A chebyshev semi-iterative approach for accelerating projective and position-based dynamics

Abstract

In this paper, we study the use of the Chebyshev semi-iterative approach in projective and position-based dynamics. Although projective dynamics is fundamentally nonlinear, its convergence behavior is similar to that of an iterative method solving a linear system. Because of that, we can estimate the "spectral radius" and use it in the Chebyshev approach to accelerate the convergence by at least one order of magnitude, when the global step is handled by the direct solver, the Jacobi solver, or even the Gauss-Seidel solver. Our experiment shows that the combination of the Chebyshev approach and the direct solver runs fastest on CPU, while the combination of the Chebyshev approach and the Jacobi solver outperforms any other combination on GPU, as it is highly compatible with parallel computing. Our experiment further shows position-based dynamics can be accelerated by the Chebyshev approach as well, although the effect is less obvious for tetrahedral meshes. The whole approach is simple, fast, effective, GPU-friendly, and has a small memory cost.

Position‐Based Simulation of Elastic Models on the GPU with Energy Aware Gauss‐Seidel Algorithm

“…Recently two novel efficient techniques were presented. One is the Chebyshev approach which was proposed by Wang [Wan15] to accelerate the iteration process during cloth and solid simulations. The other is the graph coloring method which was presented by Fratarcangeli et al [FTP16] to heavily parallelize the PBD iterations.…”

Realtime Performance‐Driven Physical Simulation for Facial Animation

“…The first obstacle to performance is the continuously changing constraints due to lips contacts that prevents efficient physical simulation techniques such as Projective Dynamics to use a pre‐factorized solver. Recent works handled changing systems by using iterative solvers such as an accelerated Jacobi [Wan15, WY16] or a parallel Gauss–Seidel [FTP16]. However, these methods are designed to take advantage of the computational power of the GPU and are likely to be less efficient on the CPU.…”

“…In particular, most volumetric forces require performing numerous SVDs, which are computationally expensive, even with the fast SVD technique of McAdams and colleagues [MST*11]. Fast polar decomposition methods have been used to simulate volumetric elasticity [RJ07, Wan15, MBCM16], however the polar decomposition cannot be used to simulate volume preservation, which is crucial to simulate the behaviour of facial skin.…”

“…The skin's elasticity is modelled using an as‐rigid‐as‐possible potential. For each tetrahedron, its potential is written as where is the matrix to form the deformation gradient of the tetrahedron with respect to the rest configuration [Wan15], and is the SVD of the deformation gradient. Intuitively, this potential constrains tetrahedron to stay close to a rotated rest configuration.…”

“…However, in the face of changing constraints, the global step matrix is no longer constant, and alternative methods have to be explored. Wang [Wan15] solves the global step system using one Jacobi iteration, and relies on Chebyshev multipliers to accelerate convergence, therefore requiring no pre‐factorization of the system. However, Jacobi iterations have no convergence guarantees for matrices that are not diagonally dominant, and the system matrix for tetrahedral strain and volume constraints contains strong off‐diagonal components.…”