SIERE: A Hybrid Semi-Implicit Exponential Integrator for Efficiently Simulating Stiff Deformable Objects
Authors
Yu Ju (Edwin) Chen, Seung Heon Sheen, Uri M. Ascher, Dinesh K. Pai
Code (coming soon)
Abstract
Physics-based simulation methods for deformable objects suffer limitations due to the conflicting requirements that are placed on them. The work horse semi-implicit (SI) backward Euler method is very stable and inexpensive, but it is also a blunt instrument. It applies heavy damping, which depends on the timestep, to all solution modes and not just to high-frequency ones. As such, it makes simulations less lively, potentially missing important animation details. At the other end of the scale, exponential methods (exponential Rosenbrock Euler (ERE)) are known to deliver good approximations to all modes, but they get prohibitively expensive and less stable for very stiff material. In this article, we devise a hybrid, semi-implicit method called SIERE that allows the previous methods SI and ERE to each perform what they are good at. To do this, we employ at each timestep a partial spectral decomposition, which picks the lower, leading modes, applying ERE in the corresponding subspace. The rest is handled (i.e., effectively damped out) by SI. No solution of nonlinear algebraic equations is required throughout the algorithm. We show that the resulting method produces simulations that are visually as good as those of the exponential method at a computational price that does not increase with stiffness, while displaying stability and damping with respect to the high-frequency modes. Furthermore, the phenomenon of occasional divergence of SI is avoided.
Citation
@article{10.1145/3410527,
author = {Chen, Yu Ju (Edwin) and Sheen, Seung Heon and Ascher, Uri M. and Pai, Dinesh K.},
title = {SIERE: A Hybrid Semi-Implicit Exponential Integrator for Efficiently Simulating Stiff Deformable Objects},
year = {2020},
issue_date = {January 2021},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {40},
number = {1},
issn = {0730-0301},
url = {https://doi.org/10.1145/3410527},
doi = {10.1145/3410527},
abstract = {Physics-based simulation methods for deformable objects suffer limitations due to
the conflicting requirements that are placed on them. The work horse semi-implicit
(SI) backward Euler method is very stable and inexpensive, but it is also a blunt
instrument. It applies heavy damping, which depends on the timestep, to all solution
modes and not just to high-frequency ones. As such, it makes simulations less lively,
potentially missing important animation details. At the other end of the scale, exponential
methods (exponential Rosenbrock Euler (ERE)) are known to deliver good approximations
to all modes, but they get prohibitively expensive and less stable for very stiff
material. In this article, we devise a hybrid, semi-implicit method called SIERE that
allows the previous methods SI and ERE to each perform what they are good at. To do
this, we employ at each timestep a partial spectral decomposition, which picks the
lower, leading modes, applying ERE in the corresponding subspace. The rest is handled
(i.e., effectively damped out) by SI. No solution of nonlinear algebraic equations
is required throughout the algorithm. We show that the resulting method produces simulations
that are visually as good as those of the exponential method at a computational price
that does not increase with stiffness, while displaying stability and damping with
respect to the high-frequency modes. Furthermore, the phenomenon of occasional divergence
of SI is avoided.},
journal = {ACM Trans. Graph.},
month = aug,
articleno = {3},
numpages = {12},
keywords = {time integration, deformable object, nonlinear constitutive material, stiffness, Physically-based simulation}
}