Affine body dynamics

Abstract

Simulating stiff materials in applications where deformations are either not significant or else can safely be ignored is a fundamental task across fields. Rigid body modeling has thus long remained a critical tool and is, by far, the most popular simulation strategy currently employed for modeling stiff solids. At the same time, rigid body methods continue to pose a number of well known challenges and trade-offs including intersections, instabilities, inaccuracies, and/or slow performances that grow with contact-problem complexity. In this paper we revisit the stiff body problem and present ABD, a simple and highly effective affine body dynamics framework, which significantly improves state-of-the-art for simulating stiff-body dynamics. We trace the challenges in rigid-body methods to the necessity of linearizing piecewise-rigid trajectories and subsequent constraints. ABD instead relaxes the unnecessary (and unrealistic) constraint that each body's motion be exactly rigid with a stiff orthogonality potential, while preserving the rigid body model's key feature of a small coordinate representation. In doing so ABD replaces piecewise linearization with piecewise linear trajectories. This, in turn, combines the best of both worlds: compact coordinates ensure small, sparse system solves, while piecewise-linear trajectories enable efficient and accurate constraint (contact and joint) evaluations. Beginning with this simple foundation, ABD preserves all guarantees of the underlying IPC model we build it upon, e.g., solution convergence, guaranteed non-intersection, and accurate frictional contact. Over a wide range and scale of simulation problems we demonstrate that ABD brings orders of magnitude performance gains (two- to three-orders on the CPU and an order more when utilizing the GPU, obtaining 10, 000× speedups) over prior IPC-based methods, while maintaining simulation quality and nonintersection of trajectories. At the same time ABD has comparable or faster timings when compared to state-of-the-art rigid body libraries optimized for performance without guarantees, and successfully and efficiently solves challenging simulation problems where both classes of prior rigid body simulation methods fail altogether.

Monolithic Friction and Contact Handling for Rigid Bodies and Fluids Using SPH

“…As a consequence, friction forces might violate the Coulomb constraint and there are no guarantees concerning the principle of maximum dissipation [Erl17]. Alternatively, smoothing the velocity–force relation eliminates the discontinuous jump of the friction force and the resulting computational burden at stick–slip transitions [PRM19, GHZ*20, LFS*20, FLS*21, MEM*20, CLL*22, LKL*22]. Even though in most approaches, the amount of smoothing is parametrized and can be reduced to get a better approximation of the true velocity–force relation at the cost of less computational stability, the friction force at zero velocity is always null and thus, these models cannot exactly reproduce stiction [PRSV16, PRM19, LKL*22].…”

“…Alternatively, smoothing the velocity–force relation eliminates the discontinuous jump of the friction force and the resulting computational burden at stick–slip transitions [PRM19, GHZ*20, LFS*20, FLS*21, MEM*20, CLL*22, LKL*22]. Even though in most approaches, the amount of smoothing is parametrized and can be reduced to get a better approximation of the true velocity–force relation at the cost of less computational stability, the friction force at zero velocity is always null and thus, these models cannot exactly reproduce stiction [PRSV16, PRM19, LKL*22]. Some approaches model static friction as linear springs which pull contacts back together over multiple simulation steps [YN06, XZB14].…”

Variational schemes and mixed finite elements for large strain isotropic elasticity in principal stretches: Closed‐form tangent eigensystems, convexity conditions, and stabilised elasticity

“…We note that, while our formulation has linear treatment of variables $$$ {\lambda}_i $$$s and $$$ {\Sigma}_{\lambda_i} $$$s it still requires a parabolic mesh. Alternate and efficient formulations for stiff materials exist, for cases where engineering accuracy is not desired 2 …”

“…(2) Simulation of rigid , stiff and inextensible systems. Like quasi‐incompressibility, stiff materials have been traditionally modeled by decomposing the deformation into a rotation and a stretch component and by penalising the latter part 2 . Intuitively, and as studies in rigid body dynamics have shown, simulating stiff materials poses the same challenge as that of incompressibility, since rigidifying a deformable system through large penalty terms eventually either blows up the numerical routine and/or the accuracy of secondary variables, which reflects in stresses remains oscillatory or at best poor.…”

“…It is worth mentioning that, finite element simulations based on stretch formulations for finitely deformable solids have become more common in geometry processing applications and specifically in surface parametrisation 18,29,32,33 . In this regard, a set of popular algorithms have emerged over the past two decades commonly referred to as the “local‐global” algorithms 2,33‐36 . On cursory examination, these algorithms have a noteworthy resemblance to P1‐P0 mixed finite elements wherein displacements and rotations (from the polar decomposition of deformation gradient) are treated as separate variables.…”