An optimal genetic synthesis of DELTA robot for a

results (curves, values tables, ISO values . . .) and to take them into account in the design model (CAD model). In order to solve these problems, many research projects have been undertaken in order to improve the integration of the mechanical model (CAD) and the Analysis model (CAE).

Automatic generation of CAD model based on CAE results Borhen Louhichi*, Abdelmajid BenAmara*, Vinçent François**, Lotfi Romdhane* *

In the last few years, some solutions have been proposed to automate the transfer of data and support the automatic modification of the analysis model when there is a modification in the CAD model [3] [4]. The inverse transfer of data (the return of the deformed geometric model deformed as a result of the CAD model) represents our main objective in this work. Indeed, the reconstruction of the deformed CAD model as a result of finite elements analysis makes it possible to guide the process of design towards simulations. In fact, the validation and the verification of the design criteria will be efficiantly performed by simulation in an integrated CAD/Analysis environment [5] [6].

Laboratoire de Génie Mécanique, Ecole Nationale d’Ingénieurs de Monastir, Monastir 5019, TUNISIA ** Département de Génie Mécanique, Université du Québec à Trois-Rivières CP 500, Trois-Rivières, Québec, G9A 5H7, Canada

Abstract Nowadays, the integration of different mechanical design process stages (Design, analysis and manufacturing) of is a necessity so as to reduce the time of design and optimization. This tendency of integration CAD/CAM/Analysis and automation of the corresponding processes requires shared data between the various tasks using an integrated product model. Our research is oriented to CAD/CAM/Analysis integration by rebuilding the CAD model (BREP) starting from the CAE results (deformed mesh).

Within a context of integrated design, the boundary conditions are captured on the CAD model, and for that reason, the deformations of the finite element model must be transmitted to the CAD model in order to take into account the new boundary conditions. The utility of the tools for rebuilding of CAD model (BREP) starting from FEA is more important for some types of simulations. Indeed, the simulation of a mechanical product using the deformed model of mechanical parts makes it possible to detect eventual design problems (collisions, frictions, etc.). We underline that actual CAD systems doesn’t allow this possibility. In fact, supported simulations are based on undeformed CAD model.

Keywords: Integration, CAD, BREP, Analysis, Finite Elements, Surface. I.

Introduction For several years, research has been brought to improve the integration of various tasks CAD, CAM and Analysis through a better communication between various analysis tools. This tendency of integration CAD/Analysis and automation of the corresponding processes requires the data sharing between the various tasks using an integrated product model. This Work consists in improving the level of integration between CAD and Analysis [1] [2] with the aim of defining a unified FEA/CAD model. This definition requires updating the Analysis model whenever the CAD model is modified and vice versa. For the first case, solutions were developed [3], but for the second case have not been treated yet and this work is a contribution toward its solution.

This project is part of a research project that consists of proposing a unique CAD/CAM/Analysis model. The main stake of this research is the elaboration of an integrated dynamic model which leads to an efficient integration between CAD, CAM and Analysis tools. In this framework, collaboration between the Mechanical Engineerig Laboratory of the ENIM Engineering School (Monastir – Tunisia) and the Mechanical Engineering Departement of the UQTR University (Trois Rivières – CANADA) is trying to improve the level of integration between CAD (geometric model) and analysis (finite-element analysis). The objectif is to reintegrate the CAD model, which was reconstructed from the finite element results (deformed mesh) into the design tool (CAD) in order to give the designer quantitative means of validation of the design model.

At present, the data exchange process between CAD tools and analysis tools (CAE) is a one-way process (data is transferred from CAD to analysis). It is done by way of neutral formats which can generate loss of data. Research work has been developed in order to improve and automate this transfer. Multiple iterations between CAD and Analysis are usually necessary to validate the design model. It is thus the responsibility of the user to transfer back the analysis

II.

Developed algorithm

The proposed algorithm is based on the BREP model features [7]. The BREP model is based on a boundary representation of CAD model. In fact, the BREP model

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describes not only geometrical information, but also topological information (faces, edges, vertices and vertices) (Figure 1). The proposed algorithm is based on two main parts: determine the topology and rebuild the geometry. We first build the BREP entities and then join them to build the deformed CAD model. BREP entities are defined using the deformed mesh boundary nodes.

loops

Deformed mesh

Identify the triangulation of each face

Surface support the face

Rebuilding deformed surfaces From triangulations Face

Rebuilding the vertex, edges and loops

loops

Edges

Deformed CAD Model Curve support the Edge Edge

Vertex

Figure 2 : Developed algorithm. The most complex step of this algorithm is the identification of the surface interpolating the extracted meshed faces (Rebuilding deformed surfaces from triangulation). The topology of deformed CAD model is the same as the undeformed CAD model.

Figure 1: BREP model entity. The general algorithm developed (figure 2), represents the different steps to be followed in order to reconstruct a CAD model from a deformed mesh. The main steps of this algorithm are the following: • Identification of the triangulation associated with each face of the model. • Reconstruction of the bearing surfaces of the faces. • Reconstruction of the loops, edges and vertices.

III.

The state of the art: from meshed surfaces to CAD surfaces

One work has been based on using Béziers triangles to evaluate a surface from a mesh. This work has to do with calculating a triangular surface (Béziers triangle) that corresponds to each triangle of the mesh, in such a way as to be continuous with the adjoining surfaces (that correspond with the neighbouring triangles of the mesh) (Figure 3). The continuity is of the type C1 or C2 [8] [9] [10] [11] [12] [13]. Evaluated surface

Figure 3 : the Béziers triangle corresponds to each triangle of the mesh. In the same context of assessment, parameterization or reconstruction of a surface from a triangulation, works were interested in the approximation of a surface by successive subdivision of the corresponding meshing [14] [15]. It is about inserting a new node between two consecutive nodes, so that every bone is divided in two and every triangle is divided in four triangles while respecting the shape of the

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meshed object. Other works are based on different diagrams of known subdivisions (Diagram of Catmull-Clark, Diagram of Loop, Diagram of Butterfly…), to refine a meshing successively in order to get a smooth surface (Figure 4) [16] [17].

This approach is developed under the Open Cascade environment that contains some functions in its API (Application Programming Interface) which permit to adhere a surface to a set of curves and points. These functions of Open Cascade are based on energetic methods [18]. It corresponds well to our case: it is about rebuilding a surface that interpolate the mesh nodes and that adheres to the curves of loop of the face that are going to be rebuilt from the mesh. The algorithm of reconstruction of a deformed face from the mesh is based on six steps (Figure. 5.):

1. To extract the triangulation corresponding to the deformed face

Figure. 4. Subdivision.

2. To determine the nodes of the edges of the face

Different research projects have been focused on the evaluation of the surface especially around one node or a triangle of the mesh. But in our work, we focus on the evaluation and the reconstruction of the surface, as a part of the BREP, using its corresponding triangulation.

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2

3. To build the curves which interpolate the points determined in the previous step 3

4. To determine the nodes Which are on the face but not on the edges 4

IV.

Reconstruction of CAD surface from triangulated surface

5. To build a surface which adheres to the rebuilt curves and points found previously 5

6. To add deformed loops 6

Figure. 5. Reconstruct faces algorithm.

V.

Results obtained : Application to the reconstruction algorithm

In the first example (Figure 6), the subject is a part of complex topology. At the time of the reconstruction, the faces are identified and reconstructed one by one, and then the edges and loops are found by the intersection of the different faces. The CAD model chosen contains cylindrical and planar faces, the identification of the triangulation’s corresponding to each of these faces is done directly (without investigating the topology), because the topology has not changed. All the reconstructed faces are of the NURBS type, except for the embedded faces, which are planar.

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deformed CAD model using the deformed mesh as a FEA results output. In the last part of this paper we have illustrated our work using a parts with topology and geometry complex. Actually some cases are resolved to rebuild the parts with simple and complex topology and we are actually working to extend our algorithm to resolve more complex parts.

Integration CAD/Analysis CAD/Analysis Analysis

CAD Pressure

Clamped

VII.

Acknowledgements

Deformed mesh

This research is part of a cooperative project between the laboratory of mechanical engineering (Lab-Ma05) of l’École Nationale d’Ingénieurs de Monastir (Tunisie) and the department of mechanical engineering of l’Université du Québec à Trois-Rivières (Canada).

Deformed CAD model

VIII.

Figure. 6. Reconstruction process. In the second example a bended beam is considered. The beam is fixed on one of its faces and we apply a bending stress to this beam (Figure 7). The FEA of this case study makes it possible to extract deformed mesh. which will be used to rebuild the faces one by one, according to the proposed algorithm. The connectivity of these various rebuilt faces makes it possible to obtain deformed CAD model (Figure 7).

[1] : Benamara A., Ifaoui N., Deneux D, "Intégration CAOCalcul – une démarche fonctionnelle intégrée", Journal Européen des Systèmes Automatisés (JESA), Vol. 34, n°2-3 – Avril 2000. [2] : N. Aifaoui, « Intégration CAO/Calcul, une approche par les features de calcul », Thèse de Doctorat, Université de Valenciennes, Juillet 2003. [3]: Francois V, Automatic meshing and remeshing methods applied to model modification in the simultaneous engineering context (paper in French), PhD. Thesis. Université Henri POINCARE, Nancy I (France). November 1998. [4] V. Francois, J.C. Cuillère « 3D Automatic remeshing applied to model modification » Computer-Aided Design, Vol 32, No7, pp 433-444, 2000. [5] Hoppe H, “Surface Reconstruction from Unorganized Points”, Ph Doctor of Philosophy University of Washington. 1994. [6] Boubekeur T, « Reconstruction de surface à l'aide de surfaces de subdivision », Master Recherche MM - Juin 2004. [7]: Mantyla M., « An introduction to solid modeling », Computer science press, 1998. [8] D J Walton and D S Meek, “A triangular G1 patch from boundary curves”, Computer Aided Design, Vol. 28, pp 113123, 1996. [9] Praveen Kashyap, “Geometric interpretation of continuity over triangular domains”, Computer Aided Geometric Design 15 (1998) 773-786. [10] J.M. Morvan, B. Thibert, On the approximation of a smooth surface with a triangular mesh. Computational Geometry 23 (2002) 337-352. [11] Shaoming Wang, “A smooth surface interpolation to 3D triangulations”, Journal of computation and applied Mathematics 163 (2004) 287-293.

Integration CAD/Analysis CAD/Analysis Analysis

CAD Pressure

Clamped

Deformed CAD model Deformed mesh

Figure. 7. Reconstruction process. VI.

Bibliography

Conclusion and perspectives

In this paper, we first underline the integration of FEA/CAD systems and then we present our work in this field. The proposed approach allows to automatically rebuild

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[12] Steven J. Owen, David R. White, Timothy J. Tautges, “Facet-based surfaces for 3D mesh generation”, 11th international meshing roundtable, September 15-18, 2002, Ithaca, New York. [13] Steven J. Owen and David R. White, “Mesh based geometry: A systematic approach to constructing geometry from a finite element mesh”, 10th International Meshing Roundtable Newport Beach, California, U.S.A. October 710, 2001. [14] B.Y. Ren, I. Hagiwara, Composite freeform surface reconstruction using recursive interpolating subdivision scheme. Computers in Industry 50 2003. [15] Xunnian Yang, Surface interpolation of meshes by geometric subdivision. Computer Aided Design, Vol 37, pp 497-508, 2005. [16] Weiyin Ma, Xiaohu Ma, Shiu-Kit Tso, Zhigeng Pan « A direct approach for subdivision surface fitting from a dense triangle mesh », Computer Aided Design, Vol. 36, pp 525536, 2004. [17] Daniel Rypl, Zdenek Bittnar « Triangulation of 3D surfaces reconstructed by interpolating subdivision », Computers and Structures, Vol. 82, pp 2093-2103, 2004. [18] www.opencascade.com.

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