Mixed finite element methods for geometric modeling using general fourth order geometric flows

Computer Aided Geometric Design

Volumn 26, Issue 4, 2009, Pages 378-395

  Xu, Guoliang ^a  

^a CHINESE ACADEMY OF SCIENCES   (China)

Author keywords

Geometric partial differential equation; Hole filling; Mixed finite element method; Surface blending; Surface denoising; Surface mesh refinement

Indexed keywords

BLENDING; COMPUTATIONAL FLUID DYNAMICS; FINITE ELEMENT METHOD; IMAGE SEGMENTATION; MESH GENERATION; SURFACE TOPOGRAPHY;

GEOMETRIC PARTIAL DIFFERENTIAL EQUATION; HOLE FILLING; MIXED FINITE ELEMENT METHOD; SURFACE BLENDING; SURFACE MESH REFINEMENT;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 63149182580     PISSN: 01678396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cagd.2008.12.004     Document Type: Article

References (36)

1. Arden, G., 2001. Approximation properties of subdivision surfaces. PhD thesis, University of Washington
2. Bajaj C., and Xu G. Anisotropic diffusion of surface and functions on surfaces. ACM Transactions on Graphics 22 1 (2003) 4-32
3. Bajaj, C., Xu, G., Warren, J., 2002. Acoustics scattering on arbitrary manifold surfaces. In: Proceedings of Geometric Modeling and Processing, Theory and Application, Japan, pp. 73-82
4. Bloor M.I.G., and Wilson M.J. Generating blend surfaces using partial differential equations. Computer-Aided Design 21 3 (1989) 165-171
5. Bloor M.I.G., and Wilson M.J. Using partial differential equations to generate free-form surfaces. Computer-Aided Design 22 4 (1990) 221-234
6. Chavel I. Riemannian Geometry - A Modern Introduction (1993), Cambridge University Press
7. Clarenz U., Diewald U., Dziuk G., Rumpf M., and Rusu R. A finite element method for surface restoration with boundary conditions. Computer Aided Geometric Design 21 5 (2004) 427-445
8. Clarenz, U., Diewald, U., Rumpf, M., 2000. Anisotropic geometric diffusion in surface processing. In: Proceedings of Viz2000, IEEE Visualization, Salt Lake City, Utah, pp. 397-405
9. Davis, J., Marschner, R., Garr, M., Levoy, M., 2002. Filling holes in complex surfaces using volumetric diffusion. In: Proceedings of the 1st International Symposium on 3D Data Processing Visualization and Transmission, pp. 428-438
10. Deckelnick K., and Dziuk G. A fully discrete numerical scheme for weighted mean curvature flow. Numerische Mathematik 91 (2002) 423-452
11. Desbrun, M., Meyer, M., Schröder, P., Barr, A.H., 1999. Implicit fairing of irregular meshes using diffusion and curvature flow. In: SIGGRAPH99, Los Angeles, USA, pp. 317-324
12. do Carmo M.P. Differential Geometry of Curves and Surfaces (1976), Prentice-Hall, Englewood Cliffs, NJ
13. do Carmo M.P. Riemannian Geometry (1992), Birkhäuser, Boston, Basel, Berlin
14. Du H., and Qin H. Direct manipulation and interactive sculpting of PDE surfaces. Computer Graphics Forum 19 3 (2000) 261-270
15. Du H., and Qin H. Dynamic PDE-based surface design using geometric and physical constraint. Graphical Models 67 1 (2005) 43-71
16. Dziuk G. An algorithm for evolutionary surfaces. Numerische Mathematik 58 (1991) 603-611
17. Giaquinta M., and Hildebrandt S. Calculus of Variations, vol. I. A Series of Comprehensive Studies in Mathematics vol. 310 (1996), Springer-Verlag, Berlin
18. Gu, X., He, Y., Qin, H., 2005. Manifold splines. In: ACM Symposium on Solid and Physical Modeling, pp. 27-38
19. Kobbelt L., Hesse T., Prautzsch H., and Schweizerhof K. Iterative mesh generation for FE-computation on free form surfaces. Engng. Comput. 14 (1997) 806-820
20. Lowe T., Bloor M., and Wilson M. Functionality in blend design. Computer-Aided Design 22 10 (1990) 655-665
21. Meyer, M., Desbrun, M., Schröder, P., Barr, A., 2002. Discrete differential-geometry operators for triangulated 2-manifolds. In: Proceedings of Visual Mathematics'02, Berlin, Germany
22. Saad Y. Iterative Methods for Sparse Linear Systems. second ed (2003), SIAM, Philadelphia, PA
23. 1 boundary conditions. In: Geometric Modeling and Processing, Hong Kong, China, pp. 251-261
24. Schneider R., and Kobbelt L. Geometric fairing of irregular meshes for free-form surface design. Computer Aided Geometric Design 18 4 (2001) 359-379
25. Stam, J., 1998. Fast evaluation of Loop triangular subdivision surfaces at arbitrary parameter values. In: SIGGRAPH '98 Proceedings, CD-ROM supplement
26. Ugail H., Bloor M., and Wilson M. Techniques for interactive design using the PDE method. ACM Transactions on Graphics 18 2 (1999) 195-212
27. Verdera, J., Caselles, V., Bertalmo, M., Sapiro, G., 2003. Inpainting surface holes. In: Proceedings of IEEE International Conference on Image Processing (ICIP), pp. 903-906
28. Xu G. Discrete Laplace-Beltrami operators and their convergence. Computer Aided Geometric Design 21 8 (2004) 767-784
29. Xu G. Surface fairing and featuring by mean curvature motions. Journal of Computational and Applied Mathematics 163 1 (2004) 295-309
30. Xu G. Interpolation by loop's subdivision functions. Journal of Computational Mathematics 23 3 (2005) 247-260
31. 1 surface modelling using fourth order geometric flows. Computer-Aided Design 38 4 (2006) 392-403
32. Xu G., Pan Q., and Bajaj C. Discrete surface modelling using partial differential equations. Computer Aided Geometric Design 23 2 (2006) 125-145
33. Xu G., and Zhang Q. Construction of geometric partial differential equations in computational geometry. Mathematica Numerica Sinica 28 4 (2006) 337-356
34. Xu G., and Zhang Q. A general framework for surface modeling using geometric partial differential equations. Computer Aided Geometric Design 25 3 (2008) 181-202
35. Zhang, Q., Xu, G., 2006. Geometric partial differential equations for minimal curvature variation surfaces. Research Report No. ICM-06-03, Institute of Computational Mathematics, Chinese Academy of Sciences
36. Zhao W., Gao S., and Lin H. A robust hole-filling algorithm for triangular mesh. The Visual Computer 23 12 (2007) 987-997