Skipping steps in deformable simulation with online model reduction

Abstract

Figure 1: We learn fast subspace models on-the-fly to accelerate deformable simulation: (Top) Ground truth frames from a 1000-frame deformable character simulation with 165,941 tetrahedra and Arruda-Boyce material which took 48 hours (174,521 seconds). (Middle) Frames from our online integrator which were generated in only 51 minutes (3,101 seconds) -56.28× faster -by learning a fast 11-dimensional subspace model on-the-fly (Bottom) Visualization of the temporal behavior of the online integrator reveal full steps (in black), reduced steps (in gray), and basis updates (red ticks). A reduced basis is detected quickly, and the first "skip" occurs at timestep 2. Only a handful of full steps are needed, and our reduced solver computes over 98% of the the steps. AbstractFinite element simulations of nonlinear deformable models are computationally costly, routinely taking hours or days to compute the motion of detailed meshes. Dimensional model reduction can make simulations orders of magnitude faster, but is unsuitable for general deformable body simulations because it requires expensive precomputations, and it can suppress motion that lies outside the span of a pre-specified low-rank basis. We present an online model reduction method that does not have these limitations. In lieu of precomputation, we analyze the motion of the full model as the simulation progresses, incrementally building a reduced-order nonlinear model, and detecting when our reduced model is capable of performing the next timestep. For these subspace steps, full-model computation is "skipped" and replaced with a very fast (on the order of milliseconds) reduced order step. We present algorithms for both dynamic and quasistatic simulations, and a "throttle" parameter that allows a user to trade off between faster, approximate previews and slower, more conservative results. For detailed meshes undergoing low-rank motion, we have observed speedups of over an order of magnitude with our method.

Subspace fluid re-simulation

“…multi-dimensional quadrature. This approach has been successfully applied to subspace material non-linearities in solid and shell mechanics Kim and James 2009;Chadwick et al 2009;Kim and James 2011]. Similar point-based approaches have also been successful, such as Monte-Carlo sampling [Baraff and Witkin 1992], and Key Point Subspace Acceleration (KPSA) [Meyer and Anderson 2007].…”

“…Trivially, if all of the grid cells are included in the cubature set, with weights equal to one, e f will be computed perfectly, but in O(N ) time. In the case of solid and shell mechanics, several works Chadwick et al 2009;Kim and James 2009] have found that that P ∝ r in practice. We will show that similar efficiencies arise in fluid mechanics.…”

Non-polynomial Galerkin projection on deforming meshes

“…An et al [2008] demonstrated model reduction of nonlinear elastic dynamics using cubature, which approximates nonlinear functions in low dimension by sampling the original functions and fitting a model to the samples. This method was also used to reduce thin shell dynamics by Chadwick et al [2009], and can be extended to support online updating of the basis [Kim and James 2009] and domain decomposition [Kim and James 2011]. Kim et al [2013] also applied this technique to fluid simulation.…”

Real-time large-deformation substructuring

“…In our work, we employ model reduction, and demonstrate that whenever the object can be decomposed into natural components, this can provide deformation-rich real-time simulations. In computer graphics, model reduction of nonlinear systems has been used for fast simulation of deformable solids [Metaxas and Terzopoulos 1992;Barbič and James 2005;Kaufman et al 2008;An et al 2008;Kim and James 2009] and fluids [Treuille et al 2006], and for fast control of such systems [Barbič et al 2009]. One drawback of these systems has been that the reduction basis is global in space.…”