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Decomposed optimization time integrator for large-step elastodynamics

Authors:

Timothy Langlois,

Chenfanfu Jiang,

Danny M. KaufmanAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 38, Issue 4

Article No.: 70, Pages 1 - 10

https://doi.org/10.1145/3306346.3322951

Published: 12 July 2019 Publication History

Abstract

Simulation methods are rapidly advancing the accuracy, consistency and controllability of elastodynamic modeling and animation. Critical to these advances, we require efficient time step solvers that reliably solve all implicit time integration problems for elastica. While available time step solvers succeed admirably in some regimes, they become impractically slow, inaccurate, unstable, or even divergent in others --- as we show here. Towards addressing these needs we present the Decomposed Optimization Time Integrator (DOT), a new domain-decomposed optimization method for solving the per time step, nonlinear problems of implicit numerical time integration. DOT is especially suitable for large time step simulations of deformable bodies with nonlinear materials and high-speed dynamics. It is efficient, automated, and robust at large, fixed-size time steps, thus ensuring stable, continued progress of high-quality simulation output. Across a broad range of extreme and mild deformation dynamics, using frame-rate size time steps with widely varying object shapes and mesh resolutions, we show that DOT always converges to user-set tolerances, generally well-exceeding and always close to the best wall-clock times across all previous nonlinear time step solvers, irrespective of the deformation applied.

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References

[1]

A. Abdelfattah, A. Haidar, S. Tomov, and J. Dongarra. 2017. Fast Cholesky factorization on GPUs for batch and native modes in MAGMA. J of Comp Sci 20 (2017).

[2]

U. M Ascher. 2008. Numerical methods for evolutionary differential equations.

Digital Library

[3]

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).

Digital Library

[4]

S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein, et al. 2011. Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends® in Machine learning 3, 1 (2011).

Digital Library

[5]

J. Brown and P. Brune. 2013. Low-rank quasi-Newton updates for robust Jacobian lagging in Newton-type methods. In Int Conf Math Comp Meth App Nucl Sci Eng.

[6]

J. C. Butcher. 2016. Numerical methods for ordinary differential equations.

[7]

I. Chao, U. Pinkall, P. Sanan, and P. Schröder. 2010. A simple geometric model for elastic deformations. ACM Trans Graph (SIGGRAPH) 29, 4 (2010).

Digital Library

[8]

Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam. 2008. Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate. ACM Trans on Mathematical Software (TOMS) 35, 3 (2008).

Digital Library

[9]

P. Deufihard. 2011. Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms.

Digital Library

[10]

V. Dolean, P. Jolivet, and F. Nataf. 2015. An introduction to domain decomposition methods: algorithms, theory and parallel implementation.

Digital Library

[11]

T. Gast, C. Schroeder, A. Stomakhin, C. Jiang, and J. M Teran. 2015. Optimization integrator for large time steps. IEEE Trans Vis Comp Graph 21, 10 (2015).

Digital Library

[12]

E. Hairer, C. Lubich, and G. Wanner. 2006. Geometric Numerical Integration.

[13]

E. Hairer, S. P Nørsett, and G. Wanner. 2008. Solving Ordinary Differential Equations I.

Digital Library

[14]

E. Hairer and G. Wanner. 1996. Solving Ordinary Differential Equations II.

[15]

F. Hecht, Y.J. Lee, J. R. Shewchuk, and J. F. O'Brien. 2012. Updated Sparse Cholesky Factors for Corotational Elastodynamics. ACM Trans Graph 31, 5 (2012).

Digital Library

[16]

J. Huang, X. Liu, H. Bao, B. Guo, and H. Shum. 2006. An efficient large deformation method using domain decomposition. Comp & Graph 30, 6 (2006).

Digital Library

[17]

C. Kane, J. E Marsden, M. Ortiz, and M. West. 2000. Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems. Int J for Numer Meth in Eng 49, 10 (2000).

[18]

G. Karypis and V. Kumar. 1998. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J on Sci Comp 20 (1998).

Digital Library

[19]

L. Kharevych, W. Yang, Y. Tong, E. Kanso, J. E Marsden, P. Schröder, and M. Desbrun. 2006. Geometric, variational integrators for computer animation. In Symp Comp Anim.

Digital Library

[20]

T. Kim and D. L James. 2012. Physics-based character skinning using multidomain subspace deformations. IEEE Trans on visualization and Comp Graph 18, 8 (2012).

Digital Library

[21]

H. Liu, N. Mitchell, M. Aanjaneya, and E. Sifakis. 2016. A scalable schur-complement fluids solver for heterogeneous compute platforms. ACM Trans Graph 35, 6 (2016).

Digital Library

[22]

T. Liu, A. W. Bargteil, J. F. O'Brien, and L. Kavan. 2013. Fast Simulation of Mass-Spring Systems. ACM Trans Graph 32, 6 (2013). Proc of ACM SIGGRAPH Asia.

Digital Library

[23]

T. Liu, S. Bouaziz, and L. Kavan. 2017. Quasi-Newton Methods for Real-Time Simulation of Hyperelastic Materials. ACM Trans Graph 36, 4 (2017).

[24]

M. Macklin, M. Müller, and N. Chentanez. 2016. XPBD: position-based simulation of compliant constrained dynamics. In Proc of the 9th Int Conf on Motion in Games.

Digital Library

[25]

S. Martin, B. Thomaszewski, E. Grinspun, and M. Gross. 2011. Example-based elastic materials. ACM Trans Graph (SIGGRAPH) 30, 4 (2011).

Digital Library

[26]

A. McAdams, A. Selle, R. Tamstorf, J. Teran, and E. Sifakis. 2011. Computing the singular value decomposition of 3X 3 matrices with minimal branching and elementary floating point operations. University of Wisconsin Madison (2011).

[27]

M. Müller, B. Heidelberger, M. Hennix, and J. Ratcliff. 2007. Position based dynamics. J. Vis. Commun. Imag Represent. 18, 2 (2007).

Digital Library

[28]

R. Narain, M. Overby, and G. E Brown. 2016. ADMM &supe; projective dynamics: fast simulation of general constitutive models. In Symp on Comp Anim.

Digital Library

[29]

JW Neuberger. 1985. Steepest descent and differential equations. J of the Mathematical Society of Japan 37, 2 (1985).

[30]

J. Nocedal and S. Wright. 2006. Numerical Optimization.

[31]

M. Ortiz and L. Stainier. 1999. The variational formulation of viscoplastic constitutive updates. Comp Meth in App Mech and Eng 171, 3--4 (1999).

[32]

M. Overby, G. E Brown, J. Li, and R. Narain. 2017. ADMM &supe; Projective Dynamics: Fast Simulation of Hyperelastic Models with Dynamic Constraints. IEEE Trans Vis Comp Graph 23, 10 (2017).

Digital Library

[33]

N. Parikh and S. Boyd. 2012. Block splitting for distributed optimization.

[34]

A. Quarteroni, A. Valli, and P.M.A. Valli. 1999. Domain Decomposition Methods for Partial Differential Equations.

[35]

T. Schneider, Y. Hu, J. Dumas, X. Gao, D. Panozzo, and D. Zorin. 2018. Decoupling simulation accuracy from mesh quality. ACM Trans Graph (2018).

Digital Library

[36]

S. Sellán, H. Y. Cheng, Y. Ma, M. Dembowski, and A. Jacobson. 2018. Solid Geometry Processing on Deconstructed Domains. CoRR (2018).

[37]

VE Shamanskii. 1967. A modification of Newton's method. Ukrainian Mathematical J 19, 1 (1967).

[38]

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).

Digital Library

[39]

B. Smith, F. De Goes, and T. Kim. 2018. Stable Neo-Hookean Flesh Simulation. ACM Trans Graph 37, 2 (2018).

Digital Library

[40]

A. Stomakhin, R. Howes, C. Schroeder, and J. M Teran. 2012. Energetically consistent invertible elasticity. In Symp Comp Anim.

Digital Library

[41]

J. Teran, E. Sifakis, G. Irving, and R. Fedkiw. 2005. Robust Quasistatic Finite Elements and Flesh Simulation. In Symp Comp Anim.

Digital Library

[42]

C. Xiao-Chuan and D. Maksymilian. 1994. Domain Decomposition Methods for Monotone Nonlinear Elliptic Problems. In Contemporary Math.

[43]

Y. Zhu, R. Bridson, and D. M. Kaufman. 2018. Blended Cured Quasi-Newton for Distortion Optimization. ACM Trans. on Graph (SIGGRAPH) (2018).

Digital Library

Cited By

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Chen YHan YChen JZhang ZMcadams ATeran J(2024)Position-Based Nonlinear Gauss-Seidel for Quasistatic HyperelasticityACM Transactions on Graphics10.1145/365815443:4(1-15)Online publication date: 19-Jul-2024
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Index Terms

Decomposed optimization time integrator for large-step elastodynamics
1. Computing methodologies
  1. Computer graphics
    1. Animation
      1. Physical simulation

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cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 38, Issue 4

August 2019

1480 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/3306346

Editor:
Olga Sorkine-Hornung
ETH Zurich

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Copyright © 2019 ACM.

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Publication History

Published: 12 July 2019

Published in TOG Volume 38, Issue 4

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Cited By

Chen ALiu ZYang YYuksel C(2024)Vertex Block DescentACM Transactions on Graphics10.1145/365817943:4(1-16)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658179
Chen YHan YChen JZhang ZMcadams ATeran J(2024)Position-Based Nonlinear Gauss-Seidel for Quasistatic HyperelasticityACM Transactions on Graphics10.1145/365815443:4(1-15)Online publication date: 19-Jul-2024
https://dl.acm.org/doi/10.1145/3658154
Yu JWang Z(2024)Super-Resolution Cloth Animation with Spatial and Temporal CoherenceACM Transactions on Graphics10.1145/365814343:4(1-14)Online publication date: 19-Jul-2024
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