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doi: 10.3934/jcd.2021021
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## Simulating deformable objects for computer animation: A numerical perspective

 Dept. Computer Science, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada

* Corresponding author

Received  March 2021 Revised  October 2021 Early access December 2021

Fund Project: The first and last authors are supported by NSERC Discovery grants 84306 and RGPIN/2017-04604 respectively. Pai's research was also supported by a Canada Research Chair and an NSERC Idea-to-Innovation grant co-sponsored by Vital Mechanics

We examine a variety of numerical methods that arise when considering dynamical systems in the context of physics-based simulations of deformable objects. Such problems arise in various applications, including animation, robotics, control and fabrication. The goals and merits of suitable numerical algorithms for these applications are different from those of typical numerical analysis research in dynamical systems. Here the mathematical model is not fixed a priori but must be adjusted as necessary to capture the desired behaviour, with an emphasis on effectively producing lively animations of objects with complex geometries. Results are often judged by how realistic they appear to observers (by the "eye-norm") as well as by the efficacy of the numerical procedures employed. And yet, we show that with an adjusted view numerical analysis and applied mathematics can contribute significantly to the development of appropriate methods and their analysis in a variety of areas including finite element methods, stiff and highly oscillatory ODEs, model reduction, and constrained optimization.

Citation: Uri M. Ascher, Egor Larionov, Seung Heon Sheen, Dinesh K. Pai. Simulating deformable objects for computer animation: A numerical perspective. Journal of Computational Dynamics, doi: 10.3934/jcd.2021021
##### References:
 [1] A. H. Al-Mohy and N. J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J. Sci. Comput., 33 (2011,488–511. doi: 10.1137/100788860.  Google Scholar [2] R. Alexander, Diagonally implicit runge-kutta methods for stiff ode's, SIAM J. Numer. Anal., 14 (1977), 1006-1021.  doi: 10.1137/0714068.  Google Scholar [3] U. Ascher, Numerical Methods for Evolutionary Differential Equations, Computational Science & Engineering, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718911.  Google Scholar [4] U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998.  Google Scholar [5] J. Awrejcewicz, D. Grzelczyk and Y. Pyryev, A novel dry friction modeling and its impact on differential equations computation and lyapunov exponents estimation, Journal of Vibroengineering, 10 (2008). Google Scholar [6] D. Baraff and A. Witkin, Large steps in cloth simulation, Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, (1998), 43–54. doi: 10.1145/280814.280821.  Google Scholar [7] J. Barbic and D. James, Real-time subspace integration for st. venant-kirchhoff deformable models, ACM Trans. Graphics, 24 (2005), 982-990.  doi: 10.1145/1186822.1073300.  Google Scholar [8] E. Boxerman and U. Ascher, Decomposing cloth, Proceedings of the 2004 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, Eurographics Association, (2004), 153–161. doi: 10.1145/1028523.1028543.  Google Scholar [9] J. C. Butcher and D. J. L. Chen, A new type of singly-implicit runge-kutta method, Appl. Numer. Math., 34 (2000), 179-188.  doi: 10.1016/S0168-9274(99)00126-9.  Google Scholar [10] D. Chen, D. I. W. Levin, W. Matusik and D. M. Kaufman, Dynamics-aware numerical coarsening for fabrication design, ACM Trans. Graph., 36 (2017), 1-15.  doi: 10.1145/3072959.3073669.  Google Scholar [11] Y. J. Chen, U. Ascher and D. K. Pai, Exponential rosenbrock-euler integrators for elastodynamic simulation, IEEE Transactions on Visualization and Computer Graphics, 24 (2018), 2702-2713.  doi: 10.1109/TVCG.2017.2768532.  Google Scholar [12] Y. J. (Edwin) Chen, D. I. W. Levin, D. M. Kaufman, U. M. Ascher and D. K. Pai, Eigenfit for consistent elastodynamic simulation across mesh resolution, Proceedings SCA, (2019), Article No. 5, 1–13. doi: 10.1145/3309486.3340248.  Google Scholar [13] Y. J. (Edwin) Chen, S. H. Sheen, U. M. Ascher and D. K. Pai, Siere: A hybrid semi-implicit exponential integrator for efficiently simulating stiff deformable objects, ACM Transactions on Graphics (TOG), 40 (2020), 1–12. doi: 10.1145/3410527.  Google Scholar [14] J. Chung and G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-$\alpha$ method, J. Applied Mech., 60 (1993), 371-375.  doi: 10.1115/1.2900803.  Google Scholar [15] P. G. Ciarlet, Three-Dimensional Elasticity, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 1. Masson, Paris, 1986.  Google Scholar [16] G. Daviet, F. Bertails-Descoubes and L. Boissieux, A hybrid iterative solver for robustly capturing coulomb friction in hair dynamics, ACM Trans. Graph., 30 (2011), 1-12.  doi: 10.1145/2024156.2024173.  Google Scholar [17] G. De Saxcé and Z. Q. Feng, The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms, Math. Comput. Modelling, 28 (1998), 225-245.  doi: 10.1016/S0895-7177(98)00119-8.  Google Scholar [18] K. 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Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7.  Google Scholar [23] E. Hairer and G. Wanner, Solving Ordinary Differential Equations Ⅱ: Stiff and Differential-Algebraic Problems, 2$^nd$ edition, Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-05221-7.  Google Scholar [24] C. Kane, J. Marsden, M. Ortiz and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Internat. J. Numer. Methods Engrg., 49 (2000), 1295-1325.  doi: 10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W.  Google Scholar [25] D. M. Kaufman, S. Sueda, D. L. James and D. K. Pai, Staggered projections for frictional contact in multibody systems, ACM Transactions on Graphics (SIGGRAPH Asia 2008), 27 (2008), 1-11.  doi: 10.1145/1457515.1409117.  Google Scholar [26] R. Kikuuwe, N. Takesue, A. Sano, H. Mochiyama and H. Fujimoto, Fixed-step friction simulation: From classical Coulomb model to modern continuous models, 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, (2005), 1009–1016. doi: 10.1109/IROS.2005.1545579.  Google Scholar [27] E. Larionov, Y. Fan and D. K Pai, Frictional Contact on Smooth Elastic Solids, ACM Transactions on Graphics, (2021), 1–17. doi: 10.1145/3446663.  Google Scholar [28] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007. doi: 10.1137/1.9780898717839.  Google Scholar [29] M. Li, Z. Ferguson, T. Schneider, T. Langlois, D. Zorin, D. Panozzo, C. Jiang and D. M. Kaufman, Incremental potential contact: Intersection- and inversion-free, large-deformation dynamics, ACM Transactions on Graphics (TOG), 39 (2020), 1-20.  doi: 10.1145/3386569.3392425.  Google Scholar [30] M. Li, D. M. Kaufman and C. Jiang, Codimensional Incremental Potential Contact, arXiv: 2012.04457, [cs], 2021. Google Scholar [31] A. Longva, F. Löschner, T. Kugelstadt, J. A. Fernández-Fernández and J. Bender, Higher-order finite elements for embedded simulation, ACM Transactions on Graphics, 39 (2020), Article No. 181, 1–14. doi: 10.1145/3414685.3417853.  Google Scholar [32] F. Loschner, A. Longva, S. Jeske, T. Kugelstadt and J. Bender, Higher order time integration for deformable solids, Computer Graphics Forum, 39 (2020), 157-169.  doi: 10.1111/cgf.14110.  Google Scholar [33] P. Lotstedt, Mechanical systems of rigid bodies subject to unilateral constraints, SIAM J. Appl. Math., 42 (1982), 281-296.  doi: 10.1137/0142022.  Google Scholar [34] P. Lotstedt and L. Petzold, Numerical solution of nonlinear differential equations with algebraic constraints i: Convergence results for backward differentiation formulas, Math. Comp., 46 (1986), 491-516.  doi: 10.2307/2007989.  Google Scholar [35] D. L. Michels, V. T. Luan and M. Tokman, A stiffly accurate integrator for elastodynamic problems, ACM Transactions on Graphics (TOG), 36 (2017), Article No.: 116, 1–14. doi: 10.1145/3072959.3073706.  Google Scholar [36] D. L. Michels and J. P. T. Mueller, Discrete computational mechanics for stiff phenomena, SIGGRAPH ASIA 2016 Courses, (2016), Article No.: 13, 1–13. doi: 10.1145/2988458.2988464.  Google Scholar [37] C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), 3-49.  doi: 10.1137/S00361445024180.  Google Scholar [38] J. Niesen and W. M. Wright, Algorithm 919: A Krylov subspace algorithm for evaluating the $\phi$-functions appearing in exponential integrators, ACM Trans. Math. Software (TOMS), 38 (2012), Art. 22, 19pp. doi: 10.1145/2168773.2168781.  Google Scholar [39] J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research. Springer-Verlag, New York, 1999. doi: 10.1007/b98874.  Google Scholar [40] D. K. Pai, K. van den Doel, D. L. James, J. Lang, J. E. Lloyd, J. L. Richmond and S. H. Yau, Scanning physical interaction behavior of 3D objects, Computer Graphics (ACM SIGGRAPH 2001 Conference Proceedings), (2001), 87–96. Google Scholar [41] D. K. Pai, A. Rothwell, P. Wyder-Hodge, A. Wick, Y. Fan, E. Larionov, D. Harrison, D. R. Neog and C. Shing, The human touch: Measuring contact with real human soft tissues, ACM Transactions on Graphics (TOG), 37 (2018), Article No.: 58, 1–12. doi: 10.1145/3197517.3201296.  Google Scholar [42] E. Sifakis and J. Barbic, FEM simulation of 3D deformable solids: A practitioner's guide to theory, discretization and model reduction, ACM SIGGRAPH 2012 Courses, (2012), Article No.: 20, 1–50 doi: 10.1145/2343483.2343501.  Google Scholar [43] B. Smith, F. de Goes and T. Kim, Stable neo-hookean flesh simulation, ACM Trans. Graph., 37 (2018), Article No.: 12, 1–15. doi: 10.1145/3180491.  Google Scholar [44] O. Sorkine and M. Alexa, As-rigid-as-possible surface modeling, Eurographics Symposium on Geometry Processing, 4 (2007), 109-116.   Google Scholar [45] M. Verschoor and A. C. Jalba, Efficient and accurate collision response for elastically deformable models, ACM Trans. Graph., 38 (2019), Article No.: 17, 1–20. doi: 10.1145/3209887.  Google Scholar [46] B. Wang, L. Wu, K. Yin, U. Ascher, L. Liu and H. Huang, Deformation capture and modelling of soft objects, ACM trans. on Graphics (SIGGRAPH), 34 (2015). Google Scholar [47] J. Wojewoda, A. Stefański, M. Wiercigroch and T. Kapitaniak, Hysteretic effects of dry friction: Modelling and experimental studies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 336 (2008), 747-765.  doi: 10.1098/rsta.2007.2125.  Google Scholar [48] H. Xu and J. Barbic, Example-based damping design, ACM Trans. Graphics, 36 (2017), Article No.: 53, 1–14. doi: 10.1145/3072959.3073631.  Google Scholar

show all references

##### References:
 [1] A. H. Al-Mohy and N. J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators, SIAM J. Sci. Comput., 33 (2011,488–511. doi: 10.1137/100788860.  Google Scholar [2] R. Alexander, Diagonally implicit runge-kutta methods for stiff ode's, SIAM J. Numer. Anal., 14 (1977), 1006-1021.  doi: 10.1137/0714068.  Google Scholar [3] U. Ascher, Numerical Methods for Evolutionary Differential Equations, Computational Science & Engineering, 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. doi: 10.1137/1.9780898718911.  Google Scholar [4] U. Ascher and L. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998.  Google Scholar [5] J. Awrejcewicz, D. Grzelczyk and Y. Pyryev, A novel dry friction modeling and its impact on differential equations computation and lyapunov exponents estimation, Journal of Vibroengineering, 10 (2008). Google Scholar [6] D. Baraff and A. Witkin, Large steps in cloth simulation, Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, (1998), 43–54. doi: 10.1145/280814.280821.  Google Scholar [7] J. Barbic and D. James, Real-time subspace integration for st. venant-kirchhoff deformable models, ACM Trans. Graphics, 24 (2005), 982-990.  doi: 10.1145/1186822.1073300.  Google Scholar [8] E. Boxerman and U. Ascher, Decomposing cloth, Proceedings of the 2004 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, Eurographics Association, (2004), 153–161. doi: 10.1145/1028523.1028543.  Google Scholar [9] J. C. Butcher and D. J. L. Chen, A new type of singly-implicit runge-kutta method, Appl. Numer. Math., 34 (2000), 179-188.  doi: 10.1016/S0168-9274(99)00126-9.  Google Scholar [10] D. Chen, D. I. W. Levin, W. Matusik and D. M. Kaufman, Dynamics-aware numerical coarsening for fabrication design, ACM Trans. Graph., 36 (2017), 1-15.  doi: 10.1145/3072959.3073669.  Google Scholar [11] Y. J. Chen, U. Ascher and D. K. Pai, Exponential rosenbrock-euler integrators for elastodynamic simulation, IEEE Transactions on Visualization and Computer Graphics, 24 (2018), 2702-2713.  doi: 10.1109/TVCG.2017.2768532.  Google Scholar [12] Y. J. (Edwin) Chen, D. I. W. Levin, D. M. Kaufman, U. M. Ascher and D. K. Pai, Eigenfit for consistent elastodynamic simulation across mesh resolution, Proceedings SCA, (2019), Article No. 5, 1–13. doi: 10.1145/3309486.3340248.  Google Scholar [13] Y. J. (Edwin) Chen, S. H. Sheen, U. M. Ascher and D. K. Pai, Siere: A hybrid semi-implicit exponential integrator for efficiently simulating stiff deformable objects, ACM Transactions on Graphics (TOG), 40 (2020), 1–12. doi: 10.1145/3410527.  Google Scholar [14] J. Chung and G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-$\alpha$ method, J. Applied Mech., 60 (1993), 371-375.  doi: 10.1115/1.2900803.  Google Scholar [15] P. G. Ciarlet, Three-Dimensional Elasticity, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 1. Masson, Paris, 1986.  Google Scholar [16] G. Daviet, F. Bertails-Descoubes and L. Boissieux, A hybrid iterative solver for robustly capturing coulomb friction in hair dynamics, ACM Trans. Graph., 30 (2011), 1-12.  doi: 10.1145/2024156.2024173.  Google Scholar [17] G. De Saxcé and Z. Q. Feng, The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms, Math. Comput. Modelling, 28 (1998), 225-245.  doi: 10.1016/S0895-7177(98)00119-8.  Google Scholar [18] K. Erleben, Rigid body contact problems using proximal operators, Proceedings of the ACM SIGGRAPH / Eurographics Symposium on Computer Animation, (2017), Article No. 13, 1–12. doi: 10.1145/3099564.3099575.  Google Scholar [19] Z. Ferguson, M. Li, T. Schneider, F. Gil-Ureta, T. Langlois, C. Jiang, D. Zorin, D. M. Kaufman and D. Panozzo, Intersection-free rigid body dynamics, ACM Transactions on Graphics (SIGGRAPH), 40 (2021). Google Scholar [20] T. F. Gast, C. Schroeder, A. Stomakhin, C. Jiang and J. M. Teran, Optimization integrator for large time steps, IEEE Trans Visualization and Computer Graphics, 21 (2015), 1103-1115.  doi: 10.1109/TVCG.2015.2459687.  Google Scholar [21] M. Geilinger, D. Hahn, J. Zehnder, M. Bacher, B. Thomaszewski and S. Coros, Add: Analytically differentiable dynamics for multi-body systems with frictional contact, ACM Transactions on Graphics (TOG), 39 (2020), 1-15.  doi: 10.1145/3414685.3417766.  Google Scholar [22] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31. Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-05018-7.  Google Scholar [23] E. Hairer and G. Wanner, Solving Ordinary Differential Equations Ⅱ: Stiff and Differential-Algebraic Problems, 2$^nd$ edition, Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-05221-7.  Google Scholar [24] C. Kane, J. Marsden, M. Ortiz and M. West, Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Internat. J. Numer. Methods Engrg., 49 (2000), 1295-1325.  doi: 10.1002/1097-0207(20001210)49:10<1295::AID-NME993>3.0.CO;2-W.  Google Scholar [25] D. M. Kaufman, S. Sueda, D. L. James and D. K. Pai, Staggered projections for frictional contact in multibody systems, ACM Transactions on Graphics (SIGGRAPH Asia 2008), 27 (2008), 1-11.  doi: 10.1145/1457515.1409117.  Google Scholar [26] R. Kikuuwe, N. Takesue, A. Sano, H. Mochiyama and H. Fujimoto, Fixed-step friction simulation: From classical Coulomb model to modern continuous models, 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, (2005), 1009–1016. doi: 10.1109/IROS.2005.1545579.  Google Scholar [27] E. Larionov, Y. Fan and D. K Pai, Frictional Contact on Smooth Elastic Solids, ACM Transactions on Graphics, (2021), 1–17. doi: 10.1145/3446663.  Google Scholar [28] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007. doi: 10.1137/1.9780898717839.  Google Scholar [29] M. Li, Z. Ferguson, T. Schneider, T. Langlois, D. Zorin, D. Panozzo, C. Jiang and D. M. Kaufman, Incremental potential contact: Intersection- and inversion-free, large-deformation dynamics, ACM Transactions on Graphics (TOG), 39 (2020), 1-20.  doi: 10.1145/3386569.3392425.  Google Scholar [30] M. Li, D. M. Kaufman and C. Jiang, Codimensional Incremental Potential Contact, arXiv: 2012.04457, [cs], 2021. Google Scholar [31] A. Longva, F. Löschner, T. Kugelstadt, J. A. Fernández-Fernández and J. Bender, Higher-order finite elements for embedded simulation, ACM Transactions on Graphics, 39 (2020), Article No. 181, 1–14. doi: 10.1145/3414685.3417853.  Google Scholar [32] F. Loschner, A. Longva, S. Jeske, T. Kugelstadt and J. Bender, Higher order time integration for deformable solids, Computer Graphics Forum, 39 (2020), 157-169.  doi: 10.1111/cgf.14110.  Google Scholar [33] P. Lotstedt, Mechanical systems of rigid bodies subject to unilateral constraints, SIAM J. Appl. Math., 42 (1982), 281-296.  doi: 10.1137/0142022.  Google Scholar [34] P. Lotstedt and L. Petzold, Numerical solution of nonlinear differential equations with algebraic constraints i: Convergence results for backward differentiation formulas, Math. Comp., 46 (1986), 491-516.  doi: 10.2307/2007989.  Google Scholar [35] D. L. Michels, V. T. Luan and M. Tokman, A stiffly accurate integrator for elastodynamic problems, ACM Transactions on Graphics (TOG), 36 (2017), Article No.: 116, 1–14. doi: 10.1145/3072959.3073706.  Google Scholar [36] D. L. Michels and J. P. T. Mueller, Discrete computational mechanics for stiff phenomena, SIGGRAPH ASIA 2016 Courses, (2016), Article No.: 13, 1–13. doi: 10.1145/2988458.2988464.  Google Scholar [37] C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), 3-49.  doi: 10.1137/S00361445024180.  Google Scholar [38] J. Niesen and W. M. Wright, Algorithm 919: A Krylov subspace algorithm for evaluating the $\phi$-functions appearing in exponential integrators, ACM Trans. Math. Software (TOMS), 38 (2012), Art. 22, 19pp. doi: 10.1145/2168773.2168781.  Google Scholar [39] J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research. Springer-Verlag, New York, 1999. doi: 10.1007/b98874.  Google Scholar [40] D. K. Pai, K. van den Doel, D. L. James, J. Lang, J. E. Lloyd, J. L. Richmond and S. H. Yau, Scanning physical interaction behavior of 3D objects, Computer Graphics (ACM SIGGRAPH 2001 Conference Proceedings), (2001), 87–96. Google Scholar [41] D. K. Pai, A. Rothwell, P. Wyder-Hodge, A. Wick, Y. Fan, E. Larionov, D. Harrison, D. R. Neog and C. Shing, The human touch: Measuring contact with real human soft tissues, ACM Transactions on Graphics (TOG), 37 (2018), Article No.: 58, 1–12. doi: 10.1145/3197517.3201296.  Google Scholar [42] E. Sifakis and J. Barbic, FEM simulation of 3D deformable solids: A practitioner's guide to theory, discretization and model reduction, ACM SIGGRAPH 2012 Courses, (2012), Article No.: 20, 1–50 doi: 10.1145/2343483.2343501.  Google Scholar [43] B. Smith, F. de Goes and T. Kim, Stable neo-hookean flesh simulation, ACM Trans. Graph., 37 (2018), Article No.: 12, 1–15. doi: 10.1145/3180491.  Google Scholar [44] O. Sorkine and M. Alexa, As-rigid-as-possible surface modeling, Eurographics Symposium on Geometry Processing, 4 (2007), 109-116.   Google Scholar [45] M. Verschoor and A. C. Jalba, Efficient and accurate collision response for elastically deformable models, ACM Trans. Graph., 38 (2019), Article No.: 17, 1–20. doi: 10.1145/3209887.  Google Scholar [46] B. Wang, L. Wu, K. Yin, U. Ascher, L. Liu and H. Huang, Deformation capture and modelling of soft objects, ACM trans. on Graphics (SIGGRAPH), 34 (2015). Google Scholar [47] J. Wojewoda, A. Stefański, M. Wiercigroch and T. Kapitaniak, Hysteretic effects of dry friction: Modelling and experimental studies, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 336 (2008), 747-765.  doi: 10.1098/rsta.2007.2125.  Google Scholar [48] H. Xu and J. Barbic, Example-based damping design, ACM Trans. Graphics, 36 (2017), Article No.: 53, 1–14. doi: 10.1145/3072959.3073631.  Google Scholar
]">Figure 1.  Deformable articulated objects: a swaying tree and a constrained jelly brick; cf. [13]
Moving tetrahedral FEM mesh for position coordinates ${\bf q}(t)$
Damping curves for the SDIRK method (solid line), TR-BDF2 (dashed) and BDF2 (dash-dot). The two DIRK methods behave similarly, and they differ significantly from BDF2
]. The cost of exponential integrators including ERE becomes prohibitive as the stiffness parameter increases. By contrast, the cost of SIERE does not grow significantly with stiffness">Figure 4.  Computational costs for a swinging armadillo simulation [13]. The cost of exponential integrators including ERE becomes prohibitive as the stiffness parameter increases. By contrast, the cost of SIERE does not grow significantly with stiffness
Plot of of the first 1000 eigenvalues of a soft body problem
Potential energy plots for different integrators applied to a soft object: BE (thick solid line), SIERE with $s = 10$ (thin solid line), STR-SBDF2ERE with $s = 10$ (dash-dot), TR-BDF2 (dotted), and SDIRK (dashed). A soft beam is fixed at its ends and is subjected to gravity. Notice that the TR-BDF2 and SDIRK energies do not decay by much, whereas BE dissipates energy quickly. SIERE is less damping than BE but still much more damping than STR-SBDF2ERE, which in turn is still more damping than the two DIRK methods
With large enough time steps, velocity based contact constraints may reject plausible steps. If a vertex in blue is constrained to have a strictly positive velocity with respect to the convex gray contact surface, then plausibly valid next-step configurations (right) may be erroneously rejected
Barrier function $b = b(x; \delta)$ for different values of $\delta$. It is used in (30) to approximate the contact force
Plot of the smoothing function $s = s(x;\epsilon)$ for different values of $\epsilon$. It is used in (34) through (32) to approximate the Coulomb friction force
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