Element-wise mixed implicit-explicit integration for stable dynamic simulation of deformable objects

Abstract

In order to evolve a deformable object in time, the underlying equations of motion have to be numerically integrated. This is commonly done by employing either an explicit or an implicit integration scheme. While explicit methods are only stable for small time steps, implicit methods are unconditionally stable. In this paper, we present a novel methodology to combine explicit and implicit linear integration approaches, based on element-wise stability considerations. First, we detect the ill-shaped simulation elements which hinder the stable explicit integration of the element nodes as a pre-computation step. These nodes are then simulated implicitly, while the remaining parts of the mesh are explicitly integrated. As a consequence, larger integration time steps than in purely explicit methods are possible, while the computation time per step is smaller than in purely implicit integration. During modifications such as cutting or fracturing, only newly created or modified elements need to be reevaluated, thus making the technique usable in real-time simulations. In addition, our method reduces problems due to numerical dissipation.

A modular framework for deformation and fracture using GPU shaders

“…It should be noted that using the explicit Euler integration caused some instability and in order to keep the simulation from becoming unstable in both deformation and fracture simulations, material parameters had to be tuned very finely and only work within strict limits. The biggest improvement that could be made to our system would therefore be to implement a more advanced numerical integration method such as the implicit Euler method, which would not only add stability to the simulation but should allow a larger number of materials to be simulated, or possibly a hybrid method as presented by Fierz et al [30]. The procedural edges methods presented here provide an initial step towards an automated system for producing edge geometry for fracture simulation.…”

Real‐time physical deformation and cutting of heterogeneous objects via hybrid coupling of meshless approach and finite element method

“…To improve this, Chaos et al proposed a combination of a geometric material model with a fully variational geometric integrator, and Fieri et al showed how to identify ill‐shaped elements hindering stable numerical time integration. Moreover, based on element‐wise stability considerations, a hybrid methodology combining explicit and implicit linear integration approaches is also proposed in .…”

Haptic Simulation of Organ Deformation and Hybrid Contacts in Dental Operations

“…Based on the Courant condition [32], [33], we now derive the stable range of the time step for damped massspring systems. We consider a linear damped massspring system to explain the stability of explicit integration.…”

“…To ensure the stability of the system, the spectral radius of the system matrix, rðAÞ :¼ maxðk 0 k; k 1 kÞ, should not be greater than one [32], [33], that is,…”