BOUNDARY REPRESENTATION MODELLING WITH LOCAL TOLERANCES

Transcription

1 BOUNDARY REPRESENTATON MODELLNG WTH LOCAL TOLERANCES David J. Jackson Parasolid Business Uni EDS Unigraphics Parker s House 6 Regen!Sree! Cambridge England ABSTRACT Convenional boundary represenaion (b-rep) solid modellers perform all compuaions o a single global olerance. For solids bounded by freeform surfaces his means ha an excessive amoun of daa may be required o define he surface o he required accuracy Given ha differen modellers invariably operae o differen modelling olerances global olerances are also a major roadblock o he robus exchange of b-rep models beween sysems This paper describes an exension o b-rep modelling o allow differen feaures in a model o be consruced o differen olerances This approach which we call oleran modelling has been implemened and released in a commercially used solid modelling kernel EDS Parasolidmodeller. A olerance is associaed wih each face edge and verex of he b-rep; subsequen modelling operaions ake accoun of hese local olerances. This enables he efficien use of consrucion echniques which can operae a a olerance appropriae o he paricular feaure. mporing combining and modelling wih feaures creaed in several differen surface and solid modelling sysems also becomes possible. This paper discusses he problems associaed wih radiional approaches o handling olerances in b-rep modellers. The principles of oleran modelling are hen inroduced and a boolean algorihm ha operaes on oleran models is described. Finally. aspecs of oher modelling operaions and daa exchange are discussed. NTRODUCTON This paper describes a ype of boundary represenaion which has recenly been implemened in he Parasolid solid modeller and which allows he modeller o represen and uilise geomery regardless of he olerance o which i was creaed and o mix geomeries of differen olerances in he same model As developmen of solid modellers has progressed much work has been done in an effor o make modelling operaions and booleans in paricular reliable. deally i should be possible Permission o copy wihou fee all or par of his maerial is graned provided fha he copies are no made or disribued for direc commercial advanage he ACM copyrigh noice and he ile of he publicaion and is dae appear and noice is given ha copying is by permission of he Associaion of Compuing Machinery.To copy oherwise or o republish requires a fee andor specific permission. Solid Modeling 95 Sal Lake Ciy Uah USA 995 ACM $3.5 o ake any wo solids which he modeller regards as valid and unie hem ino a valid solid which correcly models he seheoreic union (o an appropriae olerance). n pracise modellers only approach his level of reliabiliy especially when dealing wih free-form surfaces No only is i difficul o design reliable algorihms o inersec curves and surfaces i is even harder o incorporae hese ino a boolean algorihm which deals consisenly wih he resuls of he many inersecion and coincidence enquiries which ake place in a complex boolean operaion. Toleran modelling is a foundaion on which reliable solid modelling can be buil. By aaching local olerances o faces edges and verices he boolean algorihm can fxrform inersecions and coincidence ess o appropriae olerances and ake advanage of a richer srucure o represen he resuling solid. 2. BACKGROUND 2. Oher approaches o robusness A number of differen approaches o he problem of robus geomeric modelling have been pu forward. Exac (raional) arihmeic has been used [] bu his suffers from he build-up of complexiy of he numerical values (miigaed somewha by only evaluaing he exac values when required). Moreover i may creae small undesirable feaures in he model which do no correspond o any design inen. Poenially ambiguous configuraions can be avoided by applying small perurbaions o he geomery before he boolean [2]. Again his creaes small edges and faces and removes coincidences which were probably inenional. Symbolic reasoning sysems have been used o ensure ha cerain consisency saemens abou he various incidence or coincidence relaions beween he geomeries remain ree. n oher words if cerain relaions are compued ohers can be deduced. The difficuly here is describing all he consisency requiremens in he general case; so far his is limied o special cases [3] []. n [5] he auhors describe an algorihm which uses a hybrid CSGb-rep model o perform boolean operaions by avoiding redundancy when evaluaing he boundary of a manifold CSG model. A global olerance is used. 27

2 Adapive olerance-based approaches associae olerances wih individual poins curves and surfaces and deec ambiguous siuaions. When an ambiguiy is deeced he olerances have o be increased and some or all of he algorihm re-ron. [6] uses a CSGb-rep model whereas [7] is limied o polyhedral b-reps. 2.2 Use of olerances.> -...> Mos geomeric modelling sysems use small olerances when. _7 deermining answers o quesions of inersecion or coincidence Two poins are considered coinciden if heir disance is less han - he olerance value; wo edges are coinciden if hey are everywhere wihin olerance of each oher and so on. The olerance also deermines he accuracy o which compuaions are performed for example when converging o an Figure. Tangenial surfaces may fail o meea a higher inersecionbeweenwo curves resoluion Tolerances help in several ways: By seing he olerance several orders of magniude above he floaing poin precision available he modeller can be largely unaffeced by rounding error. The olerance prevens he creaion of very small faces and edges which would serve no pracical purpose in he represenaion of he model and cause problems for downsream applicaions.. Consisen use of olerances o guide decisions in modelling operaions can avoid inconsisen decisions leading o conflic beween opology and geomery. A larger fixed olerance is also more likely o run ino consisency problems - for example i is no possible o spli a shor edge of lengh less han imes he olerance wihou creaing edges which are oo shor or edges which do no mee up. Alhough shor edges like his would no normally be modelled deliberaely hey can easily occur in he course of modelling a complex par. Figure 2 shows an example of his where a face wih a shor edge is being spli by he verical dashed line. 7Tre modelling olerance is represened by dashed olerance regions around he edges and verices. The verex which would be creaed in he middle of he shor edge woud be coinciden wih boh ends of he edge. Some modellers use ad-hoc olerances bu mos use a single global disance which can be se by he user. 2.3 Limiaions of a single olerance The approach aken by many modellers and by Parasolid unil recenly was o choose a fixed global olerance value and o recommend ha all sysems using he modeller use he same olerance so ha models can be exchanged beween hem. n Parasolid s case his value was 5.Oe-9 meres chosen o be below he smalles realisic olerance for mos mechanical pars. The difficuly is ha here is no single olerance which will be suiable for all rwuiremens A very small value will necessiae he creaion of surfaces wih a large amoun of defining daa in operaions such as blending bu a large olerance on a model prevens he creaion of small feaures anywhere on ha model. Moreover if rimmed surfaces are impored from anoher modelling sysem hey will only mee o ha sysem s olerance. n some cases he surfaces can be exended and re-inerseced o impose any desired accuracy bu in he case of surfaces which mee smoohly his may be impossible (Figure ). may be undesirable or impracical o modify he surfaces o mee more accuraely. Consequenly he global olerance would have o be increased o encompass he wors case. -. _ Figure 2. Divfding a fa;e hrough a shor edge. 2. Toleran Mo&ing [n oleran modelling a local olerance is associaed wih every face edge and verex in he model This provides flexibiliy in represening geomeric models which enables sraighforward ransfer of models beween sysems and a foundaion for robus solid modelling. When necessary olerances are locally increased and opology merged o avoid ambiguiy. 28

3 3. OVERVEW OF TOLERANT MODELLNG 3. Model Represenaion Each verex has an associaed olerance which is a disance value. This can be he defaul olerance (normally 5.Oe-9 meres) or any larger value The olerance region around he verex is a sphere of radius equal olhe olerance cenred a [he poin of he verex An edge also has an associaed olerance. f his is greaer han he defaul he edge is called a oleran edge An edge wih defaul olerance is represened by a single 3-space curve A oleran edge is represened by a collecion of curves in he parameer spaces of he faces conneced o i. (For a manifold solid edge here will be wo such curves). The olerance region around he edge is a ube of radius equal o he edge olerance cenred on one of he parameer space curves. Thechoice of curve is arbirary bu fixed for each edge. Thus all he parameer-space curves lie in his ube. Figure 3 shows a face wih wo oleran edges. he olerance regions are shown doed Faces also have associaed olerances. which define he hickness of he face. [n Parasolid faces currenly always have he defaul olerance The verex olerance mus be a leas as grea as ha of conneced edges and he edge olerance mus be a leas as grea as ha of conneced faces 3.2 ModelConsisency nersecion and coincidence ess are always performed relaive o appropriae olerances - e.g. wo eniies inersec if hey approach wihin he sum of heir olerances Wih his inerpreaion. he usual b-rep consisency rules apply - verices mus lie on conneced edges and faces edges mus lie on conneced faces. and edges and faces mus only inersec. -= ~=. \ \ ). r L f \ i ) ~ ;.. 8 :. \ -. [ \ \; r (! ( (....!< Figure 3. Parameer-space where opology exiss. -\ ; curves lie in olerance regions n paricular he ends of he parameer-space curves on an edge mus lie inside he appropriae verex sphere. Noe however ha he curves bounding a face need no mee exacly end-o-end as ongas hey are wihin verex olerance. More precisely edges mus only inersec a heir end verices f he inersecion of he olerance ubes of wo edges is nonempy hen each conneced componen of ha inersecion mus inersec a common verex of he wo edges. Figure shows an invalid face where wo edges inersec in heir inerior and a valid face where he inersecion area is conracible ono he verex. Faces mus only inersec a verices or along edges More precisely if he inersecion of heolerance regions of wo faces is non-empy hen each conneced componen of ha inersecion mus be conracible ono a collecion of common edges or. # i L A-. ) ( J : ).? ~c \ --- l \ J h J! ;( b Figure. The lef face is invalid (self-inersecing). The righ face is valid as he inersecion is conracible o a verex. 29

4 verices. 3.3 General Principles heoreic union o he appropriae.2 Phases of he boolean olerances There are hree main olerances: reasons for he inroducion of local The boolean algorihm described here proceeds in he usual hree phases which we call imprin join and selec.. To allow he imporing of b-rep or surface models buil in oher sysems whaever heir accuracy. 2. To enable he use of modelling echniques such as blending which generae approximaed surfaces or curves wihou compromising he accuracy of he res of he model. 3. Byidenifying areas which have been approximaed o avoid performing numerical compuaions o inappropriaely high accuracy which would be slow and unreliable. Noe ha olerances are associaed wih opology no geomery Poins curves and surfaces are reaed as exac as are derived objecs such aspoin-on-surface (i.e. UVpoin surface) orrimmed curves. Tha is hey have he defaul olerance. When a geomeric es such as coincidence or inersecion is performed a olerance equal o he sum of he olerances of he opologies is used. When aopological operaion such as spliing an edge or adding a face cann~ be ~rformed wih exising olerances because eniies collide hen he local olerances are increased and redundan opology removed unil a consisen model can be creaed.. BOOLEAN OPERATONS mprin niially he wo models may have opologies which parially overlap each oher e.g. crossing edges crossing faces or verices in he middle of edges. The purpose of he imprin phase is o add opology o divide exising faces and edges so ha each opological eniy in one model inersecs eniies in he oher model only a oher opologies. The inerpreaion here is he same as if he wo had been par of he same model. The resul of he imprin is a lis of opologies on one body and maching opologies on he oher. Noe ha a verex can correspond o an edge face or even a 3-space region (cell) if he olerances so indicae. Adding opology in his way can require addiional changes o eiher body o mainain a valid b-rep. For example an edge may be spli o creae a new verex which may be wihin olerance of anoher edge. This may require he removal of a hin face which has degeneraed o an edge. Figure 5 shows an example of his - when he faces on he lef are spli by imprining he verical line wo of he new verices are coinciden. These verices need o be combined and he hin degenerae face removed as shown on he righ. Join. nroducion This secion describes a boolean algorihm which can be used on models wih local olerances. applies o convenional solid b- reps or general mixed-dimension cellular non-manifold models. The boolean operaions of unie subrac and inersec operae on wo models and perform an approximaion o he corresponding se-heoreic operaions. n a robus modeller he resul of a unie wi[l always be a valid b-rep which models he se ) * (. : ( o This phase performs wo funcions - he compression of opologies on he inpu models and he acual combinaion ino orie m~del which will in general be non-manifold and composed of a number of regions or cells bounded by faces. For example if a verex on one model coincides wih an edge in he oher so ha he edge is enirely conained wihin he olerance sphere of he verex hen ha edge needs o be removed and is ends combined before he models can be joined \ - l - * *.. ~ L.-s::: <. i : j $ s ) Figure 5. When he verical line is isnprined a hin face needs o be removed. : ~ : i! i $ 25

5 Selec Depending on he operaion (unie inersec subrac) and on he ype of regularisaion required some of he merged model will survive he res is deleed. The selecion phase deermines which eniies are kep based on insidclouside classificaion which is possible because regions mus be unambiguously inside or ouside afer he imprin phase.3 mprin The algorihm compares low-dimension opologies of he wo models before moving on o higher ones. This simplifies he capure of all geomeric relaionships is) verex-verex Deermine coinciden verices. Noe ha one verex may correspond o more han one on he oher model. b) verex-edge f a verex lies inside an edge spli he edge and make he wo correspond. Noe ha spliing he edge may resul in removal of nearby opology some of which may already have been mached o opology in he oher model. The new verex inheris he olerance of he spli edge. c) edge-edge nersec pairs of edges using he sum of he edge olerances. The edges will inersec in areas of coincidence which are parameric ranges wihin each edge. nersecions which represen an area of coincidence which conains a verex can be ignored as hey will have been found by verex-edge comparisons. Any oher inersecions will resul in spliing boh edges The requiremens on he curve inerjecor are ha an inersecion is reurned for each area of coincidence o olerance beween he wo curves. d) verex-face he oher side; and o represen each area of conac or coincidence. need no consider curves (i.e. conac areas) already known o be common by virue of edge-edge or edge-face comparison. This can be imporan since compuaion of nearangen inersecion curves can be very unsable. Trim he curves of inersecion agains he boundaries of he faces. Secions of curve which appear on boh faces are added as new (corresponding) edges. Again any spli edges need o be re-compared for coincidence wih exising edges and faces This can be done reasonably efficienly since only edges wih known common verices need be considered A he end of he imprining phase he boundary of each model has been divided up so ha each face lies inside ouside or on he boundary of he oher model.. Join When eniies of differen olerances correspond he join operaion will use he larger of he wo olerances. This may resul in removal of adjacen opology. Topology on boh models is compressed unil here is a oneo-one correspondence beween mached verices edges and faces on he wo models The wo models are hen opologically joined and he resuling regions deermined..5 Selec Afer he wo models have been combined ino one he regions faces edges and verices which are o survive are deermined. This will depend on he operaion being performed and on various oher opions. For example he boolean can opionally remove lower-dimension opology or spli up a disconneced model ino conneced componens or reurn a collecion of manifold sub-models where possible. f a verex lies in he inerior of a face (no a a verex or on an edge) add a corresponding verex o he face. e) edge-face nersec he curve of he edge wih he surface of face o ge inervals of coincidence on he curve. Since he edge has already been compared wih edges of he face we need only consider coincidence or inersecions inerior o he face and edge. For coincidence add an edge o he face (beween wo exising verices) and make i correspond o he edge. For inersecions add verices o he face and edge. f an edge is spli by his process he pieces need o be recompared wih oher edges since he smaller edges may now be coinciden when hey were no before. f) face-face nersec he surfaces of he faces. The resuling curves can hen be compared wih each face %e surface inerjecor needs o reurn curves and poin conacs sufficien o divide areas of surface which are on one side of he oher surface from hose on Figure 6. Subracing a b-spline face from a block. 25

6 .6 Example boolean Figure 6 shows an example boolean operaion where a 2- dimensional model is being subraced from a block o creae a blend. The face being subraced lies on a b-spline surface and is edges approximaely coincide wih wo of he faces of he block. The boundary edges of his face are oleran The edgejedge comparison in he imprin sage will spli he long edges of he b-spline face ino hree. The edgeface comparison will deec ha wo of hese edges are coinciden wihin olerance wih faces of he block New edges will be added o hese faces dividing hem in wo. The faceface comparison will inersec he b-spline face wih he end faces of he block creaing he end faces of he resuling blend. The join and selec phases will resul in he block being spli ino wo solids. 5. BLENDNG When a blend is creaed a olerance is supplied which is used when consrucing he blend surface and which deermines how closely i fis he neighboring faces. The edges of he blend are given a olerance deermined from his value wihou affecing olerances elsewhere in he model. The blend is added o he model by consrucing he blend face and sewing i in along he oleran edges. Figure 7 shows a variable-radius blend on he edge of a cube. %e olerances are exaggeraed so ha he wo parameer-space curves can be seen on each oleran edge. Normally when he objec is drawn only one of hese curves would be shown. 6. DATA EXCHANGE Wihou informaion on he accuracy of he model exchanging b-reps beween sysems canno be performed reliably. f accuracy is only given as a single value models can only be impored by compromising he accuracy of exising daa. Toleran modelling provides a way round his. Daa from surface modeliers can be impored as rimmed surfaces for example via CES [S] and sewn ogeher. B-rep models can be impored direcly via sandards such as STEP [9]. 6. Sewing Sewing is a way of joining models along coinciden edges and is ofen used o impor rimmed surfaces from oher sysems. The models o be sewn maybe wodimensional shees of faces or solids. f he faces enclose a volume a solid is creaed. Tds is in fac he par of he boolean algorihm which deals wih maching edges and verices; he face-face and edge-face comparisons are omied. lk provides a faser way of combining models when hey are o known o abu. As before he edges are compared o wihin sum of olerances. f wo faces do no in fac mee o he expeced olerance a model may be creaed wih a hin hole in i. The hole can be closed by assigning larger olerances o he edges and re-sewing. 6.2 STEP Figure 7. A blend showing exaggeraedly large local olerances. STEP is an emerging sandard for he exchange of produc model daa. Eniies added o he sandard recenly allow uncerainy values o be associaed wih geomeric definiions. Soluions are currenly being developed which combine he STEP framework wih oleran modelling so ha local olerance informaion can be aached o opology in he STEP file. [f geomery ypes no suppored by STEP need o be ransferred hey can be approximaed providing he approximaion olerance is aached o he relevan faces or edges. 7. FUTURE WORK 7. Face okrances Currenly face olerances always ake he defaul value in Parasolid. Face olerances are no needed in order o define a valid b-rep from rimmed surface daa bu hey are useful o avoid expensive and unreliable surface inersecions. For example if wo surfaces on wo models being unied arc approximaely coinciden or approximaely angen hen inersecing hem o an unsuiable olerance will be very difficul as many complex curves may resul. Marking one or boh of hem wih a olerance would overcome his problem. 7.2 General compresswn of opology The recursive compression of opology resuling from feaures wih differen olerances being mached has no been filly implemened bu a subse sufficien for pracical use has been developed. Furher work is required o develop a general algorihm. 252

7 7.3 Prevening olerance growh Compression normally resuls in some growh of local olerances. deally models defined o a cerain olerance should unie o creae models of he same or similar olerance. Alhough his seems no o be achievable in all cases olerance growh should be conained where possible. 8. CONCLUSON Local olerances on a b-rep model can be used as a foundaion for robus solid modelling and reliable model ransfer beween differen sysems. This echnique has been implemened in a widely used producion solid modeller and has been used o solve real modelling problems. REFERENCES [ Benouamer. M.. Michelucci D. and Peroche B. Errorfree boundary evaluaion based on a lazy raional arihmeic: a deailed implemenaion Compuer-Aided Design Vol 26 No 6 (June 99). [2] Edelsbrunner H. and Mucke E. Simulaion of simpliciy: A echnique o cope wih degenerae cases in geomeric algorihms. Proc. of hacm symposium on Comp Geomery (June 988). [3] Hoffman C. M.. HopCrof J E. and Karasick M. S. Robus se operaions on polyhedral solids. EEE Compuer Graphics and Applicaions 9 (November 989). [] Sewar A.J Local robusness and is applicaion o polyhedral inersecion. Compuaional Geomery & ApplicaionsVol No (March 99) [5] Zhu X. Fang S and Bruderlin B Obaining robus Boolean se operaions for manifold solids by avoiding and eliminaing redundancy. Proc. of 993 ACMEEE 2 Symposium on Solid Modelling and Applicaions. [6] Fang S. Bruderlin B and Zhu X. Robusness in solid modelling - a olerance-based inuiionisic approach. Compu~er-Aided Design. Vol 25 No 9 (Sepember 993). [7] Segal M. Using olerances o guaranee valid polyhedral modelling resuls Compufer Graphics 2 ( 99). [8] GES (99 ). The niial Graphics Exchange Specificaion (CES) version 5.. lgespdes Organisaion Naional nsiue of Sandards and Technology Gaihersburg MD 2899 USA. [9] STEP Sandard for he Exchange of Produc Model Daa. so

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