Stable constrained dynamics

Abstract

Figure 1: Our method improves stability and step size for the simulation of constraint-based objects subject to high tensile forces, isolated or coupled with other types of objects. Bow: stiff 3D frame, 1D inextensible string, rigid arrow ; Trampoline: soft lateral springs, inextensible textile ; Knee: complex assembly of rigid bodies and stiff unilateral springs ; Ragdoll: rigid body assembly. AbstractWe present a unification of the two main approaches to simulate deformable solids, namely elasticity and constraints. Elasticity accurately handles soft to moderately stiff objects, but becomes numerically hard as stiffness increases. Constraints efficiently handle high stiffness, but when integrated in time they can suffer from instabilities in the nullspace directions, generating spurious transverse vibrations when pulling hard on thin inextensible objects or articulated rigid bodies. We show that geometric stiffness, the tensor encoding the change of force directions (as opposed to intensities) in response to a change of positions, is the missing piece between the two approaches. This previously neglected stiffness term is easy to implement and dramatically improves the stability of inextensible objects and articulated chains, without adding artificial bending forces. This allows time step increases up to several orders of magnitude using standard linear solvers.

A multi-scale model for simulating liquid-hair interactions

“…To simulate the underlying dynamics of hair, we adopt the DER framework, and model both the cohesive and repulsive effects as constraints. We solve them with a stable constraint solver [Tournier et al 2015] alongside a modified conjugate gradient preconditioner. We choose such an implementation for simplicity, although we believe more sophisticated integrators or constraint solvers [Gornowicz and Borac 2015;Kaufman et al 2014] could also be applied for the constraints proposed in this paper.…”

“…We integrate the elastic rods in time using the stable constraint-based solver of Tournier et al [2015]. This accounts for the internal forces (stretching, bending, twisting) and external forces of cohesion/repulsion, drag, and bulk pressure.…”

“…When many hairs adhere to one another, a large number of constraints are created between individual hairs, and the resulting global interactions give rise to linear systems that are difficult to solve efficiently. Furthermore, even the construction of the stiffness matrix [Tournier et al 2015] K can have a substantial performance overhead.…”

“…Each local matrix is small, banded, and remains fixed between PCG iterations; it can therefore be easily factored at the beginning of the PCG loop, and solved in parallel with a fast banded solver. To describe in detail the full algorithm for this process requires also to summarize the method and notation of Tournier et al [2015]; for details, refer to Section S4 of our supplemental document.…”

Variational stokes

“…A similar SPD transformation was proposed by Robinson-Mosher et al [2011], albeit for constant Laplacian-based viscosity with gridaligned boundaries and no free surfaces. Analogous transformations are of course common for articulated rigid bodies and other constrained dynamics problems (e.g., [Baraff 1996;Goldenthal et al 2007;Tournier et al 2015]). …”