A survey of partial differential equations in geometric design

Visual Computer

Volumn 24, Issue 3, 2008, Pages 213-225

  González Castro, Gabriela ^a   Ugail, Hassan ^a   Willis, Philip ^b   Palmer, Ian ^a  

^a UNIVERSITY OF BRADFORD   (United Kingdom)

^b University of Bath   (United Kingdom)

Author keywords

Geometric modelling; PDE method; PDE surfaces

Indexed keywords

ALGORITHMS; COMPUTER AIDED DESIGN; COMPUTER GRAPHICS; INTERPOLATION;

GEOMETRIC MODELLING; PDE METHOD; PDE SURFACES;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 38749125911     PISSN: 01782789     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00371-007-0190-z     Document Type: Review

References (47)

1. Bajaj, C., Xu, G.: Anisotropic diffusion on surfaces and functions on surfaces. ACM Trans. Graph. 22, 4-32 (2003)
2. Barr, A.: Global and local deformations of solid primitives. Comput. Graph. 18, 21-30 (1984)
3. Bertalmio, M., Cheng, L., Osher, S., Sapiro, S.: Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2), 759-780 (2001)
4. Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting, pp. 417-424. SIGGRAPH 2000, New Orleans (2000)
5. Bloor, M.I.G., Wilson, M.J.: Generating blend surfaces using partial differential equations. Comput. Aided Des. 21(3), 165-171 (1989)
6. Bloor, M.I.G., Wilson, M.J.: Representing PDE surfac es in terms of B-splines. Comput. Aided Des. 22(6), 324-331 (1990)
7. Bloor, M.I.G., Wilson, M.J.: Using partial differential equations to generate free-form surfaces. Comput. Aided Des. 22(4), 202-212 (1990)
8. Bloor, M.I.G., Wilson, M.J.: Functionality in solids obtained from partial differential equations. Computing 8, 21-42 (1993)
9. Bloor, M.I.G., Wilson, M.J.: The efficient parametrization of generic aircraft geometry. J. Aircraft 32(6), 1269-1275 (1995)
10. Bloor, M.I.G., Wilson, M.J.: Spectral approximations to PDE surfaces. Comput. Aided Des. 28(2), 145-152 (1996)
11. Bloor, M.I.G., Wilson, M.J.: An analytic pseudo-spectral method to generate a regular 4-sided PDE surface patch. Comput. Aided Geom. Des. 22(3), 203-219 (2005)
12. Celniker, G., Gossard, D.C.: Deformable curve and surface finite-elements for free-form shape design. Comput. Graph. 25, 257-266 (1991)
13. Celniker, G., Welch, W.: Linear constraints for deformable non-uniform B-spline surfaces. In: SI3D '92: Proceedings of the 1992 symposium on Interactive 3D graphics, pp. 165-170. ACM Press, New York, NY (1992)
14. Dekanski, C.W., Bloor, M.I.G., Wilson, M.J.: The representation of marine propeller blades using the PDE method. J. Ship Res. 38(2), 108-116 (1995)
15. DeRose, T., Kass, M., Truong, T.: Subdivision surfaces in character animation, pp. 85-94. SIGGRAPH 1999, New York, NY (1999)
16. Du, H., Qin, H.: Direct manipulation and interactive sculpting of PDE surfaces. Comput. Graph. Forum 19(3), C261-C270 (2000a)
17. Du, H., Qin, H.: Dynamic PDE Surfaces with Flexible and General Geometric Constraints, pp. 213-222. Pacific Graphics 2000, Hong Kong (2000b)
18. Du, H., Qin, H.: Integrating Physics-Based Modeling with PDE Solids for Geometric Design, pp. 198-207. Pacific Graphics 2001, Tokyo (2001)
19. Du, H., Qin, H.: Interactive shape design using volumetric implicit PDEs. In: SMI '03: Proceedings of the Eighth ACM Symposium on Solid Modeling and Applications, pp. 235-246. ACM Press, New York, NY (2003)
20. Du, H., Qin, H.: A shape design system using volumetric implicit PDEs. Comput. Aided Des. 36(11), 1101-1116 (2004)
21. Du, H., Qin, H.: Dynamic PDE-based surface design using geometric and physical constraints. Graph. Models 67, 43-71 (2005)
22. Enright, D., Marschner, S., Fedkiw, R.: Animation and rendering of complex water surfaces. In: SIGGRAPH '02: Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques, pp. 736-744. ACM Press, New York, NY (2002)
23. Farin, G.: Curves and Surfaces for Computer Aided Geometric Design, a Practical Guide, 5th edn. Morgan Kaufmann, San Diego, CA (2001)
24. Farlow, S.J.: Partial Differential Equations for Scientists and Enginneers, 1st edn. Dover Publications Inc., Mineaol, NY (1993)
25. Fedkiw, R.: Simulating Natural Phenomena for Computer Graphics, Geometric Level Set Methods in Imaging, Vision and Graphics, 5th edn. Springer, New York, NY (2003)
26. Fedkiw, R., Aslam, T., Jensen, H.: Visual simulation of smoke. In: Siggraph 2001 Annual Conference, pp. 23-30. ACM, New York, NY (2001)
27. Huband, J., Li, W.: Extracting design parameters from airplane wing data by using Bloor-Wilson PDE surface model. Math. Eng. Ind. 8, 239-252 (2001)
28. Kubiesa, S., Ugail, H., Wilson, M.J.: Interactive design using higher order PDEs. Visual Comput. 20, 682-693 (2004)
29. Lin, V.C., Gossard, D.C., Light, R.A.: Variational geometry in computer-aided design. Comput. Graph. 15, 171-177 (1981)
30. Lowe, T.W., Bloor, M.I.G., Wilson, M.J.: Functionality in blend design. In: Computer Aided and Computational Design ASME, vol. 1, pp. 43-50. Butterworth, New York (1990)
31. Mémoli, F., Sapiro, G., Osher, S.: Solving variational problems and partial differential equations mapping into general target manifolds. J. Comput. Phys. 195(1), 263-292 (2004)
32. Monterde, J., Ugail, H.: On harmonic and biharmonic Bézier surfaces. Comput. Aided Geom. Des. 21(7), 607-715 (2004)
33. Monterde, J., Ugail, H.: A general fourth order PDE method to generate Bézier surfaces from the boundary. Comput. Aided Geom. Des. 23, 208-225 (2006)
34. Nguyen, D., Fedkiw, R., Jensen, H.: Physically based modeling and animation of fire. In: Siggraph 2002 Annual Conference. ACM, New York, NY (2002)
35. Qin, H., Terzopoulos, D.: D-NURBS: A physics-based framework for geometric design. IEEE Trans. Vis. Comput. Graph. 2(1), 85-96 (1996)
36. Schneider, R., Kobbelt, L.: Geometric fairing of irregular meshes for free-form surface design. Comput. Aided Geom. Des. 18(4), 359-379 (2001)
37. Terzopoulos, D., Platt, J., Barr, A., Fleischer, K.: Elastically deformable models. Comput. Graph. 21, 205-214 (1987)
38. Ugail, H.: Parametric design and optimisation of thin-walled structure for food packaging. J. Optimisation Eng. 4(4), 291-307 (2003a)
39. Ugail, H.: On the Spine of a PDE Surface, pp. 366-376. Springer, New York, NY (2003b)
40. Ugail, H.: Spine based shape parametrisation for PDE surfaces. Computing 72, 195-204 (2004)
41. Ugail, H., Bloor, M.I.G., Wilson, M.J.: Techniques for interactive design using the PDE method. ACM Trans. Graph. 18(2), 195-212 (1999)
42. Ugail, H., Wilson, M.J.: Efficient shape parametrisation for automatic design optimisation using a partial differential equation formulation. Comput. Struct. 81(29), 2601-2609 (2003)
43. Westgaard, G., Nowacki, H.: Construction of fair surfaces over irregular meshes. In: SMA '01: Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, pp. 88-98. ACM Press, New York, NY (2001)
44. Witkin, A., Fleischer, K., Barr, A.: Energy constraints on parameterized models. Comput. Graph. 21, 225-232 (1987)
45. Xu, G., Pan, Q., Bajaj, C.L.: Discrete surface modelling using partial differential equations. Comput. Aided Geom. Des. 23, 125-145 (2006)
46. You, L., Comninos, P., Zhang, J.J.: PDE blending surfaces with C2 continuity. Comput. Graph. 28(6), 895-906 (2004)
47. Zhang, J.J., You, L.: Fast surface modelling using a 6th order PDE. Comput. Graph. Forum 23(3), 311-320 (2004)