaton algorthm: ntal surfae polygonzaton; teratve smplfaton and refnement of the surfae mesh along wth sharp features reonstruton; further adaptaton of the surfae mesh aordng to onstrants dependng on the attrbutes of the ntal objet; volume (tetrahedral) mesh generaton usng a modfed advanng front method; ts subsequent adaptaton to speal FEA needs. he ntal objet attrbutes are used at all steps both for ontrollng geometry and topology of the resultng objet and for alulatng new attrbutes for the resultng ellular representaton.

2 . PROBLEM SAEMEN In ths seton, we provde a formulaton of the dsretzaton problem n terms of ntal and resultng objets along wth a set of speal requrements. We onsder the problem of dsretzaton of funtonally based heterogeneous objets wthn a hybrd ellular-funtonal representaton framework [,6] n whh objets are treated as hypervolumes (multdmensonal pont sets wth multple attrbutes) [1]..1 Intal heterogeneous objet Let D be an ntal heterogeneous objet a hypervolume expressed by a tuple: D ( G, A1,..., Ak ), where G s a 3D pont set, A s an attrbute and k s a number of attrbutes. We assume that the objet s geometry G s desrbed by the funton representaton (FRep) [7]: G F = {X = (x 1, x, x 3 ) Ω E 3, F(X) 0 }, where Ω s a modelng spae and F: Ω -> s (at least a C 0 ontnuous) real-valued defnng funton. Note that the boundary of the objet D s an mplt surfae desrbed as B F = {X = (x 1, x, x 3 ) Ω E 3, F(X) = 0 }. Eah attrbute A s defned by ts set of values N along wth a map funton S (X): Ω->N and an be represented by any of the attrbute models ntrodued n [] that dffer from eah other n the way of defnng S (X). For nstane, the funton representaton (FRep) an be used for defnng attrbutes representng eletr or thermal feld dstrbuton as well as load dstrbuton. Cellular-funtonal representaton (CFRep) s espeally sutable for desrpton of materal attrbutes. Cellular models of attrbutes (CRep) are used n fnte element alulatons for representaton of funtons desrbng smulated proesses. What s mportant n the ontext of our onsderaton of the dsretzaton problem s that attrbutes an provde a formal desrpton of requrements and onstrants mposed by FEA. hus, onstrants on the sze of elements an be expressed through the mesh densty attrbute A r whose map funton S r (X) defnes a proper element sze at eah pont X of the modelng spae Ω. Suh an attrbute an be gven by the user or an be alulated on the bass of other gven attrbutes A. here s a promsng way of defnng A r through FRep on the bass of soures [8,]. As to defnng A r through CRep, t s approprate when the sze dstrbuton funton s defned n a dsrete manner and ts values are known at the vertes of a bakground geometr omplex. Suh a omplex an desrbe ether a regular mesh speally bult for defnng the attrbute A r or an FE mesh used n the prevous steps of adaptve numeral alulatons.. Resultng heterogeneous objet Gven the ntal objet D, we are gong to buld a resultng heterogeneous objet a hypervolume D ( G, A 1,..., A m). Here, the geometry omponent G s an approxmate dsrete representaton of the ntal geometry G, and attrbutes ( A 1,..., A m) m desrbe the objet s propertes. Formally, G an be expressed as a partular ase of a model based on the ellular representaton (CRep) [,6]: G ={ X X Ω E 3, X K 3 }, 3 r where K { C ; r 0,1,,3; 1,..., Ir} s a three-dmensonal r polyhedral omplex onsstng of ells C. In onventonal terms, suh a dsrete model s alled a mesh. As we have already stated, the resultng mesh G depends not only on the ntal geometry G but also on the attrbutes A 1, A k (n partular, on the mesh densty attrbute A r ): G G ( GF, A1,..., A k ). Note that the attbutes A A 1,..., m an dffer from the ntal ones A 1,,A k n terms of ther number, set of values, and desrpton. Some of the ntal attrbutes may not have dret ounterparts n the dsrete model (e.g., there may be no need to retan the mesh densty attrbute). Other attrbutes assoated wth the hypervolume D an have the same meanngs and smlar values as ther ntal ounterparts but may be desrbed by another representatonal sheme. For nstane, the materal property beng desrbed by FRep n D an be represented by CFRep n D allowng eah ell to have ts own materal ndex. Completely new attrbutes an also appear n D. hus, suh an attrbute an desrbe normals to the ntal mplt surfae at the nodes of a mesh or an represent values of 3D ell volumes evaluated n advane and useful for speedng up FE-based alulatons. So, n general we have the resultng attrbutes defned as: Aj Aj ( G, G, A1,..., Ak ), j 1,... m..3 Problem formulaton Now we an formulate the dsretzaton problem as the problem of onverson of the ntal funtonally based heterogeneous objet D nto the objet D wth dsrete (mesh-based) geometry: D ( GF, A1,... Ak ) -> D ( G, A1,..., Am ), where G F s an FRep based model, and G s a CRep based model. he boundary of the objet D s an mplt surfae B F. As to the boundary B of D, wthn the entre dsrete model G t an be desrbed through a polyhedral omplex, whh s a boundary subomplex L of the omplex K 3, suh that L K 3 and r L { C j ;r 0,1,; 1,..., J r }. hen B ={ X X Ω E 3, X L }..4 Speal requrements Our approah s orented towards some demandng applatons suh as FEA. So, we take nto aount the followng requrements and onstrants upon the resultng dsrete strutures (surfae and volume meshes): 1. he topology of the surfae mesh B has to onform to the topology of the boundary of the ntal sold G.

3 . he surfae mesh B has to nlude all the sharp features of the surfae B; ths means that there should be onformty between ntal objet rdges and peaks, and edges and nodes desrbed by the omplex L. 3. he boundary mesh B must provde an adequate approxmaton of the underlyng mplt surfae B. o ths end, t s neessary to bound the dstane between the mesh and the ntal surfae and to lmt the devaton of normal vetors to the mesh from those to the mplt surfae. 4. here an be spef onstrants upon the shape of ells nluded n the omplexes K 3 and L, whh an be gven, for nstane, as Q(C ) <, where Q yelds a shape qualty measure based on the metr haratersts of tetrahedrons or trangles. 5. Some onstrants onerned wth proportons of adjaent ell szes n the omplexes K 3 and L an also be mposed. hey an be defned as M(C )/M(C j ) <, where M s a ertan measure of a tetrahedron (trangle) suh as the length of ts maxmal sde, or ts volume (square), or rumsrbed sphere (rle) radus. hs onstrant (along wth the prevous one) s useful for ensurng a relable tetrahedrzaton and onvergene of numeral smulaton proedures. 6. Intal objet attrbutes refletng some features of FE modelng an also result n spef onstrants upon surfae and volume meshes. Let us desrbe the followng ases: a) Gven the mesh densty attrbute A r along wth the orrespondng map funton S r (X), one an set the followng onstrant for eah element C of K 3 and L : M(S r (X),C ) <, where M s a ertan metr; b) Gven the attrbute A v along wth ts map funton S v (X), one an set the followng onstrant onerned wth the lmtaton of ths funton s varatons wthn eah C of K 3 and L : max Sv( X k ) Sv( X j ) X, X C k j.in prate, one usually ompares values of S v (X) at some referene ponts (e.g., n trangle nodes) of C. he funton S v (X) an desrbe some estmated haratersts of the proess beng modeled or just error estmatons. Attrbutes representng materal or medum propertes an also mpose onstrants of ths knd. ) Let S s (X) be a peewse ontnuous map funton defned for an attrbute A s. hen, whle deomposng the objet G, one should take nto aount that funton features, namely ts sngular ponts, have to onde wth mesh nodes, and lnes/surfaes of dsontnuty have to be exatly desrbed by 1D/D subomplexes of the omplex K 3 representng the dsrete model G. For example, suh requrements are neessary when a omputatonal doman s deomposed nto sub-areas orrespondng to dfferent materals or dfferent propertes of medum. Sngular ponts and lnes an also onform to dsposton of soures or to load applaton areas. 7. If no speal onstrants are formulated, the number of elements (ells) n omplexes K 3 and L should be mnmal. For example, an deal surfae mesh of a ube onssts of twelve trangles: two trangles per eah fae. 3. RELAED WORKS In ths seton, we gve a bref revew of works n the areas that are relevant n the ontext of our onsderaton of the dsretzaton problem: heterogeneous objet modelng, mplt surfae polygonzaton, and fnte element mesh generaton and refnement. 3.1 Heterogeneous objets modelng Partular attenton n sold modelng s pad to modelng heterogeneous objets wth multple materals and non-unform nternal materal dstrbuton. Boundary representaton, funtonally based, voxel and ellular models are used to represent suh objets. A non-manfold BRep sheme s used n [9] to subdvde an objet nto omponents made of unque materals. In the objet model proposed n [10], a fber bundle s used for general desrpton of all haratersts and attrbutes of an objet. Construtve operatons for modelng funtonally graded materals assoated wth a BRep geometr model are dsussed n [11]. Voxel arrays n volume modelng and graphs an be onsdered as dsrete attrbute models wth the default geometry represented by a boundng box. Construtve Volume Geometry (CVG) [1] utlzes voxel arrays and ontnuous salar felds for representng both geometry and photometr attrbutes (opaty, olor, et.). Issues of funtonally based modelng of volumetr dstrbuton of attrbutes are also addressed n [13-15]. Modelng heterogeneous objets as multdmensonal pont sets wth multple attrbutes s dsussed n [1]. he proposed onstrutve hypervolume model s based on funton representaton (FRep) [7] and supports unform onstrutve modelng of pont set geometry and attrbutes usng vetors of real-valued funtons of several varables. Multple materals are also represented n [3] by vetors of real-valued funtons. Dstane felds are used to model varyng materal propertes satsfyng dfferent types of onstrants predefned on the ntal objet geometry. he approah of [1] was extended n [] to dmensonally heterogeneous objets wth multple attrbutes by ombnng the funtonally based and ellular representatons nto a sngle hybrd model. In ths paper, we desrbe one of the mportant operatons n the hybrd model, namely the onverson between funtonally based and ellular heterogeneous objets. 3. Polygonzaton Exstng methods of the polygonal approxmaton (polygonzaton) of mplt surfaes nlude two major groups. he frst group, onsstng of ontnuaton algorthms, s haraterzed by ntrodung a seed loal trangulaton of the mplt surfae wth the onseutve addton of new trangles to the mesh by movng along the surfae [16-18] wth the trangle sze adapted to the loal surfae urvature [19].

4 he seond group nludes methods for generatng polygons as the result of the nterseton of the mplt surfae wth ells of a regular grd (see, for example, [0-]) or an adaptve grd [3]. he algorthms of ths group dffer n the type of grd and surfae sample ponts approxmaton. he man dsadvantages of the mentoned approahes are smoothng or uttng sharp features of the surfaes. Algorthms of sharp features extraton are presented n [4-6]. he optmzaton [6] s based on the speal vertex reloaton strategy and trangles subdvson and allows for extraton of sharp edges and peaks takng nto aount the surfae urvature. However, n the proess of optmzaton, the shapes and relatve szes of neghborng trangles are not ontrolled, whh an result n generaton of degenerate trangles. Moreover, these mesh optmzaton algorthms an produe an exessve number of trangles n the regons of not very hgh urvature, whh s also undesrable for further alulatons. hus, to satsfy the dsussed earler requrements of FEA, a speal mesh refnement whle stll preservng sharp features s needed. 3.3 Fnte element mesh generaton and refnement Issues of surfae mesh optmzaton for FEA are onsdered n detal n [5,7]. he desrbed tehnques are based on the onseutve applaton of dfferent mesh smplfaton, mesh subdvson, and mesh adaptaton proedures. Detals of suh operatons are dsussed also n [8-30] and other works. Note that n these works the surfae models are not defned n terms of analytal funtons but rather by means of trangulaton (resulted, for example, from measurements, CAD, bomedal engneerng). Durng the mesh refnement, the exat defnton of underlyng surfaes s unknown. When optmzng polygonzed mplt surfaes, we an use both approxmate and prese funtonal surfae models, whh provde for more prese alulatons of surfae haratersts and orretons of the node postons n respet to the underlyng surfae for remeshng. Issues of fnte element mesh generaton are dsussed n detal n [5]. Unstrutured mesh generaton methods are also surveyed n [31, 3]. etrahedrzaton s one of the wdely used methods of 3D dsretzaton. he man approahes to automat tetrahedral (trangular) mesh generaton nlude spatal deomposton based methods, Delaunay type methods, and advanng-front tehnques. Algorthms based on spatal deomposton are relatvely easy to mplement, but they do not allow for deteton of boundary sharp features and annot dstngush boundary enttes whh are rather lose but not dretly onneted. he boundary onnetvty onstrant s not taken nto aount n Delaunay tetrahedrzaton. So, loal mesh modfatons are neessary to ft the boundary. More aurate boundary representaton s supported by the advanng-front method. hs method starts from a doman boundary dsretzaton and marhes nto the regon to be proessed by addng one element at a tme. However, sne the method s based on loal operatons, onvergene problems may be enountered. he onvergene problem s ommon for all methods as there s no theoretal result whh an guarantee that a polyhedron wth the gven boundary trangulaton an be subdvded nto tetrahedrons wthout addng nternal ponts. In spatal deomposton methods, the onvergene problems appear when refnng small detals and sharp features. For the Delaunay type methods, the onvergene of the boundary fttng proedures has not been proven. We wll use the advanng-front tehnque as t s applable to arbtrary solds and allows us to ontrol shapes and szes of tetrahedrons durng the mesh generaton proess. In addton, we onsder a modfaton of ths tehnque for nreasng the effetveness of the tetrahedrzaton proedure for FRep solds. 3D mesh optmzaton proedures are desrbed n [33-37]. 4. GENERAL ALGORIHM OF DISCREIZAION In ths seton we provde a systemat desrpton of our dsretzaton algorthm. 4.1 General desrpton In prnple, there are two man strateges to dsretze a heterogeneous objet wth generaton of a volume mesh. he frst strategy nvolves deomposng an ntal 3D objet nto 3D elements (tetrahedrons, bloks, prsms, some ombned volume mesh) that are optmzed under requrements and onstrants desrbed n.4. Here, the boundary mesh B appears as a sde effet of the 3D ntal objet deomposton. Another approah mples that frst we deompose the surfae B of the ntal objet thus yeldng the surfae mesh B. hen, ths surfae mesh s subjeted to optmzaton and refnement to make sure that t satsfes all the requrements, and a volume mesh onformable to the refned surfae mesh an subsequently be bult. In our work, we follow the seond approah, beause most of the onstrants and requrements deal wth the boundary mesh B whose qualty s rual n the ontext of FEA. As some of the onstrants ontradt eah other, t s mportant to ensure that all the aessble teratve optmzaton proedures are performed to provde the best possble result. In addton, t s known that some effetve methods of volume mesh generaton are atually based on boundary desrptons of omputatonal domans [5]. So, we deompose the dsretzaton problem nto two relatvely ndependent sub-tasks: generaton of a mesh of the objet surfae along wth ts refnement: B B ( B, A,... A ) ; F 1 k generaton of a onformable volume mesh: G G ( B, G, A,..., A ). We use tetrahedral meshes as F 1 k the most unversal: they are suessfully used both n vsualzaton and n FEA, and they often serve as a base for buldng meshes onsstng of more omplex patterns. 4. Surfae dsretzaton Here, we desrbe how the frst task of generatng the qualty boundary mesh approxmatng the ntal objet surfae an be solved n step-by-step manner Polygonal approxmaton of the objet surfae 0 0 ( F We form a smplal omplex L L B ) representng (n aordane wth.3) an approxmate CRep based surfae model

5 B 0. Requrement 1 from.4 should be satsfed, and the subsequent steps on the surfae mesh reonstruton are suh that they preserve the surfae topology. We use a polygonzaton algorthm desrbed n [0], whh solves a problem of topologal ambgutes on the faes wth four edge-surfae nterseton ponts peular to the seond group of polygonzaton methods desrbed n 3.. We assume that the user an ontrol the preservaton of topology equvalene between the ntal mplt surfae B F and the resultng polygonal (trangular) surfae by B 0 provdng proper ntal data neessary for polygonzaton, namely the surfae boundng box and the grd resoluton. 4.. Sharp features reonstruton hs step deals wth optmzaton of the surfae mesh B 0 thus yeldng a new CRep based model, desrbed by the omplex B 1 L1 L1 ( L 0, BF ). In ths model, sharp edges and orners present n the ntal surfae B F are desrbed by the onformable 0D and 1D elements n the omplex L 1. Aordngly, the requrements 1 and from.4 are satsfed for the ellular model. o extrat B 1 sharp features from an mplt surfae oarse trangulaton, we use the algorthm desrbed n [6]. hs algorthm s based on ombnng the applaton of the followng mesh optmzaton proedures: urvature-weghted vertes resamplng; dual/prmal mesh optmzaton that nvolves projetng the trangle entrods onto the mplt surfae and movng eah vertex of the ntal surfae trangulaton to a new poston mnmzng the sum of the squared dstanes from the vertex to the planes whh are tangent to the mplt surfae at the projetons of adjaent trangles entrods. one-to-four subdvson of mesh trangles where the mesh normals have large devatons from mplt surfae normals. Our experene wth applaton of ths optmzaton algorthm shows that n most ases t does produe surfae mesh B allowng for a qualty representaton of sharp features of the ntal mplt surfae. However, the resultng mesh an have badly shaped or degenerate trangles near sharp edges and orners and may onsst of an exessve number of trangles produed by the adaptve subdvson proedure n the regons of low urvature. Consequently, we propose further refnement desrbed n the followng subsetons Surfae mesh refnement and smplfaton he objetve of ths phase s to get rd of badly shaped trangles, to make the mesh fner n the regons of hgh surfae urvature, and, on the ontrary, to make the mesh oarser n the areas where the urvature s low. he requrement for preservng sharp features should be satsfed. As an deal result, the surfaes of the tetrahedron and ube should be represented by just four and twelve trangles respetvely. 1 hus we am at buldng CRep based model B desrbed by a omplex optmzaton of the omplex L. he optmzaton proedure onssts n teratve applaton of edge swappng, edge splttng, edge ollapsng and vertex reloaton operatons. Edge splttng allows for enrhng the mesh, edge ollapsng provdes mesh smplfaton, and edge swappng and vertex reloaton operatons are used for mesh refnement. We desrbe n seton 5 the man haratersts of these operatons as well as the rtera for ther applaton. In the end, the model B desrbed by the omplex must ensure a qualty polygonal approxmaton of the mplt surfae B F under the requrements 1 4 from the subseton.4. L ( L L1, B F ), whh s obtaned as the result of L Surfae mesh adaptaton At ths step, we am at adaptng the mesh to the needs of FEA n the ontext of some partular applaton, thus produng Crep based surfae model. hs means that the requrements 4 6 B 4 from seton.4 should be satsfed. Frst, sngular lnes and ponts of attrbute funtons should be taken nto aount. We projet those lnes and ponts onto the dsretzed surfae B obtaned at the prevous step, and then make a partton of elements of the omplex L, thus yeldng a new omplex 3 3 F A L L ( L, B, 1,..., A k ) n whh the sngular lnes and ponts are desrbed by some 1D and 0D subomplexes. After ths proedure, t s useful to exeute the prevous step one agan to preserve not only geometr sharp features but also the sngular lnes and ponts of the attrbute funtons. hen, we make the adaptve mesh subdvson and refnement to satsfy the onstrants 4 6 through usng one agan edge splttng, edge swappng, and vertex reloaton operatons as well as the trangle subdvson proedure. Note that the rtera for applaton of those proedures do depend on both the attrbute funtons S (X) and the funton F(X) desrbng geometry (more detals are gven n Seton 5). Fnally, we get the omplex 4 4 ( 3, F, A L L L B 1,..., A k ) desrbng the surfae model B 4 that hopefully meets all the onstrants and requrements mportant for FE meshes D objet dsretzaton he seond task of buldng a qualty volume mesh based on the surfae mesh generated n the prevous stage s desrbed n ths seton etrahedral mesh generaton Gven the CRep based model of the surfae mesh B, desrbed by the omplex L we an now buld a CRep based L 4 model of the 3D objet G 0 G0 ( B4, GF, A1,..., A k ). o produe suh a volume mesh, we use the advanng front method brefly haraterzed n 3.3. More detals about our modfaton of that algorthm are gven n subseton 5.. he model G s B 4 0

6 desrbed by a 3D omplex K 3 0, whh s bult on the bass of the boundary omplex L C and on the ntal objet attrbutes. So, we have 3 3 K. Whle generatng suh a volume 0 K0 ( L, A1,..., Ak ) mesh, the onstrants 4-6 from.4 are taken nto aount. However, beause of the onvergene problems dsussed n 3.3, not all of those restrtons an mmedately be satsfed. So, addtonal post-remeshng may be needed to get a qualty tetrahedral mesh Volume mesh adaptaton he objetve of ths phase s to re-buld the volume mesh desrbed by the omplex K 3 0 that allows for mprovng the tetrahedra shapes under all the requrements for the mesh related to attrbutes. o sueed n solvng ths problem, we use fae swappng, tetrahedra subdvson and vertex reloaton operatons [34-37]. Some features of the mplementaton of these operatons are onsdered n 5. Note that one agan we pay attenton to preservng sharp features, and sngular ponts and lnes of the boundary surfae. As a result, we get a new omplex 3 3 K1 K1 ( L, A1,..., Ak ) defnng a CRep based model G 1 that should satsfy all the requrements rtal for FE meshes Attrbute transformatons he fnal step n dsretzaton of heterogeneous objets s onerned wth onvertng ntal attrbutes for gettng new ones: 3 A j Aj ( K1, L, A1,..., Ak ), j 1,,...,m. As we dsussed n.4, some attrbutes present n the ntal model an dsappear, others may be desrbed by another representaton sheme, and new attrbutes an appear. he attrbute onverson proedures an heavly depend on applaton spefs. Eventually, we get the sought-for dsrete model D ( G G1, A1,..., Am ) of the ntal heterogeneous objet D ( G F, A1,... A k ) wth CRep based geometry, and CRep and CFRep based attrbutes. 5. DEAILS OF KEY PROCEDURES In ths seton we present some operatons and proedures n more sgnfant detal. 5.1 Surfae mesh optmzaton Here we onsder n detal the bas mesh refnement operatons ntrodued n hese operatons are teratvely appled to the urrent mesh n order to provde an aurate peewse lnear approxmaton of the underlyng mplt surfae and to adapt the element szes to FEA requrements gven n.4. In partular, t s neessary to preserve sharp edges and orners of the surfae. For ths purpose, we ntrodue so-alled sharp funton Sh for the mesh edges and vertes. Usng the planarty estmatons [5] for eah edge e, we set: 1 ( n ( t1) n( t) 1 N ( t1) N ( t) Sh(e)=1, f and Sh(e)=0, otherwse. Here n( t1), n( t) are unt normals of the adjaent trangles t1, t orrespondngly; N ( t1), N ( t) are the mplt surfae normals at the entral ponts of t1, t; s a threshold that measures the sharpness of a feature. he mplt surfae normals are alulated as the normalzed gradents of the funton F(X) desrbng the mplt surfae. Note that a nonzero value of Sh(e) ndates that the edge e probably les on a sharp feature of the surfae. For the vertes, the sharp funton s defned as follows: Sh ( P) Sh( ), e where e s an edge ndent to the vertex P. If Sh(P) = s true, then vertex P les on a sharp edge, but the ase Sh(P)> orresponds to a orner pont. o sngle out those spkes that do not le on sharp edges, we use the heurst estmaton proposed n [4]. m n( m, n( t)) and m n( M, N ( t)) Here n(t) are unt normals of trangles t adjaent to pont P, N (t) are the mplt surfae normals at the entral ponts of t, m [ n t n 1] s the normal vetor to the plane spanned by two 0 t t0, nt normals n 1 whh enlose the largest angle, and smlarly M [ N ( t0) N ( t1)], s a threshold. For the vertes whh are found n suh a way, we set Sh(P)=3. Let us onsder all the bas mesh refnement operatons n more detal Mesh vertes reloaton hs operaton onssts of applyng two proedures: mesh smoothng and vertex movng. Mesh smoothng mproves regularty of elements sze. If V s a non-sharp vertex (.e., Sh(V )=0), then ts reloaton s desrbed as: ( V ) new ( V ) old U / U. Here s a small postve number ( < l mn, where l mn s the smallest length of all edges ndent to the pont) that lmts dsplaement of vertes, and takng t nto aount prevents the appearane of suh mesh defets as folds and self-nterseton. he movng dreton U R ( R N ) N, where U s defned by the formula [5]: N s the surfae normal at the pont. ( V ) old, 1 1 R w j Pj ( V ) old, w, w P ( ) j j j j V old Pj s a vertex havng a ommon edge wth V.

7 If the vertex V s a orner or belongs to more than two sharp edges, then ts poston does not hange. However, f the vertex V has exatly two ndent sharp edges, for example, 1 N ( e1) N ( e) e1 ( V, P1 ), e ( V, P ) and, where N ( e1), N ( e) are the surfae normals at the enters of edges e1, e, s a threshold, then a new pont ( V ) new s plaed at the enter of the ar onnetng vertes V, P 1, P. One may note that durng the mesh smoothng proess some vertes an be detahed from the ntal surfae; therefore, ther postons have to be orreted. he vertex movng proedure moves vertes towards the underlyng mplt surfae. It s organzed as an optmal searh proess n aordane wth the formula: In the proess of the fnte element mesh adaptaton, these haraterst funtons are defned wth respet to the salar and vetor attrbute funtons: W ( X ) W ( S1( X ),..., Sk ( X )) D( X ) D( S1( X ),..., Sk ( X )). In order to ontrol the qualty of trangles, we use the fnte element shape qualty measure [5]: l q( ) max, r where r k denotes the nradus of the trangle, l max s the largest edge length, and s a normalzaton oeffent so that q()=1 for an equlateral trangle. ( V ) new ( V ) r Z / Z, old * * where r arg mn F (( V ) old rz / Z ) r[0, ] wth Z F(( V ) ) F(( V ) ). old old 5.1. Edge swappng hs operaton s used to mprove the mesh elements shape qualty. Let us onsder an llustratve example n Fg. 1. We elmnate the dagonal (P 1,P ) and nstead nsert the dagonal (P 3,P 4 ) f max(1, ) < max(1, ). Fgure 1: Edge swappng hs operaton s appled only f adjaent trangles sharng the edge e are almost oplanar, so that (1 n ( t1) n( t) n 1 N and ( t1) N ( t) N where the onstants σ N and σ n are user-spefed thresholds Edge splttng hs operaton s performed for a number of reasons: to mprove geometry approxmaton, to ft attrbute funtons or to subdvde badly shaped trangles. o desrbe rtera for hoosng those edges whh are sutable for splttng, we ntrodue the followng denotatons. Let W(X) be a salar haraterst funton, and D (X ) be a vetor one. hen, n the ontext of the optmzaton proedure of geometry approxmaton, these funtons depend on the funton F(X) desrbng geometry of the ntal objet and on the gradent of F(X): W ( X ) F ( X ) D( X ) F( X ) / F( X )., Fgure : Edge splttng Our edge splttng operaton s llustrated by Fg.. he proess of mesh subdvson s teratve, and edges are proessed n the order of dereasng ther lengths. he followng estmatons are useful to measure: - qualty of elements: 1 length( e)/ Sr ( C) max( q( ), q( )) 1 - varatons of the salar haraterst funtons: max( W ( C) W ( P1 ), W ( C) W ( ) ) 1 P max( W ( C1) W ( P1 ), W ( C1) W ( P ), W ( C1) W ( P3 ) ) 3 max( W ( C) W ( P1 ), W ( C) W ( P ), W ( C) W ( P4 ) ) - varatons of the vetor haraterst funtons: 1 1 mn ( D, j{1,,3,4} j ( P), D( P )) N ( P1 ) P1 P N ( P ) P 1P P1, P 3 1 max((1 N ( C ) n( 1) ),(1 N ( C) n( ) )), where C s a enter of an edge, and C 1,C are entrods of adjanent trangles, S r s a mesh densty attrbute funton, N denotes surfae normals and n are normals of trangles. he edge e s subdvded f one of the lsted estmatons or several of them exeed the user-spefed thresholds. he splt vertex V s plaed at the poston S ( P1 ) V P1 r P1 P, S ( P ) S ( P ) r 1 r

8 where S r (X) s a mesh densty attrbute funton. hen, V s moved towards the mplt surfae aordng to the proedure desrbed n Dfferent haraterst funtons and estmatons an be used at the dfferent phases of mesh optmzaton and adaptaton proess. After a few teratons, t s reommended to exeute edge swappng and mesh smoothng operatons whh an mprove the mesh qualty. For quk mesh subdvson n the ases where an average mesh element sze s rather greater than the desred fnte element sze, we use one-to-four and one-to-three trangle subdvson oupled wth mesh smoothng and edge swappng Edge ollapsng hs operaton s used to smplfy the surfae mesh n low urvature regons. We hoose an edge e and replae t wth a vertex V (see Fg. 3a). In ths proess, two vertes P1, P are substtuted wth a new vertex V, and the trangles 1, are ollapsed to edges. edge ollapsng and the poston of a new vertex V depends on the edge vertes weghts as follows: f w(p1)=w(p)=0, then the edge e annot be ollapsed; f w(p1)=w(p)>0, then the edge e an be substtuted wth the vertex V plaed at the mdpont of e; f w(p1) > w(p) 0, then the edge e an be substtuted wth the vertex V = P ; f w(p) > w(p1) 0, then the edge e an be substtuted wth the vertex V =P 1. hen we move the new vertex V onto the mplt surfae usng the proedure desrbed n Fnally, we evaluate the qualty degradaton durng the ntended ollapse operaton usng the followng measures: 1 max ((1 N n( V ) ( C ) n( ) )) ( N ( V ) VPj ) max P jstar ( V ) VPj Here, N denotes normals of the ntal surfae, n s for normals of trangles, C s a entrod of the trangle, and =n(v) s for trangles ndent to V. a Fgure 3: Edge ollapsng We hek for ollapse andday edges n the order of nreasng length. he proess s as follows. Frst, we hek the topologal valdty of the edge ollapsng operaton for eah edge. Edge e s a anddate for substtuton wth a new vertex f the followng ondton s satsfed: Star(P1) Star(P) = {P3, P4}, where Star(P) s the set of the mesh vertes sharng a ommon edge wth the vertex P. hs restrton allows for avodng mesh degradaton n suh ases as shown n Fg.3b. Seondly, f length(e) σ mn, then the edge e s ollapsed wthout need for any addtonal analyss, and the mdpont of e s hosen as the poston of a new vertex V. Otherwse, we ontnue hekng e for a andday edge. For ths purpose, we alulate so-alled weght oeffents w(p 1 ), w(p ) for the edge vertes. We set w(p )=1, f Sh(P )=0. he values Sh(P )=1 or Sh(P ) > ndate that the vertex P s a spke or a orner, so we set w(p )=0 n ths ase. From (Sh(P )= & Sh(e)=0) t follows that there exst other sharp edges ndent to P, so we set w(p )=0. But f e s one of the two sharp edges ndent to P (.e. Sh(p)= & Sh(e)=1), then we should alulate the angle α between these edges. If os(α)<0 and (1- os(α) ) σ α, then we set w(p )=0.5, otherwse w(p )=0. Let us formulate the ondtons under whh one an onsder exeuton of the edge ollapsng operaton. he possblty of b If 1 σ t and σ n, then the orrespondng edge e s ollapsed and substtuted wth the vertex V. Constants σ mn, σ α, σ t, and σ n are user-spefed thresholds. It s reommended to ouple the edge ollapsng operaton wth mesh smoothng and edge swappng. 5. Volume mesh generaton and optmzaton 5..1 he modfaton of the advanng front method Frst, let us haraterze some rtal steps of the advanng front algorthm (brefly outlned n 3.3) as appled to tetrahedrzaton. A surfae mesh desrbed by a D omplex forms the ntal front. hen, at eah step of the algorthm an atve fae s seleted to serve as a base for buldng a new tetrahedron whose sze s alulated takng nto aount the gven mesh densty attrbute. Provded the new tetrahedron does not ross the front, t s added to the front thus formng a new one. o avod emergene of thn elements, a new tetrahedron vertex s plaed n the losest node of the mesh satsfyng the gven dstane threshold. here are dfferng strateges [5] for seleton of both the next atve fae and new tetrahedron node that prevent the proedure from generatng an nfnte seres of tetrahedrons yet yeld a volume mesh of reasonable qualty (e.g., wthout thn elements, et.). We propose a modfaton of the advanng front method that takes advantage of the fat that we deal wth the problem wthn the ellular-funtonal framework. In partular, avalablty of an exat funtonal desrpton of the objet to be subdvded an greatly smplfy the proedure of the evaluatng pont membershp relaton whh s mportant n the ontext of determnng whether a tetrahedron rosses the front. Now, let us outlne the modfed algorthm. We start from the CRep based model of the surfae mesh B represented by the

9 smplal omplex L and the funton S r (X) desrbng the mesh densty attrbute A r. Frst, some sample tetrahedral mesh overng the objet s generated. hs mesh s desrbed by a smplal omplex M 3 0. hen, elements of M 3 0 are subdvded to onform to the mesh densty attrbute A r. For ths purpose, 3D mesh optmzaton operatons (see 5..) are appled. So, the modfed mesh s represented by the smplal omplex M 3 1 = M 3 1 (M 3 0, A r ). At the next step, we sngle out the subomplex M 3 lyng ompletely nsde the subdvded objet G F. So M 3 M 3 1, M 3 G. o ensure the onvergene of the algorthm, the dstane between the boundary surfae of the objet and the mesh desrbed by the omplex M 3 must be a few tmes greater than the tetrahedron sze desrbed by the funton S r (X). Note that algebra dstanes between underlyng mesh nodes and the objet surfae are estmated usng the objet s defnng funton. herefore, M 3 nludes only those elements of M 3 1 whose nodes V satsfy the followng ondton: F(V ) k*s r (V ) 0, where k a b Fgure 4: Modfed advanng front method: a) the ntal retangular objet wth the pattern mesh subdvded aordng to a mesh densty attrbute b) the sub-mesh lyng ompletely nsde the subdvded objet ) the fnal mesh hen we defne a two-dmensonal omplex Q that represents the boundary of M 3. akng nto onsderaton that the surfae normals pont nsde, one need to hange the orentaton of Q elements. hen, ths omplex together wth the omplex L desrbng the dsretzaton of the boundary surfae form the front Q 3= L Q. Usng ths front, we subdvde the remanng sub-area, wth help of the advanng front tetrahedrzaton method and thus get the 3D omplex M 3 3= M 3 3(Q 3,S r (X)). he overall omplex K 3 0 defnng a CRep based model G s alulated as K 3 0= M 3 M 3 3. hen, we apply a few teratons of 3D mesh optmzaton to the omplex K 3 0 to ensure that we get a volume mesh of good qualty and regularty, and that all the FE related onstrants are satsfed. As a result we get a new omplex K K K, A,..., A ). 1 1 ( 0 1 k Fg. 4 shows a D llustraton of the desrbed algorthm. he ntal retangular objet wth the pattern mesh subdvded aordng to a mesh densty attrbute Sr(X) s shown n Fg 4a. Fg. 4b llustrates the sub-mesh lyng ompletely nsde the subdvded objet. he area between the ntal boundary and that sub-mesh s subdvded usng the advanng front method. he fnal mesh s shown n Fg 4. Our modfed frontal method s more effetve n ases when a onsderable part of the objet an be overed by pattern mesh elements that reman unhangeable. In these ases we redue the number of elements generated by the frontal tetrahedrzaton and 1 keep well shaped elements of the pattern mesh. Moreover our modfed algorthm allows for ombnaton of meshes of dfferent types, for example, the ntal mesh overng the objet an onsst of hexahedral elements. 5.. Volume mesh optmzaton operatons he 3D mesh optmzaton algorthm onssts of an teratve applaton of fae swappng, tetrahedral splttng and vertex repostonng tehnques. he fae swappng operaton s the 3D extenson of the edge swappng tehnque desrbed n wo neghborng tetrahedra havng a ommon fae are transformed wth the exhange of the ommon dagonal. he tetrahedra splttng operaton nludes one-to-two subdvson by bsetng the longest edge. he seleton of a tetrahedron to be splt s made on the bass of rtera smlar to those desrbed n 5.1. for the edge splttng operaton for a surfae mesh. he vertex reloaton tehnque n 3D onssts n the movement of a node towards the baryenter of the polyhedron formed by the surroundng tetrahedral mesh. Sne the surroundng polyhedron an be non-onvex, the postonng of the node dretly at the baryenter an result n overlappng tetrahedra. hs s the reason for usng an adaptve proedure wth a varable step of the movement to the baryenter. More general three dmensonal versons of swap and splt operators remesh a polyhedron formed by neghbourng tetrahedra sharng a ommon edge or a ommon vertex [5]. 6. EXAMPLES In ths seton we present several examples llustratng our algorthm for dsretzaton of funtonally based heterogeneous objets. he examples were prepared usng our orgnal software tools ntally ntended for data preproessng n omputatonal physs and allowng for the users nteratve work n a step-bystep manner. We used HyperFun modelng language [1] to defne all the models, and spealzed software tools wth the bult-n HyperFun nterpreter have been used to mplement the examples on a Pentum III 800 MHz omputer. 6.1 etrahedrzaton of FRep solds Here we llustrate the man steps of the algorthm usng the example of an FRep objet wth sharp features shown n Fgure 5. he ntal surfae mesh produed by polygonzaton s shown n Fg 5a. he surfae mesh generated n the proess of sharp features reonstruton s shown n Fg. 5b. For the purpose of FEA, ths mesh has badly shaped and degenerate trangles near sharp edges and orners, and nludes an exessve number of trangles produed by the adaptve subdvson proedure n the regons of low urvature. Subsequent mesh optmzaton and smplfaton produes the mnmal surfae trangulaton presented n Fg. 5. hs trangulaton then serves as an ntal front whle applyng our advanng front algorthm. Fg. 5d shows the resultng tetrahedrzaton. etrahedrzaton of another FRep objet havng more omplex topology s presented n Fg. 6. It s a typal CSG-lke objet wth

10 sharp edges and both flat and urvlnear faes. he result of polygonzaton and sharp features extraton s shown n Fg. 6a. he mesh generated s not ompletely sutable for subsequent FEA: one an observe badly shaped and degenerate trangles near sharp edges and orners n the surfae mesh after the sharp features extraton. he surfae mesh adaptaton to FEA requrements after applyng mesh optmzaton proedures s shown n Fg. 6b. An enlarged vew of an nternal struture of the tetrahedral mesh s shown n Fg. 6. As to omputatonal tmes, mesh dematon proess took 4. se, FE mesh adaptaton took.9 se, and tetrahedrzaton took 331 se. 6. Dsretzaton of a heterogeneous objet wth varous attrbutes hs example llustrates the nfluene of attrbutes on the mesh elements szes. Fg. 7 presents four dsretzaton varants onformng to dfferent attrbutes. We start from Fg. 7b where no attrbute has an effet. Note that the mnmal surfae trangulaton for ths objet was reated usng our mesh optmzaton and smplfaton algorthm smlar to the prevous example. hen, ths surfae mesh was subdvded to onform to the attrbutes. Fgs. 7-7e llustrate applaton of varous types of attrbute funtons. Fg. 7 shows the result of subdvson based on an attrbute nfluenng the mesh densty and defned by a pont soure plaed at a ube orner. Fgure 7d smulates mesh adaptaton near the vertal well, where the element szes are adjusted to the reservor pressure gradent. An example of a mesh ftted to an annular heatng deve s shown n Fg.7e. Mesh adaptaton proess n all these examples took no longer than 7 se. 6.3 Modelng of a mxng tank mpeller An applaton example shown n Fg. 8 s onerned wth real omputer-aded modelng of a mxng tank mpeller whh s used as the agtator of the flow omponents n a hemal reator. he mpeller whose entral body and blades are made of dfferent materals (marked by dfferent olors) s shown n Fg.8a. Fg. 8b shows the surfae mesh wth reovered sharp features. One an observe that there are many badly shaped trangles n the mesh, so the followng mesh optmzaton s neessary. Fg. 8 llustrates the result of suh optmzaton. he surfae mesh takng nto aount the materal attrbutes s shown n Fg. 8d. he mesh was splt on the blades faes aordng to the spefed attrbute. hs surfae mesh then serves as ntal data for the advanng front tetrahedrzaton. he resultng 3D mesh was used for FE thermal and stress-stran analyss. As to alulaton tmes, mesh dematon took 3.1 se, FE mesh adaptaton took 10 se, mesh adaptaton aordng to the attrbute took 7 se, and tetrahedrzaton took 634 se. 7. CONCLUSION hs work ams at makng funtonally defned solds and heterogeneous objets wth mplt surfaes avalable for pratal applatons requrng fnte element analyss and smulaton. We dsussed a dsretzaton proedure resultng n surfae and volume meshes for heterogeneous objets wth geometry and attrbutes defned usng real-valued funtons. hs proedure s an mplementaton of the funtonal to ellular models onverson operaton n the ellular-funtonal modelng framework for heterogeneous objets [,6]. In ontrast to prevous works on mplt surfae polygonzaton and volume mesh generaton, the man motvaton of ths work s generaton of meshes sutable for fnte element analyss wth onstrants mposed by the heterogeneous objet attrbutes. Let us summarze the man ontrbutons of the paper: 1) he dsretzaton problem s stated for heterogeneous objets represented as funtonally defned 3D solds (wth mplt surfaes) and attrbutes (salar felds). ) he paper aggregates dfferent tehnques nto a systemat step-by-step proedure of solvng the above problem. Some of the known tehnques have been adapted and extended to work wth the funtonally defned solds as explaned below. 3) he proposed surfae mesh optmzaton preserves sharp features of mplt surfaes, satsfes the requrements of FEA, and uses the defnng funton to do ths (see the mesh transformaton from Fg. 5b to 5). 4) he advanng front method of 3D tetrahedrzaton s extended, takng the defnng funton of the sold nto aount. We provded examples of tetrahedrzaton of objets wth omplex topology and sharp features, and llustrated mesh adaptaton to attrbutes of varous types. A pratal example of FEM generaton for a mxng tank mpeller was gven. All examples have been prepared usng software tools developed by the authors. In the ellular-funtonal model [,6], heterogeneous objets are represented as hypervolumes or multdmensonal pont set wth multple attrbutes. A multdmensonal pont sets an nlude elements of dfferent dmensons, whh an be hgher than three. In ths paper, we dealt only wth 3D objets wth attrbutes. he extenson of the proposed algorthms to dmensonally heterogeneous and tme-dependent objets s a subjet of future work. 8. ACKNOWLEDGMENS he authors would lke to thank Drs. Y. Ohtake and A. Belyaev for frutful dsussons and for the possblty to work wth ther mplementaton of the dynam meshes algorthm. Jody Vlbrandt has greatly helped wth makng the text more reader frendly. 9. REFERENCES [1] Pasko, A., Adzhev, V., Shmtt, B., Shlk, C., 001, Construtve Hypervolume Modellng, Graphal Models, a speal ssue on Volume Modelng, 63(6), pp [] Adzhev, V., Kartasheva, E., Kun,., Pasko, A., Shmtt, B., 00, Cellular-funtonal Modelng of Heterogeneous Objets, Pro. 7 th ACM Symposum on Sold Modelng and Applatons, Kunwoo Lee, N. Patrkalaks, eds., Saarbruken, Germany, ACM Press, pp [3] Bswas, A., Shapro, V., sukanov, I., 00, Heterogeneous Materal Modelng wth Dstane Felds, ehnal Report SAL 00-4, Unversty of Wsonsn-Madson USA. [4] V. Shapro, V., sukanov, I., 1999, Meshfree Smulaton of Deformng Domans, Computer Aded Desgn, 31(7), pp

11 [5] Frey, P.J., George, P.-L., 000, Mesh Generaton: Applaton to Fnte Elements, HERMES Sene Europe, OXFORD & PARIS, 814p. [6] Adzhev, V., Kartasheva, E., Kun,., Pasko, A., Shmtt, B., 00, Hybrd Cellular-funtonal Modelng of Heterogeneous Objets, Journal of Computng and Informaton Sene n Engneerng, (4), pp [7] Pasko, A., Adzhev, V., Sourn, A., Savhenko, V., 1995, Funton Representaton n Geometr Modellng: Conepts, Implementaton and Applatons, he Vsual Computer, 11(8), pp [8] Lohner, R., 1997, Automat Unstrutured Grd Generators, Fnte Elements n Analyss and Desgn, 5, pp [9] Kumar,V., Dutta, D., 1997, An Approah to Modelng Multmateral Objets, Pro. 4th Symposum on Sold Modelng and Applatons, ACM SIGGRAPH, Atlanta, pp [10] Kumar, V., Burns, D., Dutta, D., Hoffmann, C., 1999, A Framework for Objet Modelng, Computer-Aded Desgn, 31(9), pp [11] Shn, K., Dutta, D., 001, Construtve Representaton of Heterogeneous Objets, Journal of Computng and Informaton Sene n Engneerng, 1(3), pp [1] Chen, M., uker, J., 000, Construtve Volume Geometry, Computer Graphs Forum, 19(4), pp [13] Jakson,. R., Lu, H., Patrkalaks, N. M., Sahs, E. M., and Cma, M. J., 1999, Modelng and Desgnng Funtonally Graded Materal Components for Fabraton wth Loal Composton Control, Materals and Desgn, 0(/3), pp [14] Martn, W., Cohen, E., 001, Representaton and Extraton of Volumetr Attrbutes Usng rvarate Splnes: a Mathematal Framework, Pro. 6th ACM Symposum on Sold Modelng and Applatons, D. Anderson, K. Lee, eds., Ann Arbor, ACM Press, pp [15] Park, S. M., Crawford, R., Beaman, J., 001, Volumetr Mult-texturng for Funtonally Gradent Materal Representaton, Pro. 6 th ACM Symposum on Sold Modelng and Applatons, D. Anderson, K. Lee, eds., Ann Arbor, ACM Press, pp [16] Wyvll, G., MPheeters, C., Wyvll, B., 1986, Data struture for soft objets, he Vsual Computer, (4), pp [17] Bloomenthal, J., 1994, An Implt Surfae Polygonzer, Graphs Gems IV, P. Hekbert, (ed), Aadem Press, pp [18] Hartmann, E., 1998, A Marhng Method for the rangulaton of Surfaes, he Vsual Computer, 14(3), pp [19] Karkans,., Stewart, A. J., 001, Curvature-dependent rangulaton of Implt Surfaes, IEEE Computer Graphs and Applatons, 1(), pp [0] Pasko, A., Plyugn, V., Pokrovsky, V., 1986, Geometr Modelng n the Analyss of rvarate Funtons, Communatons of Jont Insttute of Nulear Researh, P , Dubna, USSR, (n Russan). Englsh translaton: 1988, Computers and Graphs, 1(3/4), pp [1] Lorensen, W., Clne, H., 1987, Marhng Cubes: a Hgh Resoluton 3D Surfae Construton Algorthm, Computer Graphs, 1(4), pp [] Nelson, G., Hamann, B., 1991, he Asymptot Deder: Resolvng the Ambguty n Marhng Cubes, Pro. Vsualzaton '91, IEEE Computer Soety Press, pp [3] Shmdt, M., 1993, Cuttng Cubes Vsualzng Implt Surfaes by Adaptve Polygonzaton, he Vsual Computer, 10(), pp [4] Kobbelt, L., Botsh, M., Shwaneke, U., Sedel, H.-P., 001, Feature Senstve Surfae Extraton from Volume Data, Pro. SIGGRAPH 001, pp [5] Ohtake, Y., Belyaev, A., Pasko, A., 001, Dynam Meshes for Aurate Polygonzaton of Implt Surfaes wth Sharp Features, Shape Modelng Internatonal 001, IEEE Computer Soety, pp [6] Ohtake Y., Belyaev A., 00, Dual/prmal Mesh Optmzaton for Polygonzed Implt Surfaes, Pro. 7 th ACM Symposum on Sold Modelng and Applatons, K. Lee and N. Patrkalaks, eds., ACM Press, Saarbruken, pp [7] Frey, P.J., Borouhak, H., 1998, Geometr Surfae Mesh Optmzaton, Computng and Vsualzaton n Sene, 1(3), pp [8] Kobbelt, L., 000, 3-Subdvson,, Pro. SIGGRAPH 000, pp [9] Garland M., Hekbert, P.S., 1997, Surfae Smplfaton Usng Error Metrs, Pro. SIGGRAPH 001, pp [30] Sheffer, A., 001, Model Smplfaton for Meshng Usng Fae Clusterng, Computer-Aded Desgn, 33(13), pp [31] Owen, S. J., 1998, A Survey of Unstrutured Mesh Generaton ehnology, Pro. 7 th Internatonal Meshng Roundtable, Dearborn, MI. [3] Sethan, J., A., 1999, Level Set Methods and Fast Marhng Methods, Cambrdge Unversty Press. [33] Frank, K., Lang, U., 000, Gradent and Curvature Approxmaton n Data-dependent Surfae Smplfaton, Computng and Vsualzaton n Sene, (4), pp [34] Fretag, L., Ollver-Gooh, C., 1997, etrahedral Mesh Improvement Usng Swappng and Smoothng, Int. J. Numer. Meth. Eng., 40, pp [35] Rvara, M., Levn, C., 199, A 3D Refnement Algorthm Sutable for Adaptve and Mult-grd ehnques, J. Comp. and Appl. Math., 8, pp [36] Lu, A., Joe, B., 1994, On the Shape of etrahedra from Bseton, Math. Comp., 63, pp [37] Lu, A., Joe, B., 1995, Qualty Loal Refnement of etrahedral Meshes Based on Bseton, SIAM J. S. Comput., 16, pp