Research Papers

A Three-Dimensional Adaptive Mesh Generation Approach Using Geometric Modeling With Multi-Regions and Parametric Surfaces

[+] Author and Article Information
Antonio C. O. Miranda

Department of Civil Engineering,
University of Brasília,
Brasília, 70910-900, Brazil
e-mail: acmiranda@unb.br

William W. M. Lira

LCCV—Laboratory of Scientific Computing and Visualization,
Technology Center,
Federal University of Alagoas,
Maceió, 57072-970, Brazil
e-mail: william@lccv.ufal.br

Joaquim B. Cavalcante-Neto

Department of Computing,
Federal University of Ceará,
Fortaleza, 60020-181, Brazil
e-mail: joaquim@lia.ufc.br

Rafael A. Sousa

Department of Civil Engineering,
Pontifical Catholic University of Rio de Janeiro,
Rio de Janeiro, 22451-900, Brazil
e-mail: rfldesousa@hotmail.com

Luiz F. Martha

Tecgraf—Computer Graphics Technology Group,
Department of Civil Engineering,
Pontifical Catholic University of Rio de Janeiro,
Rio de Janeiro, 22451-900, Brazil,
e-mail: lfm@tecgraf.puc-rio.br

1Present address: Campus A. C. Simoes, Av. Lourival Melo Mota, Cidade Universitária, Maceió, AL, Brazil. CEP: 57072-900.

Contributed by the Computers and Information Division of ASME for publication in the Journal of Computing and Information Science in Engineering. Manuscript received March 9, 2012,; final manuscript received March 14, 2013; published online April 22, 2013. Assoc. Editor: Xiaoping Qian.

J. Comput. Inf. Sci. Eng 13(2), 021002 (Apr 22, 2013) (13 pages) Paper No: JCISE-12-1042; doi: 10.1115/1.4024106 History: Received March 09, 2012; Revised March 14, 2013

This work presents a methodology for adaptive generation of 3D finite element meshes using geometric modeling with multiregions and parametric surfaces, considering a geometric model described by curves, surfaces, and volumes. This methodology is applied in the simulation of stress analysis of solid structures using a displacement-based finite element method and may be extended to other types of 3D finite element simulation. The adaptive strategy is based on an independent and hierarchical refinement of curves, surfaces, and volumes. From an initial model, new sizes of elements obtained from a discretization error analysis and from geometric restrictions are stored in a global background structure, a recursive spatial composition represented by an octree. Based on this background structure, the model's curves are initially refined using a binary partition algorithm. Curve discretization is then used as input for the refinement of adjacent surfaces. Surface discretization also employs the background octree-based refinement, which is coupled to an advancing front technique in the surface's parametric space to generate an unstructured triangulated mesh. Surface meshes are finally used as input for the refinement of adjacent volumetric domains, which also uses an advancing front technique but in 3D space. In all stages of the adaptive strategy, the refinement of curves, surface meshes, and solid meshes is based on estimated discretization errors associated with the mesh of the previous step in the adaptive process. In addition, curve and surface refinement takes curvature information into account. Numerical examples of simulation of engineering problems are presented in order to validate the methodology proposed in this work.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Mackerle, J., 2001, “Error Estimates and Adaptive Finite Element Methods: A Bibliography (1990-2000),” Eng. Comput., 18, pp. 802–914. [CrossRef]
McRae, D. S., 2000, “r-Refinement Grid Adaptation Algorithms and Issues,” Comput. Methods Appl. Mech. Eng., 189, pp. 1161–1182. [CrossRef]
Xu, G., Mourrain, B., Duvigneau, R., and Galligo, A., 2011, “Parametrization of Computational Domain in Isogeometric Analysis: Methods and Comparison,” Comput. Methods Appl. Mech. Eng., 200(23–24), pp. 2021–2031. [CrossRef]
Kallinderis, Y., and Vijayant, P., 1993, “Adaptive Refinement-Coarsening Scheme for Three-Dimensional Unstructured Meshes,” AIAA J., 43(8), pp. 1440–1447. [CrossRef]
Muthukrishnan, N., Shiakolas, P. S., Nambiar, R. V., and Lawrence, K. L., 1995, “Simple Algorithm for Adaptive Refinement of Three Dimensional Finite Element Tetrahedral Meshes,” AIAA J., 33(5), pp. 928–932. [CrossRef]
Golias, N. A., and Tsiboukis, T. D., 1994, “An Approach to Refining Three-Dimensional Tetrahedral Meshes Based on Delaunay Transformations,” Int. J. Numer. Methods Eng., 37, pp. 793–801. [CrossRef]
Golias, N. A., and Dutton, R. W., 1997, “Delaunay Triangulation and 3D Adaptive Mesh Generation,” Finite Elem. Anal. Design, 25, pp. 331–341. [CrossRef]
Lee, C. K., and Lo, S. H., 1997, “Automatic Adaptive Refinement Finite Element Procedure for 3D Stress Analysis,” Finite Elem. Anal. Design, 25, pp. 135–166. [CrossRef]
Lee, C. K., and Lo, S. H., 1999, “A Full 3D Finite Element Analysis Using Adaptive Refinement and PCG Solver With Back Interpolation,” Comput. Methods Appl. Mech. Eng., 170, pp. 39–64. [CrossRef]
Merrouche, A., Selman, A., and Knopf-Lenoir, C., 1998, “3D Adaptive Mesh Refinement,” Commun. Numer. Methods Eng., 14(5), pp. 397–407. [CrossRef]
De Cougny, H. L., and Shephard, M. S., 1999, “Parallel Refinement and Coarsening of Tetrahedral Meshes,” Int. J. Numer. Methods Eng., 46, pp. 1101–1125. [CrossRef]
Lee, C. K., and Xu, Q. X., 2005, “A New Automatic Adaptive 3D Solid Mesh Generation Scheme for Thin-Walled Structures,” Int. J. Numer. Methods Eng., 62, pp. 1519–1558. [CrossRef]
Qian, J., and Zhang, Y., 2012, “Automatic Unstructured All-Hexahedral Mesh Generation From b-Reps for Non-Manifold CAD Assemblies,” Eng. Comput., 28, pp. 345–359. [CrossRef]
Zhang, Y., Hughes, T. J., and Bajaj, C. L., 2010, “An Automatic 3D Mesh Generation Method for Domains With Multiple Materials,” Comput. Methods Appl. Mech. Eng., 199(5–8), pp. 405–415. [CrossRef] [PubMed]
Kettil, P., Ekevid, T., and Wiberg, N. E., 2003, “Towards Fully Mesh Adaptive FE-Simulations in 3D Using Multi-Grid Solver,” Comput. Struct., 81, pp. 735–746. [CrossRef]
Hughes, T. J. R., Cottrell, J. A., and Bazilevs, Y., 2005, “Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement,” Comput. Methods Appl. Mech. Eng., 194, pp. 4135–4195. [CrossRef]
Cavalcante-Neto, J. B., Martha, L. F., Menezes, I. F. M., and Paulino, G. H., 1998, “A Methodology for Self-Adaptive Finite Method Analysis Using an Object Oriented Approach,” Proceedings of the 4th World Congress on Computational Mechanics (IV WCCM), pp. 1–20.
Baehmann, P. L., and Shephard, M. S., 1989, “Adaptive Multiple-Level h-Refinement in Automated Finite Element Analysis,” Eng. Comput., 5, pp. 235–247. [CrossRef]
Cavalcante-Neto, J. B., 1998, “Mesh Generation and Error Estimative for Finite Element 3D Models With Crack,” Ph.D. thesis, Pontifical University Catholic of Rio de Janeiro, Rio de Janeiro, Brazil.
Xu, G., Mourrain, B., Duvigneau, R., and Galligo, A., 2013, “Analysis-Suitable Volume Parameterization of Multi-Block Computational Domain in Isogeometric Applications,” Comput.-Aided Des., 45(2), pp. 395–404. [CrossRef]
Zienkiewicz, O. C., and Taylor, R. L., 2000, The Finite Element Method: The Basis, 5th ed., Vol. 1, Butterworth-Heinemann, Oxford, UK.
Zienkiewicz, O. C., and Zhu, J. Z., 1992, “The Superconvergent Patch Recovery and a Posterior Error Estimates. Part 1: The Recovery Technique,” Int. J. Numer. Methods Eng., 33, pp. 1331–1364. [CrossRef]
Zienkiewicz, O. C., and Zhu, J. Z., 1992, “The Superconvergent Patch Recovery and a Posterior Error Estimates. Part 2: Error Estimates and Adaptivity,” Int. J. Numer. Methods Eng., 33, pp. 1365–1382. [CrossRef]
Farin, G., 2002, Curves and Surfaces for CAGD: A Practical Guide. The Morgan Kaufmann Series in Computer Graphics, Elsevier Science, New York.
Quadros, W. R., Vyas, V., Brewer, M., Owen, S. J., and Shimada, K., 2010, “A Computational Framework for Automating Generation of Sizing Function in Assembly Meshing via Disconnected Skeletons,” Eng. Comput., 26(3), pp. 231–247. [CrossRef]
Knuth, D., 1997, The Art of Computer Programming Volume 1. Fundamental Algorithms, Addison-Wesley, Reading, MA.
Miranda, A. C. O., and Martha, L. F., 2002, “Mesh Generation on High-Curvature Surfaces Based on a Background Quadtree Structure,” Proceedings of 11th International Meshing Roundtable, pp. 333–341.
Cavalcante-Neto, J. B., Wawrzynek, P. A., Carvalho, M. T. M., Martha, L. F., and Ingraffea, A. R., 2001, “An Algorithm for Three-Dimensional Mesh Generation for Arbitrary Regions With Cracks,” Eng. Comput., 17(1), pp. 75–91. [CrossRef]
Foley, T. A., and Nielson, G. M., 1989, Knot Selection for Parametric Spline Interpolation. Mathematical Methods in Computer Aided Geometric Design, Academic, New York.
Cavalcante-Neto, J. B., Martha, L. F., Wawrzynek, P. A., and Ingraffea, A. R., 2005, “A Back-Tracking Procedure for Optimization of Simplex Meshes,” Commun. Numer. Methods Eng., 21(12), pp. 711–722. [CrossRef]
Paulino, G. H., Menezes, I. F. M., Cavalcante-Neto, J. B., and Martha, L. F., 1999, “A Methodology for Adaptive Finite Element Analysis: Towards an Integrated Computational Environment,” Computat. Mech., 23, pp. 361–388. [CrossRef]
Boroomand, B., and Zienkiewicz, O. C., 1997, “Recovery by Equilibrium in Patches (REP),” Int. J. Numer. Methods Eng., 40, pp. 137–164. [CrossRef]
Martha, L. F., and Parente, E., Jr., 2002, “An Object-Oriented Framework for Finite Element Programming,” Proceedings of the 5th World Congress on Computational Mechanics.
Lee, C. K., and Lo, S. H., 1997, “Automatic Adaptive 3-D Finite Element Refinement Using Different-Order Tetrahedral Elements,” Int. J. Numer. Methods Eng., 40, pp. 2195–2226. [CrossRef]
Cuilliere, J., Francois, V., and Drouet, J., 2012, “Automatic 3D Mesh Generation of Multiple Domains for Topology Optimization Methods,” Proceedings of 21st International Meshing Roundtable, pp. 243–249.
Krysl, P., 1996, “Computational Complexity of the Advancing Front Triangulation,” Eng. Comput., 12, pp. 16–22. [CrossRef]
Lohner, R., and Parikh, P., 1988, “Generation of Three-Dimensional Unstructured Grids by the Advancing-Front Method,” Int. J. Numer. Methods Fluids, 8, pp. 1135–1149. [CrossRef]
Bonet, J., and Peraire, J., 1991, “An Alternating Digital Tree (ADT) Algorithm for 3D Geometric Search and Intersection Problems,” Int. J. Numer. Methods Eng., 31, pp. 1–17. [CrossRef]
Jin, H., and Tanner, R. I., 1993, “Generation of Unstructured Tetrahedral Meshes by Advancing Front Technique,” J. Numer. Methods Eng., 36, pp. 1805–1823. [CrossRef]
Moller, P., and Hansbo, P., 1995, “On Advancing Front Mesh Generation in Three Dimensions,” Int. J. Numer. Methods Eng., 38, pp. 1805–1823. [CrossRef]
Gosselin, S., and Ollivier-Gooch, C., 2011, “Tetrahedral Mesh Generation Using Delaunay Refinement With Non-Standard Quality Measures,” Int. J. Numer. Methods Eng., 87, pp. 795–820. [CrossRef]
Klingner, B. M., and Shewchuk, J. R., 2007, “Aggressive Tetrahedral Mesh Improvement,” Proceedings of 16th International Meshing Roundtable, pp. 3–23.
Alliez, P., Cohen-Steiner, D., Yvinec, M., and Desbrun, M., 2005, “Variational Tetrahedral Meshing,” ACM Trans. Graphics, 24(3), pp. 617–625. [CrossRef]


Grahic Jump Location
Fig. 1

The proposed adaptive refinement process

Grahic Jump Location
Fig. 2

Curve refinement based on curve curvature

Grahic Jump Location
Fig. 3

Approximating the curvature by circular arcs

Grahic Jump Location
Fig. 4

A hypothetical model under applied σ tension to explain the steps of octree construction

Grahic Jump Location
Fig. 5

External appearance of background octree based on discretization error analysis

Grahic Jump Location
Fig. 6

Refinement after considering curve curvatures

Grahic Jump Location
Fig. 7

Refinement after considering surface curvatures

Grahic Jump Location
Fig. 8

Refinement after considering maximum cell size at boundary cells and maximum difference of one level between adjacent cells

Grahic Jump Location
Fig. 9

Curve refinement and its corresponding binary tree

Grahic Jump Location
Fig. 17

Adaptive refinement meshes for example 2

Grahic Jump Location
Fig. 18

Adaptive refinement meshes for example 3

Grahic Jump Location
Fig. 19

Adaptive refinement meshes for example 4

Grahic Jump Location
Fig. 20

Adaptive refinement meshes for example 5

Grahic Jump Location
Fig. 11

Example 1: Short cantilever under end shear

Grahic Jump Location
Fig. 12

Example 2: L-shaped domain under horizontal uniform force (face y = 0)

Grahic Jump Location
Fig. 13

Example 3: Biaxial bending of a column footing

Grahic Jump Location
Fig. 14

Example 4: 3D Frame

Grahic Jump Location
Fig. 15

Example 5: Bike suspension rocker

Grahic Jump Location
Fig. 16

Adaptive refinement meshes for example 1

Grahic Jump Location
Fig. 10

Measurements in parametric space

Grahic Jump Location
Fig. 22

Comparison of convergence rate

Grahic Jump Location
Fig. 23

Time performance of 3D mesh generator

Grahic Jump Location
Fig. 21

Detail of mesh refinement for example 5



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In