The efficient simulation of incompressible fluids remains a difficult and open problem. Prior works often make various tradeoffs between incompressibility, stability, and cost. Yet, it is rare to obtain all three. In this paper, we introduce a novel incompressible Smoothed Particle Hydrodynamics (SPH) scheme which uses a second-order implicit descent scheme to optimize a variational energy specially formulated to approach incompressibility. We demonstrate that our method is superior in both incompressibility and stability with a minimal cost to computational budget. Furthermore, we demonstrate that our method is unconditionally stable even under extreme time steps, making it suitable for interactive applications.

Vertex Block Descent is a fast physics-based simulation method that is unconditionally stable, highly parallelizable, and capable of converging to the implicit Euler solution. We extend it using an augmented Lagrangian formulation to address some of its fundamental limitations. First, we introduce a mechanism to handle hard constraints with infinite stiffness without introducing numerical instabilities. Second, we substantially improve the convergence in the presence of high stiffness ratios. These changes we introduce allow simulating complex contact scenarios involving rigid bodies with stacking and friction, articulated bodies connected with hard constraints, including joints with limited degrees of freedom, and stiff systems interacting with soft bodies. We present evaluations using a parallel GPU implementation that can deliver real-time performance and stable simulations with low iteration counts for millions of objects interacting via collisions, various joint/attachment constraints, and springs of various stiffness. Our results show superior performance, convergence, and stability compared to the state-of-the-art alternatives.

Achieving stable simulation of constrained rigid body systems is a primary concern for many computer graphics applications, such as video games, robotic planning, and virtual reality training. In this paper, we present a novel adaptive sub-stepping scheme that achieves stable simulation by adaptively reducing the time step as needed. Our approach employs a diagonalized geometric stiffness matrix as a heuristic to determine when smaller time steps are required, and adjusts the number of sub-steps accordingly. Our method is straightforward to integrate into existing rigid body simulators, and further eliminates manually tuning the number of sub-steps required. We demonstrate the ability of our method to produce stable simulates at real-time frame rates using a number of challenging, complex examples.