Fast Corotated FEM using Operator Splitting

Abstract

In this paper we present a novel operator splitting approach for corotated FEM simulations. The deformation energy of the corotated linear material model consists of two additive terms. The first term models stretching in the individual spatial directions and the second term describes resistance to volume changes. By formulating the backward Euler time integration scheme as an optimization problem, we show that the first term is invariant to rotations. This allows us to use an operator splitting approach and to solve both terms individually with different numerical methods. The stretching part is solved accurately with an optimization integrator, which can be done very efficiently because the system matrix is constant over time such that its Cholesky factorization can be precomputed. The volume term is solved approximately by using the compliant constraints method and Gauss‐Seidel iterations. Further, we introduce the analytic polar decomposition which allows us to speed up the extraction of the rotational part of the deformation gradient and to recover inverted elements. Finally, this results in an extremely fast and robust simulation method with high visual quality that outperforms standard corotated FEMs by more than two orders of magnitude and even the fast but inaccurate PBD and shape matching methods by more than one order of magnitude without having their typical drawbacks. This enables a very efficient simulation of complex scenes containing more than a million elements.

Fast Updates for Least‐Squares Rotational Alignment

“…Based on the value of c, we name them Cayley (c = tr(M)), Cayley Conservative (c = tr(M) 2 + m T m) and Cayley Gershgorin. Note that mathematically Cayley is equivalent to [KKB18] that uses a single Newton step with quarternions. In all cases, the general result is the same.…”

“…It has also been used for more general alignment problems that cannot be solved using SVD or eigendecomposition [WV06]. Newton's method has been applied to polar decomposition [Hig86,BX08], and recently Kugelstadt et al [KKB18] also applied the Cayley transformation in this context. Their method is based on the local quadratic approximation implicitly used in the Newton step, representing rotations as quaternions.…”

Higher‐Order Time Integration for Deformable Solids

“…This enables to speed up computations by using efficient optimization methods like alternating local-global solvers [LBBK13, BML * 14], Newton's method with sophisticated line search strategies [GSS * 15], the Chebyshev Semi-Iterative approach [Wan15], ADMM [OBLN17], L-BFGS [LBK17], or domain decomposition [LGL * 19]. Moreover there are specialized material models which are well suited to be used with optimization integrators [KKB18]. It has been pointed out that there is a connection between compliant constraint systems and optimization based integration of Backward Euler [TNGF15].…”

“…The first-order Backward Euler integrator is arguably the most popular choice in computer graphics (e.g. [BW98,MG04,MTGG11,KBT17,KKB18]) owing to its reliability even in very demanding scenarios. However, its stability comes at the cost of excessive numerical dissipation, which in many cases precludes fine, local details such as folds and wrinkles in cloth, or produces strongly globally damped motion for otherwise vividly animated objects.…”

Simulation of Mooring Lines Based on Position-Based Dynamics Method

“…Müller et al [19] presented a method to simulate the dynamic behavior of inextensible hair and fur, which ensures zero stretching in a single iteration of each visual frame, and allows the real-time simulation of thousands of hair strands. Kugelstadt et al [20] proposed a novel operator splitting method for joint FEM simulation, and the method enables a very efficient simulation of complex scenes with more than a million elements. Deul et al [21], [22] put forward a method for simulating rigid bodies with orientation constraints in PBD.…”