Illoul, Amran Lounès (2008) Implementation of the constrained natural element method in 3D : application to the simulation of adiabetic shearing. PhD thesis Mécanique, Laboratoire de mécanique des systèmes et des procédés, ENSAM 2008ENAM0015 p.124.
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Abstract
This work deals with the implementation in 3D of the Constrained Natural Element Method CNEM in order to use it in the simulation of high speed shearing problems. The CNEM is a member of the large family of mesh-free methods, that is at the same time very close to the finite element method. The CNEM’s shape function is built using the constraint Voronoï diagram (the dual of the constraint Delaunay tessellation) of a set of nodes defining both the whole domain and its boundaries. The implementation comprises three principal aspects :i) the built of the constrained Voronoï diagram, ii) the Sibson-type CNEM shape functions computation, iii) the discretization of a generic variational formulation by invoking an “stabilized conforming nodal integration”, SCNI, introduced by Chen & al. in 2001. In this work, we focus especially on the two last points. In order to compute the Sibson shape function, five algorithms are presented, analyzed and compared, two of them are developed in the context of this PhD. For the integration task, a discretization strategy is proposed to handle domains with strong non-convexities. These approaches are validated on some benchmarks, first in 3D elasticity under the hypothesis of small transformations. The results are compared with the analytical solutions and with approximate finite elements results. Furthermore, a validation in large strain with plasticity effects (Taylor-bar impact) is achieved and gives reasonable results. Finally, the 3D CNEM is applied for addressing high speed shearing models. The developed simulation code (in Fortran and C++) is integrated in the LMSP software platform NESSY. NESSY aims the capitalization of the LMSP know-how in numerical simulation.
| Item Type: | PhD Thesis (PhD) |
|---|---|
| PhD Supervisor: | Chinesta, Francisco |
| Date: | 09 July 2008 |
| Board of examiners: | Boissonnat, Jean-Daniel and Cescotto, Serge and Chastel, Yvan and Chinesta, Francisco and Cueto, Elias and Lorong, Philippe and De Vuyst, Florian |
| Ecole Doctorale: | ED 432 ECOLE DOCTORALE SCIENCES DES METIERS DE L'INGENIEUR |
| Discipline: | Mécanique |
| Collection (Fonds): | Arts et Métiers ParisTech (ENSAM) |
| Institution: | ENSAM |
| Department: | Laboratoire de mécanique des systèmes et des procédés |
| Subjects: | 4. Materials Science, Mechanics and Mechanical Engineering |
| Uncontrolled Keywords: | Méthode des éléments naturels contrainte 3d (CNEM), Diagramme de Voronoï contraint (DVC), Tétraèdrisation de Delaunay contrainte (TDC), Fonction de forme Sibson, Transformations finie, Dynamique explicite, Barre de Taylor, Cisaillage adiabatique, 3D constrained natural elements method (CNEM), Constrained Voronoï diagram (CVD), Constrained Delaunay tetraedrisation (CDT), Sibson shape function, Finite Transformations, Explicit dynamics, Taylor bare impact, Adiabatic shearing. |
| ID Code: | 4035 |
| Deposited By: | illoul amran lounès |
| Deposited On: | 05 December 2008 |
References
I. Alfaro, J. Yvonnet, F. Chinesta, and E. Cueto. A study on the performance of natural neighbour-based galerkin methods. Int. J. Numer. Meth. Engng, 00 :1–28, 2006.
Biswajit Banerjee. Taylor impact tests : Detailed report. Technical report, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT 84112, USA, November 2005. Report No. C-SAFE-CD-IR-05-001. www.csafe.utah.edu/documents/C-SAFE-CD-IR-05-001.pdf.
Klaus-Jurgen Bathe. Finite Element Procedures. Prentice Hall, 1995.
J-D. Boissonnat and F. Cazals. Smooth surface reconstruction via natural neighbour interpolation of distance functions. Computational Geometry, 22 :185–203, 2002.
B. Büeler, A. Enge, and K. Fukuda. Exact volume computation for polytopes : A practical study. Polytopes - Combinatorics and Computation, 29 :131–154, 2000.
Etienne Bertin. DIAGRAMMES DE VORONOI 2D ET 3D : APPLICATIONS EN ANALYSE D’IMAGES. PhD thesis, UNIVERSITE Joseph FOURIER - GRENOBLE 1, TIMC - Institut IMAG, Janvier 1994.
T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl. Meshless methods : An overview and recent developments. Computer methods in applied mechanics and engineering, 139 :3–47, 1996.
A. Bowyer. Computing dirichlet tessellations. THE COMPUTER JOURNAL, 24(2) :162–166, 1981.
Vitali V. Belikov and Andrei Yu. Semenov. Non-sibsonian interpolation on arbitrary system of points in euclidean space and adaptive isolines generation. Applied Numerical Mathematics, 32 :371–387, 2000.
Jiun-Shyan Chen, Cheng-Tang Wu, Sangpil Yoon, and Yangu You. Stabilized conforming nodal integration for galerkin mesh-free methods. International Journal for Numerical Methods in Engineering, 50 :435–466, 2001.
Jiun-Shyan Chen, Sangpil Yoon, and Cheng-Tang Wu. Non-linear version of stabilized conforming nodal integration for galerkin mesh-free methods. International Journal for Numerical Methods in Engineering, 53 :2587– 2615, 2002.
David F. Watson. Compound signed decomposition, the core of natural neighbour interpolation in n-dimensional space.(unpublished manuscript). c 2001 D.F.Watson. http://www.iamg. org/naturalneighbour.html.
Schönhardt E. Über die zerlegung von dreieckspolyedern in tetraeder. Mathematische Annalen, 98 :309–312, 1928.
Andreas Enge(enge@lix.polytechnique.fr), Benno Büeler(bueeler@ifor.math.ethz.ch), and Komei Fukuda( komei.fukuda@ifor.math.ethz.ch). Vinci version 1.0.5, computing volumes of convex polytopes, July 2003. http://www.lix. polytechnique.fr/Labo/Andreas.Enge/Vinci.html.
D. Gonzalez, E. Cueto, M. A. Mart´ınez, and M. Doblaré. Numerical integration in natural neighbour galerkin methods. International Journal for Numerical Methods in Engineering, 60 :2077–2104, 2004.
Voronoï G.M. Nouvelles applications des paramètres continus à la théorie des formes quadratiques, deuxiéme mémoire : recherches sur les parallélloèdres primitifs. J. Reine Angew. Math, 134 :198–287, 1908. [GR96] M. Géradin and D. Rixen. Théorie des vibrations : application à la dynamique des structures. MASSON, 1996.
P. J. Green and R. Sibson. Computing dirichlet tessellation in the plane. Computer Journal, 21 :168–173, 1978.
Hisamoto Hiyoshi and Kokichi Sugihara. Vorono¨ı-based interpolation with higher continuity. In Proceedings of the 16th Annual ACM Symposium on Computational Geometry, pages 242–250, 2000.
Hang Si. Tetgen version 1.4, a quality tetrahedral mesh generator and three-dimensional delaunay triangulator, January 2006. c 2002, 2004, 2005, 2006. http://tetgen.berlios.de/.
[Int] IntelR . Math kernel library. Copyright c Intel Corporation, 2007. http: //www.intel.com/software/products/mkl.
J.C. Jaeger and N.G.W. Cook. Fundamentals of Rock Mechanics. Chapman and Hall, 1976.
Barry Joe and Cao An Wang. Duality of constrained voronoi diagrams and delaunay triangulations. Algorithmica, 9 :142–155, 1993.
Jim Lawrence. Polytope volume computation. Mathematics of Computation, 57 :259–271, 1991.
Wing Kam Liu, Sukky Jun, and Yi Fei Zhang. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 20 :1081–1106, 1995.
Philippe Mestat and Pierre Humbert. Référentiel de tests pour la vérification de la programmation des lois de comportement dans les logiciels élément finis. Bulletin des laboratoires des ponts et chaussées, 230 :23–38, 2001.
Gray L. Miller, Dafna Talmor, Shang-Hua Teng, Noel Walkington, and Han Wang. Control volume meshes using sphere packing : Generation, refinement and coarsening. In 5th International Meshing Roundtable, pages 47–61, 1996.
Delaunay Boris N. Sur la sphère vide. Izvestia Akademia Nauk SSSR, VII Seria, Otdelenie Matematicheskii i Estestvennyka Nauk, 7 :793–800, 1934. [OFTB96] D. Organ, M. Fleming, T. Terry, and T. Belytschko. Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Computational Mechanics, 18 :225–235, 1996.
B. Piper. Properties of local coordinates based on dirichlet tessellations. Computing Suppl., 8 :227–239, 1993.
Nicolas Ranc. Etude des champs de température et de déformation dans les matériaux métalliques sollicités à grande vitesse de déformation. PhD thesis, Université Paris X, Nanterre, Décembre 2004.
Rafael DALA ROSA DOS SANTOS. Projet d’expertise, cisaillage adiabatique : De la modelisation a la realite. Technical report, ENSAM Paris, 2006-2007.
F. Schmitt and H. Borouchaki. Algorithme rapide de maillage de delaunay dans rd. In journées de géométrie algorithmique, pages 131–133, 1990.
Malcolm Sambridge, Jean Braun, and Herbert McQueen. Geophysical parametrization and interpolation of irregular data using natural neighbours. Geophysical Journal International, 122 :837–857, 1995.
M. Sambridge, J. Braun, and H. McQueen. New computational methods for natural neighbour interpolation in two and three dimensions. In Computational Techniques and Applications : CTAC95, pages 685–692, 1996.
H. SI and K. GÄRTNER. Meshing piecewise linear complexes by constrained delaunay tetrahedralizations. In 14th International Meshing Roundtable, pages 147–163, 2005.
Jonathan Richard Shewchuk. A condition guaranteeing the existence of higher-dimensional constrained delaunay triangulations. In Proceedings of the fourteenth annual symposium on Computational geometry, pages 76–85, 1998.
R. Sibson. The dirichlet tessellation as an aid in data analysis. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 7, pages 14–20, 1980.
R. Sibson. A vector identity for the dirichlet tessellation. Scandinavian Journal of Statistics, 87 :151–155, 1980.
R. Sibson. A brief description of natural neighbor interpolation. In V. Barnett, editor, Interpreting Multivariate Data, pages 21–36. John Wiley and Sons Ltd, 1981.
Natarajan Sukumar. The Natural Element Methode in Solid Mechanics. PhD thesis, Northwestern University, Evanston, Illinois, June 1998. [Wat81] D.F. Watson. Computing the n-dimensional delaunay tessellation with application to voronoi polytopes. Computer Journal, 24 :167–172, 1981.
Julien Yvonnet. Nouvelles approches sans maillage basées sur la méthode des éléments naturels pour la simulation numérique des procédés de mise en forme. PhD thesis, ENSAM, LMSP, Décembre 2004.
O.C. Zienkiewicz, B. Boroomand, and J.Z. Zhu. Recovery procedures in error estimation and adaptivity. part i : Adaptivity in linear problems. Computer methods in applied mechanics and engineering, 176 :111–125, 1999.
O.C. Zienkiewicz, B. Boroomand, and J.Z. Zhu. Recovery procedures in error estimation and adaptivity. part ii : Adaptivity in nonlinear problems of elasto-plasticity behaviour. Computer methods in applied mechanics and engineering, 176 :127–146, 1999.
Amran Lounès Illoul. Mise en œuvre de la méthode des éléments naturels contrainte en 3d : application au cisaillage adiabatique. PhD thesis, ENSAM, LMSP, juillet 2008.
Table of content
1 Introduction 11
2 Présentation de l’approche CNEM 15
2.1 Méthode des éléments naturels (NEM) - Méthode des éléments naturels
contrainte (CNEM) - 16
2.2 Les fonctions de forme CNEM - 20
2.2.1 Préliminaire - 20
2.2.2 Calcul des fonctions de forme CNEM - 22
2.2.3 Calcul du gradient des fonctions de forme CNEM de type Sibson 26
2.2.4 Propriétés des fonctions de forme CNEM - 27
2.3 Intégration numérique et gradient stabilisé - 28
2.4 Conclusion - 31
3 Mise en oeuvre de la CNEM en 3d 33
3.1 Structuration adoptée pour les données du diagramme de Voronoï contraint 34
3.2 Construction du diagramme de Voronoï contraint - 37
3.2.1 Construction de la tétraèdrisation de Delaunay contrainte d’un
domaine 3d - 37
3.2.2 Passage de la tétraèdrisation de Delaunay contrainte au diagramme
de Voronoï contraint - 39
3.3 Calcul des fonctions de formes CNEM de type Sibson - 42
3.3.1 Etape-1 : Insertion du point x dans le diagramme de Voronoï contraint
existant - 42
3.3.2 Etape-2 : Calcul de la mesure du volume commun à ´cx et cv . . 45
3.3.3 Test comparatif des différents algorithmes - 58
3.3.4 Mise en évidence de phénomènes associés aux fonctions de formes
CNEM apparaissant en 3d pour des domaines non convexes . . 60
3.4 discrétisation du domaine en vue de l’intégration numérique - 67
3.4.1 Cas des domaines convexes ou faiblement non convexes - 67
3.4.2 Cas des domaines fortement non convexes - 79
3.5 Conclusion - 80
4 Validation 83
4.1 Simulations menées et résultats - 84
4.1.1 Sphère creuse sous pression - 85
5 Applications de la CNEM aux grandes transformations 97
5.1 Mise en oeuvre des grandes transformations dans Nessy - 98
5.1.1 Contexte général de la mise en oeuvre - 98
5.1.2 Formulation Lagrangienne Actualisée (FLA) - 98
5.1.3 Traitement des points additionnels - 99
5.1.4 Schéma d’intégration temporelle - 100
5.1.5 Traitement du contact - 100
5.1.6 Intégration de la relation de comportement - 100
5.2 Applications - 103
5.2.1 Barre de Taylor - 103
5.2.2 Cisaillage adiabatique - 109
5.3 Conclusion - 118
6 Conclusion 119
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