Discretization of Functionally Based Heterogeneous Objects

Transcription

1 Dscretzaton of Functonally Based Heterogeneous Objects Elena Kartasheva Insttute for Mathematcal Modelng Russan Academy of Scence Moscow, Russa Oleg Fryaznov Insttute for Mathematcal Modelng Russan Academy of Scence Moscow, 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 Mathematcal Modelng Russan Academy of Scence Moscow, Russa ABSRAC he presented approach to dscretzaton of functonally defned heterogeneous objects s orented towards applcatons assocated wth numercal smulaton procedures, for example, fnte element analyss (FEA). Such applcatons mpose specfc constrants upon the resultng surface and volume meshes n terms of ther topology and metrc characterstcs, exactness of the geometry approxmaton, and conformty wth ntal attrbutes. he functon representaton of the ntal object s converted nto the resultng cellular representaton descrbed by a smplcal complex. We consder n detal all phases of the dscretzaton algorthm from ntal surface polygonzaton to fnal tetrahedral mesh generaton and ts adaptaton to specal FEA needs. he ntal object attrbutes are used at all steps both for controllng geometry and topology of the resultng object and for calculatng new attrbutes for the resultng cellular representaton. Categores and Subject Descrptors I.3.5 [Computer Graphcs]: Computatonal Geometry and Object Modelng Curve, surface, sold, and object modelng, Physcally based modelng; I.3.6 [Computer Graphcs]: Methodology and echnques; I.3.8 [Computer Graphcs]: Applcatons. General erms Algorthms, Desgn. Keywords Heterogeneous objects, attrbutes, functon representaton, cellular representaton, volume modelng, constructve hypervolume, fnte element analyss, mesh. 1. INRODUCION In ths paper, we deal wth generaton of dscrete models for heterogeneous objects defned usng real-valued functons. Generally, heterogeneous objects have an nternal structure wth non-unform dstrbuton of materal and other attrbutes of an arbtrary nature (photometrc, physcal, statstcal, etc.), and elements of dfferent dmenson. Recently we can observe a steady Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. o copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. SM 03, June 16-0, 003, Seattle, Washngton, USA. Copyrght 003 ACM /03/0006 $5.00. nterest n functonally based geometrc models such as mplct surfaces, functonally defned solds and heterogeneous objects (see [8, 1, ] for detals). hese models provde compact and ntutve mathematcal representaton for complex heterogeneous objects, support set-theoretc and other operatons such as offsettng, blendng, and sweepng. Rapd development of hardware, computatonal algorthms, and specalzed software allow for manpulaton of such models at nteractve rates. Practcal applcatons of functonally based models n CAD/CAM/CAE and fnte element analyss (FEA) requre some key procedures to be also applcable to these models. Numercal FEA methods use dscrete models (surface and volume meshes) of geometrc objects, although meshfree analyss and smulaton methods are also emergng [34]. Algorthms for fnte element mesh generaton are well developed for boundary and spatal enumeraton representatons [8]. Meshes are also actvely used now n vsualzaton, anmaton, computatonal geometry, mage processng, and other areas. However, requrements for dscrete models n FEA are strcter than n other areas. Such requrements are formulated n terms of dscrete model topology and metrc characterstcs, exactness of the geometry approxmaton, and conformty wth ntal attrbutes. hs results from the use of meshes n FEA for approxmaton of systems of equatons of mathematcal physcs. he sze and shape of mesh elements and mesh structure serously nfluence the stablty of numercal smulaton procedures and accuracy of obtaned solutons. hat s why meshes used for vsualzaton usually do not satsfy FEA requrements and specal refnement of them s needed. he above s the motvaton for work on dscrete model generaton for functonally based surfaces, solds, and heterogeneous objects. In ths paper, we deal wth the dscretzaton problem wthn the hybrd cellular-functonal model [1] of heterogeneous objects. he functon representaton of the ntal 3D heterogeneous object s converted nto the resultng cellular representaton descrbed by a smplcal complex. We consder n detal the followng phases of the dscretzaton algorthm: ntal surface polygonzaton; teratve smplfcaton and refnement of the surface mesh along wth sharp features reconstructon; further adaptaton of the surface mesh accordng to constrants dependng on the attrbutes of the ntal object; volume (tetrahedral) mesh generaton usng a modfed advancng front method; ts subsequent adaptaton to specal FEA needs. he ntal object attrbutes are used at all steps both for controllng geometry and topology of the resultng object and for calculatng new attrbutes for the resultng cellular representaton. 145

2 . PROBLEM SAEMEN In ths secton, we provde a formulaton of the dscretzaton problem n terms of ntal and resultng objects along wth a set of specal requrements. We consder the problem of dscretzaton of functonally based heterogeneous objects wthn a hybrd cellular-functonal representaton framework [1] n whch objects are treated as hypervolumes (multdmensonal pont sets wth multple attrbutes) [8]..1 Intal heterogeneous object Let D be an ntal heterogeneous object 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 object s geometry G s descrbed by the functon representaton (FRep) [7]: G F = {X Χ = (x 1, x, x 3 ) Ω E 3, F(X) 0 }, where Ω s a modelng space and F: Ω -> R s (at least a C 0 contnuous) real-valued defnng functon. Note that the boundary of the object D s an mplct surface descrbed as B F = {X Χ = (x 1, x, x 3 ) Ω E 3, F(X) = 0 }. Each attrbute A s defned by ts set of values N R m along wth a map functon S (X): Ω->N and can be represented by any of the attrbute models ntroduced n [1] that dffer from each other n the way of defnng S (X). For nstance, the functon representaton (FRep) can be used for defnng attrbutes representng electrc or thermal feld dstrbuton as well as load dstrbuton. Cellular-functonal representaton (CFRep) s especally sutable for descrpton of materal attrbutes. Cellular models of attrbutes (CRep) are used n fnte element calculatons for representaton of functons descrbng smulated processes. What s mportant n the context of our consderaton of the dscretzaton problem s that attrbutes can provde a formal descrpton of requrements and constrants mposed by FEA. hus, constrants on the sze of elements can be expressed through the mesh densty attrbute A r whose map functon S r (X) defnes a proper element sze at each pont X of the modelng space Ω. Such an attrbute can be gven by the user or can be calculated on the bass of other gven attrbutes A. here s a promsng way of defnng A r through FRep on the bass of sources [19], [1]. As to defnng A r through CRep, t s approprate when the sze dstrbuton functon s defned n a dscrete manner and ts values are known at the vertces of a background geometrc complex. Such a complex can descrbe ether a regular mesh specally bult for defnng the attrbute A r or an FE mesh used n the prevous steps of adaptve numercal calculatons.. Resultng heterogeneous object Gven the ntal object D, we are gong to buld a resultng heterogeneous object a hypervolume D = ( G, A 1,..., A m). Here, the geometry component G s an approxmate dscrete representaton of the ntal geometry G, and attrbutes A,..., A ) ( 1 m descrbe the object s propertes. Formally, G can be expressed as a partcular case of a model based the cellular representaton (CRep) [1]: G c ={ X X Ω E 3, X K 3 }, 3 r where K = { C ; r = 0,1,,3; = 1,..., Ir} s a three-dmensonal polyhedral complex consstng of cells C r. In conventonal terms, such a dscrete model s called a mesh. As we have already stated, the resultng mesh G c depends not only on the ntal geometry G but also on the attrbutes A 1, A k (n partcular, on the mesh densty attrbute A r ): G c = Gc ( GF, A1,..., Ak ). Note that the attbutes A A 1,..., m can dffer from the ntal ones A 1,,A k n terms of ther number, set of values, and descrpton. Some of the ntal attrbutes may not have drect counterparts n the dscrete model (e.g., there may be no need to retan the mesh densty attrbute). Other attrbutes assocated wth the hypervolume D can have the same meanngs and smlar values as ther ntal counterparts but may be descrbed by another representatonal scheme. For nstance, the materal property beng descrbed by FRep n D can be represented by CFRep n D allowng each cell to have ts own materal ndex. Completely new attrbutes can also appear n D. hus, such an attrbute can descrbe normals to the ntal mplct surface at the nodes of a mesh or can represent values of 3D cell volumes evaluated n advance and useful for speedng up FE-based calculatons. So, n general we have the resultng attrbutes defned as: Aj = Aj ( G, G, A1,..., Ak ), j = 1,... m..3 Problem formulaton Now we can formulate the dscretzaton problem as the problem of converson of the ntal functonally based heterogeneous object D nto the object D wth dscrete (mesh-based) geometry: D = ( GF, A1,... Ak ) -> D = ( Gc, A1,..., Am ), where G F s an FRep based model, and G c s a CRep based model. he boundary of the object D s an mplct surface B F. As to the boundary B c of D, wthn the entre dscrete model G c t can be descrbed through a polyhedral complex, whch s a boundary subcomplex L of the complex K 3, such as L K 3 and r L = { C j ; r = 0,1,; = 1,..., J r }. hen B c ={ X X Ω E 3, X L }..4 Specal requrements Our approach s orented towards some demandng applcatons such as FEA. So, we take nto account the followng requrements and constrants upon the resultng dscrete structures (surface and volume meshes): 1. he topology of the surface mesh B has to conform to the topology of the boundary of the ntal sold G. 146

3 . he surface mesh B has to nclude all the sharp features of the surface B; ths means that there should be conformty between ntal object rdges and peaks, and edges and nodes descrbed by the complex L. 3. he boundary mesh B must provde an adequate approxmaton of the underlyng mplct surface B. o ths end, t s necessary to bound the dstance between the mesh and the ntal surface and to lmt the devaton of normal vectors to the mesh from those to the mplct surface. 4. here can be specfc constrants upon the shape of cells ncluded n the complexes K 3 and L, whch can be gven, for nstance, as Q(C ) < ε, where Q yelds a shape qualty measure based on the metrc characterstcs of tetrahedrons or trangles. 5. Some constrants concerned wth proportons of adjacent cell szes n the complexes K 3 and L can also be mposed. hey can be defned as M(C )/M(C j ) < ε, where M s a certan measure of a tetrahedron (trangle) such as the length of ts maxmal sde, or ts volume (square), or crcumscrbed sphere (crcle) radus. hs constrant (along wth the prevous one) s useful for ensurng a relable tetrahedrzaton and convergence of numercal smulaton procedures. 6. Intal object attrbutes reflectng some features of FE modelng can also result n specfc constrants upon surface and volume meshes. Let us descrbe the followng cases: a) Gven the mesh densty attrbute A r along wth the correspondng map functon S r (X), one can set the followng constrant for each element C of K 3 and L : M(S r (X),C ) < ε, where M s a certan metrc; b) Gven the attrbute A v along wth ts map functon S v (X), one can set the followng constrant concerned wth the lmtaton of ths functon s varatons wthn each C of K 3 and L. In practce, one usually compares values of S v (X) at some reference ponts (e.g., n trangle nodes) of C. he functon S v (X) can descrbe some estmated characterstcs of the process beng modeled or just error estmatons. Attrbutes representng materal or medum propertes can also mpose constrants of ths knd. c) Let S s (X) be a pecewse contnuous map functon defned for an attrbute A s. hen, whle decomposng the object G, one should take nto account that functon features, namely ts sngular ponts, have to concde wth mesh nodes, and lnes/surfaces of dscontnuty have to Cellular-functonal representaton (CFRep) s especally sutable for descrpton of materal attrbutes. Cellular models of attrbutes (CRep) are used n fnte element calculatons for representaton of functons descrbng smulated processes. be exactly descrbed by 1D/D subcomplexes of the complex K 3 representng the dscrete model G c. For example, such requrements are necessary when a computatonal doman s decomposed nto sub-areas correspondng to dfferent materals or dfferent propertes of medum. Sngular ponts and lnes can also conform to dsposton of sources or to load applcaton areas. 7. If no specal constrants are formulated, the number of elements (cells) n complexes K 3 and L should be mnmal. For example, an deal surface mesh of a cube conssts of twelve trangles: two trangles per each face. 3. RELAED WORKS In ths secton, we gve a bref revew of works n the areas that are relevant n the context of our consderaton of the dscretzaton problem: heterogeneous object modelng, mplct surface polygonzaton, and fnte element mesh generaton and refnement. 3.1 Heterogeneous objects modelng Partcular attenton n sold modelng s pad to modelng heterogeneous objects wth multple materals and non-unform nternal materal dstrbuton. Boundary representaton, functonally based, voxel and cellular models are used to represent such objects. A non-manfold BRep scheme s used n [15] to subdvde an object nto components made of unque materals. In the object model proposed n [16], a fber bundle s used for general descrpton of all characterstcs and attrbutes of an object. Constructve operatons for modelng functonally graded materals assocated wth a BRep geometrc model are dscussed n [35]. Voxel arrays n volume modelng and graphcs can be consdered as dscrete attrbute models wth the default geometry represented by a boundng box. Constructve Volume Geometry (CVG) [4] utlzes voxel arrays and contnuous scalar felds for representng both geometry and photometrc attrbutes (opacty, color, etc.). Issues of functonally based modelng of volumetrc dstrbuton of attrbutes are also addressed n [11, 1, 6]. Modelng heterogeneous objects as multdmensonal pont sets wth multple attrbutes s dscussed n [8]. he proposed constructve hypervolume model s based on functon representaton (FRep) [7] and supports unform constructve modelng of pont set geometry and attrbutes usng vectors of real-valued functons of several varables. Multple materals are also represented n [] by vectors of real-valued functons. Dstance felds are used to model varyng materal propertes satsfyng dfferent types of constrants predefned on the ntal object geometry. he approach of [8] was extended n [1] to dmensonally heterogeneous objects wth multple attrbutes by combnng the functonally based and cellular representatons nto a sngle hybrd model. In ths paper, we descrbe one of the mportant operatons n the hybrd model, namely the converson between functonally based and cellular heterogeneous objects. 3. Polygonzaton Exstng methods of the polygonal approxmaton (polygonzaton) of mplct surfaces nclude two major groups. he frst group consstng of contnuaton algorthms s characterzed by ntroducng a seed local trangulaton of the mplct surface wth the consecutve addton of new trangles to 147

4 the mesh by movng along the surface [36, 3, 10] wth the trangle sze adapted to the local surface curvature [1]. he second group ncludes methods for generatng polygons as the result of the ntersecton of the mplct surface wth cells of a regular grd (see, for example, [9, 0, ]) or an adaptve grd [31]. he algorthms of ths group dffer n the type of grd and surface sample ponts approxmaton. he man dsadvantages of the mentoned approaches are smoothng or cuttng sharp features of the surfaces. Algorthms of sharp features extracton are presented n [14, 3, 4]. he optmzaton [4] s based on the specal vertex relocaton strategy and trangles subdvson and allows for extracton of sharp edges and peaks takng nto account the surface curvature. However, n the process of optmzaton, the shapes and relatve szes of neghborng trangles are not controlled whch can result n generaton of degenerate trangles. Moreover, these mesh optmzaton algorthms can produce an excessve number of trangles n the regons of not very hgh curvature, whch s also undesrable for further calculatons. hus, to satsfy the dscussed earler requrements of FEA, a specal mesh refnement whle stll preservng sharp features s needed. 3.3 Fnte element mesh generaton and refnement Issues of surface mesh optmzaton for FEA are consdered n detal n [8, 7]. he descrbed technques are based on the consecutve applcaton of dfferent mesh smplfcaton, mesh subdvson, and mesh adaptaton procedures. Detals of such operatons are dscussed also n [13, 9, 33] and other works. Note that n these works the surface models are not defned n terms of analytcal functons but rather by means of trangulaton (resulted, for example, from measurements, CAD, bomedcal engneerng). Durng the mesh refnement, the exact defnton of underlyng surfaces s unknown. When optmzng polygonzed mplct surfaces, we can use both approxmate and precse functonal surface models, whch provde for more precse calculatons of surface characterstcs and correctons of the node postons n respect to the underlyng surface for remeshng. Issues of fnte element mesh generaton are dscussed n detal n [8]. Unstructured mesh generaton methods are also surveyed n [5, 3]. etrahedrzaton s one of the wdely used methods of 3D dscretzaton. he man approaches to automatc tetrahedral (trangular) mesh generaton nclude spatal decomposton based methods, Delaunay type methods, and advancng-front technques. Algorthms based on spatal decomposton are relatvely easy to mplement, but they do not allow for detecton of boundary sharp features and cannot dstngush boundary enttes whch are rather close but not drectly connected. he boundary connectvty constrant s not taken nto account n Delaunay tetrahedrzaton. So, local mesh modfcatons are necessary to ft the boundary. More accurate boundary representaton s supported by the advancng-front method. hs method starts from a doman boundary dscretzaton and marches nto the regon to be processed by addng one element at a tme. However, snce the method s based on local operatons, convergence problems may be encountered. he convergence problem s common for all methods as there s no theoretcal result whch can guarantee that a polyhedron wth the gven boundary trangulaton can be subdvded nto tetrahedrons wthout addng nternal ponts. In spatal decomposton methods, the convergence problems appear when refnng small detals and sharp features. For the Delaunay type methods, the convergence of the boundary fttng procedures has not been proven. We wll use the advancng-front technque as t s applcable to arbtrary solds and allows us to control shapes and szes of tetrahedrons durng the mesh generaton process. In addton, we consder a modfcaton of ths technque for ncreasng the effectveness of the tetrahedrzaton procedure for FRep solds. 3D mesh optmzaton procedures are descrbed n [5, 6, 30, 17, 18]. 4. GENERAL ALGORIHM OF DISCREIZAION In ths secton we provde a systematc descrpton of our dscretzaton algorthm. 4.1 General descrpton In prncple, there are two man strateges to dscretze a heterogeneous object wth generaton of a volume mesh. he frst strategy nvolves decomposng an ntal 3D object nto 3D elements (tetrahedrons, blocks, prsms, some combned volume mesh) that are optmsed under requrements and constrants descrbed n.4. Here, the boundary mesh B appears as a sde effect of the 3D ntal object decomposton. Another approach mples that frst we decompose the surface B of the ntal object thus yeldng the surface mesh B. hen, ths surface mesh s subjected to optmsaton and refnement to make sure that t satsfes all the possble requrements followng whch one can buld a volume mesh conformable to the refned surface mesh. In our work, we follow the second approach, because most of the constrants and requrements gven n.4 deal wth the boundary mesh B whose qualty s crucal n the context of FEA. As some of the constrants contradct each other, t s mportant to ensure that all the accessble teratve optmsaton procedures are performed to provde the best possble result. In addton, t s known [8] that some effectve methods of volume mesh generaton are actually based on boundary descrptons of computatonal domans. So, we decompose the dscretzaton problem nto two relatvely ndependent sub-tasks: generaton of a mesh of the object surface along wth ts refnement: B = B ( B, A,... A ) ; c c F 1 k generaton of a conformable volume mesh: G c = Gc ( Bc, GF, A1,..., Ak ). We use tetrahedral meshes as we consder them the most unversal: they are successfully used both n vsualzaton and n FEA, and they often serve as a base for buldng meshes consstng of more complex patterns. 4. Surface dscretzaton Here, we descrbe how the frst task of generatng the qualty boundary mesh approxmatng the ntal object surface can be solved n step-by-step manner. 148

5 4..1 Polygonal approxmaton of the object surface 0 0 ( F We form a smplcal complex L = L B ) representng (n accordance wth.3) an approxmate CRep based surface model B c0. Requrement 1 from.4 should be satsfed, and the subsequent steps on the surface mesh reconstructon are such that they preserve the surface topology. We use a polygonzaton algorthm descrbed n [9], whch solves a problem of topologcal ambgutes on the faces wth four edge-surface ntersecton ponts pecular to the second group of polygonzaton methods descrbed n 3.. We assume that the user can control the preservaton of topology equvalence between the ntal mplct surface B F and the resultng polygonal (trangular) surface B c0 by provdng proper ntal data necessary for polygonzaton, namely the surface boundng box and the grd resoluton. 4.. Sharp features reconstructon hs step deals wth optmzaton of the surface mesh B c0 thus yeldng a new CRep based model B c1, descrbed by the complex L 1 = L1 ( L 0, BF ). In ths model, sharp edges and corners present n the ntal surface B F are descrbed by the conformable 0D and 1D elements n the complex L 1. Accordngly, the requrements 1 and from.4 are satsfed for the cellular model B c1. o extract sharp features from an mplct surface coarse trangulaton, we use the algorthm descrbed n [4]. hs algorthm s based on combnng the applcaton of the followng mesh optmzaton procedures: curvature-weghted vertces resamplng; dual/prmal mesh optmzaton that nvolves projectng the trangle centrods onto the mplct surface and movng each vertex of the ntal surface trangulaton to a new poston mnmzng the sum of the squared dstances from the vertex to the planes whch are tangent to the mplct surface at the projectons of adjacent trangles centrods. one-to-four subdvson of mesh trangles where the mesh normals have large devatons from mplct surface normals. Our experence wth applcaton of ths optmzaton algorthm shows that n most cases t does produce surface mesh B c 1 allowng for a qualty representaton of sharp features of the ntal mplct surface. However, the resultng mesh can have badly shaped or degenerate trangles near sharp edges and corners and may consst of an excessve number of trangles produced by the adaptve subdvson procedure n the regons of low curvature. Consequently, we propose further refnement descrbed n the followng subsectons Surface mesh refnement and smplfcaton he objectve of ths phase s to get rd of badly shaped trangles, to make the mesh fner n the regons of hgh surface curvature, and, on the contrary, to make the mesh coarser n the areas where the curvature s low. he requrement for preservng sharp features should be satsfed. As an deal result, the surfaces of the tetrahedron and cube should be represented by just four and twelve trangles respectvely. hus we am at buldng CRep based model B c descrbed by a complex L = L ( L1, B ), whch s obtaned as the result of F optmzaton of the complex L 1. he optmzaton procedure conssts n teratve applcaton of edge swappng, edge splttng, edge collapsng and vertex relocaton operatons. Edge splttng allows for enrchng the mesh, edge collapsng provdes mesh smplfcaton, and edge swappng and vertex relocaton operatons are used for mesh refnement. We descrbe n secton 5 the man characterstcs of these operatons as well as the crtera for ther applcaton. In the end, the model B c descrbed by the complex L must ensure a qualty polygonal approxmaton of the mplct surface B F under the requrements 1 4 from the subsecton Surface mesh adaptaton At ths step, we am at adaptng the mesh to the needs of FEA n the context of some partcular applcaton, thus producng Crep based surface model B c4. hs means that the requrements 4 6 from secton.4 should be satsfed. Frst, sngular lnes and ponts of attrbute functons should be taken nto account. We project those lnes and ponts onto the dscretzed surface B c obtaned at the prevous step, and then make a partton of elements of the complex L, thus yeldng a new complex L 3 = L 3 ( L, B F, A1,..., A k ) n whch the sngular lnes and ponts are descrbed by some 1D and 0D subcomplexes. After ths procedure, t s useful to execute the prevous step once agan to preserve not only geometrc sharp features but also the sngular lnes and ponts of the attrbute functons. hen, we make the adaptve mesh subdvson and refnement to satsfy the constrants 4 6 through usng once agan edge splttng, edge swappng, and vertex relocaton operatons as well as the trangle subdvson procedure. Note that the crtera for applcaton of those procedures does depend on both the attrbute functons S (X) and the functon F(X) descrbng geometry (more detals are gven n Secton 5). Fnally, we get the complex L 4 = L 4 ( L 3, B F, A1,..., A k ) descrbng the surface model B c4 that hopefully meets all the constrants and requrements mportant for FE meshes D object dscretzaton he second task of buldng a qualty volume mesh based on the surface mesh generated n the prevous stage s descrbed n ths secton etrahedral mesh generaton Gven the CRep based model of the surface mesh B c = B c 4, descrbed by the complex L c = L 4 we can now buld a CRep based model of the 3D object G = G ( B, G, A,..., A ). o c0 c0 c4 F 1 k 149

6 produce such a volume mesh, we use the advancng front method brefly characterzed n 3.3. More detals about our modfcaton of that algorthm are gven n subsecton 5.. he model G c0 s descrbed by a 3D complex K 3 0, whch s bult on the bass of the boundary complex L C and on the ntal object attrbutes. So, we have 3 3 K 0 = K0 ( Lc, A1,..., Ak ). Whle generatng such a volume mesh, the constrants 4-6 from.4 are taken nto account. However, because of the convergence problems dscussed n 3.3, not all of those restrctons can mmedately be satsfed. So, addtonal post-remeshng may be needed to get a qualty tetrahedral mesh Volume mesh adaptaton he objectve of ths phase s to re-buld the volume mesh descrbed by the complex K 3 0 that allows for mprovng the tetrahedra shapes under all the requrements for the mesh related to attrbutes. o succeed n solvng ths problem, we use face swappng, tetrahedra subdvson and vertex relocaton operatons ([6], [30], [17], [18]). Some features of the mplementaton of these operatons are consdered n 5. Note that once agan we pay attenton to preservng sharp features, and sngular ponts and lnes of the boundary surface. As a result, we get a new complex K 1 = K1 ( K1, Lc, A1,..., Ak ) defnng a CRep based model G c 1 that should satsfy all the requrements crtcal for FE meshes Attrbute transformatons he fnal step n dscretzaton of heterogeneous objects s concerned wth convertng ntal attrbutes for gettng new ones: 3 A = A K, L, A,..., A ), j=1,,,m. As we dscussed n.4, some j j ( 1 c 1 k attrbutes present n the ntal model can dsappear, others may be descrbed by another representaton scheme, and new attrbutes can appear. he attrbute converson procedures can heavly depend on applcaton specfcs. Eventually, we get the sought-for dscrete model D = ( Gc = Gc1, A1,..., Am ) of the ntal heterogeneous object D = ( GF, A1,... Ak ) wth CRep based geometry, and CRep and CFRep based attrbutes. 5. DEAILS OF BASIC PROCEDURES In ths secton we present some operatons and procedures n more sgnfcant detal. 5.1 Surface mesh optmzaton Here we consder n detal the basc mesh refnement operatons ntroduced n hese operatons are teratvely appled to the current mesh n order to provde an accurate pecewse lnear approxmaton of the underlyng mplct surface and to adapt the element szes to FEA requrements gven n.4. In partcular, t s necessary to preserve sharp edges and corners of the surface. For ths purpose, we ntroduce so-called sharp functon Sh for the mesh edges and vertces. Usng the planarty estmatons [8] for each edge e, we set: ( 1+ ( n ( t1), n( t)) 1+ ( ( t1), N ( t)) Sh(e)=1, f ϕ and ϕ Sh(e)=0, otherwse. N Here n( t1), n( t) are unt normals of the adjacent trangles t1, t correspondngly; N ( t1), N ( t) are the mplct surface normals at the central ponts of t1, t; ϕ s a threshold that measures the sharpness of a feature. he mplct surface normals are calculated as the normalzed gradents of the functon F(X) descrbng the mplct surface. Note that a nonzero value of Sh(e) ndcates that the edge e probably les on a sharp feature of the surface. For the vertces, the sharp functon s defned as follows: Sh ( P) = Sh( ), e where e s an edge ncdent to the vertex P. If Sh(P) = s true, then vertex P les on a sharp edge, but the case Sh(P)> corresponds to a corner pont. o sngle out those spkes that do not le on sharp edges, we use the heurstc estmaton proposed n [14]. m n( m, n( t)) τ & m n( M, N ( t1)) τ. Here n(t) are unt normals of trangles t adjacent to pont P, N (t) are the mplct surface normals at the central ponts of t, m = [ n t 0 nt1] s the normal vector to the plane spanned by two normals n t0, nt1 whch enclose the largest angle, and smlarly M = [ N ( t0) N ( t1)], τ s a threshold. For the vertces whch are found n such a way, we set Sh(P)=3. Let us consder all the basc mesh refnement operatons n more detal Mesh vertces relocaton hs operaton conssts n applcaton of two procedures: 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 relocaton s descrbed as: ( V ) new = ( V ) old + λu / U. Here λ s a small postve number (λ < l mn, where l mn s the smallest length of all edges ncdent to the pont) that lmts dsplacement of vertces, and takng t nto account prevents the appearance of such mesh defects as folds and self-ntersecton. he movng drecton U = R ( R N ) N, where U s defned by the formula [3]: N s the surface normal at the pont ( V ) old, 1 1 R = w j Pj ( V ) old, w = w P ( ) j j j j V, old 150

7 Pj s a vertex havng a common edge wth V. If the vertex V s a corner or belongs to more than two sharp edges, then ts poston does not change. However, f the vertex V has exactly two ncdent sharp edges, for example, 1 + ( N ( e1), N ( e)) e1 = ( V, P1 ), e = ( V, P ) and σ, where N ( e1), N ( e) are the surface normals at the centers of edges e1, e, σ s a threshold, then a new pont ( V ) new s placed at the center of the arc connectng vertces V, P 1, P. One should pay attenton that durng the mesh smoothng process some vertces can be detached from the ntal surface; therefore, ther postons have to be corrected. he vertex movng procedure moves vertces towards the underlyng mplct surface. It s organzed as an optmal search process n accordance wth the formula: W ( X ) = F ( X ) D( X ) = F( X ) / F( X ). In the process of the fnte element mesh adaptaton, these characterstc functons are defned wth respect to the scalar and vector attrbute functons: W ( X ) = W ( S1( X ),..., Sk ( X )) D( X ) = D( S1( X ),..., Sk ( X )). In order to control the qualty of trangles, we use the fnte element shape qualty measure [8]: l q( ) = α, max r where r k denotes the nradus of the trangle, l max s the largest edge length, and α s a normalzaton coeffcent so that q()=1 for an equlateral trangle. ( V ) = ( V ) + r Z / Z, new old * * where r = arg mn F (( V ) old + rz / Z ) r [0, λ] 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 consder an llustratve example n Fg. 1. We elmnate the dagonal (P 1,P ) and nstead nsert the dagonal (P 3,P 4 ) f max(β1, β) < max(α1, α). Fgure 1: Edge swappng hs operaton s appled only f adjacent trangles sharng the edge e are almost coplanar, so that (1 + ( n ( t1), n( t)) σ n 1+ ( N ( t1), N ( t)), and σ N, where the constants σ N and σ n are user-specfed thresholds Edge splttng hs operaton s performed for a number of reasons: to mprove geometry approxmaton, to ft attrbute functons or to subdvde badly shaped trangles. o descrbe crtera for choosng those edges whch are sutable for splttng, we ntroduce the followng denotatons. Let W(X) be a scalar characterstc functon, and D (X ) be a vector one. hen, n the context of the optmzaton procedure of geometry approxmaton, these functons depend on the functon F(X) descrbng geometry of the ntal object and on the gradent of F(X): Fgure : Edge splttng Our edge splttng operaton s llustrated by Fg.. he process of mesh subdvson s teratve, and edges are processed n the order of decreasng ther lengths. We use the followng estmatons to measure value varatons of the characterstc functons: τ 1 = length( e)/ Sr ( C) τ = max( q( 1 ), q( )) ε = max( W ( C) ( P1 ), W ( C) ( ) ) 1 P ε = max( W ( C1) ( P1 ), W ( C1) ( P ), W ( C1) ( P3 ) ) ε 3 = max( W ( C) ( P1 ), W ( C) ( P ), W ( C) ( P4 ) ) 1 mn ( D, j {1,,3,4} j ν = 1 ν ( P), D( P )) ( N ( P1 ), P1 P ) + ( N ( P ), P1 P ) = P1, P ν 3 = max((1 ( N ( C1), n( 1)) ),(1 ( N ( C), n( )) )) where C s a center of an edge, and C 1,C are centrods of adjancent trangles, S r s a mesh densty attrbute functon, N denotes surface normals and n are normals of trangles. he edge e s subdvded f one of the lsted estmatons or several of them exceed the user-specfed thresholds. he splt vertex V s placed at the poston 151

8 S ( P1 ) V = P1 + r P1 P, S ( P ) + S ( P ) r 1 r where S r (X) s a mesh densty attrbute functon. hen, V s moved towards the mplct surface accordng to the procedure descrbed n Dfferent characterstc functons and estmatons can be used at the dfferent phases of mesh optmzaton and adaptaton process. After a few teratons, t s recommended to execute edge swappng and mesh smoothng operatons whch can mprove the mesh qualty. For quck mesh subdvson n the cases 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 coupled wth mesh smoothng and edge swappng Edge collapsng hs operaton s used to smplfy the surface mesh n low curvature regons. We choose an edge e and replace t wth a vertex V (see Fg. 3a). In ths process, two vertces P1, P are substtuted wth a new vertex V, and the trangles 1, are collapsed to edges. Let us formulate the condtons under whch one can consder executon of the edge collapsng operaton. he possblty of edge collapsng and the poston of a new vertex V depends on the edge vertces weghts as follows: f w(p1)=w(p)=0, then the edge e cannot be collapsed; f w(p1)=w(p)>0, then the edge e can be substtuted wth the vertex V placed at the mdpont of e; f w(p1) > w(p) 0, then the edge e can be substtuted wth the vertex V = P ; f w(p) > w(p1) 0, then the edge e can be substtuted wth the vertex V =P 1. hen we move the new vertex V onto the mplct surface usng the procedure descrbed n Fnally, we evaluate the qualty degradaton durng the ntended collapse operaton usng the followng measures: θ 1 = max ((1 ( N = nc( V ) θ = max ( N ( C ), n( )) )) ( V ), VP ) VP Pj Star ( V ) j j Here, N denotes normals of the ntal surface, n s for normals of trangles, C s a centrod of the trangle, and =nc(v) s for trangles ncdent to V. If θ 1 σ t and θ σ n, then the correspondng edge e s collapsed and substtuted wth the vertex V. Constants σ mn, σ α, σ t, and σ n are user-specfed thresholds. It s recommended to couple the edge collapsng operaton wth mesh smoothng and edge swappng. a b Fgure 3: Edge collapsng We check for collapse canddacy edges n the order of ncreasng length. he process s as follows. Frst, we check the topologcal valdty of the edge collapsng operaton for each edge. Edge e can be substtuted wth a new vertex f the followng condton s satsfed: Star(P1) Star(P) = {P3, P4}, where Star(P) s the set of the mesh vertces sharng a common edge wth the vertex P. hs restrcton allows for avodng mesh degradaton n such cases as shown n Fg.3b. Secondly, f length(e) σ mn, then the edge e s collapsed wthout need for any addtonal analyss, and the mdpont of e s chosen as the poston of a new vertex V. Otherwse, we contnue checkng e for a canddacy edge. For ths purpose, we calculate so-called weght coeffcents w(p 1 ), w(p ) for the edge vertces. We set w(p )=1, f Sh(P )=0. he values Sh(P )=1 or Sh(P ) > ndcate that the vertex P s a spke or a corner, so we set w(p )=0 n ths case. From (Sh(P )= & Sh(e)=0) t follows that there exst other sharp edges ncdent to P, so we set w(p )=0. But f e s one of the two sharp edges ncdent to P (.e. Sh(p)= & Sh(e)=1), then we should calculate the angle α between these edges. If cos(α)<0 and (1- cos(α) ) σ α, then we set w(p )=0.5, otherwse w(p )=0. 5. Volume mesh generaton and optmzaton 5..1 he modfcaton of the advancng front method Frst, let us characterze some crtcal steps of the advancng front algorthm (brefly outlned n 3.3) as appled to tetrahedrzaton. A surface mesh descrbed by a D complex forms the ntal front. hen, at each step of the algorthm an actve face s selected to serve as a base for buldng a new tetrahedron whose sze s calculated takng nto account the gven mesh densty attrbute. Provded the new tetrahedron does not cross the front, t s added to the front thus formng a new one. o avod emergence of thn elements, a new tetrahedron vertex s placed n the closest node of the mesh satsfyng the gven dstance threshold. here are dfferng strateges [8] for selecton of both the next actve face and new tetrahedron node that prevent the procedure from generatng an nfnte seres of tetrahedrons yet yeld a volume mesh of reasonable qualty (e.g., wthout thn elements, etc.). We propose modfcaton of the advancng front method that takes advantage of the fact that we deal wth the problem wthn the cellular-functonal framework. In partcular, avalablty of an exact functonal descrpton of the object to be subdvded can greatly smplfy the procedure of the evaluatng pont membershp relaton whch s mportant n the context of determnng whether a tetrahedron crosses the front. 15

9 Now, let us outlne the modfed algorthm. We start from the CRep based model of the surface mesh B c represented by the smplcal complex L c and the functon S r (X) descrbng the mesh densty attrbute A r. Frst, a regular tetrahedral mesh coverng the object s generated. hs mesh s descrbed by a smplcal complex M 3 0. hen, elements of M 3 0 are subdvded to conform 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 smplcal complex M 3 1 = M 3 1 (M 3 0, A r ). At the next step, we sngle out the subcomplex M 3 lyng completely nsde the subdvded object G F. So M 3 M 3 1, M 3 G. o ensure the convergence of the algorthm, the dstance between the boundary surface of the object and the mesh descrbed by the complex M 3 must be a few tmes greater than the tetrahedron sze descrbed by the functon S r (X). herefore, M 3 ncludes only those elements of M 3 1 whose nodes V satsfy the followng condton: F(V ) k*s r (V ) 0, where k a b c Fgure 4: Modfed advancng front method: a) the ntal rectangular object wth the pattern mesh subdvded accordng to a mesh densty attrbute b) the sub-mesh lyng completely nsde the subdvded object c) the fnal mesh hen we defne a two-dmensonal complex Q that represents the boundary of M 3. hs complex together wth the complex L c descrbng the dscretzaton of the boundary surface form the front Q 3= L c Q. Usng ths front, we subdvde the remanng sub-area wth help of the advancng front tetrahedrzaton method and thus get the 3D complex M 3 3= M 3 3(Q 3,S r (X)). he overall complex K 3 0 defnng a CRep based model G c1 s calculated as K 3 0= M 3 M 3 3. hen, we apply a few teratons of 3D mesh optmzaton to the complex K 3 0 to ensure that we get a volume mesh of good qualty and regularty, and that all the FE related constrants are satsfed. As a result we get a new complex K 1 = K1 ( K1, Lc, A1,..., Ak ). Fg. 4 shows a D llustraton of the descrbed algorthm. he ntal rectangular object wth the pattern mesh subdvded accordng to a mesh densty attrbute Sr(X) s shown n Fg 4a. Fg. 4b llustrates the sub-mesh lyng completely nsde the subdvded object. he area between the ntal boundary and that sub-mesh s subdvded usng the advancng front method. he fnal mesh s shown n Fg 4c. Our modfed frontal method s more effectve n cases when a consderable part of the object can be covered by pattern mesh elements that reman unchangeable. In these cases we reduce the number of elements generated by the frontal tetrahedrzaton and keep well shaped elements of the pattern mesh. Moreover our modfed algorthm allows for combnaton of meshes of dfferent types, for example, the ntal mesh coverng the object can consst of hexahedral elements. 5.. Volume mesh optmzaton operatons he 3D mesh optmzaton algorthm conssts n the teratve applcaton of face swappng, tetrahedral splttng and vertex repostonng technques. he face swappng operaton s the 3D extenson of the edge swappng technque descrbed n wo neghborng tetrahedra havng a common face are transformed wth the exchange of the common dagonal. he tetrahedra splttng operaton ncludes one-to-two subdvson by bsectng the longest edge. he selecton of a tetrahedron to be splt s made on the bass of crtera smlar to those descrbed n 5.1. for the edge splttng operaton for a surface mesh. he vertex relocaton technque n 3D conssts n the movement of a node towards the barycenter of the polyhedron formed by the surroundng tetrahedral mesh. Snce the surroundng polyhedron can be non-convex, the postonng of the node drectly at the barycenter can result n overlappng tetrahedra. hs s the reason for usng an adaptve procedure wth a varable step of the movement to the barycenter. More general three dmensonal versons of swap and splt operators remesh a polyhedron formed by neghbourng tetrahedra sharng a common edge or a common vertex [8]. 6. EXAMPLES In ths secton we present several examples llustratng our algorthm for dscretzaton of functonally based heterogeneous objects. he examples were prepared usng our orgnal software tools ntally ntended for data preprocessng n computatonal physcs and allowng for the users nteractve work n a step-bystep manner. We used HyperFun modelng language [8] to defne all the models, and specalzed software tools wth the bult-n HyperFun nterpreter have been used to mplement the examples on a Pentum III 800 MHz computer. 6.1 etrahedrzaton of FRep solds Here we llustrate the man steps of the algorthm usng the example of an FRep object wth sharp features shown n Fgure 5. he ntal surface mesh produced by polygonzaton s shown n Fg 5a. he surface mesh generated n the process of sharp features reconstructon s shown n Fg. 5b. For the purpose of FEA, ths mesh has badly shaped and degenerate trangles near sharp edges and corners, and ncludes an excessve number of trangles produced by the adaptve subdvson procedure n the regons of low curvature. Subsequent mesh optmzaton and smplfcaton produces the mnmal surface trangulaton presented n Fg. 5c. hs trangulaton then serves as an ntal front whle applyng our advancng front algorthm. Fg. 5d shows the resultng tetrahedrzaton. etrahedrzaton of another FRep object havng more complex topology s presented n Fg. 6. It s a typcal CSG-lke object wth sharp edges and both flat and curvlnear faces. he result of 153

10 polygonzaton and sharp features extracton s shown n Fg. 6a. he mesh generated s not completely sutable for subsequent FEA: one can observe badly shaped and degenerate trangles near sharp edges and corners n the surface mesh after the sharp features extracton. he surface mesh adaptaton to FEA requrements after applyng mesh optmzaton proced ures s shown n Fg. 6b. An enlarged vew of an nternal structure of the tetrahedral mesh s shown n Fg. 6c. As to computatonal tmes, mesh decmaton process took 4. sec, FE mesh adaptatontook.9 sec, and tetrahedrzaton took 331 sec. 6. Dscretzaton of a heterogeneous object wth varous attrbutes hs example llustrates the nfluence of attrbutes on the mesh elements szes. Fg. 7 presents four dscretzaton varants conformng to dfferent attrbutes. We start from Fg. 7b where no attrbute has an effect. Note that the mnmal surface trangulaton for ths object was created usng our mesh optmzaton and smplfcaton algorthm smlar to the prevous example. hen, ths surface mesh was subdvded to conform to the attrbutes. Fgs. 7c-7e llustrate applcaton of varous types of attrbute functons. Fg. 7c shows the result of subdvson based on an attrbute nfluencng the mesh densty and defned by a pont source placed at a cube corner. Fgure 7d smulates mesh adaptaton near the vertcal well, where the element szes are adjusted to the reservor pressure gradent. An example of a mesh ftted to an annular heatng devce s shown n Fg.7e. Mesh adaptaton process n all these examples took no longer than 7 sec. 6.3 Modelng of a mxng tank mpeller An applcaton example shown n Fg. 8 s concerned wth real computer-aded modelng of a mxng tank mpeller whch s used as the agtator of the flow components n a chemcal reactor. he mpeller whose central body and blades are made of dfferent materals (marked by dfferent colors) s shown n Fg.8a. Fg. 8b shows the surface mesh wth recovered sharp features. One can observe that there are many badly shaped trangles n the mesh, so the followng mesh optmzaton s necessary. Fg. 8c llustrates the result of such optmzaton. he surface mesh takng nto account the materal attrbutes s shown n Fg. 8d. he mesh was splt on the blades faces accordng to the specfed attrbute. hs surface mesh then serves as ntal data for the advancng front tetrahedrzaton. he resultng 3D mesh was used for FE thermal and stress-stran analyss. As to calculaton tmes, mesh decmaton took 3.1 sec, FE mesh adaptaton took 10 sec, mesh adaptaton accordng to the attrbute took 7 sec, and tetrahedrzaton took 634 sec. 7. CONCLUSION hs work ams at makng functonally defned solds and heterogeneous objects wth mplct surfaces avalable for practcal applcatons requrng fnte element analyss and smulaton. We dscussed a dscretzaton procedure resultng n surface and volume meshes for heterogeneous objects wth geometry and attrbutes defned usng real-valued functons. hs procedure s consdered as an mplementaton of the functonal to cellular models converson operaton n the cellular-functonal modelng framework for heterogeneous objects [1]. In contrast to prevous works on mplct surface polygonzaton and volume mesh generaton, the man motvaton of ths work s generaton of meshes sutable for fnte element analyss wth constrants mposed by the heterogeneous object attrbutes. Let us summarze the man contrbutons of the paper: 1) he dscretzaton problem s stated for heterogeneous objects represented as functonally defned 3D solds (wth mplct surfaces) and attrbutes (scalar felds). ) he paper aggregates dfferent technques nto a systematc step-by-step procedure of solvng the above problem. Some of the known technques have been adapted and extended to work wth the functonally defned solds as explaned below. 3) he proposed surface mesh optmzaton preserves sharp features of mplct surfaces, satsfes the requrements of FEA, and uses the defnng functon to do ths (see the mesh transformaton from Fg. 5b to 5c). 4) he advancng front method of 3D tetrahedrzaton s extended takng the defnng functon of the sold nto account. We provded examples of tetrahedrzaton of objects wth complex topology and sharp features, and llustrated mesh adaptaton to attrbutes of varous types. A practcal 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 cellular-functonal model [1], heterogeneous objects are represented as hypervolumes or multdmensonal pont sets wth multple attrbutes. A multdmensonal pont sets can nclude elements of dfferent dmensons, whch can be hgher than three. In ths paper, we dealt only wth 3d objects wth attrbutes. he extenson of the proposed algorthms to the dmensonally heterogeneous and tme-dependent objects s a subject of future work. 8. ACKNOWLEDGMENS he authors would lke to thank Drs. Y. Ohtake and A. Belyaev for frutful dscussons and for the possblty to work wth ther mplementaton of the dynamc meshes algorthm. Jody Vlbrandt has greatly helped wth makng the text more reader frendly. 9. REFERENCES [1] Adzhev, V., Kartasheva, E., Kun,., Pasko, A., Schmtt, B. Cellular-functonal modelng of heterogeneous objects, Proc. ACM Sold Modelng and Applcatons 00 Symposum, Saarbryucken, Germany, Ed. Kunwoo Lee, N. 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12 a b c d Fgure 5: An example of dscretzaton of a functonally based object wth sharp features: a) polygonzaton of the ntal object surface; b) a surface mesh after sharp features reconstructon; c) the optmal surface mesh; d) the fnal tetrahedral tessellaton. a b c Fgure 6: An example of dscretzaton of a complex FRep object: a) surface mesh wth reconstructed sharp features (599 trangles); b) surface mesh after FE adaptaton (7794 trangles); c) cut of tetrahedral mesh (enlarged vew) (43059 tetrahedra) a b c d e Fgure 7. An nfluence of attrbutes on the mesh elements szes: a) surface mesh wth reconstructed sharp features (776 trangles); b) the mnmal surface trangulaton (1 trangles); c-e) mesh adaptaton to attrbutes of varous types (05, 701, 615 trangles, respectvely) a b c d Fgure 8: Modelng of an mpeller: a) mpeller consstng of varous materals); b) surface mesh wth reconstructed sharp features (846 trangles; c) surface mesh after FE adaptaton (88 trangles) ; d) enlarged fragment of a surface mesh conformng the materal attrbute 156