We introduce a practical iterative solver for mass-spring systems which can be trivially mapped to massively parallel architectures, in particular GPUs.We employ our solver for the interactive animation of virtual cloth and show that it is computationally fast, robust and scalable, making it suitable for real-time graphics applications. Under the assumption that the input system is represented by a quadrangular network of masses connected by springs, we first partition the particles into two independent sets. Then, during the animation, the dynamics of all the particles belonging to each set is computed in parallel. This enables a full Gauss-Seidel iteration in just two parallel steps, leading to an approximated solution of large mass-spring systems in a few milliseconds. We use our solver to accelerate the solution of the popular Projective Dynamics framework, and compare it with other common iterative solvers in the current literature.

Fast Quadrangular Mass-Spring Systems using Red-Black Ordering

]]>Damping is an important ingredient in physics-based simulation of deformable objects. Recent work introduced new fast simulation methods such as Position Based Dynamics and Projective Dynamics. Explicit velocity damping methods currently used in conjunction with Position Based Dynamics or Projective Dynamics are simple and fast, but have some limitations. They may damp global motion or non-physically transport velocities throughout the simulated object. More advanced damping models do not have these limitations, but are slow to evaluate, defeating the benefits of fast solvers such as Projective Dynamics. We present a new type of damping model specifically designed for Projective Dynamics, which provides the quality of advanced damping models while adding only minimal computing overhead. The key idea is to define damping forces using Projective Dynamics’ Laplacian matrix. In a number of simulation examples we show that this damping model works very well in practice. When used with a modified Projective Dynamics solver that uses a non-dissipative implicit midpoint integrator, our damping method provides fully user-controllable damping, allowing the user to quickly produce visually pleasing and vivid animations.

]]>Peridynamics is a formulation of the classical elastic theory that is targeted at simulating deformable objects with discon-

tinuities, especially fractures. Till now, there are few studies that have been focused on how to model general hyperelastic

materials with peridynamics. In this paper, we target at proposing a general strain energy function of hyperelastic materials

for peridynamics. To get an intuitive model that can be easily controlled, we formulate the strain energy density function as a

function parameterized by the dilatation and bond stretches, which can be decomposed into multiple one-dimensional functions

independently. To account for nonlinear material behaviors, we also propose a set of nonlinear basis functions to help design

a nonlinear strain energy function more easily. For an anisotropic material, we additionally introduce an anisotropic kernel to

control the elastic behavior for each bond independently. Experiments show that our model is flexible enough to approximately

regenerate various hyperelastic materials in classical elastic theory, including St.Venant-Kirchhoff and Neo-Hookean materi

Reformulating Hyperelastic Materials with Peridynamic Modeling

]]>We propose a method for accurately simulating dissipative forces in deformable bodies when using optimization-based integrators. We represent such forces using dissipation functions which may be nonlinear in both positions and velocities, enabling us to model a range of dissipative effects including Coulomb friction, Rayleigh damping, and power-law dissipation. We propose a general method for incorporating dissipative forces into optimization-based time integration schemes, which hitherto have been applied almost exclusively to systems with only conservative forces. To improve accuracy and minimize artificial damping, we provide an optimization-based version of the second-order accurate TR-BDF2 integrator. Finally, we present a method for modifying arbitrary dissipation functions to conserve linear and angular momentum, allowing us to eliminate the artificial angular momentum loss caused by Rayleigh damping.

]]>We present a strong fluid-rigid coupling for SPH fluids and rigid bodies with particle-sampled surfaces. The approach interlinks the iterative pressure update at fluid particles with a second SPH solver that computes artificial pressure at rigid body particles. The introduced SPH rigid body solver models rigid-rigid contacts as artificial density deviations at rigid body particles. The corresponding pressure is iteratively computed by solving a global formulation which is particularly useful for large numbers of rigid-rigid contacts. Compared to previous SPH coupling methods, the proposed concept stabilizes the fluid-rigid interface handling. It significantly reduces the computation times of SPH fluid simulations by enabling larger time steps. Performance gain factors of up to 58 compared to previous methods are presented. We illustrate the flexibility of the presented fluid-rigid coupling by integrating it into DFSPH, IISPH and a recent SPH solver for highly viscous fluids. We further show its applicability to a recent SPH solver for elastic objects. Large scenarios with up to 90M particles of various interacting materials and complex contact geometries with up to 90k rigid-rigid contacts are shown. We demonstrate the competitiveness of our proposed rigid body solver by comparing it to Bullet.

Interlinked SPH Pressure Solvers for Strong Fluid-Rigid Coupling

]]>Accurate high-resolution simulation of cloth is a highly desired computational tool in graphics applications. As single resolution simulation starts to reach the limit of computational power, we believe the future of cloth simulation is in multi-resolution simulation. In this paper, we explore nonlinearity, adaptive smoothing, and parallelization under a full multigrid (FMG) framework. The foundation of this research is a novel nonlinear FMG method for unstructured meshes. To introduce nonlinearity into FMG, we propose to formulate the smoothing process at each resolution level as the computation of a search direction for the original high-resolution nonlinear optimization problem. We prove that our nonlinear FMG is guaranteed to converge under various conditions and we investigate the improvements to its performance. We present an adaptive smoother which is used to reduce the computational cost in the regions with low residuals already. Compared to normal iterative solvers, our nonlinear FMG method provides faster convergence and better performance for both Newton’s method and Projective Dynamics. Our experiment shows our method is efficient, accurate, stable against large time steps, and friendly with GPU parallelization. The performance of the method has a good scalability to the mesh resolution, and the method has good potential to be combined with multi-resolution collision handling for real-time simulation in the future.

]]>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.

]]>We present a new structure-preserving numerical scheme for solving the Euler–Poincaré Differential (EPDiff) equation on arbitrary triangle meshes. Unlike existing techniques, our method solves the difficult non-linear EPDiff equation by constructing energy preserving, yet fully explicit, update rules. Our approach uses standard differential operators on triangle meshes, allowing for a simple and efficient implementation. Key to the structure-preserving features that our method exhibits is a novel numerical splitting scheme. Namely, we break the integration into three steps which rely on linear solves with a fixed sparse matrix that is independent of the simulation and thus can be pre-factored. We test our method in the context of simulating concentrated reconnecting wavefronts on flat and curved domains. In particular, EPDiff is known to generate geometrical fronts which exhibit wave-like behavior when they interact with each other. In addition, we also show that at a small additional cost, we can produce globally-supported periodic waves by using our simulated fronts with wavefronts tracking techniques. We provide quantitative graphs showing that our method exactly preserves the energy in practice. In addition, we demonstrate various interesting results including annihilation and recreation of a circular front, a wave splitting and merging when hitting an obstacle and two separate fronts propagating and bending due to the curvature of the domain.

An Explicit Structure Preserving numerical scheme for EPDiff

]]>We propose a novel discrete scheme for simulating viscous thin films at real-time frame rates. Our scheme is based on a new formulation of the gradient flow approach, that leads to a discretization based on local stencils that are easily computable on the GPU. Our approach has physical fidelity, as the total mass is guaranteed to be preserved, an appropriate discrete energy is controlled, and the film height is guaranteed to be non-negative at all times. In addition, and unlike all existing methods for thin films simulation, it is fast enough to allow realtime interaction with the flow, for designing initial conditions and controlling the forces during the simulation.

]]>Many strategies exist for optimizing non-linear distortion energies in geometry and physics applications, but devising an approach that achieves the convergence promised by Newton-type methods remains challenging. In order to guarantee the positive semi-definiteness required by these methods, a numerical eigendecomposition or approximate regularization is usually needed. In this paper, we present analytic expressions for the eigensystems at each quadrature point of a wide range of isotropic distortion energies. These systems can then be used to project energy Hessians to positive semi-definiteness analytically. Unlike previous attempts, our formulation provides compact expressions that are valid both in 2D and 3D, and does not introduce spurious degeneracies. At its core, our approach utilizes the invariants of the stretch tensor that arises from the polar decomposition of the deformation gradient. We provide closed-form expressions for the eigensystems for all these invariants, and use them to systematically derive the eigensystems of any isotropic energy. Our results are suitable for geometry optimization over flat surfaces or volumes, and agnostic to both the choice of discretization and basis function. To demonstrate the efficiency of our approach, we include comparisons against existing methods on common graphics tasks such as surface parameterization and volume deformation.

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