Automated generation of FEA models through idealization operators

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2000; 49:83}108

Automated generation of FEA models throughidealization operators

L. Fine*,s, L. Remondinit and J.-C. LeonA

¸aboratoire des Sols, Solides, Structures ;MR-CNRS 5521, BP 53 38041 Grenoble Cedex, France

SUMMARY

This paper describes an automated idealization process of FEA models monitored by a mechanical criterion.This process, based on transformations of a polyhedral geometry, allows large geometric modi"cationsincluding topology modi"cations (such as hole removal). Polyhedra are used as input geometry for modeladaptation purposes because of their signi"cant freedom for shape modi"cations compared with CSG orB-Rep models. These polyhedra act as intermediate models to form the adapted input geometry required fora "nite element mesh generation.

These idealizations are carried out through a vertex removal process which transforms the geometry ofa part while preserving it within a discrete envelope de"ned around its initial geometry. This envelopeis obtained from a mechanical criterion which can be based either on an a posteriori error estimatoror on a priori estimation. The use of such a criterion ensures that all geometric transformations whichkeep the geometry of the part within this envelope do not signi"cantly change the results of the FEAIn addition to geometric transformations strictly respecting this envelope, operators used for idealiza-tion are also able to transfer speci"c data (like boundary conditions) from the initial geometry tothe idealized geometry. Such operators allow an extended automation of geometry simpli"cationand idealization processes and ensure that the new geometry is more suited to the element sizerequirements of the mesh generation process. This approach is illustrated and validated through anexample. Copyright ( 2000 John Wiley & Sons, Ltd.

KEY WORDS: geometric adaptation; idealization operators; polyhedral model; vertex removal; discretecurvature; mechanical criterion

1. INTRODUCTION

In various engineering applications, "nite element analysis (FEA) has become the standardtechnique for evaluating the physical performance of structural systems. During mechanicalanalysis, such processes are often restricted to validation phases of a design process because of the

*Correspondence to: Lionel Fine, Laboratoire des Sols, Solides, Structures UMR-CNRS 5521, BP 53 38041 GrenobleCedex, France

sE-mail: lionel. "[email protected]: [email protected]: [email protected]

Received 15 April 1999Copyright ( 2000 John Wiley & Sons, Ltd. Revised 4 August 1999

time needed to generate the mechanical model. Indeed, the generation of FEA models needssigni"cant expertise from engineers to avoid wasting computer resources. In the past decades,with limitations in computational power and software, engineers were forced to idealize theirdesign as 2D plane stress, plane strain, or axisymetric analyses. Recent developments of softwareand hardware allow them to perform "nite element analyses directly from 3D design models.Marc Halpern [1] presents an extensive study on industrial requirements and practices in "niteelement meshing and shows that, between 1991 and 1996, advances in mesh generation, improve-ments in CAD-FEA integration and the impact of adaptivity and error control have reduced themedian time requirement by 27 per cent to prepare models and by 48 per cent to perform the fullanalysis. In addition, he asserts that the state-of-art is not yet able to satisfy the requirements ofthe early design practice. Therefore, signi"cant advances are still required, in particular in theautomation of the generation of e$cient FEA models directly from CAD models.

To deal with the dynamic behaviour of an automation process, the approach developed herefocuses on the integration of adaptation and idealization processes of FEA models (see Section 2).It is based on polyhedral representation of parts which should allow signi"cant freedom for shapemodi"cations compared with other geometric models such as CSG or B-Rep ones. Sucha geometric representation is also able to cope with all the data required for the de"nition of FEAmodels (see Section 3). These capabilities, combined with e$cient geometric and mechanicalcriteria (see Section 4) allow to monitor geometry transformations in order to obtain idealizedgeometries more suited for the "nite element analysis considered (see Section 5). Applications onfew examples of this adaption process are presented in Section 6.

2. TOWARD AUTOMATED FEA ANALYSIS

2.1. Di+ering needs of CAD and FEM models

CAD and FEM are two signi"cantly di!erent disciplines; their respective needs from a part modeldi!er substantially: a CAD objective is to build the geometry of models through surface or solidrepresentations whereas an FEM objective is to analyse the mechanical behaviour of thesemodels.

CAD models are mainly used to de"ne the geometry of parts during the design process.Through the evolution of this process, di!erent levels of representation are generated. During theearly steps, coarse representations (like non-manifold models or volume models bounded byelementary surfaces (plane, cylinders, etc.)) can be su$cient to de"ne the basic geometry (includ-ing functional areas and main bulk) of a part. At later steps, CAD models used for manufacturingmust incorporate all the geometric details of a part, thus more complex representations (likeB-Rep models bounded by free form surfaces) can be required.

FEM models are dedicated to the physical analysis (mechanical, thermal, etc.) of a part.Hence, the most accurate geometric representation of the part is not necessarily suited forthis phase. Indeed, FEA is subjected to two strong constraints: the desired accuracy ofphysical phenomena modelled within parts and the computation time needed to obtain thisaccuracy.

To reach a good compromise between accuracy and computation time, it is necessary toproduce an e$cient FEA model. Therefore, the automatic generation of an FEA model froma CAD model cannot be thought of as a basic process utilizing CAD data only. Indeed, various

84 L. FINE, L. REMONDINI AND J.-C. LEON

Copyright ( 2000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2000; 49:83}108

problems could appear:

(1) generating a mesh from CAD data produces an FEA model geometry close to CADgeometry which is not generally satisfactory from computation aspects. Indeed, CADmodels include various minor details relevant for design and manufacturing but irrelevantand even harmful to the analysis. Irrelevant details, while not contributing signi"cantlyto the analysis results, can strongly constrain and even a!ect the quality of the mesharound these details. The concept of detail is closely linked to the parameters of ananalysis. E!ectively, small geometric features (like holes, thin ribs) may be the locationof critical stress concentrations under speci"c load cases and may have a negligiblee!ect under other con"gurations. Therefore, details are not only small geometricfeatures with respect to the size of the model but they can also cover large areas of themodel. This status depends on the in#uence of these zones on the behaviour of the part fora given load case. To ensure an e$cient analysis, such details must be removed from theFEA model,

(2) a CAD model can be also badly suited to an FEA Indeed, speci"c areas of the initial 3Dgeometry may need to be represented by 2D or 1D elements (like shell or beam elements) inthe FEA model. This involves large modi"cations of CAD models, such as a dimensionalreduction.

Analysts often avoid these problems by generating their own idealized and adapted geometricmodels. However, they still have to take care to build a coherent and unambiguous geometricmodel. These tasks need expertise and time. Thus, an automated generation of analysis modelsfrom CAD data is critical and requires an e$cient process of geometric adaption in order toreduce the complexity of the FEA without signi"cantly changing its results.

2.2. Geometry simpli,cation and adaptation for FEA models

Former work in the "eld of detail removal and geometry adaptation can be characterized by theapproaches of:

(1) She!er [2], who developed an automated scheme for detailed removal and geometryclean-up using virtual topology. She de"ned, through a clustering algorithm based on themodel geometry, an alternative topology of the input model. This topology is based ona subdivision of model face envelopes into regions of restricted curvature and distancedeviation. This concept of virtual topology is used to de"ne a set of automatic andsemi-automatic tools used to pre-process CAD models in order to make them more suitedfor mesh generation,

(2) Dabke [3], who uses feature recognition to perform detail suppression from CSG models.This technique, based on heuristic knowledge, is able to support "nite element idealization.It requires the de"nition of a set of insigni"cant features to be detected and removed fromthe model. Such a technique relies on a possible mapping between details and features,which does not always exist.

(3) Dey [4], who developed local mesh modi"cation operators to perform the automatedidenti"cation and elimination of small geometric features from an initial mesh. Theidenti"cation process is based on the detection of small edges, which create small angles oftriangular faces or tetrahedral elements that have extremely large dihedral angles. Theirelimination is obtained through an edge collapse operator. Tests are performed on the

AUTOMATED GENERATION OF FEA MODELS 85


resulting geometry and topology to ensure that no dimensional reduction has beengenerated in the modi"ed mesh. Such a technique was developed to improve the quality ofmeshes to maintain optimal solution convergence rate.

(4) Mobley [5], who uses an object oriented approach for CAD geometry defeaturing. Hisalgorithms deal with near tangencies, coincident edges, curve accuracy, and small geometri-cal features. Two general categories of defeaturing are presented: geometry-based defeatur-ing and "nite element model-based defeaturing.

(5) Armstrong [6], Donaghy [7], who use data provided by the medial axis transform (MAT)for adapting and idealizing geometry. MAT produces a skeleton of a geometric representa-tion which can be analysed to identify features. These features are then adapted or idealizedthrough Euler operators applied on the model topology.

The approaches described above deal with either CAD models (Dabke, Mobley, Armstrong,Donaghy) or FEA models (Dey, She!er) and present various ways to automate the adaptationand idealization of an FEA model geometry. To increase the global automation of the adaptationprocess during FEA computation, it is necessary to ensure that all the transformations performeddo not signi"cantly change the FEA results. Therefore, it is necessary to have adaptation andidealization operators not only managed by the geometric or topological descriptions of a partbut also by data linked with the desired analysis (like boundary conditions, mechanical behav-iour). To this aim, the next section presents a way to integrate automated geometry adaptationinto an FEA process.

2.3. An approach to automated geometry adaptation

Like the methods presented in Section 2.2, the current work focuses on an automated process forgeometry adaptation of FEA models.

Two directions have been considered to incorporate adaptation phases within an FE analysisprocess (see Section 2.4):

(1) a priori approach: during the pre-processing step when the engineer de"nes the initial meshfor the FE analysis,

(2) a posteriori approach: during the FEA computation loop. In this case, adaptation stands asan automatic process modifying the geometry of the FEA model much in the same waya posteriori mesh adaptation does.

To perform such an adaptation it is necessary to use:

(1) a geometric model allowing large shape modi"cations,(2) criteria having a mechanical meaning and ensuring that all the geometric transformations

applied do not signi"cantly change the results of FEA computations. They should staycoherent with the objectives of the analysis,

(3) geometric operators capable of performing general geometric transformations.

Here, geometric adaptation is performed on a polyhedral representation of the part. Sucha representation is de"ned using a polyhedral model based on an unstructured triangulation ofthe part. This polyhedron can be obtained through various ways such as CAD models or scannedmodels triangulations. In any case, the polyhedral representation obtained is used as a geometric



representation from which meshes (triangular, tetrahedral, hexahedral, etc.) can be generated.Therefore in this document a triangulation of the geometry is called &polyhedron' and the "niteelement model built from this geometry is called &mesh'.

Compared with B-Rep or CSG models, a polyhedral model, which can be generated from CADenvironments, has many advantages in the present context. On one hand, it is an appropriatebasic representation for mesh generation. On the other hand, such a geometric representationallows signi"cant freedom for shape modi"cations whereas CSG or B-Rep models use theconcept of elementary features (plane area, cylindrical area, etc.), which are di$cult to handlewhen general transformations should take place. Consequently, the adaptation operators are freeto modify a polyhedral representation of a part without any constraint of feature preservationrelated to the input model. Thus, they are well suited to carry out all the required modi"cationson the initial geometry.

The analysis of this geometry is composed of two steps. The "rst step is based on thecomputation of discrete curvatures of each area of the polyhedron. This analysis helps locate thegeometric details, i.e. the areas which would need a high mesh density because of their geometricshape (see Section 4.1). The second step is based on a discrete envelope de"ned around the inputpolyhedron. This envelope de"nes the accuracy of the discretization needed to get signi"cantresults for an FEA Hence, this criterion describes the mechanical behaviour of the load casestudied (see Section 4.2).

Both criteria de"ned above are used to monitor the adaptation operators. These operators arebased on a vertex removal process and are used to perform all geometric transformationsincluding idealization (see Section 5.5). Some results, including an example of geometry adapta-tion and FEA computation are shown in Section 6.

2.4. Geometric adaptation integrated into an FEA process

Interactive a priori adaptation: The aim of a priori treatments for geometry adaptation is toprovide an easy way to perform geometry transformations interactive functions. Among thecriteria used to monitor these transformations is the expected level of discretization used for themesh (see Section 4.2). Such a discretization can be described by a map of element sizes and isde"ned a priori by the analyst during the pre-processing step of the FEA. Thus, the analyst is freeto use directly either the map of element sizes of the mesh that he wants to generate or to builda speci"c map dedicated only to geometry adaptation.

Automatic a posteriori adaptation: The automated process for geometry adaptation presentedhere, can be considered as a complement of another process of FEA model optimization: theautomated adaptation of meshes. This adaptation process is inserted into a repeated computationof FEA which exploits an a posteriori error estimator to provide, at each iteration, the dataneeded to de"ne a new mesh producing better FEA results [8, 9]. The iterative process stopswhen the desired accuracy of FEA results is reached. Figure 1 shows the global scheme of such aniterative FEA computation.

During this process, the a posteriori error estimator incorporates the generation of a map whichde"nes in each area of the part a new and better suited size of elements. The error estimator isused as a mechanical criterion to monitor the mesh adaptation (see Section 4.2). In the approachdeveloped here the geometry adaptation process is thought as a process which adapts thegeometric representation of a part in the same way a mesh is adapted. Thus, at each iteration,



Figure 1. A posteriori mesh adaptation process.

Figure 2. A posteriori geometry and mesh adaptation.

a new adapted geometry becomes the new geometry for the generation of the next mesh. Anillustration of the insertion of such a geometry adaptation concept in an FEA computation isgiven in Figure 2.



3. DATA REQUIRED FOR THE ADAPTATION OF AN FEA MODEL

The adaptation operators must consider all the data required for the model geometry and all thedata of the FEA model that could interact with the adaptation process.

3.1. Geometric data

The input data of the adaptation process is a polyhedral model of a part. Such a geometric modelcan describe all the di!erent shapes required for an FEA model like volume (by the way of closedsurfaces), open surfaces or non-manifold geometries (for idealized models) and can be easily usedin mesh generation process.

For each of these di!erent categories (surfaces, closed surfaces, non-manifold geometries),adaptation algorithms must behave in a speci"c way. Hence, an identi"cation of all thesecategories is necessary before the adaptation process starts. This identi"cation is performed bymeans of a classi"cation of the edges and nodes of the polyhedron into speci"c classes. Threedi!erent classes have been de"ned: boundary class, surface and non-manifold classes.

The classi"cation of each edge and node is performed as follows:Classi"cation of the edges:

(1) an edge connected to one face only is classi"ed as a boundary edge,(2) an edge connected to two faces is exactly classi"ed as a surface edge,(3) an edge connected to more than two faces is classi"ed as a non-manifold edge.

Classi"cation of the nodes:

(1) a node connected to two boundary edges only is classi"ed as a boundary node,(2) a node connected to surface edges only is classi"ed as a surface node,(3) a node connected at least to one non-manifold edge is classi"ed as a non-manifold

node.

This classi"cation forms the basic data required to manipulate polyhedral geometries. The nextsection shows that another set of classes is required to take care of speci"c data linked with anFEA model.

3.2. FEA data

FEA models incorporate data strongly linked with their geometric description. Two categories ofdata can be distinguished:

(1) data which participate in the geometric description of the FEA model. Indeed, FEAmodels include speci"c geometric concepts using planes of symmetry or axi-symmetry tode"ne the geometry of the FEA model. Thus, adaptation operators must take care of suchentities;

(2) data which de"ne speci"c areas of the object (like an area where a pressure is applied duringa structural analysis or where a heat source is applied during a thermal analysis). Such areasde"ned on the initial geometry must still exist on the adapted one. Moreover, values(pressure value, etc.) usually exist and are linked with these areas. Hence, these values mustalso be transferred onto the adapted geometry.



These data have led to the de"nition of a second level of classi"cation for the nodes and edgesof the polyhedron. Thus, nodes and edges can be classi"ed as:

(1) symmetry entity when they are located on or at the boundary of a symmetry areas,(2) domain entity when they are located on or at the boundary of speci"c areas.

4. IDENTIFICATION OF IRRELEVANT DETAILS

The identi"cation of irrelevant details is performed through two steps. The "rst step is based onthe analysis of the discrete curvatures of the input geometry and leads to the identi"cation ofgeometric details, i.e. the areas which would need high mesh density because of their geometricshape. The second step based on the computed or the estimated mechanical behaviour of the part(for a speci"c FEA), identi"es which geometric details could be removed without signi"cantlychanging the FEA results.

4.1. Geometric analysis

The geometric analysis is directly performed on each vertex of the polyhedral geometry. The&curvature' analysis is carried out using a geometric criterion based on the discrete absolutecurvature of each star-shaped area (such an area designates the set of faces placed around a givenvertex) of the surface. This criterion can be compared to the formulation of the strain energy ofa thin plate limited to small strains with a Poisson coe$cient equal to zero. For a C2 continuoussurface, this strain energy can be computed at a point on the surface with one of the followingequations:

E"PP(s)

(k21#k2

2) dS"PP

(s)

(4H2!2K) dS (1)

where k1

and k2

are the principal curvatures at each point, H stands for the mean curvature(k

1#k

2)/2 and K is the Gaussian curvature k

1.k

2.

In the context of a piecewise linear geometry, the expressions of the mean and Gaussiancurvatures were introduced by Alexandrov [10] and further developed by Calladine [11] andBoix [12]. Here, the polyhedron geometry is used to approximate the values of H and K. At eachvertex of a polyhedron, the discrete &absolute curvature' can be de"ned as follows:

E"4H2!2K (2)

4.1.1. Expression of the discrete Gaussian curvature. Calladine de"ned the discrete Gaussiancurvature at a vertex J by the following equation:

Kp"

2n!+iai

(1/3)+iai

(3)

where aiis the angle of the face F

imeeting at vertex J and a

iis the area of the face F

iconnected at

the vertex J (Figure 3).



Figure 3. Geometric parameters in#uencing thediscrete Gaussian curvature at a vertex J.

Figure 4. Geometric parameters in#uencing thediscrete mean curvature on a polyhedral surface.

Boix has modi"ed this equation to improve its convergence towards the value of the Gaussiancurvature of a continuous surface associated with the polyhedron at the same point.

To this end, the following equation takes into account the shape of each face Fi:

Kp"

2n!+iai

(1/2)+iai!(1/8) +

icot (a

i) l2

i

(4)

In this equation, liis the length of the edge of the face F

iwhich is opposite to the vertex J (see

Figure 3).With this equation, Boix has proved that the value of K

pconverges even if the triangulation

around vertex J is not regular, i.e. each face is not equilateral.

4.1.2. Expression of the discrete mean curvature. The equation of the discrete mean curvaturealong an edge JM is given by

He"12(n!angle (F

k,F

k`1)) ) l

k,k`1(5)

and can be aggregated at vertex J by summing half of the contribution of each edge connected atthat vertex. The resulting equation is

HP"

(1/4)+i(n!angle (F

k, F

k`1)) ) l

k,k`1(1/3)+

iai

(6)

where Fkare the faces connected at the vertex J, l

k,k`1is the length of the common edge JM of the

two adjacent faces Fkand F

k`1and angle (F

k,F

k`1) is the angle between faces F

kand F

k`1(see

Figure 4).



Figure 5. Equivalent meshes around vertex J.

Figure 6. Example of mesh con"guration: (a) initial mesh around vertex J; (b) &invariant'mesh around vertex J.

4.1.3. Concept of *invariant+ of a discrete curvature. The discrete curvatures values Kpand H

pare

local approximations which are meaningful in areas where a C2 surface can be de"ned.Therefore, when a polyhedron includes skew areas (which is common for mechanical parts),these values lose their accuracy. Indeed, the values of K

pand H

pon such areas depend

on the number and the dimension of the edges and faces connected at the vertex where thesediscrete curvatures are computed. As an illustration, Figures 5 (a) and 5 (b) show two di!erentmeshes representing the same geometry (a corner of box). These two meshes lead to di!erentvalues of discrete curvatures K

pand H

pdue to the change of the geometry of the faces around

vertex J.To analyse each area of the object with the same criteria, it is necessary to ensure that the values

of Kpand H

pare totally independent of the number and the dimensions of the faces connected at

a vertex but depend only on the local geometry of the object. To obtain such a property, thestar-set of faces around a vertex is replaced by a set of circular sectors (see Figures 6(a) and 6(b)).Each sector has the same length: ¸

*/7and this length stays constant for every vertex of the

polyhedron. With this new set of areas around a vertex, the closed curve around this vertex, hasa length equal to

Ci"¸

inv.a

i(7)

where aiis the angle of face F

iat vertex J.



Figure 7. Areas of a part with large values ofEinv (light grey colours).

Figure 8. Map of element sizes.

Based on the new areas, Hp, K

pand E

plead to the de"nition of new parameters H

*/7, K

*/7, which

are independent of the local con"guration of the polyhedron. Equations (4) and (6) of Hp

andK

pnow produce

Kinv"2n!+

iai

(1/4)¸2inv (+i(a

i)!(1/2)+

icot (a

i) a2

i)

(8)

and

Hinv"+

k(n!angle (F

k, F

k`1))

(2/3)¸inv +i(a

i)

(9)

Equations (8) and (9) allow to compute a new parameter E*/7

based on Equation (2), which isinvariant for equivalent meshes. Computing E

*/7at each vertex indicates areas where a shape

modi"cation could take place. Figure 7 shows such areas on a part. In this "gure, light greycolours indicate large values of E

*/7, i.e. areas with high curvature (including sharp areas).

4.2. Geometric and mechanical features

To identify the geometric details which could be removed, it is essential to characterize or toestimate the mechanical behaviour of the part in the context of load cases and objectives ofa given structural analysis. One interesting mechanical criterion is the map of element sizes usedto generate a mesh. This map de"nes for each area of an object the most appropriate size ofelements to model accurately the stress or the strain "eld for a given FEA. It could also beinterpreted as a minimum level of discretization of the object geometry required for a FE analysis.Therefore, comparison between the target element sizes and the size of the edges de"ning the



Figure 9. Case study.

input polyhedron can identify which details have no signi"cant e!ect on the mechanical behav-iour of the part.

According to Sections 2.2 and 2.3 above, this element size map could be speci"ed either duringan a priori step using the analyst expertise or directly computed from an a posteriori analysis of anFEA. This computation can be based either on the deviation from the local equilibrium equationswhich are not exactly satis"ed [8] or on errors in stress continuity across element edges [9].Figure 8 shows an example of such a map for a particular FEA whose boundary conditions arede"ned in Figure 9. In Figure 8, a value of element size has been assigned to each vertex of theinput polyhedron and is displayed as a sphere. The radius of each sphere is proportional to theelement sizes de"ned in the size map. This map is the result of an a posteriori error analysis led bythe LMT [9] (Laboratoire de MeH canique et de Technologie, Cachan, France).

Figure 8 can also be seen as a geometric envelope which bounds the geometric transformationswithout signi"cantly modifying the FEA result. Indeed, using this envelope ensures that anadapted polyhedron lying within this envelope should allow a mesh generation process toproduce elements compatible with the target map size.

5. OPERATORS FOR GEOMETRIC TRANSFORMATIONS

Several approaches for the simpli"cation of polyhedra have already been developed [13}16].Among them, the approach proposed by VeH ron [16] has the advantage to ensure, through an&error zone' concept, that a user speci"ed distance between the initial and the simpli"ed polyhedracan be preserved during the node removal process (see Section 5.1).

Node removal operators developed by VeH ron are well adapted for small geometric simpli"ca-tions like noise reduction functions. However, in the current approach, the node removal



operators must be able to allow large geometric transformations (up to idealization) ensuringa global shape preservation (see Sections 5.2 and 5.3).

Moreover VeH ron's approach, like the others, focuses only on the geometric aspect of thesimpli"cation and is not able to take care of speci"c data like FEA ones which should bemanipulated in the present work. Therefore, a recon"guration of VeH ron's remeshing operators isnecessary (see Section 5.4).

5.1. Simplixcation of a polyhedron and the *error zone+ concept

The simpli"cation algorithms developed by VeH ron are based on two main steps:

(1) A vertex removal process linked with remeshing operators (see Figures 10(a) and 10(b)). Atthe initialization of the simpli"cation process, all the nodes of the polyhedron are con-sidered as candidate nodes. Then, they are sorted using decreasing values of the &invariant'discrete curvature E

*/7and they are tried one by one during the simpli"cation process.

This procedure leads to a simpli"cation process which starts with areas considered asgeometrical &details'.

For each node removal candidate, a 3D contour polygon, built with the nodes surround-ing this candidate node is extracted from the polyhedron. This contour is used by theremeshing operators to generate a new local mesh which replaces the initial faces in thepolyhedron (Figure 10(b)). The remeshing operators are designed to handle the FEA dataattached to the polyhedron, i.e. they are able to manage the boundary, surface andnon-manifold classes de"ned in Section 3.1.

The choice of the remeshing scheme applied to the 3D polygon is an important step of thesimpli"cation process. Indeed, the choice of the new faces generated is very important topreserve the overall shape of the object. VeH ron has developed two remeshing criteria [17].The "rst one, called equilateral remeshing, favours the generation of faces as equilateral aspossible. The second one, called height remeshing, generates, as a priority, faces havingedges as close as possible to the removed vertex. Such a criterion tends to generate facesstretched in the direction of the minimum curvature of the initial geometry. As a result, itprovides a better preservation of the input geometry and increases the level of decimation.However, these criteria become quickly unadapted when large geometric transformationsare required in complex areas (like saddle point areas), as described in Section 5.2.

(2) A discrete error and error inheritance concept ensuring error bounded simpli"cations. Thegeometric transformations are monitored by an &error zone' concept. During the initializa-tion step of the simpli"cation process, a spherical error zone is attached to each vertex ofthe input polyhedron. All these spheres de"ne a discrete envelope around the inputgeometry where the simpli"ed geometry must lie (Figure 11).

To ensure that all the new faces generated through the remeshing operators stay withinthis discrete envelope, it is necessary to check that each error zone attached to the inputpolyhedron intersects at least one face or one boundary edge of the simpli"ed polyhedron.

This test is performed after each local remeshing operation through an error inheritanceprocess. The "rst step of this process is to assign to each face and each boundary edge all theerror zones intersecting them. Thus, a dependency list of errors is attached to each face andto each boundary edge (see Figure 12(a)). At the initialization stage, the dependency lists aresolely made up from the error zones attached to the vertices de"ning each face or eachboundary edge.



Figure 10. (a) Initial mesh with candidate vertex to beremoved; (b) new mesh without the candidate vertex.

Figure 11. Principle of the envelope concept ona 2D example.

Figure 12. The error inheritance concept for the error zones: (a) initial dependencies; (b) updated dependencies.



Figure 13. An example of unacceptable decimated polyhedron: (a) initial polyhedron; (b) anenlarged of a area before vertex decimation; (c) an enlarged of a area of the decimated

polyhedron where a notch appears.

The second step, consists in extracting the list of the error zones participating in theremoval of the candidate vertex. This list is built as the union of the dependency lists of theerror zones attached to the faces and boundary edges involved in a node removal.

During this last step, if the removal of the candidate vertex can be carried out, i.e. theremeshing operators are able to generate a new local mesh which is acceptable fromgeometric and topological criteria to ensure its conformity, an update of the dependencylists associated to each face and each boundary edge newly created takes place (see Figure12(b)). If one of the error zones cannot be inserted into a dependency list because it isintersecting no face or boundary edge, the process is stopped and the candidate vertex is notremoved.

This scheme, using the remeshing operators and error zones inheritance de"ned above, hasshown its e$ciency for small geometric simpli"cations, i.e. when the size of the error zones issmaller than the length of the edges connected to a vertex. At the opposite, when largertransformations are faced and particularly when transformations must take place into locallycomplex areas of the polyhedron, like saddle point areas, these operators are not e$cient enoughto provide an acceptable simpli"ed polyhedron. Figures 13(a) and 13(b) show an example of sucha con"guration which can appear after a simpli"cation process has been performed.

The next section presents a new remeshing operator and an add-on to the inheritance conceptto provide a better shape preservation even with large simpli"cation con"gurations of complexgeometric areas.

5.2. Operators for large geometric transformations

5.2.1. A new shape preserving remeshing operator. The height remeshing scheme, developed byVeH ron, even if it preserves as much as possible the local shape of a polyhedron, does not alwaysprovide satisfactory results from a user point of view. Indeed, some 3D contour polygoncon"gurations (extracted during the node removal process) may have a local geometric settingthat leads to the generation of new faces which do not preserve the initial geometry. A typicalexample is encountered when the node to be removed lies on the boundary of a hole of a thinplane. Figures 14(a) and 14(b) illustrate such a case. Figures 14(a) shows the initial polyhedronand Figure 14(b) depicts the simpli"ed polyhedron where a notch has appeared and the topboundary of the hole is not restored.



Figure 14. (a) Candidate vertex located on a boundary of a hole; (b) new con"gurationof faces creating a notch.

Figure 15. (a) Dihedral angles between adjacent faces; (b) new faces de"ned fromthe discrete mean curvature criterion.

To avoid such di$culties, a new remeshing operator has been developed which takes care ofthe local discrete curvatures of the polyhedron around the candidate vertex. First of all,a polyhedral surface orientation is performed such that the normal of each face points outwards ifthe input polyhedron is closed or the orientation is user selected if the input polyhedron is open.Then an &invariant' discrete mean curvature, H

jknv, is evaluated on each edge k of the star-shapedarea of the faces connected to the candidate node with

Hkinv"

12(n!b

k)¸inv (10)

where ¸*/7

is constant and chosen equal to 1 to produce

Hkinv"

12(a

k!n) (11)

when using the relationship between bkand a

k(see Figure 15(a)).



These values can be seen as a geometric characteristic which are assigned to each nodeof the 3D contour polygon (vertices 1}7 in Figure 15(a)). The remeshing operator uses thesevalues to create the new edges in order to restore the main direction of curvature of the initialstar-set.

To this aim, the "rst step consists in determining the local concavity or convexity of thepolyhedron at vertex J. This is achieved using the sign of H

p*/7at vertex J (H

p*/7'0: convexity,

Hp*/7

(0 concavity)The second step is to sort the vertices of the 3D contour polygon by increasing values of

Hj*/7

when polyhedron is locally convex or by decreasing values when the con"guration isconcave.

The last step leads to the generation of edges using the previously ordered list of vertices.A "rst edge is created from the "rst vertices of the list if this edge does not already exist intothe 3D contour polygon. If it exists, the second vertex of the edge (i.e. of the list) is replacedby the "rst vertex found in the list which does not de"ne an edge of the 3D contour polygon.Then, tests are performed to validate this edge from topologic and geometric aspectsaccording to the test developed by VeH ron [17]. As a result, two adjacent 3D contour polygons arecreated.

This remeshing process is repeated for each 3D contour polygon created until all the newcontours are reduced to triangles.

This way, all the edges generated allow to preserve as much as possible the initial object shape.During this process, the determination of the "rst edge built is a signi"cant step of the remeshingprocess because of its great in#uence on the curvature of the new mesh. In the example of Figure15(b)), the main edge built is the edge connecting vertex 1 to vertex 4. The choice of the otheredges is less critical.

5.2.2. An extension to the inheritance concept. The use of the above remeshing operatorprovides satisfactory geometric simpli"cations even during large geometric transforma-tions. Nonetheless, it happens to be insu$cient in saddle point areas where there are stillproblems with the discrete envelope restoration criterion. In fact, the new mesh does notnecessary lie within this envelope. In such areas, the intersection test between the faces and theerror zones is not su$cient to preserve the discrete envelope concept and parts of faces mayviolate the distance criterion. As shown by the new remeshing operator, the edges of theremeshing scheme have a great in#uence on the polyhedron geometry. Hence, an inheritanceconcept needs to be applied to edges connecting two nodes of the 3D contour polygon havinga negative discrete Gaussian curvature (K

p(0). The edges are called &saddle edges'. To this end,

a dependency list is attached to each edge connecting nodes with negative Gaussian curvature, i.e.a saddle edge. When one of these edges is removed during a simpli"cation process, all errors of thedependency list must intersect at least one edge connecting two vertices with a negative Gaussiancurvature (see Figures 16(a) and 16(b)). If this condition is not satis"ed the candidate vertex is notremoved.

These dependency lists are initialized with the error zones of the vertices de"ning each saddleedge. The inheritance process is propagated using an union operator between dependency lists ofsaddle edges.

The use of the new remeshing operator associated with the new inheritance concept leads toproduce a simpli"ed polyhedra which strictly "ts into the geometric envelope de"ned by the errorzones and preserves as much as possible the overall shape of the object.



Figure 16. (a) Initial dependencies; (b) updated dependencies including lists linked main edge.

5.3. Idealization operators

Until now, the work performed strictly uses as input a polyhedral geometry and FEA data. Noextra data like CAD data (contour lines, geometric features, and so on) is input. Therefore, all thedata used by the idealization operators set-up are extracted by means of an analysis of thepolyhedron and the FEA data.

Two levels of idealization operators have been distinguished: the "rst one is based on a holeremoval concept and does not change the geometric manifold of the input polyhedron, the secondone concentrates on the dimensional changes of the input model to reduce tri-dimensional(respectively bi-dimensional) areas to bi- or mono-dimensional (respectively to mono-dimen-sional) areas.

The discrete envelope concept, de"ned in Section 4.2, only provides data able to identify areasthat should be considered as irrelevant details for the FEA model. These areas which maycorrespond to holes of some topological con"guration is obtained and if the simpli"cationprocess faces topologic modi"cations the decimation goes on. Hence, dimensional reductionoperations are out of the scope of the present operators and the present work focuses only ona hole removal process.

This operator is based on the following hypotheses:

(1) the holes removed are geometric details, i.e. all the errors zones attached to the nodes of theinput polyhedron which de"nes these holes have larger radii than the radii of these holes,

(2) holes considered are through holes which necessarily belong to a closed polyhedronde"ning a volume.

Within this context, the hole removal process can be performed automatically.The main step of this operator is the identi"cation of the edges which describes the boundary of

the hole. According to the dimension of the error zones attached to the hole, a "rst step of thesimpli"cation leads to the reduction of a hole to its basic polyhedral shape, i.e. a prism withtriangular base (see Figure 17). This result can be obtained for holes with convex boundaries only.More general con"gurations have not yet been treated. This step of the idealization process is thelast con"guration which preserves the two-manifold property of the polyhedron. Removing any



Figure 17. Initial hole lying in the input polyhedron and basic polyhedral modelobtained after simpli"cation.

Figure 18. A loop of edges which does not de"nea through hole boundary.

Figure 19. An example of closed path open-ing a closed polyhedron.

node located on a hole boundary would produce a non-manifold polyhedron, something which isnot acceptable.

After this "rst step, the edge loops de"ning the boundaries of the holes can be extracted fromthe simpli"ed polyhedron. An edge loop de"ning a hole boundary is a closed sequence of threeedges whose adjacent faces are all distinct (see Figure 18). This is a strictly topological criterionwhich provides two loops per hole.

However, this criterion is not su$cient to explicitly characterize a hole boundary as depicted inFigure 18. Therefore, to determine if an edge loop is a hole boundary, this operator usesa topological property of polyhedra with through holes: there exist closed paths distinct from theedge loops characterized above which can open a closed polyhedron without generating twoseparated parts (see Figure 19).

This property can be applied to a polyhedron to sort the edge loops and retain only thosewhich de"ne hole boundaries. Such paths can be constructed from faces through an advancingfront technique. A front is initiated from any edge of an edge loop. This edge initiates a front withany of its adjacent faces. From this face, the front is propagated using one of its adjacent faces.This front is propagated until the new faces incorporated into it either cross the edge loop whichhas initiated this front or incorporate a face which owns the initial edge of the front. The lastcon"guration validates the initial edge loop as a hole boundary (see Figure 20).

The closed path built necessarily incorporates at least one or another edge of a distinct edgeloop. All the edge loops having only one edge into the closed path is an edge loop of a throughhole (see Figure 21).

Each subset of edge loops thus identi"ed is used to build two distinct subsets S1

and S2

of facescovering entirely the polyhedron. One of these subsets represents itself, the other one being therest of the polyhedron.



Figure 20. An example of closed path laid of faces. Figure 21. Identi"cation of a couple of edge loopsde"ning a hole.

Figure 22. A beam with 3 holes: (a) initial polyhedron; (b) simpli"ed polyhedron without topologic changes.

Computing the area covered by each subset Siallows to identify the subset which characterizes

the hole. Indeed, the subset Sihaving the smallest area is necessarily located inside the other one,

hence it de"nes the polyhedral representation of the hole which needs to be removed.More complex con"gurations with holes intersecting themselves can be handled. Here, only the

basic principle of the algorithm has been described.When each hole has been identi"ed and characterized by a connected subset of faces S

i, the

idealization operator removes all the nodes, edges, and faces de"ning this subset and generatestwo new faces based on each edge loop de"ning each hole boundary close to it. Figures 22(a) and22(b) show an example of polyhedron simpli"cation reducing a polygonal approximation ofcircular holes to triangular prism (Figure 22(b)). This simpli"ed polyhedron hole removalalgorithm has been applied to produce a new polyhedron without hole (Figure 23(a)). After thesetopologic changes have been performed, the hole identi"cation process described above has beenperformed and the other simpli"cation step is carried out to obtain the most simpli"ed polyhed-ron possible (Figure 23(b)).

To improve the e$ciency of this operator, the number of edges in an edge loop de"ning a holeboundary has been raised to four. Thus, boundary edges of a hole can be found among loops of



Figure 23. Polyhedron with holes removed: (a) polyhedron after the hole removal process; (b) mostsimpli"ed polyhedron after a new simpli"cation step.

four edges not connected to a pair of adjacent faces. Then, the identi"cation and hole removalprocesses are similar to the previous treatments.

This automatic identi"cation of holes cannot be generated for schemes having more than fouredges. Indeed, there is a great number of such schemes which may require too much computertime to identify hole boundaries. Input data from CAD. models or from user speci"cations inaddition to speci"c treatments form the basis of further developments required to handle moregeneral hole con"gurations (like holes with non-convex boundaries).

The idealization operator presented above performs an automatic hole removal using the&error zone' concept and does not rely on any added data attached to the input polyhedron.Topologic data combined with the decimation algorithm behaviour are su$cient to locate andremove the holes.

5.4. Operators for handling FEA data

Speci"c data from FEA models can also require speci"c treatments within the adaptationoperators. As an example, an area of the input polyhedron subjected to pressure must still exist onthe adapted geometry.

Such speci"c data could be linked with:

(1) three-dimensional zones of the FEA model, like areas submitted to a magnetic "eld,(2) two-dimensional areas of the FEA model, like pressure areas, planes of symmetry,(3) one-dimensional areas of the FEA model like a linear distribution of loads,(4) point areas of the FEA model, like a concentrated load.

Using a polyhedral geometry, it is only possible to manage two-dimensional, one-dimensionaland point areas. Thus, con"gurations with three-dimensional areas are not part of the presentwork.



Figure 24. Extraction of speci"c contours.

When performing the adaptation process over two-dimensional areas, if a candidate vertexof the node removal process is located on the boundary polygon of a speci"c area, twospeci"c contours must be built from the start-set of faces that need to be remeshed.Each contour characterizes a di!erent category of areas where distinct types or values ofattributes are attached (see Figure 24). The remeshing process is then applied independently oneach contour.

When one-dimensional areas are part of the input polyhedron, if a candidate vertex of the noderemoval process is located on a speci"c polygon of the polyhedron, a new edge is built to preservethe topology of this line.

If a candidate vertex is a speci"c vertex linked with a point area, it cannot be removed.Both two and one-dimensional areas attached to attributes can be monitored with tolerances

(error zones) and an inheritance process applied on the edges de"ning the speci"c line or theborder line of these areas.

All these speci"c areas are characterized by numerical or vector data of the FEA Hence, theadaptation operators must also be able to transfer these data to the new geometry while keepingtheir signi"cance. In the case of a pressure load, the resultant of the load must be preserved.

At present, operators only deal with the logical data de"ning the areas where the boundaryconditions data are settled. The purpose of these operators is to preserve, during the adaptationprocess, the areas where the data are de"ned according to geometric tolerances. Indeed, the use oflow values for the local tolerances implies a low evolution of these areas and thus ensures thata modi"cation of these areas has a small in#uence on the FE computation.



Figure 25. Top area of the part: (a) with holes; (b) without holes.

New operators dealing both with areas and values linked with speci"c data such as boundaryconditions are going to be developed in order to allow a complete and accurately controlledadaptation of the model.

6. APPLICATIONS

The example of geometry adaptation developed here is based on the structural analysis of thepart presented in Figure 9. According to the element sizes map de"ned in Section 4.2 andto the boundary conditions de"ned for this analysis, a geometry adaptation has beencarried out.

The "rst step of this adaptation was to remove the two small holes on the top of the part. Toachieve this goal automatically, a "rst simpli"cation operator is applied to reduce holes to theirbasic shape (see Section 5.3) because the size of their radii is not signi"cant compared with thesizes of the elements required in these areas. From this "rst simpli"cation, &detail' holes areautomatically detected and removed by the idealization operator. Figures 25(a) and 25(b) showthe top of the part with and without these holes.

The second step was to apply adaptation operators on the polyhedron using the map ofelement sizes as a set of error zones according to the principle of Section 5.1. Figure 26 shows theresults of this adaptation.

The adaption process has signi"cantly modi"ed the top of the part and all the blending areas ofthe part has become coarser than the initial one.



Figure 26. Adapted geometry of the part illustrated in Figure 9.

Figure 27. Wireframe representation of the adapted geometry including the boundary conditions.

This new geometry stays coherent with respect to the FEA initially de"ned since the boundaryconditions have been transferred by the adaption operators during adaption process (seeFigure 27).

To evaluate the e$ciency of the adaption, an FEA has been conducted for both the initial andthe adapted part geometries. For each geometry, a mesh was generated according to the map ofelement sizes de"ned above. Figure 28 depicts the mesh generated from the initial geometry.A high density of elements is required to mesh the areas where the two holes are located.

Figure 29 depicts the mesh generated from the adapted part geometry. This new geometryallows to have a better adequacy to the element sizes map because high densities of elementsappear only in areas de"ned by this map.

Figures 30(a) and 30(b) illustrate the FEA results computed from the initial and adaptedgeometries. The map of Von Mises stresses of the FEA computations shows that stress values inhighly stressed areas stay close for both models (564 and 545 MPa). Hence, the adapted model isable to provide an acceptable representation of the mechanical behaviour of the part.

Slight di!erences between the stress "elds of the initial and adapted parts appears whencomparing Figures 30(a) and 30(b). These di!erences can be explained by the fact that thenumerical value of the pressure is identical for both models whereas the areas are not identical.This means that the boundary conditions slightly di!er since the adaptation process performedhas not included the transfer of this mechanical parameter.



Figure 28. Mesh from the initial part geometry. Figure 29. Mesh from theadapted geometry.

Figure 30. Map of Von Mises stresses from FEA computation: (a) map from the initial geometry;(b) map from the adapted geometry.

7. CONCLUSIONS

In this paper we have described how a polyhedron can be e$ciently used for the geometricadaptation of an FEA model. Indeed, such a model, based on a discrete representation of thegeometry, o!ers signi"cant advantages. On the one hand, it allows large geometric transforma-tions and, on the other hand, it can include and handle all data from FEA which are required forthe adaptation process.

The adaptation process set-up can be either interactive to let engineers perform their owntransformations on the model or automatic to adapt the geometry of the model during loops ofFEA computations. In each case, a mechanical criterion, linked with a map of element sizes, isused to monitor the transformations ensuring that the adapted model provides accurate FEA



results. Such a process, reducing complexity of FEA models, leads to reductions of time andhardware required to perform an analysis and could therefore add #exibility to FEA tools toimprove their integration into the design process.

The mechanical criterion used to perform the automatic adaptation of FEA models is based onan a posteriori error estimator using tetrahedral "nite elements. Therefore, the data extractedfrom this estimator can be directly used to monitor geometric transformations of polyhedraltriangulations of a model. The use of an error estimator based on another type of element, such ashexahedral elements, will require a modi"cation of our adaptation process. Applying the adapta-tion process proposed to shell structures using an a posteriori approach would need errorestimators which have not yet been proposed.

In the same way, even if the current mechanical criterion allows geometric idealization like holeremoval, it does not allow dimensional reduction (like medial axis extraction). New geometric andmechanical criteria are required to provide such idealizations while ensuring that the new modelsare still suited to the objectives of the FEA. Such criteria are under focus for our future work.

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Automated generation of FEA models through idealization operators