PERSISTENT NAMING FOR PARAMETRIC MODELS

Transcription

1 PERSISTENT NAMING FOR PARAMETRIC MODELS Dago AGBODAN, David MARCHEIX and Guy PIERRA Laboratory of Applied Computer Siene (LISI) National Shool of Engineers in Mehanis and Aeronautis (ENSMA) Téléport 2 1 avenue Clément Ader BP Futurosope Chasseneuil edex FRANCE agbodan, marheix, pierra@ensma.fr ABSTRACT Nowadays, many ommerial CAD systems support history-based, onstraint-based and feature-based modelling. The use of these new apabilities raises the issue of persistent naming whih refers to the problem of identifying entities in an initial parametri model and mathing them in the re-evaluated model. The goal of this paper in to propose a naming mehanism and an hierarhial struture enabling to identify topologial entities and to apprehend the "design intent". Keywords : CAD/CAM, geometri modelling, parametris, feature taxonomy. 1. INTRODUCTION Stati solid modelling systems (B-rep, CSG, et.) largely used in the Computer Aided Design (CAD) area are more and more replaed by dynami modelling systems (known as history-based, onstraintbased and feature-based modellers) whih allow both to express and to reord oneptual designs and "design intents". These dynami modelling systems are often gathered under the term of parametri modelling systems. A parametri model is omposed of a representation of an objet, of a set of parameters (haraterising the objet) and of a list of onstraints (equations or funtions) applied to the objet. By extension, a parametri modeller is a system for geometri design whih preserves not only the expliit geometry of the designed objet (alled "parametri objet" or "urrent instane"), but also the set of onstrutive gestures used to design it (alled "design proess" or "parametri speifiation"). This two-fold data struture enables rapid modifying by re-evaluation. But when re-evaluation leads to topologial modifiations, referenes (between entities) used in the onstrutive gestures are diffiult to math in the new ontext, giving results different from those expeted. A persistent naming system, robust regarding some topologial modifiation, proves useful to preserve, from a reevaluation to another, referenes between topologial entities. It is the problem known as "persistent naming" or "topologial naming" [Kripa95], [Capoy96]. The persistent naming mehanism should enable, on the one hand, unambiguous identifiation of geometri and topologial entities of the parametri model (1 st issue) in order "to find" them in the reevaluated model (2 nd issue) and on the other hand should enable to represent the "design intent" or rather to represent different semantis that might be expressed by the designer (3 rd issue). This paper mainly fouses on the 1 st and the 3 rd issue. It is strutured as follows. In setion two, we give a detailed aount of the major issues about parametri modelling. The third setion disusses some pre-existing works, essentially two of the main works about topologial naming. Eah of these works only partially addresses 1 st and 2 nd issue. None addresses the 3 rd one. We introdue, in setion four, an alternative approah. In setion five, we propose a new feature taxonomy, suited to our approah. Finally, in setion six and seven we preisely desribe our naming mehanism as well as the struture enabling to handle suh a mehanism. 2. MAJOR ISSUES The main problem for parametri re-evaluation is to haraterise geometri and topologial entities of a parametri model. Charaterising entities onsists in giving them a name at design time and "finding

2 them" again at re-evaluation time (i.e. mathing entities of the initial model and entities of the reevaluated model.) Let us take the example of Figure 1 to illustrate this problem. Initial model Re-evaluated model e swept blok horizontal slot vertial slot round edge e e e 1 e Figure 1 : Naming and mathing problems. In the above example the initial model is designed by means of a parametri speifiation ontaining four suessive onstrutive gestures. The fourth one onsists of rounding edge "e". If the initial model is saved after this fourth step, the urrent instane no longer ontains edge "e" : it was removed by the rounding funtion. Thus the rounding funtion whih has edge "e" as input parameter annot any longer be represented in the parametri speifiation part of the model. Therefore "names" are needed to represent the entities referened in the parametri speifiation whether or not they exist in the urrent instane. Moreover eah onstrutive gesture reates a number of entities whih have to be distinguished and therefore named, to be referened by further onstrutive gestures, even if the same number of entities exist in all possible re-evaluation (no topologial hange). The problem is even more omplex for parametri models, of whih the entities and the number of entities hange from one evaluation to another. Let us return to the above example, but this time in the re-evaluated model. We notie that, at step 3, the edge "e" has been split into edges "e 1 " and "e 2 ". Thus at step 4 the problem is to determine whih edge(s) has(ve) to be rounded. The problem is to identify, i.e. to math, edge "e" with edges "e 1 " and "e 2 " despite topology hanges. Thus, when reevaluation leads to topology hanges a new issue is to math two different strutures. It is thus neessary to have, in addition to the naming mehanism (1 st issue), a robust mathing mehanism (2 nd issue) regarding re-evaluation. Initial model e Re-evaluated models OR (a) (b) Figure 2 : Different semantis. The third identified problem is to be able to express different semantis that apture the "design intent" by using high level abstrations (aggregates of geometri/topologial entities). For example, the designer might want to express that a feature is applied on the global shell rather than on a partiular fae. Figure 2 illustrates this problem. The objet is designed in three steps : reation of the blok by extruding a polygonal ontour, reation of the ylinder on the blok, then rounding the edge between the blok and the ylinder. Aording to whether the rounding funtion was expressed between the ylinder and fae 1.1 (ase 1) or between the ylinder and the upper shell omposed of faes 1.1 and 1.2 (ase 2), the re-evaluated model is different beause the "design intent" is different. In the first ase one obtains model (a) where the round disappeared sine it annot be made any more between the ylinder and fae 1.1. In the seond ase one obtains model (b) where the round always exists sine it ould be made between the ylinder and the upper shell. To support these two different semantis, the naming mehanism should provide for naming and mathing high level entities suh as shells. 3. RELATED WORK Following the pioneer work of Hoffmann and Juan [Hoffm93], over the last few years several authors have analysed the internal struture of parametri data models, proposing some editable representations [Hoffm93], [Pierr94], [Solan94], [Pierr96], [Laakk96], disussing their underlying mathematial strutures [Pierr94], [Ragho98], desribing the problems, either of the semanti of modelling operations [Hoffm93], [Chen94], [Agbod99] or of onstraint management [Bouma93]. Most of them disussed parametri modelling in terms of reation (but not re-evaluation). Several naming sheme and persistent naming mehanisms have also been proposed. In partiular Kripa [Kripa94] and Chen [Chen95] proposed solutions to address some of the problems mentioned in setion 2. The first essentially developed a mathing algorithm whereas the seond foused rather on the unambiguous entity naming. 3.1 Kripa Kripa [Kripa94] fouses on the name mathing. He proposes an API (Appliation Programming Interfae) enapsulating its topologial identifiation system and guaranteeing the persistene of the names using a table of orrespondene between an entity of the initial model and one or more entities of the reevaluated model. He proposes an interesting struture for identifiation of any topologial entities based on fae history (reations, splits, merges and deletions of faes) and a omplex name mathing algorithm. Kripa's Topologial ID System onsists of 3 parts. First a fae graph allowing both a naming of any faes and a name mathing during reevaluation. Seond a table reording names for the only entities that are referened in the parametri

3 speifiation (edges and verties are named in terms of their adjaent faes). This table also ontains pointers to the geometry (urrent instane) more some information for the mathing algorithm. Third the geometry of the designed objet. During eah reevaluation all the faes, as well as every referened entity in the parametri speifiation, are mathed with the new ones. Split faes are named in terms of all adjaent faes, but Kripa does not explain how the initial faes are named. Moreover, during re-evaluation, the proposed mathing algorithm establishes some priority rules, independent of the designer s ation, making the result of the re-evaluation unpreditable. Another limitation is that in its approah, Kripa preserves a opy of the geometri models at eah step of the onstrution proess. This speeds up the re-evaluation but it would require a memory spae whih is not ompatible with the size of the real models used in CAD. Beside the limitation onerning the persistent naming problem, Kripa's model does not deal with the semanti representation problem stated as 3 rd issue in setion Chen Chen proposes a model [Chen95] whih is omposed of two representations. For the first one, he uses an editable representation, alled Erep [Hoffm93], whih is an unevaluated, high-level, generative, textual representation, independent of any underlying ore modeller. It abstrats the design operations, ontains the parametri speifiation and stores all entities by name. The seond representation, evaluated and modeller dependent, ontains the geometry (the urrent instane). The link between these two representations is obtained by a name shema whih establishes the link between the geometri entities of the geometri model and the generi names (persistent) of the unevaluated model. Chen defines a preise struture for naming invariant entities 1, partiularly, for sweep operations. Every entity in a sweep is named by referene to the orresponding soure entity of the swept 2D ontour and the onstrutive gesture. He also proposes an identifiation tehnique for ontingent entities 2 based on topologial adjaenies and feature orientation. In the Chen approah every ontingent entity is named. 1 - An invariant entity is a geometri or topologial entity whih an be, ompletely and unambiguously, haraterised by the struture of a onstrutive gesture and its input parameters, independently of involved values [Agbod99]. 2 - A ontingent entity a geometri or topologial entity that results from an interation between the pre-existing geometri model and invariant entities resulting from a partiular onstrutive gesture [Agbod99]. Unambiguous names are generated by omposition of topologial adjaenies. Unfortunately, Chen has mainly studied the naming problem at the onstrution stage of the parametri model, and has almost not studied the re-evaluation stage. The mathing mehanism is not fully defined. 4. OVERVIEW OF OUR APPROACH Our approah is similar to Kripa s one. The main differenes are that we use a shell graph as basis of the persistent naming and that we use an hierarhial aggregation struture to apture, at various levels, the "design intent". Our approah deals with the three issues stated in setion 2. We propose : A naming mehanism that onsists of two parts. First an oriented shell graph (see setion 6). Like the Kripa s fae graph, our shell graph provides for traing shell evolution. Following [Agbod99] we distinguish invariant and ontingent shells and we propose to identify, unambiguously and uniquely, both the invariant faes and shells (see setion 7.1), then the ontingent faes and shells (see setion 7.2). Seond, a table where edges, paths and verties referened in the parametri speifiation are named in terms of their adjaent faes or shells (1 st issue) A mathing mehanism, outlined in setion 7.2, where nodes of the shell graph are mathed from the initial design shell graph onto the urrent reevaluation shell graph. Other topologial entities are mathed by referene to the shell graph. A definition, for eah kind of feature, of the various semantis that might be expressed by a designer. This is done through a taxonomy (see setion 5) where eah kind of feature is assoiated with a predefined hierarhial aggregation struture (3 rd issue). 5. A FEATURE TAXONOMY In order, to identify invariant shells in any form feature, we need a feature taxonomy. Indeed, these invariant shells do not hange, whatever be the instaniation or whatever be the re-evaluation of the feature. They will onstitute entry nodes of the shell graph. Most of urrent feature taxonomies propose to interpret form features from a mahining point of view or from a speifi appliation area point of view. These taxonomies do not fous on strutural invariants and are thus not suited to our problem. We propose a feature taxonomy based on their invariant struture. By invariant struture, we mean both the initial intrinsi topologial struture of feature (before interation with the objet s geometry) and the topologial struture resulting from the interation with this existing geometry. For example, we onsider as invariant the fat that systematially the final fae of a through_hole is deleted when the feature (the through_hole) interat with the geometry of the objet (i.e. when the hole is

4 made in the objet). The invariant struture depends on how the feature is designed. For example, for extrusion and revolution features invariants are the initial shell (i), the lateral shell (l) and the final shell (f). The lateral shell (depending on the partiular topology of the extruded ontour) an be strutured in sub-shells (l 1 and l 2 ) whih themselves an be strutured. For example, in right, left and bottom shells (l d, l g, l fd ) for a rossing_flat_slot. The rossing_flat_slot example represented below in Figure 3, illustrates well the invariant struture of a feature. Note that some entities (i, l 1, f) disappear systematially. rossing_flat_ slot i* f* l l 1 * l 2 l d lfd l g Figure 3 : Invariant feature struture example : rossing_flat_slot. Our feature taxonomy (see Figure 4) is based on the feature taxonomy proposed by Shah and Mäntylä [Shah95] and the lassifiation proposed in the STEP standard [Iso94]. We lassified these features aording to their strutural invariants whih we have extrat for eah one of them. Primitive features. Suh feature are predefined features. They are haraterised by their global shell and their faes. Transition features. Suh features are features joining several shells suh as hamfer or rounding. They are haraterised by their global shell. Basi volume features. These features result from an extrusion or a revolution. They are haraterised by their initial, lateral and final shells. They are lassified aording to the result of their interation with the geometry. For example a blind_hole, obtained by extruding a irle, is haraterised by Feature Primitive feature Blok (global shell + Cylinder numbered faes) Transition feature () Basi volume feature l i f Basi surfae feature Texture feature ( ) Container feature Rounding Chamfer (loop i, l, loop f) Threading Knurl i l f i* l f i* l f* 1 i 1 * l 1 2 f 1 i 2 * l 2 f 2 pattern assembly Extruded_feature Revolved_feature_partial (toroidal,spherial) Boss Blind_hole Closed_poket Through_hole Boss_to Revolved_feature_total (toroidal,spherial) i 1 * 1 l 1 f 1 * i 2 * 2 l 2 f 2 Blind_ounter_sunk_hole Blind_ounter_bore_hole i l f i* l f i* l f* l 1 * l 2 l 1 * l 2 l 1 * l 2 Internal_slot Rounded_end Step Open_poket External_slot Crossing_slot i 1 * 1 l 1 f 1 i 2 * 2 l 2 f 2 * i 1 * 1 l 1 f 1 * i 2 * these three shells (i, l, f) and by the fat that the initial shell disappears from the geometry. Basi surfaes features. They are all the surfaes obtained by sweeping a loop. They are haraterised by their initial and final loop and their lateral shell. To these four basi lasses one may add two other lasses. Container features gathers some partiular feature aggregations : speifi aggregates (blind_ounter_sunk_hole, ), eah one assoiated with a partiular invariant shells struture, repetitive aggregates (pattern) and unstrutured aggregates (assembly) that onsist only of basi faes without invariant shells struture. Texture features (e.g. knurl, threading, ) do not introdue new topologial entities and are not relevant for our purpose. The terminology used in this taxonomy, in partiular names of the features, has been borrowed both from Shah and Mäntylä [Shah95] and from the STEP standard [Iso94]. 6. SHELL GRAPH i* l f* l 1 * l 2 l d l g l fd Through _ounter_sunk_hole Through _ounter_bore_hole i = initial shell l = lateral shell f = final shell * = deleted shell due to interation with the pre-existing geometry Figure 4 : Feature taxonomy (aording to their invariant struture). 2 l 2 f 2 * i l f i* l f l 1 * l 2 l 1 * l 2 l d l g l fd l d l g l fd Internal_flat_slot Internal_o_slot External_flat_slot External_o_slot Crossing_flat_slot Crossing_o_slot Shells are defined as shell aggregates. On the lowest level of the hierarhy eah shell represents a fae. The shells are : Hierarhial. They are strutured in shells, subshells, et., in order to be able to apprehend the geometry at various levels of granularity, and thus to express different semantis. This struture is defined for eah lass of feature (a global shell and invariant sub-shells resulting from our feature taxonomy. Conneted. Eah shell is omposed of jointed sub-shells. The objetive of the shell graph is to represent the history (split and deletion, merge is urrently not allowed) of the invariant initial struture (itself onneted). Thus, a new graph node represents a new onneted part whih appears during a split and enables to identify it and to trae it. Overlapping. A subshell an be part of several shells. The overlapping hierarhial struture enables, during onstrution, the reation of any (onneted) shell aggregate. This presents both a flexible and an extremely powerful designation mehanism. Espeially for referening aggregates stemming from the union of a feature with some parts of the objet (see setion 6.2 for details and see shell C3 in

5 Figure 5 for an example). 6.1 Interest of the shell graph There are two interests in using shells and a shell graph : entity aggregation and entity traability (with the aim of mathing). Aggregation (shell interest). The goal is, for the designer, to be able to referene geometry at various levels of granularity. Faes are low level entities, having often few meaning for the designer. Shells make it possible referene meaningful high level abstration. For example, in an extrusion there is a single lateral shell whih is omposed of several lateral faes. The designer may referene this shell for example for mathing a fillet. Traability (shell graph interest). The goal is to be able to follow the shell evolution in order to be able, during model design, to identify the involved shells, then, during re-evaluation, to identify the effetive shells (in the urrent instane) orresponding to the referened shell. In addition to the possibility of referening eah shell represented in the graph, the shell graph last allows another level of aggregation. The aggregation of all the shells having had a ommon anestor. Thus, for example, the abstration of lateral shell 2.l in Figure 5 will exist even in ase of split of this lateral shell sine at eah level of aggregation the history is preserved. 6.2 Graph struture The graph stores the feature struture (aggregation) and the shell history (traability). Eah onstrutive gesture an be broken up into two steps. The first step is the speifiation of the isolated feature. It orresponds to the invariant struture (defined in our feature taxonomy). This invariant feature struture is represented by hierarhial links between shells. It represents the entry nodes of the shell graph. The seond step is the interation with the pre-existing geometry whih produes ontingent entities. Those entities result from the hanges in the initial hierarhial struture and the pre-existing geometry. The hierarhial struture is always represented by hierarhial links. The shell evolutions are desribed by historial links. Eah node of the graph is identified extrusion nd shell level 1 st shell level C f 1.l 1.i extruded_feature, no interation with the geometry 1.4 C f 2.l 2.i 2.l1 2.l2 2.2 extrusion first extruded_ feature Null interation with the geometry : only deletions 1.4 C3 = aggregation_ feature, no interation with the geometry Figure 5 : Shell graph example. by two elements. The first one is the onstrution step number and the seond one is an identifier desribed in setion Hierarhial links extrusion C The feature struture (desribed in setion 5) is represented in the graph by the hierarhial links. Beside the representation of the feature s invariant struture, this hierarhial links enables to represent new aggregates that might appear during the design proess. Figure 5 gives an example. An internal_slot is applied on the final shell (1.f) of an extruded blo (extruded_feature). A shell C3 an be generated (for example on expliit request of the designer whih wishes to handle this aggregate) in order to be able to name and thus to address the aggregate orresponding to the union of the slot (C2) and the shell arrying this slot (1.f). C3 thus has two hierarhial links to 1.f and to C2. Moreover, we an notie that this example illustrates the ase of shell overlapping presented in setion 6 sine the shell 1.f belongs now to two distint shells C1 and C3. The hierarhial struture evolves with the modifiations of the shells. Thus, in the graph example, illustrated in Figure 5, the remaining lateral shell (2.l2) whih is omposed of the shells 2.3, 2.4 and 2.5 is ut into two shells 4.13 and These last, are in turn, omposed of the result of the split of the shells 2.3, 2.4 and 2.5, so that, at eah step, there exist a (modified) hierarhial struture. A property of this graph, whih signifiantly simplifies its management is that the hierarhial links need to be desribed only at the level of the historial leafs of the graph. To prove this property we study for two suessive steps and for two su f 4.l 4.i 4.l1 4.l seond extruded_ feature Null Hierarhial links Historial links interation with the geometry : deletions and splits

6 essive hierarhial levels (here lowest level of shells faes and first level of aggregation, but an be extended to any suessive hierarhial levels) all the possible ombinations of entity histories. The history is redued to split. Indeed merge is not allowed and entity deletion (in fat, pointers to NULL) does not hange the historial links of the graph. f 1.1 f f 1.1 f 1.2 f 1.1 f 1.2 f 1.1 f 1.2 not split not split split split shell not split f 1.1 split split split faes 1.1 f 1.1 f 1.2 f 1.1 f 2.1 f 2.2 f f f 1.1 f 2.1 f 2.2 f 1.2 f 1.2 hierarhial links f 2.3 f 2.4 deleted hierarhial links historial links Figure 6 : Hierarhial links evolution. It is possible to dedue from these four ases that the knowledge of the hierarhial struture at step i is suffiient to know the struture at step i -1. Indeed, it is suffiient to ompare the onstrution step numbers (reation order) to know when a shell was reated. For example, in the seond graph of Figure 6, the fae f 2.1 is a sub-shell of 1.1. The fae f 2.1 whih was reated after 1.1 thus results from a shell (f 1.1 ) whih was sub-shell of 1.1. This reasoning, applied to eah of the four ases, and extended to any step i and i -1, makes it possible to generalise the property. For example, in the (partial) graph represented in the table below, if the shell B is the Parent in the Hierarhy of shell D (PH i (D) = B) at step i, it is possible to dedue from the reation order of D and B (n (D) and n (B)), whih was the Parent in the Hierarhy of D (or C as the ase may be) at step i -1. Step i Step i -1 PH i (D)=B n (D)<n (B) if and n (B)=i then PH i-1 (D)=A A B n (D)=n (B) if C D and n (B)=i then PH i-1 (C)=A if n (D)>n (B) and n (D)=i then PH i-1 (C)=B It is possible to find the hierarhial struture of step i -1, knowing the one of step i. A reurrene allows to onlude that it is possible to preserve the hierarhial links only at the level of the historial leafs. Note that apability to ompute the state of the old graph, at any step, is essential to make the mathing with the graph resulting from the re-evaluation Historial links For eah graph s node and in partiular for eah leafs of the hierarhial struture, we shall trae its A C A C A C B D evolution. There are 3 possibilities for this modifiation (merge is not allowed) : Deletion. The interation between the geometry of feature and the geometry of the objet on whih it is applied, deletes a ertain number of shells (of the feature and/or the objet). These shells remain present in the graph, but without any ounter part in the geometry. The historial links point to NULL. Modifiation. By modifiation, we mean any evolution whih preserves the shells onnetions. We onsider that suh a modified shell is equal to the shell before modifiation. Indeed, if at step i a shell is perfetly identified, and if at the step i +1 this shell is modified (exept split), it remains perfetly identified in the graph. Thus, shell modifiations are not represented in the graph Split. There is a split when a shell is ut into several disonneted shells. This split is represented in the graph by the historial links. Thus, a ut shell points on the resulting sub-shells and inversely. Historial links enable to trae the topologial entities evolution (split). The history is preserved at all levels of granularity (faes, shells, aggregates of shells, ). This gives to the model an expression power higher than the one of Kripa s model. Indeed, when a shell is ut into several piees during a onstrutive gesture, it is possible to have aess not only to the shell (represented in the hierarhy) but also to eah shell piee (represented in the history). Those historial links are mainly used for graph traversal during name mathing. 6.3 Node struture Eah node represents a shell whih exists or has existed in the model. All the shells without outgoing historial links exist in the geometry. Eah node is omposed of : A name omposed of the onstrution step number and another identifier whih haraterises uniquely the shells (see setion 7 below). Pointers to the predeessor and suessor nodes representing the historial links. Pointers to the parent and the hild nodes representing the hierarhial links. A list of adjaent faes for split shells. 7. ENTITY NAMING The entity (verties, edges, paths and shells) identifiation is done by referene to faes (lowest level shells, i.e. leafs of the hierarhial struture). It is thus neessary to be able to name these faes in a unique and deterministi way. Generally, the identifiation of an entity is based on unhanging elements whih haraterise it in a unique way. In a parametri model, what never hanges is the onstrution

7 proess 3. Therefore, fae naming is done by means of the onstrution step number (reation order) and by means of another identifier whih haraterises eah fae in a unique way. The problem is to define this identifier whih haraterises them in a unique way within eah onstrution step. For eah onstrution step, we onsider that there are two phases. Firstly, the reation of the feature where all the feature s entities, i.e. its invariants, and in partiular the lowest level shells must be named. Seondly, the feature positioning within the existing geometry. This interation with the existing geometry leads to modifiation and deletion of existing shells and to reation of new (ontingent) shells. These ontingent shells must also be named. Therefore there are two types of naming to implement : one for invariant shells and another for ontingent shells. 7.1 Invariant shells For the invariant shells, two ases are to be distinguished. First, the lowest level shells that are faes (leafs of the hierarhial struture), and seond higher level shells Lowest level invariant shells Identifiation of the lowest level invariant shells (invariant faes), is done by an integer number whose alulus is based on topology. This naming is robust regarding feature re-evaluation. We do not allow modifiation of the topology of features. In partiular, for swept and revolved feature, hanges of the 2D ontours topology are not allowed. Thus, this "topologial" naming is robust regarding reevaluation. Our topologial naming is as follows. We begin from a starting oedge 4 and we make a radial traversal around the edges, then a normal traversal following the edges of eah fae ontour. The traversal whih is unique (ompared to the starting oedge) and whih over the whole objet makes it possible to assign a number to eah fae (see Figure 7). The traversal is done as follows. Do a normal traversal of the path ontaining the oedge of the urrent fae. For eah oedge do a radial traversal of all adjaent faes. If these faes are not numbered, assign a number and enqueue them. 3 We onsider the modifiation of the onstrution proess as a model edition and not as a model reevaluation. 4 The features are defined, in addition to their parameters, by a starting oedge. For example, in an extruded blok, the first edge of the 2D ontour defines the starting oedge. At the end of the path traversal take the first enqueued fae. Repeat the previous traversal, starting from the oedge shared by this fae and the fae with the smallest number n i fae number traversal order radial traversal normal traversal Figure 7 : Topologial traversal. The name of the lowest level invariant shells is thus omposed of the onstrution step number and the fae number resulting from the traversal Upper level invariant shells Invariant shells, others than the lowest level ones, are aggregations of lower level shells. The invariant shells an be named by means of the list of shells whih ompose them, and by transitivity, by means of the list of the lowest level shells whih ompose them. These last are perfetly identified and the shell list is unique ; the shells are thus identified in a unique way by the list of the lowest level shells. The name of upper level invariant shells is thus omposed of the onstrution step number and the list of the names of the lowest level shells whih ompose the shell. 7.2 Contingent shells The name of the ontingent shells is omposed of the onstrution step number and an iterative number. This number (arbitrary, but unique for eah onstrution step) will be assigned again after reevaluation during the mathing (graph mathing 5 ). This number, insuffiient to allow mathing, is assoiated with the list of the adjaent faes to the shell. This list enables to distinguish two sub-shells resulting from the same shell. The list is omposed of the lowest level shells (leafs of the hierarhial struture faes ) whih are adjaent to the split shell. This information will be used, during reevaluation, for ontingent shell mathing. Although the omplete mathing mehanism goes beyond the sope of this paper, we shall present its outlines. Its priniple onsists of : keeping the initial shell graph. The whole reevaluation proess is supported by this graph. 5 There are two stages during the mathing : (i) the graph mathing, i.e. reonstruting the graph and mathing the nodes. (ii) the entity mathing, i.e. determining the new entities involved in the parametri speifiation.

8 Construting during re-evaluation, a seond shell graph, parallel to the initial shell graph. At eah onstrution step, all the entries of the seond graph are ompared to the nodes (shells) of the initial graph. For eah mathed shell, the same iterative number will be assigned. For the shells whih are not mathed, a new iterative number (whih do not exist in the initial graph) will be assigned Mathing the entities involved in the parametri speifiation. For eah entity a mathing algorithm, similar to the one proposed by Kripa, omparing the seond graph and the initial graph, is applied. Eah old entity, whih do not math exatly a new one, is replaed by one or several new entities based on omparison of "anestors". 8. CONCLUSION With the development of parametri modelling systems, the persistent naming problem beomes more and more important to enable topology hange during model re-evaluation. Therefore, robust naming mehanism and mathing mehanism have to be developed. In this paper, we have proposed : A naming mehanism (1 st issue) where we distinguish invariant entities and ontingent entities. An hierarhial struture whih enables to express different semantis (3 rd issue). In order to identify feature invariants, a new taxonomy based on the intrinsi and semanti struture of eah feature was introdued. A shell graph reording this hierarhial struture and the evolution of the shells. The graph will enable us to math ontingent shells during re-evaluation (2 nd issue). This approah offers three prinipal advantages. Firstly it enables to unambiguously identify topologial entities of an initial parametri model. Seondly it enables to math entities between the initial model and the re-evaluated model, in spite of the multiple topologial variations. Finally, suh a mehanism enables the apture of different semantis that might be expressed by the designer. The work presented in this paper is still under development. We are atually implementing our model in the Cas.Cade development environment, testing various mathing solutions in ase of large topologial hanges. ACKNOWLEDGEMENTS The authors are grateful to Pasal Lienhardt and Laurent Fuhs for several useful workshops and a number of insightful disussions. REFERENCES [Agbod99] Agbodan,D, Marheix,D, Pierra,G : A Data Model Arhiteture For Parametris in Journal for Geometry and Graphis, Vol.3, N.1, pp.17-38,1999. [Capoy96] Capoyleas,V Chen,X Hoffmann,C.M. : Generi naming in generative, onstraint-based design in Computer Aided-Design, Vol.28 N 1 pp.17-26, [Chen95] Chen,X : Representation, Evaluation and Editing of Feature-Based and Constraint-Based design, Ph.D. thesis, Department of Computer Sienes, Purdue University, West Lafayette, Indiana, [Hoffm93] Hoffmann,C.M., Juan,R : EREP : an editable high-level representation for geometri design and analysis in Tehnial Report CER-92-24, Department of Computer Sienes, Purdue University, West Lafayette, Indiana, [Iso94] ISO FDIS :1999 : Industrial Automation Systems and Integration Produt Data Representation and Exhange Part 224 : Appliation protool : Mehanial produt definition for proess planing using mahining features, ISO, Geneva, [Kripa95] Kripa,J : A mehanism for persistently naming topologial entities in history-based parametri solid models (Topologial ID System) in Proeedings of Solid Modeling 95, Salt Lake City, Utha USA, pp.21-30, [Laakk96] Laakko,T, Mäntylä,M : Inremental onstraint modelling in a feature modelling system» in Computer Graphis forum, Vol.15, N 3, EUROGRAPHICS 96, Poitiers, Frane, pp , [Pierr94] Pierra,G, Potier,J.C., Girard,P : The EBP system : Example Based Programming for parametri design, Workshop on Graphi and Modelling In Siene and Tehnology, in : Springer Verlag Series, Coimbra, June [Pierr96] Pierra,G, Ait-Ameur,Y, Besnard,F, Girard,P, Potier,J.C. : A general framework for parametri produt model within STEP and Part Library in European Conferene Produt Data Tehnology, London, April [Ragho98] Raghotama,S, Shapiro,V : Boundary Representation Deformation in Parametri Solid Modeling in ACM Transations on Graphis, Vol.17, N 4, pp , Otober [Shen94] Shenk,D, Wilson,P : Information Modelling The EXPRESS Way, Oxford University Press, [Shah95] Shah,J.J., Mäntylä,M : Parametri and feature-based CAD/CAM : Conepts, Tehniques, Appliations, John Wiley and Sons In., July [Solan94] Solano,L, Brunet,P : Construtive Constraint-based model for parametri CAD systems in Computer-Aided Design, Vol.26, N 8, pp , 1994.