Physics-Based Animation

Daria Nogina, Silvia Sellán

Smoothed Particle Hydrodynamics (SPH) simulations rely on accurately and efficiently modeling fluid-solid interactions. However, particle-based coupling strategies introduce non-deterministic discretization errors, and implicit methods achieve high accuracy at the cost of expensive numerical integration. We introduce Tube Maps, a drop-in replacement for SPH boundary density computation that achieves accuracy comparable to implicit methods while dramatically reducing their computational cost. Our key observation is that the boundary density integral is fully determined by the local surface geometry near a fluid particle’s closest point. By expressing this geometry in tubular coordinates, we reduce the original three-dimensional integral to a one-dimensional closed-form expression that can be evaluated in constant time. We thus eliminate numerical quadrature and reduce boundary handling costs by one to three orders of magnitude, enabling fast and accurate SPH simulations with time-varying curved solids.

Tube Maps: Fast SPH Boundary Handling in Tubular Coordinates

José Antonio Fernández-Fernández, Fabian Löschner, Lukas Westhofen, Andreas Longva, Jan Bender

Optimization time integrators are effective at solving complex multi-physics problems including deformable solids with non-linear material models, contact with friction, strain limiting, etc. For challenging problems, Newton-type optimizers are often used, which necessitates first- and second-order derivatives of the global non-linear objective function. Manually differentiating, implementing, testing, optimizing, and maintaining the resulting code is extremely time-consuming, error-prone, and precludes quick changes to the model, even when using tools that assist with parts of such pipeline. We present SymX, an open source framework that computes the required derivatives of the different energy contributions by symbolic differentiation, generates optimized code, compiles it on-the-fly, and performs the global assembly. The user only has to provide the symbolic expression of each energy for a single representative element in its corresponding discretization and our system will determine the assembled derivatives for the whole simulation. We demonstrate the versatility of SymX in complex simulations featuring different non-linear materials, high-order finite elements, rigid body systems, adaptive discretizations, frictional contact, and coupling of multiple interacting physical systems. SymX’s derivatives offer performance on par with SymPy, an established off-the-shelf symbolic engine, and produces simulations at least one order of magnitude faster than TinyAD, an alternative state-of-the-art integral solution.

SymX: Energy-based Simulation from Symbolic Expressions

Yuqi Meng, Yihao Shi, Kemeng Huang, Zixuan Lu, Ning Guo, Taku Komura, Yin Yang, Minchen Li

We present an efficient B-spline finite element method (FEM) for cloth simulation. While higher-order FEM has long promised higher accuracy, its adoption in cloth simulators has been limited by larger computational costs while generating results with similar visual quality. Our contribution is a full algorithmic pipeline that makes cloth simulation using quadratic B-spline surfaces faster than standard linear FEM in practice while consistently improving accuracy and visual fidelity. Using quadratic B-spline basis functions, we obtain a globally C¹-continuous displacement field that supports consistent discretization of both membrane and bending energies, effectively reducing locking artifacts and mesh dependence common to linear elements. To close the performance gap, we introduce a reduced integration scheme that separately optimizes quadrature rules for membrane and bending energies, an accelerated Hessian assembly procedure tailored to the spline structure, and an optimized linear solver based on partial factorization. Together, these optimizations make high-order, smooth cloth simulation competitive at scale, yielding an average 2× speedup over linear FEM. Extensive experiments demonstrate improved accuracy, wrinkle detail, and robustness, including contact-rich scenarios, relative to linear FEM and recent higher-order approaches. Our method enables realistic wrinkling dynamics across a wide range of material parameters and supports practical garment animation, providing a new promising spatial discretization for high-quality cloth simulation.

Efficient B-Spline Finite Elements for Cloth Simulation

Zhiyong He, Dewen Guo, Minghao Guo, Yili Zhao, Wojciech Matusik, Hao Su, Chenfanfu Jiang, Peter Yichen Chen, Yin Yang

Simulating large-scale articulated assemblies poses a significant challenge due to the numerical stiffness and geometric complexity of jointed structures. Conventional rigid body solvers struggle with the high nonlinearity induced by rotation parameterization. This difficulty becomes more pronounced for multiple two-way-coupled bodies. This paper introduces a novel framework that leverages the linear kinematic mapping of Affine Body Dynamics (ABD). As ABD targets near-rigid objects, the constitutive variations of different materials become negligible, which justifies a co-rotational approach to isolate geometric nonlinearities of the system. This insight enables the use of constant system matrices that can be pre-factorized throughout the simulation, even with fully implicit integration schemes. To manage the high DOF counts of large-scale systems, we map primal body coordinates onto a compact dual space defined by minimal joint degrees of freedom. By solving the resulting KKT systems, our method ensures exact constraint enforcement and physically accurate motion propagation. We provide a suite of specialized solvers tailored for diverse joint topologies, including chains, trees, closed loops, and irregular networks. Experimental results show that our approach achieves interactive rates for systems with hundreds of thousands of bodies on a single CPU core, while maintaining excellent stability at large time steps.

M-ABD: Scalable, Efficient, and Robust Multi-Affine-Body Dynamics

Nicholas Mcdonald, Guillaume Cordonnier

Mountainous terrains evolve over geological timescales through erosion processes driven by the complex interplay of transported quantities such as water, sediment, and rockfall. A key challenge in erosion modeling is the simultaneous simulation of transport and erosive processes, which differ in temporal scales by several orders of magnitude. We address this challenge with a novel, parallel, stochastic particle-based method capable of simulating transport over geological timescales. Our approach relaxes the strong assumptions on velocity required by prior works (e.g., based on the Stream Power Law), enabling a new erosion model grounded in a more general form of momentum conservation. We demonstrate that our scheme accurately solves the underlying conservation laws and avoids artifacts common in previous works. Furthermore, we show that our new erosion model captures multiscale geomorphological features, producing coherent basin structures and dynamic phenomena such as braided rivers, meanders, and deltas.

Stochastic geomorphological transport for terrain erosion simulation

Xiaoyu Xiao, Haoxiang Wang, Xiaokang Yang, Mathieu Desbrun, Wei Li

Simulating the dynamic, multiscale interactions between granular materials and multiphase fluids remains a significant computational challenge in computer graphics, as the visual complexity of such mixtures arises from strongly coupled small-scale structures. We present a novel, physically-based simulation framework for sand-water-air mixtures that couples a Lattice Boltzmann Method (LBM) for weakly-compressible two-phase fluids with a Material Point Method (MPM) for granular sand. Our approach is built upon a unified continuum formulation that expresses the governing equations for both fluid phases (air and water) and the granular medium within a consistent framework. To accurately capture the transition of sand from a dry, friction-dominated state to a soaked, sticky medium, we introduce a water retention model that describes how liquid infiltrates and is retained within the granular structure. Furthermore, we enforce volume conservation of the fluids within the mixture, ensuring numerical stability and physical realism. Our robust coupling mechanism enables the simulation of complex phenomena such as sand mobilization, transport, settling, and erosion across a wide range of density ratios. We demonstrate the efficiency of our method through several challenging scenarios, including the breaching of sand-walled basins, sediment-laden flows, and the erosive collapse of sand structures.

Volume-Preserving LBM-MPM Coupling for Air-Water-Sand Mixtures

Juntian Zheng, Zhaofeng Luo, Minchen Li

We introduce a barrier-free optimization framework for non-penetration elastodynamic simulation that matches the robustness of Incremental Potential Contact (IPC) while overcoming its two primary efficiency bottlenecks: (1) reliance on logarithmic barrier functions to enforce non-penetration constraints, which leads to ill-conditioned systems and significantly slows down the convergence of iterative linear solvers; and (2) the time-of-impact (TOI) locking issue, which restricts active-set exploration in collision-intensive scenes and requires a large number of Newton iterations. We propose a novel second-order constrained optimization framework featuring a custom augmented Lagrangian solver that avoids TOI locking by immediately incorporating all requisite contact pairs detected via CCD, enabling more efficient active-set exploration and leading to significantly fewer Newton iterations. By adaptively updating Lagrange multipliers rather than increasing penalty stiffness, our method prevents stagnation at zero TOI while maintaining a well-conditioned system. We further introduce a constraint filtering and decay mechanism to keep the active set compact and stable. A comprehensive set of experiments demonstrates the efficiency, robustness, finite-step termination, and first-order time integration accuracy of our method under a cumulative TOI-based termination criterion. With a GPU-optimized simulator design, our method achieves an up to 103x speedup over GIPC on challenging, contact-rich benchmarks – scenarios that were previously tractable only with barrier-based methods.

Robust and Efficient Penetration-Free Elastodynamics without Barriers

Alvin Shi, Florence Bertails-Descoubes, A.M. Darke, Theodore Kim

Due to its highly curved geometry, tightly coiled hair is challenging to model and edit using standard position-based tools. In this work we propose using material curvatures and twists to analyze and edit tightly coiled hair styles. Our method relies on the geometry of super-helices, primitives parametrized by piecewise constant curvatures and twists, whose helical geometry naturally resembles a coiled hair strand. Using this curvature/twist space, we introduce new editing tools that allow us to expand, shrink, “ruffle”, interpolate or guide the position of coiled hair in a natural way. We present analytical expressions for geometry and gradients that allow our method to run efficiently and without the need for any training data. We successfully apply our tools to highly coiled simulated hairs, as well as those generated procedurally.

Curvature Space Editing of Highly-Coiled Hair

Runze Zhang, Bo Ren

The intricate motion arising from fluid–boundary interactions is visually compelling, yet notoriously difficult and computationally expensive to simulate in the presence of complex boundaries. Accurately resolving boundary geometry requires body-fitted grids constructed via cut-cell methods, which often leads to poorly conditioned linear systems in the pressure projection stage and, consequently, prohibitive computational cost. We present FastVEM, an efficient boundary-conforming fluid simulation framework that enables high-fidelity flow–boundary interaction at substantially reduced cost. Computational efficiency is achieved through a coordinated, top-down design spanning numerical discretization, grid construction, and linear solvers. FastVEM adopts a Virtual Element Method (VEM) discretization to robustly
enforce incompressibility and boundary conditions on irregular body-fitted grids, and employs a VEM polynomial-space Particle-in-Cell scheme for advection. Complementing this discretization, a convexity-preserving cut-cell strategy is introduced to construct simulation-friendly body-fitted grids. To accelerate pressure projection, we develop a Galerkin geometric multigrid solver featuring a diffusion-free prolongation operator that prevents coarse-level matrix densification, along with a nested, boundary-aware grid hierarchy that ensures well-posed placement of coarse-level degrees of freedom. Compared to prior cut-cell–based fluid simulators, FastVEM speeds up the computationally dominant pressure projection stage by up to 100×, while robustly handling even more challenging boundary geometries.

Fast VEM Fluid Simulation

Ruolan Li, Yanrui Xu, Yalan Zhang, and Jiri Kosinka, Alexandru C. Telea, Jian Chang, Jian Jun Zhang, Xiaojuan Ban, Xiaokun Wang

Simulating the interactions between fluids and porous media has attracted significant attention in computer graphics. A key challenge in this domain is modeling the Poro-Elasto-Capillary (PEC) coupling effect which describes the intricate interplay of three physical phenomena in soft porous materials: pore-structure evolution, elastic deformation, and wetting driven by capillary pressure. These phenomena collectively govern dynamic behavior such as the softening and fracturing of biscuits upon water absorption or the swelling of cellulose sponges due to liquid infiltration. Most existing simulation methods model porous media either as static grids or as solid particles with augmented water content attributes, failing to capture the full spectrum of PEC-driven effects due to the lack of physical modeling for elasticity, dynamic porosity changes, and capillary interactions. We propose a multiphase particle-based framework to holistically simulate PEC coupling effects with porous media. We develop a physics-driven model that captures elasticity and dynamic pore-structure evolution under capillary action, enabling realistic simulation of softening and swelling. We derive a saturation-aware pressure Poisson equation to enforce fluid incompressibility within and around the porous medium, ensuring accurate capillary-driven flow while preserving mass and momentum. Finally, we propose a representative elementary volume-based formulation to unify the modeling of homogeneous macro-porous media and cavity-embedded structures, enhancing the representation of pore-scale PEC effects. Comparisons with prior work and real footage show the advantages of our approach in achieving visually realistic fluid-porous media interactions.

Multiphase Particle-Based Simulation of Poro-Elasto-Capillary Effects