Physics-Based Animation

Kevin You, Juntian Zheng, Minchen Li

Dynamic simulation of elastic bodies is a longstanding task in engineering and computer graphics. In graphics, numerical integrators like implicit Euler and BDF2 are preferred due to their stability at large time steps, but they tend to dissipate energy uncontrollably. In contrast, symplectic methods like implicit midpoint can conserve energy but are not unconditionally stable and fail on moderately stiff problems. To address these limitations, we propose a general class of numerical integrators for Hamiltonian problems which are symplectic on linear problems, yet have superior stability on nonlinear problems. With this, we derive a novel energy-controllable time integrator, A-search, a simple modification of implicit Euler that can follow user-specified energy targets, enabling flexible control over energy dissipation or conservation while maintaining stability and physical fidelity. Our method integrates seamlessly with barrier-type energies and allows for inversion-free and penetration-free guarantees, making it well-suited for handling large deformations and complex collisions. Extensive evaluations over a wide range of material parameters and scenes demonstrate that A-search has biases to keep energy in low frequency motion rather than dissipation, and A-search outperforms traditional methods such as BDF2 at similar total running times by maintaining energy and leading to more visually desirable simulations.

Energy-Controllable Time Integration for Elastodynamic Contact

Caoliwen Wang, Minghao Guo, Siyuan Chen, Heng Zhang, Mengdi Wang, Xingyu Ni, Hanson Sun, Kunyi Wang, Zherong Pan, Kui Wu, Lingjie Liu, Yin Yang, Chenfanfu Jiang, Taku Komura, Wojciech Matusik, Peter Yichen Chen

A unified simulator that can model diverse physical phenomena without solver-specific redesign is a long-standing goal across simulation science. We present a learning-based particle simulator built on a single transformer architecture to model cloth, elastic solids, Newtonian and non-Newtonian fluids, granular materials, and molecular dynamics. Our model follows a prediction-correction design on a shared Lagrangian particle representation. An explicit predictor first advances particles under the known external forces, producing an intermediate state that captures externally driven motion but not inter-particle interactions. A learned corrector then predicts the residual position and velocity updates through three stages: a particle tokenizer that encodes local particle-particle, particle-boundary, and topology-guided interactions; a super-token encoder that hierarchically merges particle tokens into a compact set of super tokens via alternating self-attention and token merging; and a super-token decoder that lifts these super tokens back to particle resolution through cross-attention to predict per-particle position and velocity corrections. Progressive token merging reduces the attention cost at successive encoder layers by halving the token count at each level, and the decoder communicates through the compact super-token set rather than full particle-to-particle attention. Across the six dynamics categories, the same architecture generalizes to unseen materials, boundary configurations, initial conditions, and external forces. We further demonstrate downstream interactive control, inverse design, and learning from real-world manipulation data, reducing the need for per-phenomenon solver engineering.

WorldParticle: Unified World Simulation of Lagrangian Particle Dynamics via Transformer

Xukun Luo, Xiao Chen, Yuzhong Guo, Ying Qiao, Wenchen Wang, Xiaowei He

We propose Semi-Implicit Pairwise Descent (SIPD), a unified nonlocal pairwise framework for simulating large-scale hyperelastic materials involving complex contact and friction. By reformulating the Finite Element Method (FEM) equations of motion into a pairwise force representation from a non-local perspective, our approach avoids costly Hessian computations, leading to a reduction in per-iteration computational overhead. Furthermore, we propose an analytical projection strategy for projecting our Hessian-free coefficient matrices to positive semi-definiteness. And we treat contact and friction as a unified anisotropic elastic energy, allowing for a seamless integration into the elastic solver framework. We mathematically prove that our method is unconditionally stable and numerically convergent. Experimental results demonstrate that SIPD achieves real-time performance for million-scale simulations even under intricate contact and friction conditions.

Semi-Implicit Pairwise Descent for Nonlocal Continuum Mechanics

Xingyu Ni, Jiong Chen^†, Siyuan Chen, Caoliwen Wang, Mathieu Desbrun, Taku Komura

Poisson equations arise pervasively in computer graphics, yet efficiently solving them on unbounded domains remains a long-standing challenge. Existing approaches attempting to model boundary conditions at infinity either incur prohibitive computational costs by expanding the simulation domain, sacrifice accuracy through artificial boundary conditions, or lack the flexibility to handle varying coefficients. In this paper, we introduce a simple and practical method for solving generalized Poisson equations on finite domains with open boundaries, enabling exact enforcement of asymptotic conditions at infinity. Our method couples a finite-volume interior solver with a boundary-element formulation of the exterior problem, enforcing continuity of both the solution and its normal derivative across an artificial boundary. Inspired by Johnson–Nédélec coupling and interface relaxation, we develop a partitioned, iterative scheme based on damped fixed-point iterations, which leverages fast, inexact boundary-element solves for the exterior problem and efficient sparse solves for the interior. Theoretical analysis and empirical tests confirm that our method provides superior accuracy and performance compared to existing ad-hoc solutions across a variety of graphics applications.

Efficient Open Boundary Poisson Solves

Zhiyang Dou, Ang Zhao, Chen Peng, Minghao Guo, Haixu Wu, Cheng Lin, Yuan Liu, Junfeng Yao, Xiaohu Guo, Wenping Wang, Wojciech Matusik

Rigid-body interpenetration frequently occurs in procedurally assembled and generated scenes and must be removed before downstream applications such as physical simulation. We present S4R (Scaling for Rigid-Body Interpenetration Resolution), a scale-continuation method for static interpenetration repair. S4R first uniformly shrinks each body about a fixed reference center to a small initial scale, at which the layout is penetration-free, and then restores full scale through a sequence of minimum-norm convex contact quadratic programs (QPs) that target the linearized separation margin during continuation. Resolution thereby replaces one deep correction with a sequence of shallow-contact subproblems. A conservative scale-event bound and frozen-witness gap predictions cut the number of exact mesh queries; the continuation then ends with a full-scale evaluator check and bounded tail refinement. We evaluate S4R on Kubric [Greff et al. 2022], HY3D-Bench [Team Hunyuan3D et al. 2026], and Thingi10K [Zhou and Jacobson 2016] using a shared mesh-level evaluator and a unified per-scene timing protocol. In the main comparisons on all three benchmarks, up to 5000 bodies, S4R reaches zero reported penetration with displacement that stays small and nearly independent of scene size, and at the lowest wall time within each hardware tier among the compared methods. A GPU implementation extends these results to large-scale scenes. Our code and data can be found on our project page: https://frank-zy-dou.github.io/projects/S4R/index.html.

S4R: Scaling for Rigid-Body Interpenetration Resolution

Joel Wretborn, Alexey Stomakhin, Christopher Batty

We present a novel method for simulating large-scale, high density-ratio two-phase immiscible flows. Our approach adopts a two-fluid mixture formulation discretized on a shared Eulerian grid, in which phase fractions are represented using Lagrangian material points. We carefully adapt established closure models to this hybrid framework to capture the exchange of forces between phases, enabling the smooth transition between stratified (grid-resolved) and dispersed (sub-grid) flow regimes without recourse to secondary solvers. The resulting volumetric coupling forces are jointly integrated in a pressure projection, yielding a strongly coupled scheme that remains stable under large time steps and in highly dynamic scenarios. By combining these features, our method provides a unified, multi-scale discretization capable of capturing immiscible two-phase flows covering a large range of spatial scales in a single simulation. We illustrate our method’s capabilities on a diverse set of multi-scale scenarios, including a massive waterfall, a waterbombing airplane, large underwater bubbles, a geyser, and crashing ocean waves.

A Unified Multi-Scale Method for Simulating Two-Phase Immiscible Flows

Yuhao Quan, Hui Wang, Weile Lian, Zhi Wang, Xubo Yang

This paper presents a highly efficient and robust differentiable fluid framework centered on a novel surrogate gradient method that utilizes the flow map structural advantages. Our key insight reveals a significant misalignment between computational intensity and gradient importance during the backward pass. Specifically, we identify a physical duality within the adjoint process, revealing that the cross-step connections inherent in the flow map act as dominant gradient “highways” that propagate sensitivities over long horizons with high fidelity. Leveraging these insights, we develop a surrogate gradient model that retains these critical connections while pruning redundant adjoint computations in a physics-informed manner. Integrated with tailored acceleration techniques, our framework is successfully applied to diverse, challenging optimization tasks characterized by long time horizons and rich vorticity. Results demonstrate significant speedups and memory reductions while maintaining nearly-identical gradients compared to the full-gradient baseline.

DiffSurFlow: Efficient and Robust Differentiable Fluid Optimization via Surrogate Strategy on Flow Map

Jan Bender, Stefan Rhys Jeske, Timna Böttcher, Fabian Löschner

In recent years, the Smoothed Particle Hydrodynamics (SPH) approach has been increasingly used for multiphase simulations involving interactions between diverse materials. A critical component of an SPH simulator is the pressure solver, which not only facilitates the simulation of compressible or incompressible fluids but also handles contact by preventing penetration between different materials. Currently, most SPH simulations in computer graphics employ implicit dual pressure solvers such as PBF, IISPH, or DFSPH. However, these solvers often exhibit instability when simulating high density ratios. Furthermore, they are difficult to strongly couple with many existing methods for non-pressure forces, which typically utilize primal formulations. Consequently, pressure and non-pressure solvers are often only weakly coupled, which can lead to stability issues. We present a novel implicit primal SPH pressure solver designed for multiphase simulations. Our method enables stable simulation of multiple interacting materials with large density ratios. We show that our solver robustly handles ratios of up to 1:1000 (e.g., air-water interactions) which was not possible with previous implicit SPH pressure solvers. Moreover, we demonstrate how our solver allows for strong coupling with existing implicit simulation methods for viscosity, elasticity, and surface tension. Overall, our strong coupling significantly improves stability in complex multiphase simulations involving fluids, highly viscous materials, and deformable solids.

Primal SPH Solver for Strongly Coupled Multiphase Simulations with High Density Ratios

Hui Wang, Jinjin He, Xubo Yang

We present a novel particle flow map framework for the high-fidelity simulation of complex two-phase flows. Our method is built upon a unified Lagrangian formulation in which a shared flow map jointly governs the phase interface evolution and the underlying fluid dynamics. For interface tracking, our particles drive a particle-flow-map-based level set equipped with a hybrid reinitialization strategy, effectively preserving sub-grid geometric features while ensuring robust topological stability. For two-phase dynamics, we introduce an impulse-based solver that reformulates the impulse path integration to depend solely on the continuous velocity field, enabling efficient and accurate handling of interfacial discontinuities without artificial smoothing. By leveraging the particle flow map’s inherently low-dissipation tracking of both geometry and dynamics, our framework achieves enhanced geometric accuracy and physical fidelity relative to existing two-phase solvers. Our framework faithfully captures the intricate interplay between intense vortical motion and complex interface geometry, as evidenced by the reproduction of a broad range of challenging phenomena, including interacting bubble rings, breaking waves, and whirlpool drainage.

Two-phase Impulse Fluid on Particle Flow Map