Fast simulation of mass-spring systems

Abstract

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. AbstractWe 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.

Nonsmooth Developable Geometry for Interactively Animating Paper Crumpling

“…The technique we present in Section 6.3 treats the preservation of edge lengths as a geometric problem while Liu et al [2013] simulate a mass-spring system in a physical perspective with their approach called Fast Simulation of Mass-Spring Systems (FSMS). In the optimization method of FSMS, the edge length is set as a hard constraint.…”

Virtual Reality Surgery Simulation: A Survey on Patient Specific Solution

“…Mass-spring [5,10,36], a simple and inexpensive scheme takes Newtons second law of motion as the theory basis to first compute the velocity from accelerations and then the positions from velocities, which is hard to tune the spring constants to get desired behavior and leads to overshooting problem.…”

3D Character Animation: A Brief Review

“…Non-rigid deforming parts such as hair or clothes is also another interest of this area (30)(31)(32). Velocity, bending, stretching of mass-spring systems play a key role in motion of this deformation.…”