“Analytic Eigensystems for Isotropic Distortion Energies” by Smith, Goes and Kim

  • ©Breannan Smith, Fernando de Goes, and Theodore Kim

Conference:


Type:


Title:

    Analytic Eigensystems for Isotropic Distortion Energies

Session/Category Title:   Deformation and FEM


Presenter(s)/Author(s):



Abstract:


    Many strategies exist for optimizing non-linear distortion energies in geometry and physics applications, but devising an approach that achieves the convergence promised by Newton-type methods remains challenging. In order to guarantee the positive semi-definiteness required by these methods, a numerical eigendecomposition or approximate regularization is usually needed. In this article, we present analytic expressions for the eigensystems at each quadrature point of a wide range of isotropic distortion energies. These systems can then be used to project energy Hessians to positive semi-definiteness analytically. Unlike previous attempts, our formulation provides compact expressions that are valid both in 2D and 3D, and does not introduce spurious degeneracies. At its core, our approach utilizes the invariants of the stretch tensor that arises from the polar decomposition of the deformation gradient. We provide closed-form expressions for the eigensystems for all these invariants, and use them to systematically derive the eigensystems of any isotropic energy. Our results are suitable for geometry optimization over flat surfaces or volumes, and agnostic to both the choice of discretization and basis function. To demonstrate the efficiency of our approach, we include comparisons against existing methods on common graphics tasks such as surface parameterization and volume deformation.

References:


    1. M. Alexa, D. Cohen-Or, and D. Levin. 2000. As-rigid-as-possible shape interpolation. In Proceedings of SIGGRAPH. 157–164.
    2. S. S. An, T. Kim, and D. L. James. 2008. Optimizing cubature for efficient integration of subspace deformations. ACM Trans. Graph. 27, 5 (2008).
    3. Uri M. Ascher and Linda R. Petzold. 1998. Computer Methods for Ordinary Differential Equations and Differential-algebraic Equations. Vol. 61. SIAM.
    4. David Baraff and Andrew Witkin. 1998. Large steps in cloth simulation. In Proceedings of SIGGRAPH. 43–54.
    5. David Baraff, Andrew Witkin, and Michael Kass. 2003. Untangling cloth. ACM Trans. Graph. 22, 3 (2003).
    6. J. Barbič and Doug L. James. 2005. Real-time subspace integration for St. Venant-Kirchhoff deformable models. ACM Trans. Graph. 24, 3 (2005), 982–990.
    7. J. Barbič and Y. Zhao. 2011. Real-time large-deformation substructuring. ACM Trans. Graph. 30, 4 (2011).
    8. G. L. Bernstein, C. Shah, C. Lemire, Z. Devito, M. Fisher, P. Levis, and P. Hanrahan. 2016. Ebb: A DSL for physical simulation on CPUs and GPUs. ACM Trans. Graph. 35, 2 (2016).
    9. D. Bommes, H. Zimmer, and L. Kobbelt. 2009. Mixed-integer quadrangulation. ACM Trans. Graph. 28, 3 (2009).
    10. J. Bonet and R. D. Wood. 2008. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press.
    11. M. Botsch, L. Kobbelt, M. Pauly, P. Alliez, and B. Lévy. 2010. Polygon Mesh Processing. AK Peters.
    12. S. Bouaziz, M. Deuss, Y. Schwartzburg, T. Weise, and M. Pauly. 2012. Shape-up: Shaping discrete geometry with projections. Comput. Graph. Forum 31, 5 (2012), 1657–1667.
    13. S. Bouaziz, S. Martin, T. Liu, L. Kavan, and M. Pauly. 2014. Projective dynamics: Fusing constraint projections for fast simulation. ACM Trans. Graph. 33, 4 (2014).
    14. R. Bridson, R. Fedkiw, and J. Anderson. 2002. Robust treatment of collisions, contact and friction for cloth animation. In ACM Trans. Graph. 21, 3 (2002), 594–603.
    15. I. Chao, U. Pinkall, P. Sanan, and P. Schröder. 2010. A simple geometric model for elastic deformations. ACM Trans. Graph. 29, 4 (2010).
    16. R. Chen and O. Weber. 2017. GPU-accelerated locally injective shape deformation. ACM Trans. Graph. 36, 6 (2017).
    17. S. Claici, M. Bessmeltsev, S. Schaefer, and J. Solomon. 2017. Isometry-aware preconditioning for mesh parameterization. Comp. Graphics. Forum 36, 5 (2017), 37–47.
    18. M. Eigensatz and M. Pauly. 2009. Positional, metric, and curvature control for constraint-based surface deformation. Comput. Graph. Forum 28, 2 (2009), 551–558.
    19. X.-M. Fu and Y. Liu. 2016. Computing inversion-free mappings by simplex assembly. ACM Trans. Graph. 35, 6 (2016).
    20. X.-M. Fu, Y. Liu, and B. Guo. 2015. Computing locally injective mappings by advanced MIPS. ACM Trans. Graph. 34, 4 (2015).
    21. G. H. Golub and C. F. Van Loan. 2012. Matrix Computations. Vol. 3. JHU Press.
    22. G. Guennebaud, B. Jacob, et al. 2010. Eigen v3. Retrieved from http://eigen.tuxfamily.org.
    23. D. Harmon, E. Vouga, R. Tamstorf, and E. Grinspun. 2008. Robust treatment of simultaneous collisions. ACM Trans. Graph. 27, 3 (2008), 23:1–23:4.
    24. K. Horman and G. Greiner. 1999. MIPS: An efficient global parameterization method. In Curve and Surface Design. 153–162.
    25. Intel. 2018. Math Kernel Library. Retrieved from https://software.intel.com/en-us/mkl.
    26. G. Irving, J. Teran, and R. Fedkiw. 2004. Invertible finite elements for robust simulation of large deformation. In SIGGRAPH/Eurog. Symp. on Comp. Anim. 131–140.
    27. A. Jacobson, I. Baran, L. Kavan, J. Popović, and O. Sorkine. 2012. Fast automatic skinning transformations. ACM Trans. on Graphics 31, 4 (2012).
    28. F. Kjolstad, S. Kamil, J. Ragan-Kelley, D. I. W. Levin, S. Sueda, D. Chen, E. Vouga, D. M. Kaufman, G. Kanwar, W. Matusik, and S. Amarasinghe. 2016. Simit: A language for physical simulation. ACM Trans. Graph. 35, 2 (2016).
    29. T. G. Kolda and B. W. Bader. 2009. Tensor decompositions and applications. SIAM Rev. 51, 3 (2009), 455–500.
    30. S. Z. Kovalsky, M. Galun, and Y. Lipman. 2016. Accelerated quadratic proxy for geometric optimization. ACM Trans. Graph. 35, 4 (2016).
    31. L. Liu, L. Zhang, Y. Xu, C. Gotsman, and S. J. Gortler. 2008. A local/global approach to mesh parameterization. Computer Graphics Forum 27, 5 (2008), 1495–1504.
    32. T. Liu, S. Bouaziz, and L. Kavan. 2017. Quasi-Newton methods for real-time simulation of hyperelastic materials. ACM Trans. Graph. 36, 3 (2017).
    33. J. E Marsden and T. JR Hughes. 1994. Mathematical Foundations of Elasticity. Dover Publications.
    34. 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 (2011).
    35. R. Narain, M. Overby, and G. E. Brown. 2016. ADMM ⊇ projective dynamics: Fast simulation of general constitutive models. In Proc. of the ACM SIGGRAPH/Eurog. Symp. on Comp. Anim. 21–28.
    36. J. Nocedal and S. J. Wright. 2006. Numerical Optimization. Springer.
    37. T. Papadopoulo and M. I. A. Lourakis. 2000. Estimating the Jacobian of the Singular Value Decomposition: Theory and Applications. Springer, 554–570.
    38. F. Pighin and J. P. Lewis. 2007. Practical least-squares for computer graphics. In ACM SIGGRAPH Courses. 1–57.
    39. M. Rabinovich, R. Poranne, D. Panozzo, and O. Sorkine-Hornung. 2017. Scalable locally injective mappings. ACM Trans. Graph. 36, 2 (2017).
    40. A. Shtengel, R. Poranne, O. Sorkine-Hornung, S. Z. Kovalsky, and Y. Lipman. 2017. Geometric optimization via composite majorization. ACM Trans. Graph. 36, 4 (2017).
    41. E. Sifakis and J. Barbic. 2012. FEM simulation of 3D deformable solids: A practitioner’s guide to theory, discretization and model reduction. In ACM SIGGRAPH Courses.
    42. B. Smith, F. de Goes, and T. Kim. 2018. Stable Neo-Hookean flesh simulation. ACM Trans. Graph. 37, 2 (2018).
    43. J. Smith and S. Schaefer. 2015. Bijective parameterization with free boundaries. ACM Trans. Graph. 34, 4 (2015).
    44. O. Sorkine and M. Alexa. 2007. As-rigid-as-possible surface modeling. In Eurog. Symposium on Geometry Processing, Vol. 4.
    45. A. Stomakhin, R. Howes, C. Schroeder, and J. M. Teran. 2012. Energetically consistent invertible elasticity. In ACM SIGGRAPH/Eurog. Symp. Comp. Anim. 25–32.
    46. M. Tang, D. Manocha, and R. Tong. 2010. Fast continuous collision detection using deforming non-penetration filters. In Proceedings of I3D. ACM, 7–13.
    47. J. Teran, E. Sifakis, G. Irving, and R. Fedkiw. 2005. Robust quasistatic finite elements and flesh simulation. In ACM SIGGRAPH/Eurog. Symp. on Comp. Anim. 181–190.
    48. C. D. Twigg and Z. Kačić-Alesić. 2010. Point cloud glue: Constraining simulations using the procrustes transform. In ACM SIGGRAPH/Eurog. Symp. on Comp. Anim. 45–54.
    49. C. von Tycowicz, C. Schulz, H.-P. Seidel, and K. Hildebrandt. 2013. An efficient construction of reduced deformable objects. ACM Trans. Graph. 32, 6 (2013).
    50. H. Wang. 2015. A Chebyshev semi-iterative approach for accelerating projective and position-based dynamics. ACM Trans. Graph. 34, 6 (2015).
    51. Audrey Wong, David Eberle, and Theodore Kim. 2018. Clean cloth inputs: Removing character self-intersections with volume simulation. In ACM SIGGRAPH Talks. Article 42, 2 pages.
    52. H. Xu, F. Sin, Y. Zhu, and J. Barbič. 2015. Nonlinear material design using principal stretches. ACM Trans. Graph. 34, 4 (2015).

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