Physics-Based Animation

Xudong Feng, Huamin Wang, Yin Yang, Weiwei Xu

Real-world fabrics, composed of threads and yarns, often display complex stress-strain relationships, making their homogenization a challenging task for fast simulation by continuum-based models. Consequently, existing homogenized yarn-level models frequently struggle with numerical stability without line search at large time steps, forcing a trade-off between model accuracy and stability. In this paper, we propose a neural-assisted homogenized constitutive model for simulating yarn-level cloth. Unlike analytic models, a neural model is advantageous in adapting to complex dynamic behaviors, and its inherent smoothness naturally mitigates stability issues. We also introduce a sector-based warm-start strategy to accelerate the data collection process in homogenization. This model is trained using collected strain energy datasets and its accuracy is validated through both qualitative and quantitative experiments. Thanks to our model’s stability, our simulator can now achieve two-orders-of-magnitude speedups with large time steps compared to previous models.

Neural-Assisted Homogenization of Yarn-Level Cloth

Jean Jouve, Victor Romero, Rahul Narain, Laurence Boissieux, Theodore Kim, Florence Bertails-Descoubes

Feathers exhibit a highly anisotropic behaviour, governed by their complex hierarchical microstructure composed of individual hairs (barbs) clamped onto a spine (rachis) and attached to each other through tiny hooks (barbules). Previous methods in computer graph- ics have approximated feathers as strips of cloth, thus failing to cap- ture the particular macroscopic nonlinear behaviour of the feather surface (vane). To investigate the anisotropic properties of a feather vane, we design precise measurement protocols on real feather samples. Our experimental results suggest a linear strain-stress relationship of the feather membrane with orientation-dependent coefficients, as well as an extreme ratio of stiffnesses in the barb and barbule direction, of the order of 10 4 . From these findings we build a simple continuum model for the feather vane, where the vane is represented as a three-parameter anisotropic elastic shell. However, implementing the model numerically reveals severe lock- ing and ill-conditioning issues, due to the extreme stiffness ratio between the barb and the barbule directions. To resolve these is- sues, we align the mesh along the barb directions and replace the stiffest modes with an inextensibility constraint. We extensively validate our membrane model against real-world laboratory mea- surements, by using an intermediary microscale model that allows us to limit the number of required lab experiments. Finally, we enrich our membrane model with anisotropic bending, and show its practicality in graphics-like scenarios like a full feather and a larger-scale bird. Code and data for this paper are available at https://gitlab.inria.fr/elan-public-code/feather-shell/.

Modelling a feather as a strongly anisotropic elastic shell

Raphaël Charrondière, Sébastien Neukirch, Florence Bertails-Descoubes

Thin elastic ribbons represent a class of intermediary objects lying in-between thin elastic plates and thin elastic rods. Although the two latter families of thin structures have received much interest from the Computer Graphics community over the last decades, ribbons have seldom been considered and modelled numerically so far, in spite of a growing number of applications in Computer Design. In this paper, starting from the reduced developable ribbon models recently popularised in Soft Matter Physics, we propose a both accurate and efficient algorithm for computing the statics of a thin elastic ribbon. Inspired by the super-clothoid model for thin elastic rods, our method relies on compact ribbon elements whose normal curvature varies linearly with respect to arc length s, while their geodesic torsion is quadratic in s. In contrast however, for the sake of efficiency our algorithm avoids building a fully reduced kinematic chain and instead treats each element independently, gluing them only at the final solving stage through well-chosen bilateral constraints. Thanks to this mixed variational strategy, which yields a banded Hessian, our algorithm recovers the linear complexity of low-order models while preserving the quadratic convergence of curvature-based models. As a result, our approach is scalable to a large number of elements, and suitable for various boundary conditions and unilateral contact constraints, making it possible to handle challenging scenarios such as confined buckling experiments or Möbius bands with contact. Remarkably, our mixed algorithm proves an order of magnitude faster compared to Discrete Element Ribbon models of the literature while achieving, in a few seconds only, high accuracy levels that remain out of reach for such low-order models. Additionally, our numerical model can incorporate various ribbon energies, including the Ribext model for quasi-developable ribbons recently introduced in Physics, which allows to transition smoothly between a rectangular Kirchhoff rod and a (developable) Sadowsky ribbon. Our numerical scheme is carefully validated against demanding experiments of the Physics literature, which demonstrates its accuracy, efficiency, robustness, and versatility. Our MERCI code is publicly available at https://gitlab.inria.fr/elan-public-code/merci for the sake of reproducibility and future benchmarking.

Merci: Mixed curvature-based elements for computing equilibria of thin elastic ribbons

Zhiqi Li, Barnabás Börcsök, Duowen Chen, Yutong Sun, Bo Zhu, Greg Turk,

This paper introduces a novel Lagrangian fluid solver based on covector flow maps. We aim to address the challenges of establishing a robust flow-map solver for incompressible fluids under complex boundary conditions. Our key idea is to use particle trajectories to establish precise flow maps and tailor path integrals of physical quantities along these trajectories to reformulate the Poisson problem during the projection step. We devise a decoupling mechanism based on path-integral identities from flow-map theory. This mechanism integrates long-range flow maps for the main fluid body into a short-range projection framework, ensuring a robust treatment of free boundaries. We show that our method can effectively transform a long-range projection problem with integral boundaries into a Poisson problem with standard boundary conditions — specifically, zero Dirichlet on the free surface and zero Neumann on solid boundaries. This transformation significantly enhances robustness and accuracy, extending the applicability of flow-map methods to complex free-surface problems.

Lagrangian Covector Fluid with Free Surface

Honglin Chen, Hsueh-Ti Derek Liu, David I.W. Levin, Changxi Zheng, Alec Jacobson

Volume-preserving hyperelastic materials are widely used to model near-incompressible materials such as rubber and soft tissues. However, the numerical simulation of volume-preserving hyperelastic materials is notoriously challenging within this regime due to the non-convexity of the energy function. In this work, we identify the pitfalls of the popular eigenvalue clamping strategy for projecting Hessian matrices to positive semi-definiteness during Newton’s method. We introduce a novel eigenvalue filtering strategy for projected Newton’s method to stabilize the optimization of Neo-Hookean energy and other volume-preserving variants under high Poisson’s ratio (near 0.5) and large initial volume change. Our method only requires a single line of code change in the existing projected Newton framework, while achieving significant improvement in both stability and convergence speed. We demonstrate the effectiveness and efficiency of our eigenvalue projection scheme on a variety of challenging examples and over different deformations on a large dataset.

Stabler Neo-Hookean Simulation: Absolute Eigenvalue Filtering for Projected Newton

Junwei Zhou, Duowen Chen, Molin Deng, Yitong Deng, Yuchen Sun, Sinan Wang, Shiying Xiong, Bo Zhu

We propose a novel Particle Flow Map (PFM) method to enable accurate long-range advection for incompressible fluid simulation. The foundation of our method is the observation that a particle trajectory generated in a forward simulation naturally embodies a perfect flow map. Centered on this concept, we have developed an Eulerian-Lagrangian framework comprising four essential components: Lagrangian particles for a natural and precise representation of bidirectional flow maps; a dual-scale map representation to accommodate the mapping of various flow quantities; a particle-to-grid interpolation scheme for accurate quantity transfer from particles to grid nodes; and a hybrid impulse-based solver to enforce incompressibility on the grid. The efficacy of PFM has been demonstrated through various simulation scenarios, highlighting the evolution of complex vortical structures and the details of turbulent flows. Notably, compared to NFM, PFM reduces computing time by up to 49 times and memory consumption by up to 41%, while enhancing vorticity preservation as evidenced in various tests like
leapfrog, vortex tube, and turbulent flow.

Eulerian-Lagrangian Fluid Simulation on Particle Flow Maps

Octave Crespel , Emile Hohnadel, Thibaut Métivet, Florence Bertails-Descoubes

Computer Graphics has a long history in the design of effective algorithms for handling contact and friction between solid objects. For the sake of simplicity, most methods rely on low-order primitives such as line segments or triangles, both for the detection and the response stages. In this paper we carefully analyse, in the case of fibre systems, the impact of such choices on the retrieved contact forces. We highlight the presence of force artifacts due to the low-order geometry used for contact detection, that appear all the more visible as the fibre geometry at contact is curved. To remove such artifacts we develop an accurate detection scheme between two smooth curves, relying on an efficient adaptive pruning strategy. We use this algorithm to detect contact between super-helices at high precision, allowing us to recover, in the range of wavy to highly curly fibres, much smoother force profiles than with a classical segment-based strategy. Furthermore, our approach offers better scaling properties in terms of efficiency vs. precision compared to segment-based approaches, making it attractive for applications where accurate and reliable forces are desired. Finally, we demonstrate the robustness and accuracy of our fully high-order approach on a challenging hair combing scenario.

Contact detection between curved fibres: high order makes a difference

Yizhou Chen, Yushan Han, Jingyu Chen, Joseph Teran

Position based dynamics is a powerful technique for simulating a variety of materials. Its primary strength is its robustness when run with limited computational budget. We develop a novel approach to address problems with PBD for quasistatic hyperelastic materials. Even though PBD is based on the projection of static constraints, PBD is best suited for dynamic simulations. This is particularly relevant since the efficient creation of large data sets of plausible, but not necessarily accurate elastic equilibria is of increasing importance with the emergence of quasistatic neural networks. Furthermore, PBD projects one constraint at a time. We show that ignoring the effects of neighboring constraints limits its convergence and stability properties. Recent works have shown that PBD can be related to the Gauss-Seidel approximation of a Lagrange multiplier formulation of backward Euler time stepping, where each constraint is solved/projected independently of the others in an iterative fashion. We show that a position-based, rather than constraint-based nonlinear Gauss-Seidel approach solves these problems. Our approach retains the essential PBD feature of stable behavior with constrained computational budgets, but also allows for convergent behavior with expanded budgets. We demonstrate the efficacy of our method on a variety of representative hyperelastic problems and show that both successive over relaxation (SOR) and Chebyshev acceleration can be easily applied.

Position-Based Nonlinear Gauss-Seidel for Quasistatic Hyperelasticity

Leticia Mattos Da Silva, Oded Stein, Justin Solomon

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains

Shusen Liu, Xiaowei He, Yuzhong Guo, Yue Chang, Wencheng Wang

Tensile instability is one of the major obstacles to particle methods in fluid simulation, which would cause particles to clump in pairs under tension and prevent fluid simulation to generate small-scale thin features. To address this issue, previous particle methods either use a background pressure or a finite difference scheme to alleviate the particle clustering artifacts, yet still fail to produce small-scale thin features in free-surface flows. In this paper, we propose a dual-particle approach for simulating incompressible fluids. Our approach involves incorporating supplementary virtual particles designed to capture and store particle pressures. These pressure samples undergo systematic redistribution at each time step, grounded in the initial positions of the fluid particles. By doing so, we effectively reduce tensile instability in standard SPH by narrowing down the unstable regions for particles experiencing tensile stress. As a result, we can accurately simulate free-surface flows with rich small-scale thin features, such as droplets, streamlines, and sheets, as demonstrated by experimental results.

A Dual-Particle Approach for Incompressible SPH Fluids