“Regularized kelvinlets: sculpting brushes based on fundamental solutions of elasticity” by Goes and James

  • ©Fernando de Goes and Doug L. James




    Regularized kelvinlets: sculpting brushes based on fundamental solutions of elasticity


Session Title: Mappings and Deformations



    We introduce a new technique for real-time physically based volume sculpting of virtual elastic materials. Our formulation is based on the elastic response to localized force distributions associated with common modeling primitives such as grab, scale, twist, and pinch. The resulting brush-like displacements correspond to the regularization of fundamental solutions of linear elasticity in infinite 2D and 3D media. These deformations thus provide the realism and plausibility of volumetric elasticity, and the interactivity of closed-form analytical solutions. To finely control our elastic deformations, we also construct compound brushes with arbitrarily fast spatial decay. Furthermore, pointwise constraints can be imposed on the displacement field and its derivatives via a single linear solve. We demonstrate the versatility and efficiency of our method with multiple examples of volume sculpting and image editing.


    1. J. Ainley, S. Durkin, R. Embid, P. Boindala, and R. Cortez. 2008. The method of images for regularized Stokeslets. J. of Computational Physics 227, 9 (2008), 4600–4616. Google ScholarDigital Library
    2. A. Angelidis, M. P. Cani, G. Wyvill, and S. King. 2006. Swirling-Sweepers: Constant volume modeling. Graph. Models 68, 4 (2006), 324âĂŞ–332.Google Scholar
    3. A. Angelidis, G. Wyvill, and M. P. Cani. 2004. Sweepers: Swept user-defined tools for modeling by deformation. In Proc. of Shape Modeling Applications. IEEE, 63–73.Google Scholar
    4. Autodesk. 2016. Maya User Guide. (2016). https://autodesk.com/maya.Google Scholar
    5. J. R. Barber. 2010. Elasticity. Solid Mechanics and Its Applications, Vol. 172. Springer.Google Scholar
    6. J. Barbič and D. L. James. 2005. Real-Time Subspace Integration for St. Venant-Kirchhoff Deformable Models. ACM Trans. Graph. 24, 3 (2005), 982–990. Google ScholarDigital Library
    7. J. Barbič and Y. Zhao. 2011. Real-time Large-deformation Substructuring. ACM Trans. Graph. 30, 4, Article 91 (2011). Google ScholarDigital Library
    8. J. C. Bazin, C. Plüss (Kuster), G. Yu, T. Martin, A. Jacobson, and M. Gross. 2016. Physically Based Video Editing. Computer Graphics Forum (2016).Google Scholar
    9. M. Ben-Chen, O. Weber, and C. Gotsman. 2009. Variational Harmonic Maps for Space Deformation. ACM Trans. Graph. 28, 3 (2009). Google ScholarDigital Library
    10. C. A. Brebbia, J. C. F. Telles, and L. Wrobel. 2012. Boundary element techniques: theory and applications in engineering. Springer Science & Business Media.Google Scholar
    11. C. Brezinski and M. R. Zaglia. 2013. Extrapolation methods: Theory and practice. Studies in Computational Mathematics (Book 2), Vol. 2. Elsevier.Google Scholar
    12. M. P. Cani and A. Angelidis. 2006. Towards Virtual Clay. In ACM SIGGRAPH Courses. 67–83. Google ScholarDigital Library
    13. A. T. Chwang and T. Y.-T. Wu. 1975. Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows. J. of Fluid Mechanics 67, 04 (1975), 787–815. Google ScholarCross Ref
    14. S. Coquillart. 1990. Extended Free-form Deformation: A Sculpturing Tool for 3D Geometric Modeling. In ACM SIGGRAPH. 187–196.Google Scholar
    15. F. Cordier, Y. Gingold, E. Entem, M. P. Cani, and K. Singh. 2016. Sketch-based Modeling. In Eurographics Tutorials. Google ScholarDigital Library
    16. R. Cortez. 2001. The Method of Regularized Stokeslets. SIAM J. on Scientific Computing 23, 4 (2001), 1204–1225. Google ScholarDigital Library
    17. R. Cortez, L. Fauci, and A. Medovikov. 2005. The method of regularized Stokeslets in three dimensions: Analysis, validation, and application to helical swimming. Physics of Fluids 17 (2005). Google ScholarCross Ref
    18. R. W. Cottle. 1974. Manifestations of the Schur complement. Linear Algebra and its Applications 8, 3 (1974), 189–211. Google ScholarCross Ref
    19. G. Dewaele and M. P. Cani. 2004. Interactive Global and Local Deformations for Virtual Clay. Graph. Models 66, 6 (2004). Google ScholarDigital Library
    20. E. Ferley, M. P. Cani, and J.-D. Gascuel. 2001. Resolution Adaptive Volume Sculpting. Graph. Models 63, 6 (2001). Google ScholarDigital Library
    21. S. F. Frisken, R. N. Perry, A. P. Rockwood, and T. R. Jones. 2000. Adaptively sampled distance fields: a general representation of shape for computer graphics. In ACM SIGGRAPH. 249–254. Google ScholarDigital Library
    22. T. A. Galyean and J. F. Hughes. 1991. Sculpting: An Interactive Volumetric Modeling Technique. In ACM SIGGRAPH. 267–274.Google ScholarDigital Library
    23. F. Hahn, S. Martin, B. Thomaszewski, R. Sumner, S. Coros, and M. Gross. 2012. Rig-space Physics. ACM Trans. Graph. 31, 4 (2012). Google ScholarDigital Library
    24. G. J. Hancock. 1953. The Self-Propulsion of Microscopic Organisms through Liquids. Proc. of the Royal Soc. of London A: Mathematical, Physical and Engineering Sciences 217, 1128 (1953), 96–121. Google ScholarCross Ref
    25. A. Jacobson, I. Baran, L. Kavan, J. Popović, and O. Sorkine. 2012. Fast Automatic Skinning Transformations. ACM Trans. Graph. 31, 4 (2012). Google ScholarDigital Library
    26. A. Jacobson, I. Baran, J. Popović, and O. Sorkine. 2011. Bounded Biharmonic Weights for Real-time Deformation. ACM Trans. Graph. 30, 4, Article 78 (2011). Google ScholarDigital Library
    27. D. L. James and D. K. Pai. 1999. ArtDefo: Accurate Real Time Deformable Objects. In ACM SIGGRAPH. 65–72.Google Scholar
    28. D. L. James and D. K. Pai. 2003. Multiresolution Green’s Function Methods for Interactive Simulation of Large-scale Elastostatic Objects. ACM Trans. Graph. 22, 1 (2003), 47–82. Google ScholarDigital Library
    29. P. Joshi, M. Meyer, T. DeRose, B. Green, and T. Sanocki. 2007. Harmonic Coordinates for Character Articulation. ACM Trans. Graph. 26, 3 (2007). Google ScholarDigital Library
    30. L. Kavan and O. Sorkine. 2012. Elasticity-Inspired Deformers for Character Articulation. ACM Trans. on Graph. 31, 6 (2012). Google ScholarDigital Library
    31. Lord Kelvin. 1848. Note on the integration of the equations of equilibrium of an elastic solid. Cambridge and Dublin Mathematical Journal 3 (1848), 87–89.Google Scholar
    32. Y. Lipman, D. Cohen-Or, R. Gal, and D. Levin. 2007. Volume and Shape Preservation via Moving Frame Manipulation. ACM Trans. Graph. 26, 1, Article 5 (2007). Google ScholarDigital Library
    33. Y. Lipman, D. Levin, and D. Cohen-Or. 2008. Green Coordinates. ACM Trans. Graph. 27, 3, Article 78 (2008). Google ScholarDigital Library
    34. N. Magnenat-Thalmann, R. Laperrière, and D. Thalmann. 1988. Joint-dependent Local Deformations for Hand Animation and Object Grasping. In Proc. on Graphics Interface. 26–33.Google Scholar
    35. A. McAdams, Y. Zhu, A. Selle, M. Empey, R. Tamstorf, J. Teran, and E. Sifakis. 2011. Efficient Elasticity for Character Skinning with Contact and Collisions. ACM Trans. Graph. 30, 4, Article 37 (2011). Google ScholarDigital Library
    36. G. Nielson, H. Hagen, and H. Müller. 1997. Scientific Visualization. IEEE Computer Society.Google Scholar
    37. M. Pauly, D. K. Pai, and L. J. Guibas. 2004. Quasi-rigid Objects in Contact. In Proc. of the 2004 ACM SIGGRAPH/Eurographics Symp. on Computer Animation. Eurographics Association, 109–119. Google ScholarDigital Library
    38. N. Phan-Thien and S. Kim. 1994. Microstructures in elastic media : principles and computational methods. Oxford U. Press.Google Scholar
    39. P. Podio-Guidugli and A. Favata. 2014. Elasticity for Geotechnicians. Springer. Google ScholarCross Ref
    40. J. Reinders. 2007. Intel Threading Building Blocks. O’Reilly & Associates, Inc.Google Scholar
    41. T. W. Sederberg and S. R. Parry. 1986. Free-form Deformation of Solid Geometric Models. In ACM SIGGRAPH. 151–160. Google ScholarDigital Library
    42. R. Setaluri, Y. Wang, N. Mitchell, L. Kavan, and E. Sifakis. 2014. Fast grid-based nonlinear elasticity for 2D deformations. In Proc. of the ACM SIGGRAPH/Eurographics Symp. on Computer Animation. 67–76.Google Scholar
    43. K. Singh and E. Fiume. 1998. Wires: A Geometric Deformation Technique. In ACM SIGGRAPH. 405–414. Google ScholarDigital Library
    44. W. S. Slaughter. 2002. The Linearized Theory of Elasticity. Birkhäuser Basel. Google ScholarCross Ref
    45. O. Sorkine and M. Alexa. 2007. As-rigid-as-possible Surface Modeling. In Proc. of the Fifth Eurographics Symposium on Geometry Processing. 109–116.Google Scholar
    46. W. von Funck, H. Theisel, and H. P. Seidel. 2006. Vector field based shape deformations. ACM Trans. on Graph. 25, 3 (2006), 1118–1125. Google ScholarDigital Library
    47. Wolfram Research Inc. 2016. Mathematica Version 11. Wolfram Research Inc.Google Scholar

ACM Digital Library Publication: