Non-polynomial Galerkin projection on deforming meshes

Abstract

Figure 1: Our method enables reduced simulation of fluid flow around this flying bird over 2000 times faster than the corresponding full simulation and reduced radiosity computation in this architectural scene over 113 times faster than the corresponding full radiosity. AbstractThis paper extends Galerkin projection to a large class of nonpolynomial functions typically encountered in graphics. We demonstrate the broad applicability of our approach by applying it to two strikingly different problems: fluid simulation and radiosity rendering, both using deforming meshes. Standard Galerkin projection cannot efficiently approximate these phenomena. Our approach, by contrast, enables the compact representation and approximation of these complex non-polynomial systems, including quotients and roots of polynomials. We rely on representing each function to be model-reduced as a composition of tensor products, matrix inversions, and matrix roots. Once a function has been represented in this form, it can be easily model-reduced, and its reduced form can be evaluated with time and memory costs dependent only on the dimension of the reduced space.

Simulating articulated subspace self-contact

“…The main issue is that per-element polar decompositions must be performed, which are non-trivial to formulate as a tensor. The extensions introduced by Stanton et al [2013] do not appear to provide any assistance in this case.…”

“…They were subsequently extended to support the polynomial systems that arise in deformable body and fluid simulations [Barbič and James 2005;Treuille et al 2006;Wicke et al 2009], and most recently to algebraic systems [Stanton et al 2013]. In all of these cases, the subspace method successfully accelerated the underlying simulation by several orders of magnitude.…”

Subspace fluid re-simulation

“…Treuille et al [2006] pioneered subspace methods for fluids in computer graphics, and subsequent work showed how to generalize the approach to modular tiles [Wicke et al 2009]. Most recently, this work was generalized from reduced polynomials to general reduced algebraic functions [Stanton et al 2013]. The eigenfunction work of DeWitt et al [2012] could also be viewed as a subspace method, albeit one where basis functions are obtained via static analysis, not from the SVD of existing simulation data.…”

“…Promoting the tensor to 4th or 5th order may seem promising, but these would only capture higher order polynomial effects on static stencils; it does not address the fact that the locations change. While generalizations to algebraic functions are presented in [Stanton et al 2013], the underlying static stencil problem remains. Clearly, a different approach is needed.…”

“…This swap is discussed in detail in [Stanton et al 2013]. Further details on the implementation and computation of the necessary SVDs are available in §3.4.…”