Center for Human Modeling and Simulation

Document Type

Conference Paper

Date of this Version


Publication Source

ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH Asia)





Start Page

Article No. 214




We describe a scheme for time integration of mass-spring systems that makes use of a solver based on block coordinate descent. This scheme provides a fast solution for classical linear (Hookean) springs. We express the widely used implicit Euler method as an energy minimization problem and introduce spring directions as auxiliary unknown variables. The system is globally linear in the node positions, and the non-linear terms involving the directions are strictly local. Because the global linear system does not depend on run-time state, the matrix can be pre-factored, allowing for very fast iterations. Our method converges to the same final result as would be obtained by solving the standard form of implicit Euler using Newton’s method. Although the asymptotic convergence of Newton’s method is faster than ours, the initial ratio of work to error reduction with our method is much faster than Newton’s. For real-time visual applications, where speed and stability are more important than precision, we obtain visually acceptable results at a total cost per timestep that is only a fraction of that required for a single Newton iteration. When higher accuracy is required, our algorithm can be used to compute a good starting point for subsequent Newton’s iteration.

Copyright/Permission Statement

© T. Liu et al 2013. This is the author's version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in ACM Transactions on Graphics,


time integration, implicit Euler method, mass-spring systems



Date Posted: 13 January 2016