Anisotropic elasticity for inversion-safety and element rehabilitation

Abstract

We present an analysis of anisotropic hyperelasticity, specifically transverse isotropy, that obtains closed-form expressions for the eigendecompositions of many common energies. We then use these to build fast and concise Newton implementations. We leverage our analysis in two separate applications. First, we show that existing anisotropic energies are not inversion-safe, and contain spurious stable states under large deformation. We then propose a new anisotropic strain invariant that enables the formulation of a novel, robust, and inversion-safe energy. The new energy fits completely within our analysis, so closed-form expressions are obtained for its eigensystem as well. Secondly, we use our analysis to rehabilitate badly-conditioned finite elements. Using this method, we can robustly simulate large deformations even when a mesh contains degenerate, zero-volume elements. We accomplish this by swapping the badly-behaved isotropic direction with a well-behaved anisotropic term. We validate our approach on a variety of examples.

References

1. V. Alastrué, E. Peña, M. Martínez, and M. Doblaré. 2008. Experimental study and constitutive modelling of the passive mechanical properties of the ovine infrarenal vena cava tissue. J. of Biomechanics 41, 14 (2008), 3038--3045.
2. M. Alexa, D. Cohen-Or, and D. Levin. 2000. As-rigid-as-possible Shape Interpolation. In Proceedings of SIGGRAPH. 157--164.
3. D. Baraff and A. Witkin. 1998. Large Steps in Cloth Simulation. In Proceedings of SIGGRAPH. 43--54.
4. 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.
5. A. Bargteil, C. Wojtan, J. Hodgins, and G. Turk. 2007. A Finite Element Method for Animating Large Viscoplastic Flow. ACM Trans. Graph. 26, 3 (2007).
6. T. Belytschko, W. Liu, B. Moran, and K. Elkhodary. 2013. Nonlinear finite elements for continua and structures. John Wiley & Sons.
7. S. Blemker, P. Pinsky, and S. Delp. 2005. A 3D model of muscle reveals the causes of nonuniform strains in the biceps brachii. J. of Biomechanics 38, 4 (2005), 657--665.
8. J. Bonet and R. D. Wood. 2008. Nonlinear continuum mechanics for finite element analysis. Cambridge University Press.
9. 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).
10. J. Cai. 2016. Simulating Deformable Models with Anisotropic Materials. Ph.D. Dissertation. Nanyang Technological University.
11. G. Chagnon, M. Rebouah, and D. Favier. 2015. Hyperelastic energy densities for soft biological tissues: a review. J. of Elasticity 120, 2 (2015), 129--160.
12. J. Chen, H. Bao, T. Wang, M. Desbrun, and J. Huang. 2018. Numerical Coarsening Using Discontinuous Shape Functions. ACM Trans. Graph. 37, 4, Article 120 (July 2018), 12 pages.
13. P. Ciarletta, I. Izzo, S. Micera, and F. Tendick. 2011. Stiffening by fiber reinforcement in soft materials: a hyperelastic theory at large strains and its application. Journal of the Mechanical Behavior of Biomedical Materials 4, 7 (2011), 1359--1368.
14. C. Forest, H. Delingette, and N. Ayache. 2002. Removing tetrahedra from a manifold mesh. In Proceedings of Computer Animation. 225--229.
15. T. Gast, C. Schroeder, A. Stomakhin, C. Jiang, and J. Teran. 2015. Optimization Integrator for Large Time Steps. IEEE Transactions on Visualization and Computer Graphics 21, 10 (2015), 1103--1115.
16. G. H. Golub and C. F. Van Loan. 2012. Matrix computations. Vol. 3. JHU Press.
17. C. Gonzalez-Ochoa, D. Eberle, and R. Dressel. 2002. Dynamic Simulation of Wing Motion on "Reign of Fire". In ACM SIGGRAPH Sketches. 174--174.
18. G. Holzapfel. 2005. Similarities between soft biological tissues and rubberlike materials. In Constitutive Models for Rubber IV, Vol. 4. 607.
19. G. Holzapfel, T. Gasser, and R. Ogden. 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity and the Physical Science of Solids 61, 1--3 (2000), 1--48.
20. C. Horgan and G. Saccomandi. 2005. A new constitutive theory for fiber-reinforced incompressible nonlinearly elastic solids. Journal of the Mechanics and Physics of Solids 53, 9 (2005), 1985--2015.
21. Y. Hu, Q. Zhou, X. Gao, A. Jacobson, D. Zorin, and D. Panozzo. 2018. Tetrahedral Meshing in the Wild. ACM Trans. Graph. 37, 4, Article 60 (July 2018), 14 pages.
22. H. Iben. 2007. Generating Surface Crack Patterns. Ph.D. Dissertation. University of California, Berkeley.
23. G. Irving, R. Kautzman, G. Cameron, and J. Chong. 2008. Simulating the Devolved: Finite Elements on WALL-E. In ACM SIGGRAPH Talks. Article 54, 1 pages.
24. 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.
25. A. Jacobson, Z. Deng, L. Kavan, and J. P. Lewis. 2014. Skinning: Real-time Shape Deformation. In ACM SIGGRAPH Courses. Article 24, 24:1--24:1 pages.
26. P. M. Knupp. 2003. Algebraic mesh quality metrics for unstructured initial meshes. Finite Elements in Analysis and Design 39, 3 (2003), 217--241.
27. D. Koschier, J. Bender, and N. Thuerey. 2017. Robust extended Finite Elements for Complex Cutting of Deformables. ACM Trans. Graph. 36, 4, Article 55 (July 2017).
28. F. Labelle and J. Shewchuk. 2007. Isosurface Stuffing: Fast Tetrahedral Meshes with Good Dihedral Angles. ACM Trans. Graph. 26, 3, Article 57 (July 2007).
29. S. Lee and D. Terzopoulos. 2006. Heads Up!: Biomechanical Modeling and Neuromuscular Control of the Neck. ACM Trans. Graph. 25, 3 (July 2006), 1188--1198.
30. S. Lee, R. Yu, J. Park, M. Aanjaneya, E. Sifakis, and J. Lee. 2018. Dexterous Manipulation and Control with Volumetric Muscles. ACM Trans. Graph. 37, 4, Article 57 (July 2018), 13 pages.
31. Y. Lee, M. Park, T. Kwon, and J. Lee. 2014. Locomotion Control for Many-muscle Humanoids. ACM Trans. Graph. 33, 6, Article 218 (Nov. 2014), 11 pages.
32. Y. Li and J. Barbič. 2015. Stable anisotropic materials. IEEE Transactions on Visualization and Computer Graphics 21, 10 (2015), 1129--1137.
33. Y. Li, H. Xu, and J. Barbič. 2017. Enriching Triangle Mesh Animations with Physically Based Simulation. IEEE Transactions on Visualization and Computer Graphics 23, 10 (2017), 2301--2313.
34. T. Liu, S. Bouaziz, and L. Kavan. 2017. Quasi-Newton Methods for Real-Time Simulation of Hyperelastic Materials. ACM Trans. Graph. 36, 3 (2017).
35. P.-L. Manteaux, C. Wojtan, R. Narain, S. Redon, F. Faure, and M.-P. Cani. 2017. Adaptive Physically Based Models in Computer Graphics. Comput. Graph. Forum 36, 6 (Sept. 2017), 312--337.
36. 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).
37. A. Milne, M. McLaughlin, R. Tamstorf, A. Stomakhin, N. Burkard, M. Counsell, J. Canal, D. Komorowski, and E. Goldberg. 2016. Flesh, Flab, and Fascia Simulation on Zootopia. In ACM SIGGRAPH Talks. Article 34, 2 pages.
38. N. Molino, Z. Bao, and R. Fedkiw. 2004. A Virtual Node Algorithm for Changing Mesh Topology During Simulation. ACM Trans. Graph. 23, 3 (Aug. 2004), 385--392.
39. N. Molino, R. Bridson, J. Teran, and R. Fedkiw. 2003. A Crystalline, Red Green Strategy for Meshing Highly Deformable Objects with Tetrahedra.. In International Meshing Roundtable. 103--114.
40. D. Nolan, A. Gower, M. Destrade, R. Ogden, and J. McGarry. 2014. A robust anisotropic hyperelastic formulation for the modelling of soft tissue. Journal of the Mechanical Behavior of Biomedical Materials 39 (2014), 48--60.
41. G. Picinbono, J. Lombardo, H. Delingette, and N. Ayache. 2000. Anisotropic elasticity and force extrapolation to improve realism of surgery simulation. In IEEE International Conference on Robotics and Automation. IEEE, 596--602.
42. Y. Qiu. 2018. Spectra, a C++ Library For Large Scale Eigenvalue Problems. https://spectralib.org.
43. K. Rupp, P. Tillet, F. Rudolf, J. Weinbub, A. Morhammer, T. Grasser, A. Jüngel, and S. Selberherr. 2016. ViennaCL, Linear Algebra Library for Multi-and Many-Core Architectures. SIAM Journal on Scientific Computing 38, 5 (2016), 412--439.
44. P. Sachdeva, S. Sueda, S. Bradley, M. Fain, and D. K. Pai. 2015. Biomechanical Simulation and Control of Hands and Tendinous Systems. ACM Trans. Graph. 34, 4, Article 42 (July 2015).
45. S. Saito, Z. Zhou, and L. Kavan. 2015. Computational Body building: Anatomically-based Modeling of Human Bodies. ACM Trans. Graph. 34, 4, Article 41 (July 2015).
46. T. Schneider, Y. Hu, J. Dumas, X. Gao, D. Panozzo, and D. Zorin. 2018. Decoupling Simulation Accuracy from Mesh Quality. ACM Trans. Graph. (2018).
47. J. Shewchuk. 2002. What is a good linear finite element? interpolation, conditioning, anisotropy, and quality measures. University of California at Berkeley 73 (2002).
48. B. Smith, F. de Goes, and T. Kim. 2018. Stable Neo-Hookean Flesh Simulation. ACM Trans. Graph. 37, 2 (2018).
49. B. Smith, F. de Goes, and T. Kim. 2019. Analytic Eigensystems For Isotropic Distortion Energies. ACM Trans. Graph. (2019).
50. O. Sorkine and M. Alexa. 2007. As-rigid-as-possible surface modeling. In Eurog. Symposium on Geometry processing, Vol. 4.
51. A. Stomakhin, R. Howes, C. Schroeder, and J. M. Teran. 2012. Energetically Consistent Invertible Elasticity. In ACM SIGGRAPH/Eurog. Symp. Comp. Anim. 25--32.
52. J. Teran, S. Blemker, V. Hing, and R. Fedkiw. 2003. Finite volume methods for the simulation of skeletal muscle. In SIGGRAPH/Eurog. Symp. on Comp. Anim. 68--74.
53. 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.
54. Y. Tong, S. Lombeyda, A. N. Hirani, and M. Desbrun. 2003. Discrete Multiscale Vector Field Decomposition. ACM Trans. Graph. 22, 3 (July 2003), 445--452.
55. B. Wang, Y. Zhao, and J. Barbič. 2017. Botanical Materials Based on Biomechanics. ACM Trans. Graph. 36, 4, Article 135 (July 2017), 13 pages.
56. H. Wang and Y. Yang. 2016. Descent Methods for Elastic Body Simulation on the GPU. ACM Trans. Graph. 35, 6 (2016).
57. J. Weiss, B. Maker, and S. Govindjee. 1996. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Computer Methods in Applied Mechanics and Engineering 135, 1--2 (1996), 107--128.
58. C. Wojtan and G. Turk. 2008. Fast viscoelastic behavior with thin features. ACM Trans. Graph. 27, 3 (2008), 47.
59. A. Wong, D. Eberle, and T. Kim. 2018. Clean Cloth Inputs: Removing Character Self-intersections with Volume Simulation. In ACM SIGGRAPH Talks. Article 42.
60. H. Xu, F. Sin, Y. Zhu, and J. Barbič. 2015. Nonlinear Material Design Using Principal Stretches. ACM Trans. Graph. 34, 4 (2015).

Index Terms

• Computing methodologies

  • Computer graphics

    • Animation

      • Physical simulation