Shaking a cup of wine or other fluids in virtual environments is engaging but has been limited by challenges in delivering real-time haptic feedback for liquid collisions. ShakeSense is a haptic rendering system that integrates electrotactile stimulation with physics-based simulation to deliver immersive feedback for liquid dynamics in handheld containers. It employs a high-density electrode array to deliver dynamic tactile sensations, conveying friction and pressure changes on the user’s fingerpad. A dedicated end-to-end pipeline computes fingerpad forces from liquid-container-finger interactions, ensuring feedback aligns with natural fluid movement. Two studies evaluated ShakeSense’s performance and user perception. Study 1 showed that electrotactile patterns were distinguishable across directions, and synchronizing container movement with stimulation enhanced perceived force changes. Study 2 demonstrated that ShakeSense effectively simulated liquid motion, capturing multidimensional, coordinated interactions, and outperformed conventional Center-of-Mass approaches. Overall, ShakeSense provides clear, fine-grained tactile feedback for fluid interactions.

Abstract While semi‐analytical boundary handling techniques have proven effective for modeling particle‐based fluid‐solid interactions, they can become unstable when applied to mesh boundaries undergoing dynamic motion or featuring complex, sharp geometries. We propose a novel semi‐analytical energy model for boundary handling that unifies fluid simulation and boundary interactions within a variational framework. The model comprises two key components: a semi‐analytical bulk energy formulation that mitigates particle deficiency issues in the evaluation of bulk energy, and a nonlocal contact potential that effectively prevents particle penetration into boundaries. Both energy terms are naturally compatible with the Semi‐Implicit SPH (SISPH), and a unified Hessian‐free solver combined with reduced‐order collision detection enables an efficient and stable GPU‐based implementation for both fluid dynamics and nonlinear fluid‐solid interactions. Furthermore, the unified treatment of fluid bulk energy and boundary energy via the semi‐analytical formulation robustly corrects penetrations in practice, even under severe compression scenarios involving complex moving boundaries. Compared with existing semi‐analytical boundary treatments, our method is more robust under fast boundary motion and strong compression. Across challenging benchmarks with sharp features, narrow gaps, and moving meshes, it remains stable and penetration‐free where prior methods often fail.

Abstract In simulating fluids using position‐based dynamics, the accuracy and robustness depend on numerous numerical parameters, including the time step size, iteration count, and particle size, among others. This complexity can lead to unpredictable control of simulation behaviors. In this paper, we first reformulate the problem of enforcing fluid compressibility/incompressibility into an nonlinear optimization problem, and then introduce a semi‐implicit successive substitution method (SISSM) to solve the nonlinear optimization problem by adjusting particle positions in parallel. In contrast to calculating an intermediate variable, such as pressure, to enforce fluid incompressibility within the position‐based dynamics (PBD) framework, the proposed semi‐implicit approach eliminates the necessity of such calculations. Instead, it directly employs successive substitution of particle positions to correct density errors. This method exhibits reduced dependency to numerical parameters, such as particle size and time step variations, and improves consistency and stability in simulating fluids that range from highly compressible to nearly incompressible. We validates the effectiveness of applying a variety of different techniques in accelerating the convergence rate.

Prior research has demonstrated the efficacy of balanced trees as spatially adaptive grids for large-scale simulations. However, state-of-the-art methods for balanced tree construction are restricted by the iterative nature of the ripple effect, thus failing to fully leverage the massive parallelism offered by modern GPU architectures. We propose to reframe the construction of balanced trees as a process to merge N -balanced Minimum Spanning Trees ( N -balanced MSTs) generated from a collection of seed points. To ensure optimal performance, we propose a stack-free parallel strategy for constructing all internal nodes of a specified N -balanced MST. This approach leverages two 32-bit integer registers as buffers rather than relying on an integer array as a stack during construction, which helps maintain balanced workloads across different GPU threads. We then propose a dynamic update algorithm utilizing refinement counters for all internal nodes to enable parallel insertion and deletion operations of N -balanced MSTs. This design achieves significant efficiency improvements compared to full reconstruction from scratch, thereby facilitating fluid simulations in handling dynamic moving boundaries. Our approach is fully compatible with GPU implementation and demonstrates up to an order-of-magnitude speedup compared to the state-of-the-art method [Wang et al. 2024]. The source code for the paper is publicly available at https://github.com/peridyno/peridyno.

This paper pushes the performance of cloth simulation, making the simulation interactive even for high-resolution garment models while keeping every triangle untangled. The penetration-free guarantee is inspired by the interior point method, which converts the inequality constraints to barrier potentials. We propose a major overhaul of this modality within the projective dynamics framework by leveraging an adaptive weighting mechanism inspired by barrier formulation. This approach does not depend on the distance between mesh primitives, but on the virtual life span of a collision event and thus keeps all the vertices within feasible region. Such a non-distance barrier model allows a new way to integrate collision resolution into the simulation pipeline. Another contributor to the performance boost comes from the subspace reuse strategy. This is based on the observation that low-frequency strain propagation is near orthogonal to the deformation induced by collisions or self-collisions, often of high frequency. Subspace reuse then takes care of low-frequency residuals, while high-frequency residuals can also be effectively smoothed by GPU-based iterative solvers. We show that our method outperforms existing fast cloth simulators by at least one order while producing high-quality animations of high-resolution models.

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 article, 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.

Real-time simulation of hyperelastic membranes like cloth still faces a lot of challenges, such as hyperplasticity modeling and contact handling. In this study, we propose projective peridynamics that uses a local-global strategy to enable fast and robust simulation of hyperelastic membranes with contact. In the global step, we propose a semi-implicit strategy to linearize the governing equation for hyperelastic materials that are modeled with peridynamics. By decomposing the first Piola-Kirchhoff stress tensor into a positive and a negative part, successive substitutions can be taken to solve the nonlinear problems. Convergence is guaranteed by further addressing the overshooting problem. Since our global step solve requires no energy summation and dot product operations over the entire problem, it fits into GPU implementation perfectly. In the local step, we further present a GPU-friendly gradient descent method to prevent interpenetration by solving an optimization problem independently. Putting the global and local solves together, experiments show that our method is robust and efficient in simulating complex models of membranes involving hyperelastic materials and contact.

In simulating viscous incompressible SPH fluids, incompressibility and viscosity are typically solved in two separate stages. However, the interference between pressure and shear forces could cause the missing of behaviors that include preservation of sharp surface details and remarkable viscous behaviors such as buckling and rope coiling. To alleviate this problem, we introduce for the first time the semi-implicit method for pressure linked equations (SIMPLE) into SPH to solve incompressible fluids with a broad range viscosity. We propose to link incompressibility and viscosity solvers, and impose incompressibility and viscosity constraints iteratively to gradually remove the interference between pressure and shear forces. We will also discuss how to solve the particle deficiency problem for both incompressibility and viscosity solvers. Our method is stable at simulating incompressible fluids whose viscosity can range from zero to an extremely high value. Compared to state-of-the-art methods, our method not only produces realistic viscous behaviors, but is also better at preserving sharp surface details.

Granular media is the second-most-manipulated substance on Earth, second only to water. However, simulation of granular media is still challenging due to the complexity of granular materials and the large number of discrete solid particles. As we know, dry granular materials could form a hybrid state between a fluid and a solid, therefore we propose a two-layer model and divide the simulation domain into a dilute layer, where granules can move freely as a fluid, and a dense layer, where granules act more like a solid. Motivated by the shallow water equations, we derive a set of shallow sand equations for modeling dry granular flows by depth-integrating three-dimensional governing equations along its vertical direction. Unlike previous methods for simulating a 2D granular media, our model does not restrict the depth of the granular media to be shallow anymore. To allow efficient fluid-solid interactions, we also present a ray casting algorithm for one-way solid-fluid coupling. Finally, we introduce a particle-tracking method to improve the visual representation. Our method can be efficiently implemented based on a height field and is fully compatible with modern GPUs, therefore allows us to simulate large-scale dry granular flows in real time.

Abstract The treatment of solid boundary conditions remains one of the most challenging parts in the SPH method. We present a semi‐analytical approach to handle complex solid boundaries of arbitrary shape. Instead of calculating a renormalizing factor for the particle near the boundary, we propose to calculate the volume integral inside the solid boundary under the local spherical frame of a particle. By converting the volume integral into a surface integral, a computer aided design (CAD) mesh file representing the boundary can be naturally integrated for particle simulations. To accelerate the search for a particle's neighboring triangles, a uniform grid is applied to store indices of intersecting triangles. The new semi‐analytical solid boundary handling approach is integrated into a position‐based method [MM13] as well as a projection‐based [HWW*20] to demonstrate its effectiveness in handling complex boundaries. Experiments show that our method is able to achieve comparable results with those simulated using ghost particles. In addition, since our method requires no boundary particles for deforming surfaces, our method is flexible enough to handle complex solid boundaries, including sharp corners and shells.

Abstract We introduce a new biorthogonal wavelet approach to creating a water‐tight surface defined by an implicit function, from a finite set of oriented points. Our approach aims at addressing problems with previous wavelet methods which are not resilient to missing or nonuniformly sampled data. To address the problems, our approach has two key elements. First, by applying a three‐dimensional partial integration, we derive a new integral formula to compute the wavelet coefficients without requiring the implicit function to be an indicator function. It can be shown that the previously used formula is a special case of our formula when the integrated function is an indicator function. Second, a simple yet general method is proposed to construct smooth wavelets with small support. With our method, a family of wavelets can be constructed with the same support size as previously used wavelets while having one more degree of continuity. Experiments show that our approach can robustly produce results comparable to those produced by the Fourier and Poisson methods, regardless of the input data being noisy, missing or nonuniform. Moreover, our approach does not need to compute global integrals or solve large linear systems.

Abstract Peridynamics is a formulation of the classical elastic theory that is targeted at simulating deformable objects with discontinuities, 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 materials.

Unified simulation of versatile elastoplastic materials and different dimensions offers many advantages in animation production, contact handling, and hardware acceleration. The unstructured particle representation is particularly suitable for this task, thanks to its simplicity. However, previous meshless techniques either need too much computational cost for addressing stability issues, or lack physical meanings and fail to generate interesting deformation behaviors, such as the Poisson effect. In this paper, we study the development of an elastoplastic model under the state-based peridynamics framework, which uses integrals rather than partial derivatives in its formulation. To model elasticity, we propose a unique constitutive model and an efficient iterative simulator solved in a projective dynamics way. To handle plastic behaviors, we incorporate our simulator with the Drucker-Prager yield criterion and a reference position update scheme, both of which are implemented under peridynamics. Finally, we show how to strengthen the simulator by position-based constraints and spatially varying stiffness models, to achieve incompressibility, particle redistribution, cohesion, and friction effects in viscoelastic and granular flows. Our experiments demonstrate that our unified, meshless simulator is flexible, efficient, robust, and friendly with parallel computing.

Abstract Gradient‐domain compositing has been widely used to create a seamless composite with gradient close to a composite gradient field generated from one or more registered images. The key to this problem is to solve a Poisson equation, whose unknown variables can reach the size of the composite if no region of interest is drawn explicitly, thus making both the time and memory cost expensive in processing multi‐megapixel images. In this paper, we propose an approximate projection method based on biorthogonal Multiresolution Analyses (MRA) to solve the Poisson equation. Unlike previous Poisson equation solvers which try to converge to the accurate solution with iterative algorithms, we use biorthogonal compactly supported curl‐free wavelets as the fundamental bases to approximately project the composite gradient field onto a curl‐free vector space. Then, the composite can be efficiently recovered by applying a fast inverse wavelet transform. Considering an n‐pixel composite, our method only requires 2n of memory for all vector fields and is more efficient than state‐of‐the‐art methods while achieving almost identical results. Specifically, experiments show that our method gains a 5× speedup over the streaming multigrid in certain cases.

We propose a semi-Lagrangian method for multiphase interface tracking. In contrast to previous methods, our method maintains an explicit polygonal mesh, which is reconstructed from an unsigned distance function and an indicator function, to track the interface of arbitrary number of phases. The surface mesh is reconstructed at each step using an efficient multiphase polygonization procedure with precomputed stencils while the distance and indicator function are updated with an accurate semi-Lagrangian path tracing from the meshes of the last step. Furthermore, we provide an adaptive data structure, multiphase distance tree, to accelerate the updating of both the distance function and the indicator function. In addition, the adaptive structure also enables us to contour the distance tree accurately with simple bisection techniques. The major advantage of our method is that it can easily handle topological changes without ambiguities and preserve both the sharp features and the volume well. We will evaluate its efficiency, accuracy and robustness in the results part with several examples.

We introduce a novel framework for tracking multiphase interfaces with explicit contouring technique. In our framework, an unsigned distance function and an additional indicator function are used to represent the multiphase system. Our method maintains the explicit polygonal meshes that define the multiphase interfaces. At each step, distance function and indicator function are updated via semi-Lagrangian path tracing from the meshes of the last step. Interface surfaces are then reconstructed by polygonization procedures with precomputed stencils and further smoothed with a feature-preserving non-manifold smoothing algorithm to stay in good quality. Our method is easy to be implemented and incorporated into multiphase simulation, such as immiscible fluids, crystal grain growth and geometric flows. We demonstrate our method with several level set tests, including advection, propagation, etc., and couple it to some existing fluid simulators. The results show that our approach is stable, flexible, and effective for tracking multiphase interfaces.

Smoothed particle hydrodynamics (SPH) is efficient, mass preserving, and flexible in handling topological changes. However, sparsely sampled thin features are difficult to simulate in SPH-based free surface flows, due to a number of robustness and stability issues. In this article, we address this problem from two perspectives: the robustness of surface forces and the numerical instability of thin features. We present a new surface tension force scheme based on a free surface energy functional, under the diffuse interface model. We develop an efficient way to calculate the air pressure force for free surface flows, without using air particles. Compared with previous surface force formulae, our formulae are more robust against particle sparsity in thin feature cases. To avoid numerical instability on thin features, we propose to adjust the internal pressure force by estimating the internal pressure at two scales and filtering the force using a geometry-aware anisotropic kernel. Our result demonstrates the effectiveness of our algorithms in handling a variety of sparsely sampled thin liquid features, including thin sheets, thin jets, and water splashes.

Abstract Enforcing fluid incompressibility is one of the time‐consuming aspects in SPH. In this paper, we present a local Poisson SPH (LPSPH) method to solve incompressibility for particle based fluid simulation. Considering the pressure Poisson equation, we first convert it into an integral form, and then apply a discretization to convert the continuous integral equation to a discretized summation over all the particles in the local pressure integration domain determined by the local geometry. To control the approximation error, we further integrate our local pressure solver into the predictive‐corrective framework to avoid the computational cost of solving a pressure Poisson equation globally. Our method can effectively eliminate the large density deviations mainly caused by the solid boundary treatment and free surface topological change, and show advantage of a higher convergence rate over the predictive‐corrective incompressible SPH (PCISPH).

Simulating solid-fluid coupling with the classical meshless methods is an difficult issue due to the lack of the Kronecker delta property of the shape functions when enforcing the essential boundary conditions. In this work, we present a novel staggered meshless method to overcome this problem. We create a set of staggered particles from the original particles in each time step by mapping the mass and momentum onto these staggered particles, aiming to stagger the velocity field from the pressure field. Based on this arrangement, an new approximate projection method is proposed to enforce divergence-free on the fluid velocity with compatible boundary conditions. In the simulations, the method handles the fluid and solid in a unified meshless manner and generalizes the formulations for computing the viscous and pressure forces. To enhance the robustness of the algorithm, we further propose a new framework to handle the degeneration case in the solid-fluid coupling, which guarantees stability of the simulation. The proposed method offers the benefit that various slip boundary conditions can be easily implemented. Besides, explicit collision handling for the fluid and solid is avoided. The method is easy to implement and can be extended from the standard SPH algorithm in a straightforward manner. The paper also illustrates both one-way and two-way couplings of the fluids and rigid bodies using several test cases in two and three dimensions.

Abstract We propose a meshless method to simulate brittle fracture. For brittle solids, stress computation can be difficult because brittle materials generally require small timesteps which bring about heavy computational burden. Furthermore, treating the brittle objects as deformable bodies will cause inevitable visual artifact. We treat the brittle objects as fully rigid bodies and solve the brittle stress distribution with Meshless Local Petrov‐Galerkin as a quasistatic problem, so visual artifact disppears and no timestep restriction exists. As a meshless framework, our method has the advantage of easy‐resampling around high stress areas to improve computation accuracy. To generate fractured pieces, unlike previous methods which explicitly track the crack propagation, we also present a novel damage based model. Our model supports user‐control of the fracture pattern which is especially useful when simulating anisotropic materials such as glass or wood. Results show that our meshless framework is physically feasible and user controllable. Copyright © 2011 John Wiley & Sons, Ltd.