Fast Deformation Method for a 2D Shape

Transcription

1 Fast Deformation Method for a 2D Shape Guillaume Trouvain ab, Vincent Gagnol a, Emmanuel Duc a, Jean-François Sancho b a. Clermont Université, Institut Pascal, UBP-CNRS-IFMA, Campus des Cézeaux 24 Avenue des Landais, AUBIERE - FRANCE b. Centre de Technologies Michelin, ZI de Ladoux, Cébazat Résumé : La modélisation de pièces déformables est un sujet de recherche important. Que ce soit dans le domaine 2D ou 3D, deux approches existent pour modéliser une pièce : la CAO et la Méthode des Eléments Finis (MEF). Tout d abord, la CAO est utilisée pour modéliser des pièces très rapidement sans prendre en compte leurs caractéristiques physiques ce qui donne des résultats peu fidèles à la réalité. Ensuite, la MEF décrit de façon plus exacte la déformation d une pièce mais son temps de calcul est trop important. Dans cet article, une méthode pour modéliser la déformation d une pièce prenant en compte ses caractéristiques physiques est proposée. Cette méthode s appuie sur la déformation d un maillage 2D qui recouvre entièrement la section de la pièce étudiée. La déformation du maillage est régie par une loi issue de la loi matériau qui caractérise cette pièce. Enfin les résultats sont obtenus en temps réel. Abstract : Nowadays, the modeling of deformable bodies is an important research topic. In the 2D or 3D domain, two approaches exist to model a body : CAD and the Finite Element Method (FEM). On the one hand, CAD is used to design bodies very quickly without taking into account mechanical characteristics. Thus deformation results are not correct compared to reality. On the other hand, FEM describes more accurately the deformation of a body but its computation time is too long to provide a real-time model. That is why, there is a strong demand from industry to develop a realistic and interactive deformation method. In this article, a method to model the deformation of a body cross-section integrating material characteristics is proposed. The method is based on the deformation of a geometric mesh which covers the 2D body cross-section. This deformation is based on a law defined from the material characterization and results are obtained in real-time. Mots clefs : Process Modeling ; Deformation ; Real time 1 Introduction This article deals with the modeling of deformation using an industrial CAD environment. The main issue with the geometrical modeling of deformable bodies is that the position and the shape of the body depend on the plastic or hyperelastic behavior of the material. Thus, the mechanical problem must be solved in order to compute the final shape of the body. In particular, gravity, material stiffness and contact between bodies influence shape of the deformed body. Two usual approaches are used to compute the geometry of the deformable body. On the one hand, bodies can be designed very quickly thanks to CAD ; however mechanical characteristics are not taken into account. On the other hand, the Finite Element Method describes more accurately the deformation of a body. However, computation time is too long to enable an interactive model, and the method requires specific skills to be performed correctly. To conclude, neither approach is completely satisfactory, because of both accuracy issues and computing time issues is an industrial environment. Thus, a fast and exact CAD mechanical model is required for deformable bodies nowadays, particularly in the context of the deformation simulation of elastomer bodies. A fast and precise method allows the cost of the modeling process to 1

2 be reduced. Particularly, one of the most important advantages is the reduction in computation time. Moreover, this method enables an approximate final result to be obtained and allows a greater number of processes to take place at the same time. This can lead to a smaller number of real tests and trials to validate the solution. This article presents a geometrical modeling method in an 2D environment, taking the mechanical behavior of bodies into account. The goal is to compute a representation of the deformation in real-time, using a geometrical method. The considered body material characteristics are governed by a predefined mechanical law (ML) determined from the body material characteristics and a geometrical boundary curve (BC) associated with a nominal deformation. To compute the shape of the body, the contours of the cross-section are divided into lower and an upper curve. Then a mesh is created from the lower curve to include the whole cross-section and also the upper curve. The mesh is deformed according to the predefined deformation law ML. Finally, the upper curve is placed on the deformed mesh. Thus, a representation of the deformed body is obtained. In the second section, different methods from the literature dedicated to the deformation of a body are described and compared to criteria on accuracy and computation time. In the third section, the deformation method is presented more exactly, describing the various steps. These steps are presented in Figure 1. In the fourth section, an example is explained to illustrate the method. 2 Environment Figure 1 Steps modelization In this paper, the goal is to compute a fast and accurate deformation model in a CAD environment. The methods identified in the literature are divided into two categories : physically-based methods and geometrical methods. These methods are explained and compared with respect to the constraints presented above. Physically-based methods were described for the first time by Terzepoulos et al [7] [8]. These methods are used to represent physical phenomena with different degrees of speed and precision. Among these methods is the Boundary Element Method (BEM) [6] exists. BEM enables animations in real-time [3][4]. Nevertheless, the most important drawback of this method is the restriction to linear behavior of materials. Another method, called Mass-Spring, is based on the discretization of the body using a mesh of particles, connected by springs which can be characterized from the physical characteristics of body [1] [2]. This method is used in cases where a real-time physical deformation model is required. However, a physical approximation must be admissible, due to the complex characterization in order to compute the deformation in a manner which is incompatible with our industrial constraints. The Finite Element Method (FEM) is the most suitable method to take into account the physical characteristics. However, the computation time required with FEM excludes the possibility of interactive animations, which is an important industrial requirement in our case. Among the geometrical methods, Sederberg et Parry [5] present an approach called Free Form Deformation (FFD) to deform a shape from its curve control points, by distorting the environment, to produce any type of deformation. However, despite its interactivity, the precision of the results given by this method does not correspond to the requirements mentioned previously. Morphing is another type of geometrical deformation. Wolberg [9]lists the different types of morphing 2

3 which are described in the literature. Morphing allows an initial geometrical shape to be interpolated in order to obtain a final geometrical shape. This interpolation is based on the deformation of the contours of the studied form. However, even if morphing is based on the deformation of contours, this method cannot be used here because the final form must be known. Thus, this method cannot be used to estimate the final shape. Considering that no method found in the literature, whether physically-based or geometrical, responds our functional criteria, a new method is developed. 3 Geometrical modeling by fast finite elements 3.1 Method and algorithm proposed In order to illustrate the proposed method, an example of deformation on a boundary geometrical curve (GC) with the body shown in Figure 2 is taken. 3.2 Definition of the profile The deformation algorithm requires the body cross-section to be divided into two parts : a lower curve and an upper curve, as described in Figure 2. Figure 2 Definition of the lower and upper curves 3.3 Smoothing and discretization of the curve After the cross-section is divided into two parts, the lower curve is isolated. Points on the curve are computed according to a discretization step respecting a maximum admissible chordal error (Figure 3). Figure 3 Smoothing of the lower curve 3.4 Mesh of the initial form The minimum discretization step S min of the lower curve is used to determine the size of all the mesh elements (Figure 5). Moreover the predefined mechanical law (ML) f is defined from the material properties of the body and is applied to the studied shape. The cross-section size is computed using equation 1, where Se t is the initial section and Se t+1 the final section : Se t+1 = f(se t ) (1) 3

4 In order to mesh the initial cross-section (Figure 2)), the first element is hung on one of the extremities of the lower curve. The other elements are placed one by one until they reach the other extremity. The first layer of the mesh is obtained in this way. Figure 4 and 5 illustrate the proposed algorithm. Let us consider the case of an element j on the layer i. As presented in Figure 5, Nodes 1, 2 and 4 are defined arbitrarily. Node 1 and Node 2 correspond to Node 4 and Node 3, respectively, of the element j 1 of the same layer. Node 4 corresponds to Node 3 of element j of the layer i 1. The segment 1 2 is computed to be normal to the inferior layer i 1.The node 3 is defined along the segment 1-2 according to the predictable height 4 3. Figure 4 Numbering of the nodes and reference size of an element Figure 5 Computation of an element At the end of each layer (Figure 6), a smoothing by an optimization algorithm to obtain a continuous curve is performed, in order to obtain a workable mesh (Figure 7). The final layer and the mesh are completed when all the characteristic points are included in the mesh. Figure 6 Layer before smoothing 3.5 Design of the empty final mesh Figure 7 Mesh of the initial form In order to deform the cross-section, an empty mesh is computed from the BC curve (Figure 11),composed of the same number of layers and the same number of elements per layer as the initial mesh. The size of the elements can be modified according to the predefined mechanical law ML, if a modification of the area (or volume in 3D) is required. About the size of elements, it depends on the law of deformation Equation 1 which is apply to the studied case. Figure 8 Final Mesh 4

5 3.6 Correspond between the initial form and the final form After computing both meshes, the position of each characteristic point of the superior curve is located in the initial mesh, as presented in 9. Then the cross-section of the defined body is obtained by placing each characteristic point, belonging the upper curve of the initial cross-section in the final mesh (Figure 10) Figure 9 Position of a point in an element Figure 10 Computation of the final form 4 Applications The Figure 11 presents an example of the described method. In Step 1 of Figure??, the user defines the lower and upper curve of the studied body. Next, in Step 2, an initial mesh is computed from the discretization of the geometrical boundary curve. Step 3 shows the computation of the empty final mesh on the deformed boundary curve BC according to the predefined mechanical law ML. Finally, in Step 4, characteristics points are located in the initial mesh and respectively placed in the final mesh. In this case, the cross-section must be reduced by 20% (this constraint is used to define the mechanical law) ; the geometrical boundary curve is a line and the deformed geometrical boundary curve is a concave curve. At initial step, the area of the section is 4379mm 2. To verify the model, the theoretical surface is computed and compared to the result given by the proposed deformation method. The results obtained are 3503mm 2 by theoretical computation and 3500mm 2 with the presented deformation method. The computed error is 0.1% and the result is obtained into 0.05s. This deformation method can therefore be considered as a real-time method. Figure 11 Application 5

6 5 Conclusions In this paper, we treat the problem of the geometrical modeling of deformable bodies in real-time. The goal is to compute the shape of the body, according to a mechanical law of deformation. To obtain a fast solution, we propose to associate a geometrical mesh with the complex shape of the body. Different geometrical computations permit to associate the shape profile with the mesh. Then, a specific mechanical law is used to deform the mesh and to compute the final shape according to the precedent association. The main interest of the approach is to separate the geometric modeling, the meshing and the mechanical deformation, which is only applied to a very simple meshing that permits to reduce significantly the computation time. Further works will be dedicated to the modeling of a complex body and composed by a set of different deformable bodies. Particularly, the problem of the contact between bodies will be considered. Références [1] BAUDET, V Modélisation et simulation paramétrable d objets déformables. Application aux traitements des cancers pulmonaires Université Lyon 1 209p [2] FLECHON, E Simulation d objets déformables. Rapport de stage Recherche Université Lyon 1 43p [3] JAMES, Doug L. and PAI, Dinesh K ARTDEFO : Accurate Real Time Deformatable Objects Proceedings of ACM SIGGRAPH 99 Computer Graphics [4] PACCINI,A Développement d un modèle par éléments finis 3D appliqué à la simulation d opérations chirurgicales des tissus mous Ecole des Mines de Paris 126p [5] SEDERBERG,T.W. and PARRY,S.R Free-Form Deformation of Solid Geometric Models SIGGRAPH [6] TANG,Y.M. and HOU,A.F. and HUI,K.C Comparison of FEM and BEM for interactive object simulation Computer Aided Design [7] TERZOPOULOS,D., FLEISHER,K Deformable models The Visual Computer [8] TERZOPOULOS,D. and PLATT,J. and BARR,A Deformable models Computer & Graphics [9] WOLBERG,G Image morphing : a survey The Visual Computer