Physics-Based Animation

Yiheng Wu, Kai Bai, Xiaopei Liu

Two-phase flows are ubiquitous in nature, exhibiting complex fluid-fluid interactions that challenges numerical simulators. To accurately and efficiently solve two-phase flows, grid-based methods have been widely adopted. Navier-Stokes (NS) solvers consume a small memory footprint, but simultaneously achieving both high performance and low numerical dissipation remains a significant challenge. In contrast, lattice Boltzmann solvers are efficient and have low numerical dissipation, yet they remain memory-intensive, even with state-of-the-art moment-encoding schemes. To date, the simultaneous attainment of high accuracy, exceptional efficiency, and a low memory footprint remains a major challenge in the field. In this paper, we propose a novel two-phase flow solver that achieves this objective. Our work is motivated by extending gas-kinetic scheme (GKS), which is adapted to handle nearly incompressible flows. To allow stable and accurate two-phase flow simulations, we systematically derive a coupled formulation of the GKS method and the phase-field model, incorporating novel mathematical constructs. Combined with robust boundary treatments and specialized techniques for handling turbulent flows, this results a unified framework capable of efficiently simulating two-phase flows, even those with large density contrasts and high Reynolds numbers. Since our formulation is explicit, it achieves exceptional performance when optimized on GPU, making it the fastest kinetic two-phase flow solver to date. Additionally, as it is derived from GKS, it obviates the need to store distribution functions. Thus, it has a small memory footprint, competitive with, or even lower than, that of many NS solvers. As a result, our solver can efficiently simulate complex two-phase flow dynamics at high resolutions using a single commodity GPU. We validate the accuracy of our solver via several benchmark tests, compare its performance with recent methods in various aspects, and demonstrate its capability to replicate a broad range of two-phase flow phenomena, encompassing both typical and large-scale scenarios.

An Extended Full GKS Formulation for High-Efficiency and Low-Memory Two-Phase Flow Simulation

Shih-Yu Lai, Sung-Han Tien, Jui-I Huang, Yen-Chen Tseng, Yi-Ting Chiu, Siyuan Luo, Ziqiu Zeng, Fan Shi, Peter Yichen Chen, Tiantian Liu, Yu-Lun Liu, Bing-Yu Chen

Differentiable simulation of soft bodies is a foundation for system identification, trajectory optimization, and Real2Sim transfer. Yet, existing methods such as the differentiable Projective Dynamics (DiffPD) struggle when faced with heterogeneous materials with extreme stiffness contrasts, hyperelasticity under large deformations, and contact-rich interactions, which are common scenarios in the real world. We present DiffPhD, a unified GPU-accelerated differentiable Projective Dynamics framework for heterogeneous materials that tackles these intertwined challenges simultaneously. Our key insight is a careful integration of: (i) stiffness-aware projective weights to embed heterogeneity into the global system; (ii) trust-region eigenvalue filtering lifted to the backward pass for stable hyperelastic gradients and a type-II Anderson Acceleration scheme with dual-gate convergence to stabilize forward iteration under large stiffness contrasts; and (iii) a unified GPU pipeline that reuses a single sparse factor across forward, backward, and contact computations, with stiffness-amplified Rayleigh damping folded into the same factor for heterogeneity-aware dissipation at zero recurring cost. DiffPhD delivers analytic adjoints—machine-precision on the elastic path, bounded and quantified under contact—with up to an order-of-magnitude speedup over prior differentiable solvers. Crucially, this speedup does not come at the cost of stability: DiffPhD remains convergent on stiffness contrasts up to 100× where prior PD solvers degrade. This unlocks end-to-end gradient-based optimization on regimes previously bottlenecked by either solver fragility or per-iteration cost—shell–joint composite creatures, soft characters wielding stiff weapons, and soft-gripper robotic manipulation—all handled within a single forward–backward pass.

DiffPhD: A Unified Differentiable Solver for Projective Heterogeneous Materials in Elastodynamics with Contact-Rich GPU-Acceleration

Sinan Wang, Ruicheng Wang, Taiyuan Zhang, Fan Feng, Jinjin He, Yuchen Sun, Zhiqi Li, Bo Zhu

Simulating large-scale free-surface water by coupling a localized 3D fluid solver to a cheaper 2D surface model has long faced a mismatch in wave dynamics: efficient 2D wave models used in graphics are typically either linear or non-dispersive. These models are fast, simple, and accurate for calm, small-amplitude seas, but coupling them with strongly nonlinear 3D solvers produces visible reflections and artifacts at the 2D–3D interface. We address this problem by introducing a nonlinear and dispersive 2D wave model based on the canonical Zakharov formulation. Its Hamiltonian structure, in which the surface elevation and surface potential form a canonical pair (η, ψ) governed by the wave energy, enables a canonically consistent two-way coupling scheme, allowing information to pass smoothly across the 2D–3D interface. Our 2D solver reduces mean wave-height error by 1.7–5× over SWE, BEM, and Airy baselines while running more than 10³× faster than BEM; it achieves greater nonlinear accuracy and coupling fidelity than SWE and Airy, with minor losses in speed and stability. Coupling it with a 3D Navier–Stokes solver yields a full system that suppresses visible seam artifacts across a range of experiments, including dispersion-matching and Kelvin-wake tests, and runs over 4× faster than a pure GPU NB-FLIP simulation on the same domain.

Hamiltonian Two-Way Coupling of Nonlinear Waves and 3D Flows

Huanyu Chen, Yumeng He, Jernej Barbič

We present a thin-shell energy formulation rigorously derived from first volumetric principles of 3D solid mechanics, overcoming limitations of prior methods that rely on geometric heuristics or restrictive assumptions such as those in Kirchhoff-Love shells. Our approach supports arbitrary nonlinear isotropic materials, reproduces volumetric behavior exactly under small deformations, and remains accurate for large deformations. Kirchhoff-Love thin shells constrain material lines normal to the mid-surface to remain straight, normal, and unstretched, suppressing through-thickness relaxation. As a result, they fail to reproduce the behavior of the corresponding 3D volumetric material such as Poisson contraction or bending-induced through-thickness nonlinearities, even under small deformations. We analytically derive the leading small-deformation through-thickness modes of an initially flat thin shell from volumetric elasticity: a linear normal mode that scales the local thickness, a quadratic normal mode that captures bending-induced thickness variation, and higher modes. We use the lowest modes as a compact kinematic ansatz for large-deformation thin-shell simulation, with a scalar thickness variable rho and an optional, statically condensable bending amplitude zeta. Starting from an arbitrary isotropic hyperelastic energy density psi, we derive a closed-form shell energy by expanding the deformation gradient through the thickness and analytically integrating the volumetric energy. The resulting energy separates stretching and bending contributions, avoids both volumetric meshing and through-thickness quadrature, and depends only on mid-surface quantities (the first fundamental form and the shape operator), together with rho and zeta. We provide robust implementation-ready formulas for the energy, gradient, and Hessian, including systematic treatment of removable singularities that arise when principal stretches coincide. Experiments show close agreement with volumetric simulations across diverse materials and scenarios, greatly surpassing KL shells, while retaining the computational efficiency of thin shells.

Beyond Kirchhoff-Love: A Volumetric Approach to Thin-Shell Mechanics

Joel Wretborn, Marcus Schoo, Noh-hoon Lee, Christopher Batty, Alexey Stomakhin

This work addresses the challenges of distributing large physics-based simulations often encountered in the visual effects industry. These simulations, based on partial differential equations, model complex phenomena such as free surface liquids, flames, and explosions, and are characterized by domains whose shapes and topologies evolve rapidly. In this context, we propose a novel partitioning algorithm employing optimal transport—which produces a power diagram—and designed to handle a vast variety of simulation domain shapes undergoing rapid changes over time. Our Power partitioner ensures an even distribution of computational tasks, reduces inter-node data exchange, and maintains temporal consistency, all while being intuitive and artist-friendly. To quantify partitioning quality we introduce two metrics, the surface index and the temporal consistency index, which we leverage in a range of comparisons on real-world film production data, showing that our method outperforms the state of the art in a majority of cases.

A practical partitioner for distributed simulations on sparse dynamic domains using optimal transport

Rudresh Veerkhare, Hang Yin, Albert Chern

We introduce a simulation method for point vortices on surfaces with arbitrary topology. We show that incorporating the dynamics of the harmonic components on a non-simply-connected surface is equivalent to introducing a new fluid invariant that is conserved over time. This invariant admits an elegant formulation in terms of the divisor class of point vortices, viewed as a divisor on a Riemann surface. To clarify this connection, we establish a nontrivial bridge between fluid dynamics and the theory of divisors in algebraic geometry. By exploiting this invariant, our simulation method becomes a simple modification of the vortex particle-on-mesh method, which extends its applicability to surfaces with arbitrary topology.

Divisor Dynamics for Surface Fluid Simulation

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