Abstract We introduce mixed super‐circles, a position‐curvature formulation of the original dynamic 2D super‐helix model. Compared to the latter, purely curvature‐based model – the so‐called chained formulation –, the mixed formulation that we propose here drastically reduces the algorithmic complexity of the solving scheme – from quadratic to quasi‐linear – and simplifies the handling of positional constraints, including contacts. As such, it recovers the main advantages of classical position‐based models, while at the same time preserving the high‐order convergence of curvature‐based models, hence offering an interesting trade‐off. Furthermore, the smooth, piecewise circular arc representation of super‐circles allows to avoid the spurious jumps in contact forces that are difficult to get rid of with position‐based models. Our model is validated quantitatively against demanding mechanical tests involving contact, friction, snapping, and vibrations. Moreover, its versatility, robustness and efficiency are demonstrated through several interactive dynamic scenarios featuring multiple planar elastic rods subject to various types of boundary conditions and constraints. The corresponding source code, Circonflex, is freely delivered to the research community under the GNU GPL v3 licence.

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 article, starting from the reduced developable ribbon models [Sadowsky 1929 ; Wunderlich 1962 ] recently popularised in Soft Matter Physics, we propose a both accurate and efficient algorithm for computing the statics of thin elastic ribbons. 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 high-order 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 Elastic 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 [Audoly and Neukirch 2021 ], 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.

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 and versatility, 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 artifacts in the force response that are tightly related to the low-order geometry used for contact detection. Our analysis draws upon thorough comparisons between the high-order super-helix model and the low-order discrete elastic rod model. These reveal that when coupled to a low-order, segment-based detection scheme, both models yield spurious jumps in the contact force profile. Moreover, these artifacts are shown to be all the more visible as the geometry of fibres at contact is curved. In order to remove such artifacts we develop an accurate high-order detection scheme between two smooth curves, which relies 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 during sliding motion than with a classical segment-based strategy. Furthermore, we show that 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.

Measuring contact friction in soft-bodies usually requires a specialised physics bench and a tedious acquisition protocol. This makes the prospect of a purely non-invasive, video-based measurement technique particularly attractive. Previous works have shown that such a video-based estimation is feasible for material parameters using deep learning, but this has never been applied to the friction estimation problem which results in even more subtle visual variations. Because acquiring a large dataset for this problem is impractical, generating it from simulation is the obvious alternative. However, this requires the use of a frictional contact simulator whose results are not only visually plausible, but physically-correct enough to match observations made at the macroscopic scale. In this paper, which is an extended version of our former work A. H. Rasheed, V. Romero, F. Bertails-Descoubes, S. Wuhrer, J.-S. Franco, and A Lazarus, "Learning to measure the static friction coefficient in cloth contact," in Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit., 2020, pp. 9909-9918, we propose to our knowledge the first non-invasive measurement network and adjoining synthetic training dataset for estimating cloth friction at contact, for both cloth-hard body and cloth-cloth contacts. To this end we build a protocol for validating and calibrating a state-of-the-art frictional contact simulator, in order to produce a reliable dataset. We furthermore show that without our careful calibration procedure, the training fails to provide accurate estimation results on real data. We present extensive results on a large acquired test set of several hundred real video sequences of cloth in friction, which validates the proposed protocol and its accuracy.

We introduce a selected set of protocols inspired from the Soft Matter Physics community in order to validate Computer Graphics simulators of slender elastic structures possibly subject to dry frictional contact. Although these simulators were primarily intended for feature film animation and visual effects, they are more and more used as virtual design tools for predicting the shape and deformation of real objects; hence the need for a careful, quantitative validation. Our tests, experimentally verified, are designed to evaluate carefully the predictability of these simulators on various aspects, such as bending elasticity, bend-twist coupling, and frictional contact. We have passed a number of popular codes of Computer Graphics through our benchmarks by defining a rigorous, consistent, and as fair as possible methodology. Our results show that while some popular simulators for plates/shells and frictional contact fail even on the simplest scenarios, more recent ones, as well as well-known codes for rods, generally perform well and sometimes even better than some reference commercial tools of Mechanical Engineering. To make our validation protocols easily applicable to any simulator, we provide an extensive description of our methodology, and we shall distribute all the necessary model data to be compared against.

Measuring friction coefficients between cloth and an external body is a longstanding issue in mechanical engineering, never yet addressed with a pure vision-based system. The latter offers the prospect of simpler, less invasive friction measurement protocols compared to traditional ones, and can vastly benefit from recent deep learning advances. Such a novel measurement strategy however proves challenging, as no large labelled dataset for cloth contact exists, and creating one would require thousands of physics workbench measurements with broad coverage of cloth-material pairs. Using synthetic data instead is only possible assuming the availability of a soft-body mechanical simulator with true-to-life friction physics accuracy, yet to be verified. We propose a first vision-based measurement network for friction between cloth and a substrate, using a simple and repeatable video acquisition protocol. We train our network on purely synthetic data generated by a state-of-the-art frictional contact simulator, which we carefully calibrate and validate against real experiments under controlled conditions. We show promising results on a large set of contact pairs between real cloth samples and various kinds of substrates, with 93.6% of all measurements predicted within 0.1 range of standard physics bench measurements.

Projective dynamics was introduced a few years ago as a fast method to yield an approximate yet stable solution to the dynamics of nodal systems subject to stiff internal forces. Previous attempts to include contact forces in that framework considered adding a quadratic penalty energy to the global system, which however broke the simple - constant matrix - structure of the global linear equation, while failing to treat contact in an implicit manner. In this paper we propose a simple yet effective method to integrate in a unified and semi-implicit way contact as well as dry frictional forces into the nested architecture of Projective dynamics. Assuming that contacts apply to nodes only, the key is to split the global matrix into a diagonal and a positive matrix, and use this splitting in the local step so as to make a good prediction of frictional contact forces at next iteration. Each frictional contact force is refined independently in the local step, while the original efficient structure of the global step is left unchanged. We apply our algorithm to cloth simulation and show that contact and dry friction can be captured at a reasonable precision within a few iterations only, hence one order of magnitude faster compared to global implicit contact solvers of the literature.

Cloth dynamics plays an important role in the visual appearance of moving characters. Properly accounting for contact and friction is of utmost importance to avoid cloth-body and cloth-cloth penetration and to capture typical folding and stick-slip behavior due to dry friction. We present here the first method able to account for cloth contact with exact Coulomb friction, treating both cloth self-contacts and contacts occurring between the cloth and an underlying character. Our key contribution is to observe that for a nodal system like cloth, the frictional contact problem may be formulated based on velocities as primary variables, without having to compute the costly Delassus operator. Then, by reversing the roles classically played by the velocities and the contact impulses, conical complementarity solvers of the literature can be adapted to solve for compatible velocities at nodes. To handle the full complexity of cloth dynamics scenarios, we have extended this base algorithm in two ways: first, towards the accurate treatment of frictional contact at any location of the cloth, through an adaptive node refinement strategy; second, towards the handling of multiple constraints at each node, through the duplication of constrained nodes and the adding of pin constraints between duplicata. Our method allows us to handle the complex cloth-cloth and cloth-body interactions in full-size garments with an unprecedented level of realism compared to former methods, while maintaining reasonable computational timings.

We propose an inverse strategy for modeling thin elastic shells physically, just from the observation of their geometry. Our algorithm takes as input an arbitrary target mesh, and interprets this configuration automatically as a stable equilibrium of a shell simulator under gravity and frictional contact constraints with a given external object. Unknowns are the natural shape of the shell (i.e., its shape without external forces) and the frictional contact forces at play, while the material properties (mass density, stiffness, friction coefficients) can be freely chosen by the user. Such an inverse problem formulates as an ill-posed nonlinear system subject to conical constraints. To select and compute a plausible solution, our inverse solver proceeds in two steps. In a first step, contacts are reduced to frictionless bilateral constraints and a natural shape is retrieved using the adjoint method. The second step uses this result as an initial guess and adjusts each bilateral force so that it projects onto the admissible Coulomb friction cone, while preserving global equilibrium. To better guide minimization towards the target, these two steps are applied iteratively using a degressive regularization of the shell energy. We validate our approach on simulated examples with reference material parameters, and show that our method still converges well for material parameters lying within a reasonable range around the reference, and even in the case of arbitrary meshes that are not issued from a simulation. We finally demonstrate practical inversion results on complex shell geometries freely modeled by an artist or automatically captured from real objects, such as posed garments or soft accessories.

The papers in this special section were presented at the 14th Annual ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA 2015), which was held in Los Angeles, California on August 7-9, 2015.

We present a new continuum-based method for the realistic simulation of large-scale free-flowing granular materials. We derive a compact model for the rheology of the material, which accounts for the exact nonsmooth Drucker-Prager yield criterion combined with a varying volume fraction. Thanks to a semi-implicit time-stepping scheme and a careful spatial discretization of our rheology built upon the Material-Point Method, we are able to preserve at each time step the exact coupling between normal and tangential stresses, in a stable way. This contrasts with previous approaches which either regularize or linearize the yield criterion for implicit integration, leading to unrealistic behaviors or visible grid artifacts. Remarkably, our discrete problem turns out to be very similar to the discrete contact problem classically encountered in multibody dynamics, which allows us to leverage robust and efficient nonsmooth solvers from the literature. We validate our method by successfully capturing typical macroscopic features of some classical experiments, such as the discharge of a silo or the collapse of a granular column. Finally, we show that our method can be easily extended to accommodate more complex scenarios including two-way rigid body coupling as well as anisotropic materials.

Thin elastic filaments in real world such as vine tendrils, hair ringlets or curled ribbons often depict a very smooth, curved shape that low-order rod models --- e.g., segment-based rods --- fail to reproduce accurately and compactly. In this paper, we push forward the investigation of high-order models for thin, inextensible elastic rods by building the dynamics of a G 2 -continuous piecewise 3D clothoid: a smooth space curve with piecewise affine curvature. With the aim of precisely integrating the rod kinematic problem, for which no closed-form solution exists, we introduce a dedicated integration scheme based on power series expansions. It turns out that our algorithm reaches machine precision orders of magnitude faster compared to classical numerical integrators. This property, nicely preserved under simple algebraic and differential operations, allows us to compute all spatial terms of the rod kinematics and dynamics in both an efficient and accurate way. Combined with a semi-implicit time-stepping scheme, our method leads to the efficient and robust simulation of arbitrary curly filaments that exhibit rich, visually pleasing configurations and motion. Our approach was successfully applied to generate various scenarios such as the unwinding of a curled ribbon as well as the aesthetic animation of spiral-like hair or the fascinating growth of twining plants.

In the latest years, considerable progress has been achieved for accurately acquiring the geometry of human hair, thus largely improving the realism of virtual characters. In parallel, rich physics-based simulators have been successfully designed to capture the intricate dynamics of hair due to contact and friction. However, at the moment there exists no consistent pipeline for converting a given hair geometry into a realistic physics-based hair model. Current approaches simply initialize the hair simulator with the input geometry in the absence of external forces. This results in an undesired sagging effect when the dynamic simulation is started, which basically ruins all the efforts put into the accurate design and/or capture of the input hairstyle. In this paper we propose the first method which consistently and robustly accounts for surrounding forces---gravity and frictional contacts, including hair self-contacts---when converting a geometric hairstyle into a physics-based hair model. Taking an arbitrary hair geometry as input together with a corresponding body mesh, we interpret the hair shape as a static equilibrium configuration of a hair simulator, in the presence of gravity as well as hair-body and hair-hair frictional contacts. Assuming that hair parameters are homogeneous and lie in a plausible range of physical values, we show that this large underdetermined inverse problem can be formulated as a well-posed constrained optimization problem, which can be solved robustly and efficiently by leveraging the frictional contact solver of the direct hair simulator. Our method was successfully applied to the animation of various hair geometries, ranging from synthetic hairstyles manually designed by an artist to the most recent human hair data automatically reconstructed from capture.

Abstract Piecewise clothoids are 2D curves with continuous, piecewise linear curvature. Due to their smoothness properties, they have been extensively used in road design and robot path planning, as well as for the compact representation of hand‐drawn curves. In this paper we present the Super‐Clothoid model, a new mechanical model that for the first time allows for the computing of the dynamics of an elastic, inextensible piecewise clothoid. We first show that the kinematics of this model can be computed analytically depending on the Fresnel integrals, and precisely evaluated when required. Secondly, the discrete dynamics, naturally emerging from the Lagrange equations of motion, can be robustly and efficiently computed by performing and storing formal computations as far as possible, recoursing to numerical evaluation only when assembling the linear system to be solved at each time step. As a result, simulations turn out to be both interactive and stable, even for large displacements of the rod. Finally, we demonstrate the versatility of our model by handling various boundary conditions for the rod as well as complex external constraints such as frictional contact, and show that our model is perfectly adapted to inverse statics. Compared to lower‐order models, the super‐clothoid appears as a more natural and aesthetic primitive for bridging the gap between 2D geometric design and physics‐based deformation.

We focus on the challenging problem of simulating thin elastic rods in contact, in the presence of friction. Most previous approaches in computer graphics rely on a linear complementarity formulation for handling contact in a stable way, and approximate Coulombs's friction law for making the problem tractable. In contrast, following the seminal work by Alart and Curnier in contact mechanics, we simultaneously model contact and exact Coulomb friction as a zero finding problem of a nonsmooth function. A semi-implicit time-stepping scheme is then employed to discretize the dynamics of rods constrained by frictional contact: this leads to a set of linear equations subject to an equality constraint involving a nondifferentiable function. To solve this one-step problem we introduce a simple and practical nonsmooth Newton algorithm which proves to be reasonably efficient and robust for systems that are not overconstrained. We show that our method is able to finely capture the subtle effects that occur when thin elastic rods with various geometries enter into contact, such as stick-slip instabilities in free configurations, entangling curls, resting contacts in braid-like structures, or the formation of tight knots under large constraints. Our method can be viewed as a first step towards the accurate modeling of dynamic fibrous materials.

Dry friction between hair fibers plays a major role in the collective hair dynamic behavior as it accounts for typical nonsmooth features such as stick-slip instabilities. However, due the challenges posed by the modeling of nonsmooth friction, previous mechanical models for hair either neglect friction or use an approximate smooth friction model, thus losing important visual features. In this paper we present a new generic robust solver for capturing Coulomb friction in large assemblies of tightly packed fibers such as hair. Our method is based on an iterative algorithm where each single contact problem is efficiently and robustly solved by introducing a hybrid strategy that combines a new zero-finding formulation of (exact) Coulomb friction together with an analytical solver as a fail-safe. Our global solver turns out to be very robust and highly scalable as it can handle up to a few thousand densely packed fibers subject to tens of thousands frictional contacts at a reasonable computational cost. It can be conveniently combined to any fiber model with various rest shapes, from smooth to curly. Our results, visually validated against real hair motions, depict typical hair collective effects and greatly enhance the realism of standard hair simulators.

2d animation is a traditional but fascinating domain that has recently regained popularity both in animated movies and video games. This paper introduces a method for automatically converting a smooth sketched curve into a 2d dynamic curve at stable equilibrium under gravity. The curve can then be physically animated to produce secondary motions in 2d animations or simple video games. Our approach proceeds in two steps. We first present a new technique to fit a smooth piecewise circular arcs curve to a sketched curve. Then we show how to compute the physical parameters of a dynamic rod model (super-circle) so that its stable rest shape under gravity exactly matches the fitted circular arcs curve. We demonstrate the interactivity and controllability of our approach on various examples where a user can intuitively setup efficient and precise 2d animations by specifying the input geometry.

Abstract Thin elastic rods such as cables, phone coils, tree branches, or hair, are common objects in the real world but computing their dynamics accurately remains challenging. The recent Super‐Helix model, based on the discrete equations of Kirchhoff for a piecewise helical rod, is one of the most promising models for simulating non‐stretchable rods that can bend and twist. However, this model suffers from a quadratic complexity in the number of discrete elements, which, in the context of interactive applications, makes it limited to a few number of degrees of freedom ‐ or equivalently to a low number of variations in curvature along the mean curve. This paper proposes a new, recursive scheme for the dynamics of a Super‐Helix, inspired by the popular algorithm of Featherstone for serial multibody chains. Similarly to Featherstone's algorithm, we exploit the recursive kinematics of a Super‐Helix to propagate elements inertias from the free end to the clamped end of the rod, while the dynamics is solved within a second pass traversing the rod in the reverse way. Besides the gain in linear complexity, which allows us to simulate a rod of complex shape much faster than the original approach, our algorithm makes it straightforward to simulate tree‐like structures of Super‐Helices, which turns out to be particularly useful for animating trees and plants realistically, under large displacements.

This paper explores a sketch-based interface for quickly yet accurately creating visually realistic hair for virtual characters. Recently, physically-based models have proved successful for generating a wide variety of hair types, but they do not provide a straightforward method for designing target hairstyles. The contribution of this paper is to propose a user-friendly method for controlling such a physically-based model, requiring no specific knowledge of mechanics or hair styling: the user sketches example hair strands over a side view of the character's head, or alternatively annotates a picture of real hair viewed from the side serving as a reference. We show how the sketch can be used to infer the geometric and mechanical properties of the hair strands, to adjust the shape of the scalp, and to generate an adequate hair volume. Our method is validated on a wide variety of hair styles, from straight to curly and short to long hair.

Physical simulation has emerged as a compelling animation technique, yet current approaches to coupling simulations of fluids and solids with irregular boundary geometry are inefficient or cannot handle some relevant scenarios robustly. We propose a new variational approach which allows robust and accurate solution on relatively coarse Cartesian grids, allowing possibly orders of magnitude faster simulation. By rephrasing the classical pressure projection step as a kinetic energy minimization, broadly similar to modern approaches to rigid body contact, we permit a robust coupling between fluid and arbitrary solid simulations that always gives a well-posed symmetric positive semi-definite linear system. We provide several examples of efficient fluid-solid interaction and rigid body coupling with sub-grid cell flow. In addition, we extend the framework with a new boundary condition for free-surface flow, allowing fluid to separate naturally from solids.

Realistic hair modeling is a fundamental part of creating virtual humans in computer graphics. This paper surveys the state of the art in the major topics of hair modeling: hairstyling, hair simulation, and hair rendering. Because of the difficult, often unsolved problems that arise in all these areas, a broad diversity of approaches are used, each with strengths that make it appropriate for particular applications. We discuss each of these major topics in turn, presenting the unique challenges facing each area and describing solutions that have been presented over the years to handle these complex issues. Finally, we outline some of the remaining computational challenges in hair modeling.

Simulating human hair is recognized as one of the most difficult tasks in computer animation. In this paper, we show that the Kirchhoff equations for dynamic, inextensible elastic rods can be used for accurately predicting hair motion. These equations fully account for the nonlinear behavior of hair strands with respect to bending and twisting. We introduce a novel deformable model for solving them: each strand is represented by a Super-Helix , i.e., a piecewise helical rod which is animated using the principles of Lagrangian mechanics. This results in a realistic and stable simulation, allowing large time steps. Our second contribution is an in-depth validation of the Super-Helix model, carried out through a series of experiments based on the comparison of real and simulated hair motions. We show that our model efficiently handles a wide range of hair types with a high level of realism.