Article

Drucker-prager elastoplasticity for sand animation

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Abstract

We simulate sand dynamics using an elastoplastic, continuum assumption. We demonstrate that the Drucker-Prager plastic flow model combined with a Hencky-strain-based hyperelasticity accurately recreates a wide range of visual sand phenomena with moderate computational expense. We use the Material Point Method (MPM) to discretize the governing equations for its natural treatment of contact, topological change and history dependent constitutive relations. The Drucker-Prager model naturally represents the frictional relation between shear and normal stresses through a yield stress criterion. We develop a stress projection algorithm used for enforcing this condition with a non-associative flow rule that works naturally with both implicit and explicit time integration. We demonstrate the efficacy of our approach on examples undergoing large deformation, collisions and topological changes necessary for producing modern visual effects.

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... The grid, then, transfers the updated information back to the particles while it remains undeformed. Some recent lines of research have proved MPM as one of the most accurate and efficient methods for granular flow modeling [13,4,15,16,17,18]. Furthermore, instability and non-conserved angular momentum issues observed in the earlier variants including smoothed particle hydrodynamics (SPH) [19], fluid implicit particle (FLIP) [20], and particle-in-cell (PIC) [21] are addressed and fixed in MPM by using proper particle-grid transfer schemes [22,23]. ...
... However, the middle (visco)plastic regime is more challenging where the granular material flows somewhat like a liquid. For many years, for this regime, plastic models have been developed by combining various yield criteria and plastic flow rules [29,30,16]. In contrast, recently, from fluid mechanics literature, viscoplastic models have made progress by invoking a fluid-like flow approach with an appropriate yield criterion [31,32]. ...
... Plastic models can suffer from rate-independency [33] and in some cases, they may have issues with modeling strain hardening [16]. Viscoplastic models eliminate some numerical difficulties associated with plastic models, such as hardening [36]. ...
Preprint
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The accurate and efficient modeling of granular flows and their interactions with external bodies is an open research problem. Continuum methods can be used to capture complexities neglected by terramechanics models without the computational expense of discrete element methods. Constitutive models and numerical solvers are the two primary aspects of the continuum methods. The viscoplastic size-dependent non-local granular fluidity (NGF) constitutive model has successfully provided a quantitative description of experimental flows in many different configurations in literature. This research develops a numerical approach, within a hyperelasticity framework, for implementing the dynamical form of NGF in three-dimensional material point method (3D MPM, an appropriate numerical solver for granular flow modeling). This approach is thermodynamically consistent to conserve energy, and the dynamical form includes the nonlocal effect of flow cessation. Excavation data, both quantitative measurements and qualitative visualization, are collected experimentally via our robotic equipment to evaluate the model with respect to the flow geometry as well as interaction forces. The results are further compared with the results from a recent modified plastic Drucker-Prager constitutive model, and in other configurations including wheel-soil interactions, a gravity-driven silo, and Taylor-Couette flow.
... The grid, then, transfers the updated information back to the particles while it remains undeformed. Some recent lines of research have proved MPM as one of the most accurate and efficient methods for granular flow modeling [6,[12][13][14][15][16]. Furthermore, instability and non-conserved angular momentum issues observed in the earlier variants including smoothed particle hydrodynamics (SPH) [17], fluid implicit particle (FLIP) [18], and particle-in-cell (PIC) [19] are addressed and fixed in MPM by using proper particle-grid transfer schemes [20,21]. ...
... For instance, Eqs. (12) to (14) compute the impulse from a particle ( m p Δv c p ) on a rigid body as shown in Fig. 5(left). The opposite impulses from a rigid body ( m r Δv c r ) can be derived with a similar approach for various contact conditions (e.g. ...
... The plastic Drucker-Prager (DP) constitutive model (with projection) [14] is also used to show the MPM gravity sensitivity to constitutive models. Similarly to Figs. 9, 14 compares the normalized angular velocity profiles but this time between MPM with DP model and the experiment. ...
Article
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One of the major challenges in space exploration robotics is understanding the interactions between robot wheels and planetary terrains consisting of granular regolith in reduced gravity. A key factor, and the focus of this work, is the effect of gravity. Experimental results from the literature for a Taylor Couette cell, with granular material between a rotating inner and a fixed outer concentric cylinder, flown aboard a reduced-g aircraft are taken as a baseline. In this research, granular flow is modeled using continuum methods to capture complexities neglected by terramechanics models without the computational expense of discrete element method (DEM). This research identifies material point method (MPM) as an appropriate continuum solver to model granular flows under the influence of gravity, as it generates both absolute velocity values and trends more consistent with the variable-g experiments than do analytical methods or finite element method (FEM). The modeling of stress-free particles in the shear band by MPM generates a pressure field that leads to the desirable results. This research focuses on quasi-static and intermediate flow regimes, as a survey of the present and past rovers shows these are the most important regimes in planetary applications. Improved flow modeling can contribute to advancing future robot wheel design and mobility control.
... snow, jelly). However, the continuum force-based material point method (MPM) has already been shown [5,8,9,41,42] as one of the most accurate and efficient methods for our purpose i.e. granular flow modeling. MPM [41] has also been recently used to produce sand simulations for the purpose of training a simulator [26]. ...
... However, the intermediate (viscoplastic) regime is more challenging where the granular material flows most like a liquid. For many years, for this regime, plastic models have been developed by combining various yield criteria and plastic flow rules [42,56]. Plastic models can suffer from rate-independency [57] and in some cases, they may have issues with modeling strain hardening [42]. ...
... For many years, for this regime, plastic models have been developed by combining various yield criteria and plastic flow rules [42,56]. Plastic models can suffer from rate-independency [57] and in some cases, they may have issues with modeling strain hardening [42]. In contrast, recently, from fluid mechanics literature, viscoplastic models have been made progress invoking a fluid-like flow approach with an appropriate yield criterion [57][58][59]. ...
Conference Paper
Maintaining traction remains one of the challenges that space exploration rovers encounter while roving on Martian or Lunar deformable terrains. Such terrains consist of granular regolith under reduced gravity conditions. Real-time simulation of wheel-soil interactions, that accurately takes gravity effects into account, can improve rovers’ online mobility control. The research problem investigated in this paper is the development of a machine learning-based wheel-on-soil simulation model. Machine learning enables efﬁcient and fast mapping of simulation inputs to outputs, trained using high-ﬁdelity (non-realtime) models validated by experiments. The training data is produced by a continuum method comprising a modern constitutive model, nonlocal granular ﬂuidity (NGF), and a state-of-the-art numerical solver, material point method (MPM). Machine learning techniques, including graph network-based simulator (GNS) and principal component analysis (PCA), are proposed to learn efﬁcient mappings. The important aspects that must be captured include the traction forces on the wheel and the behavior of the underlying granular ﬂows. Hence, the experimental data includes force measured using an instrumented single-wheel testbed and subsurface soil motion observed with a high-speed camera and analyzed with optical ﬂow.
... Note that with implicit Euler, we have φ n+1 = φ n + ∆tV n+1 , which also allows us to reformulate Eq. 23 in terms of velocities. Note that our derivation has assumed pure hyperelasticity since the implicit treatment of plasticity cannot be easily formulated as a minimization problem Klár et al. [2016]. For plastic materials, we follow Jiang et al. [2016b] and treat the plastic flow explicitly through a return mapping of the elastic deformation gradient at the end of each time step. ...
... Venant-Kirchhoff energy with logarithmic strain was adopted for simplified derivation of plasticity Klár et al. [2016], the fixed corotated energy was designed for robust treatment Note the overlapping energies when principal stretches σ 1 and σ 2 are below 1, illustrating that compression is left undegraded in our approach. Conversely, we see that as damage increases (c decreases) the tensile stress gets significantly degraded. ...
... Modeling plasticity and plastic flow has already shown to be especially useful in graphics for modeling the flow of granular materials like sand Klár et al. [2016], snow Stomakhin et al. [2013], wet sand Pradhana et al. [2017], and continuum materials like foam Yue et al. [2015]. Naturally, computationally modeling plasticity has also seen extensive After these graphics focused works, we present a modification of our NACC plasticity that enables its application to the real-world modeling of large-scale glacial calving and ice-ocean interaction (including the dangerous ensuing tsunami waves) Wolper et al. ...
Thesis
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Material fracture surrounds us every day from tearing off a piece of fresh bread to dropping a glass on the floor. Modeling this complex physical process has a near limitless breadth of applications in everything from computer graphics and VFX to virtual surgery and geomechanical modeling. Despite the ubiquity of material failure, it stands as a notoriously difficult phenomenon to simulate and has inspired numerous efforts from computer graphics researchers and mechanical engineers alike, resulting in a diverse set of approaches to modeling the underlying physics as well as discretizing the branching crack topology. However, most existing approaches focus on meshed methods such as FEM or BEM that require computationally intensive crack tracking and re-meshing procedures. Conversely, the Material Point Method (MPM) is a hybrid meshless approach that is ideal for modeling fracture due to its automatic support for arbitrarily large topological deformations, natural collision handling, and numerous successfully simulated continuum materials. In this work, we present a toolkit of augmented Material Point Methods for robustly and efficiently simulating material fracture both through damage modeling and through plastic softening/hardening. Our approaches are robust to a multitude of materials including those of varying structures (isotropic, transversely isotropic, orthotropic), fracture types (ductile, brittle), plastic yield surfaces, and constitutive models. The methods herein are applicable not only to the needs of computer graphics (efficiency and visual fidelity), but also to the engineering community where physical accuracy is key. Most notably, each approach has a unique set of parametric knobs available to artists and engineers alike that make them directly deployable in applications ranging from animated movie production to large-scale glacial calving simulation.
... Die Material Point Method (MPM) hat sich in der Computergrafik als äußerst fähige Simulationsmethode erwiesen, die in der Lage ist ansonsten schwierig zu animierende Materialien zu modellieren [1,2]. Abgesehen von der Simulation einzelner Materialien stellt die Simulation mehrerer Materialien und ihrer Interaktion weitere Herausforderungen bereit. ...
... The Material Point Method (MPM) has proven to be a very capable simulation method in computer graphics that is able to model materials that were previously very challenging to animate [1,2]. Apart from simulating singular materials, the simulation of multiple materials that interact with each other introduces new challenges. ...
... Their model can simulate many different types of snow, which has previously been very difficult to model [1]. MPM further proved very effective in the simulation of granular materials, i.e. for sand animation [2,5], as well as viscoelastic fluids, foams and sponges [6,7]. There even exist more advanced formulations of MPM that can be used for cloth simulations [8]. ...
Thesis
Full-text available at: https://kola.opus.hbz-nrw.de/frontdoor/index/index/docId/2129 The Material Point Method (MPM) has proven to be a very capable simulation method in computer graphics that is able to model materials that were previously very challenging to animate [1, 2]. Apart from simulating singular materials, the simulation of multiple materials that interact with each other introduces new challenges. This is the focus of this thesis. It will be shown that the self-collision capabilities of the MPM can naturally handle multiple materials interacting in the same scene on a collision basis, even if the materials use distinct constitutive models. This is then extended by porous interaction of materials as in [3], which also integrates easily with MPM. It will furthermore be shown that regular single-grid MPM can be viewed as a subset of this multi-grid approach, meaning that its behavior can also be achieved if multiple grids are used. The porous interaction is generalized to arbitrary materials and freely changeable material interaction terms, yielding a ﬂexible, user-controllable framework that is independent of speciﬁc constitutive models. The framework is implemented on the GPU in a straightforward and simple way and takes advantage of the rasterization pipeline to resolve write-conﬂicts, resulting in a portable implementation with wide hardware support, unlike other approaches such as [4].
... Tampubolon et al. [36] simulated the interaction of sand and water mixtures using the MPM and obtained encouraging results. For the porous material property, Klár et al. [37] used the improved Drucker-Prager plastic flow model with volume correction. For the MPM implementation, Arduino et al. [38] and Jassim et al. [39] examined that momentum exchange using the two-grid MPM for the multi-species interaction. ...
... It is used to measure how the material has locally rotated and deformed due to its motion [36]. Plasticity is represented by factoring the deformation gradient into elastic and plastic parts as F = F E F P [37]. ...
... Tampubolon and Angeles [36] modified the model of Klár et al. [37] to include cohesive stresses for a porous material and water mixture. Therefore, the amount of cohesion varies with the saturation of water in the mixture. ...
Article
Full-text available
Dam embankment breaches caused by overtopping or internal erosion can impact both life and property downstream. It is important to accurately predict the amount of erosion, peak discharge, and the resulting downstream flow. This paper presents a new model based on the material point method to simulate soil and water interaction and predict failure rate parameters. The model assumes that the dam consists of a homogeneous embankment constructed with cohesive soil, and water inflow is defined by a hydrograph using other readily available reach routing software. The model uses continuum mixture theory to describe each phase where each species individually obeys the conservation of mass and momentum. A two-grid material point method is used to discretize the governing equations. The Drucker–Prager plastic flow model, combined with a Hencky strain-based hyperelasticity model, is used to compute soil stress. Water is modeled as a weakly compressible fluid. Analysis of the model demonstrates the efficacy of our approach for existing examples of overtopping dam breach, dam failures, and collisions. Simulation results from our model are compared with a physical-based breach model, WinDAM C. The new model can capture water and soil interaction at a finer granularity than WinDAM C. The new model gradually removes the granular material during the breach process. The impact of material properties on the dam breach process is also analyzed.
... The Material Point Method (MPM) family of discretizations [Sulsky et al. 1994], such as Fluid Implicit Particle (FLIP) [Brackbill et al. 1988] and Particle-in-Cell (PIC) [Sulsky et al. 1995], emerged as an effective choice for simulating various materials and gained popularity in visual effects (VFX) for providing high-fidelity physics simulations of snow [Stomakhin et al. 2013], sand [Daviet and Bertails-Descoubes 2016;Klár et al. 2016], phase change [Gao et al. 2018b;Stomakhin et al. 2014], viscoelasticity [Ram et al. 2015;Su et al. 2021;Yue et al. 2015], viscoplasticity ], elastoplasticity [Gao et al. 2017], fluid structure interactions , fracture [Hegemann et al. 2013;Wolper et al. 2019], fluid-sediment mixtures [Gao et al. 2018a;], baking and cooking [Ding et al. 2019], and diffusion-driven phenomena [Xue et al. 2020]. In contrast to Lagrangian mesh-based methods, such as the Finite Element Method (FEM) [Sifakis and Barbic 2012;Zienkiewicz et al. 1977], and pure particle-based methods, such as Smoothed Particle Hydrodynamics (SPH) [Desbrun and Gascuel 1996;Liu et al. 2008], MPM merges the advantages of both Lagrangian and Eulerian approaches and automatically supports dynamic topology changes such as material splitting and merging. ...
... The seminal work of Zhu and Bridson [Zhu and Bridson 2005] first introduced the FLIP method for sand simulation. Subsequent works further explored its strength in simulating a broader spectrum of material behaviors including snow [Stomakhin et al. 2013], granular materials [Daviet and Bertails-Descoubes 2016;Gao et al. 2018a;Klár et al. 2016;, foam [Ram et al. 2015;Yue et al. 2015], complex fluids Gao et al. 2017], cloth, hair and fiber collisions [Fei et al. 2018[Fei et al. , 2017Jiang et al. 2017a], fracture [Wolper et al. 2020[Wolper et al. , 2019 and phase change [Gao et al. 2018b;Stomakhin et al. 2014;Su et al. 2021]. We also note the related works of [McAdams et al. 2009] for hair simulation, [Sifakis et al. 2008] for cloth simulation, [Narain et al. 2010] for sand simulation and [Patkar et al. 2013] for bubble simulation, which bear similarities to MPM due to their hybrid nature. ...
Preprint
We present an arbitrary updated Lagrangian Material Point Method (A-ULMPM) to alleviate issues, such as the cell-crossing instability and numerical fracture, that plague state of the art Eulerian formulations of MPM, while still allowing for large deformations that arise in fluid simulations. Our proposed framework spans MPM discretizations from total Lagrangian formulations to Eulerian formulations. We design an easy-to-implement physics-based criterion that allows A-ULMPM to update the reference configuration adaptively for measuring physical states including stress, strain, interpolation kernels and their derivatives. For better efficiency and conservation of angular momentum, we further integrate the APIC[Jiang et al. 2015] and MLS-MPM[Hu et al. 2018] formulations in A-ULMPM by augmenting the accuracy of velocity rasterization using both the local velocity and its first-order derivatives. Our theoretical derivations use a nodal discretized Lagrangian, instead of the weak form discretization in MLS-MPM[Hu et al. 2018], and naturally lead to a "modified" MLS-MPM in A-ULMPM, which can recover MLS-MPM using a completely Eulerian formulation. A-ULMPM does not require significant changes to traditional Eulerian formulations of MPM, and is computationally more efficient since it only updates interpolation kernels and their derivatives when large topology changes occur. We present end-to-end 3D simulations of stretching and twisting hyperelastic solids, splashing liquids, and multi-material interactions with large deformations to demonstrate the efficacy of our novel A-ULMPM framework.
... As one of the most promising discretization choices in physics-based simulation, MPM has been used for simulating numerous materials and diverse phenomena. Prior work includes snow [Gaume et al. 2018;Stomakhin et al. 2013], granular materials [Daviet and Bertails-Descoubes 2016;Gao et al. 2018b;Klár et al. 2016;], viscoelastic solids , cloth [Fei et al. 2018;Montazeri et al. 2019], hair [Fei et al. 2018;, and non-Newtonian fluids and foam [Nagasawa et al. 2019;Ram et al. 2015;Yue et al. 2015Yue et al. , 2018. Additionally, other complex phenomena have been simulated with MPM including melting [Gao et al. 2018b;Stomakhin et al. 2014], baking , topological changes and fracture Wolper et al. 2019;Wretborn et al. 2017], multiple-material interaction [Gao et al. 2018a;Han et al. 2019;Yan et al. 2018], frictional contact and collision [Ding and Craig 2019], etc. ...
... We showcase a suite of simulations with various materials to demonstrate the scalability of our multi-GPU MPM algorithm. The following constitutive models with plasticity are implemented to demonstrate the applicability of our methods to diverse materials: 1) fixed corotated [Stomakhin et al. 2012] to simulate elastic jello, 2) Non-Associated Cam-Clay (NACC) [Wolper et al. 2019] to reproduce soil and concrete, 3) Drucker-Prager elastoplasticity [Klár et al. 2016] for sand animation, and 4) weakly compressible fluid ] to generate water. All timings and spatial resolution settings are summarized in Table 5. ...
... In this work, we rely on the finite strain elastoplastic MPM formulation proposed by Klár et al. (39). Figure 1 displays the computational procedure for a time integration step. ...
... • The mathematical foundations of the proposed model are not new (36,39,72). Yet, its application to complex full-scale real-world events has been hampered for decades due to high computational costs. ...
Preprint
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Alpine mass movements can generate process cascades involving different materials including rock, ice, snow, and water. Numerical modelling is an essential tool for the quantification of natural hazards, but state-of-the-art operational models reach their limits when facing unprecedented or complex events. Here, we advance our predictive capabilities for process cascades on the basis of a three-dimensional numerical model, coupling fundamental conservation laws to finite strain elastoplasticity. Through its hybrid Eulerian-Lagrangian character, our approach naturally reproduces fractures and collisions, erosion/deposition phenomena, and multi-phase interactions, which finally grant very accurate simulations of complex dynamics. Four benchmark simulations demonstrate the physical detail of the model and its applicability to real-world full-scale events, including various materials and ranging through four orders of magnitude in volume. In the future, our model can support risk-management strategies through predictions of the impact of potentially catastrophic cascading mass movements at vulnerable sites.
... It was later extended to Poly-PIC ] and finally unified as MLS-MPM ]. Various materials were successfully handled by MPM, including snow [Stomakhin et al. 2013], sand [Klár et al. 2016], cloth ] and thin shell ]. There have been some studies focusing on the performance improvement of MPM. ...
... where = tr( ) I is mean stress, = − is deviator stress, represents cohesion, and represents friction. Klár et al. [2016] used this model to animate sand. We adopt a similar return mapping augmented with cohesion handling like in ]. ...
... The choice of using a model based on Hencky strain, as opposed to e.g. the Green strain, makes the elastoplastic numerical integration scheme more convenient (Mast, 2013). The elastic model presented here has been successfully used in various publications on material point method modeling (Mast, 2013;Klár et al., 2016;Gaume et al., 2018). ...
... The background grid, which is considered regular and structured, has no hard boundary in the sense that a grid node is only activated by particles in its vicinity at each time step. Although there exist several variations of MPM, we have resorted to explicit time integration and a weighted combination of the particle-in-cell (PIC) and fluid-implicitparticle (FLIP) method for grid-particle interpolation, as presented and used by, e.g., Klár et al. (2016), Gaume et al. (2018). In Algorithm 1 we outline in details the steps of the MPM used in this work, and in Algorithm 2 more details are provided on the computation of the return mapping procedure involved with the strain softening Drucker-Prager criterion outlined in the previous section. ...
Article
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Porous brittle solids evidence complex mechanical behavior, where localized failure patterns originate from mechanical processes on the microstructural level. In order to investigate the failure mechanics of porous brittle solids, we outline a general stochastic and numerical microstructure-based approach. To this end, we generate random porous microstructures by level-cutting Gaussian random fields, and conduct numerical simulations using the material point method. This allows investigating both small and large deformation characteristics of irregular porous media where a segmentation into grains and bonds is ambiguous. We demonstrate the versatility of our approach by examining elasticity and failure as a function of a wide range of porosities, from 20% to 80%. Observing that onset of failure can be well described through the second order work, we show that the stress at failure follows a power law similar to that of the elastic modulus. Moreover, we propose that the failure envelope can be approximated by a simple quadratic fitting curve, and that plastic deformation appears to be governed by an associative plastic flow rule. Finally, large deformation simulations reveal a transition in the mode of localization of the deformation, from compaction bands for highly porous samples to shear bands for denser ones.
... Although these methods can successfully improve the transfer accuracy between the particles and grid, the angular momentum is still not conserved (Jiang et al., 2015). To address this problem, Klár et al. (2016) applied an affine transfer (Jiang et al., 2015) to sand simulations. In addition, a more general form of the affine transfer algorithm was developed using the weighted least squares (WLS) approximation (Hu et al., 2018). ...
Article
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This study presents a new particle-to-surface frictional contact algorithm for the material point method (MPM) to simulate the interaction between material points and rigid bodies. Although a grid-based multivelocity field technique is a common method that works well with MPM, there are several disadvantages in developing such simulations: (1) The technique needs additional treatment to prevent early contact and satisfy the impenetrability condition. (2) The method performs poorly in the contact detection of rigid bodies, as it depends on the arrangement of their constituent material points. To overcome these problems, the proposed algorithm uses the penalty method based on particle-to-surface contact. However, the possibility of physical quantities being transferred from the particles to grid nodes located within the rigid bodies arises since a particle can be very close to the surface. The weighted least squares approximation could effectively handle this problem, without forfeiting the partition-of-unity property while constructing the shape function, and is thus incorporated into the MPM framework in this study. The proposed algorithm is validated by comparing numerical simulations with analytical solutions and FEM results. The numerical results show that the proposed algorithm produces reasonable results for frictional contact phenomena.
... Doing so, and reformulating transfer functions accordingly, results in a method which is more similar to classical PIC, but which preserves angular momentum and avoids dissipation to a degree similar to FLIP. APIC has been adopted by several recent PIC-like implementations, and has been expanded upon by other researchers [22,34,31,37]. ...
Article
Within computational continuum mechanics there exists a large category of simulation methods which operate by tracking Lagrangian particles over an Eulerian background grid. These Lagrangian/Eulerian hybrid methods, descendants of the Particle-In-Cell method (PIC), have proven highly effective at simulating a broad range of materials and mechanics including fluids, solids, granular materials, and plasma. These methods remain an area of active research after several decades, and their applications can be found across scientific, engineering, and entertainment disciplines. This thesis presents a GPU driven PIC-like simulation framework created using the Vulkan® API. Vulkan is a cross-platform and open-standard explicit API for graphics and GPU compute programming. Compared to its predecessors, Vulkan offers lower overhead, support for host parallelism, and finer grain control over both device resources and scheduling. This thesis harnesses those advantages to create a programmable GPU compute pipeline backed by a Vulkan adaptation of the SPgrid data-structure and multi-buffered particle arrays. The CPU host system works asynchronously with the GPU to maximize utilization of both the host and device. The framework is demonstrated to be capable of supporting Particle-in-Cell like simulation methods, making it viable for GPU acceleration of many Lagrangian particle on Eulerian grid hybrid methods. This novel framework is the first of its kind to be created using Vulkan® and to take advantage of GPU sparse memory features for grid sparsity.
... Return mapping and its gradients Following Klár et al. (2016) and Gao et al. (2017), we implement the return mapping as a 3D projection process on the singular values of the deformation gra-dients of each particle. This means we need a singular value decomposition (SVD) process on the particles' deformation gradients, and we provide the pseudocode of this process in Appendix A. For backpropagation, we need to evaluate gradients of SVD. ...
Preprint
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Simulated virtual environments serve as one of the main driving forces behind developing and evaluating skill learning algorithms. However, existing environments typically only simulate rigid body physics. Additionally, the simulation process usually does not provide gradients that might be useful for planning and control optimizations. We introduce a new differentiable physics benchmark called PasticineLab, which includes a diverse collection of soft body manipulation tasks. In each task, the agent uses manipulators to deform the plasticine into the desired configuration. The underlying physics engine supports differentiable elastic and plastic deformation using the DiffTaichi system, posing many under-explored challenges to robotic agents. We evaluate several existing reinforcement learning (RL) methods and gradient-based methods on this benchmark. Experimental results suggest that 1) RL-based approaches struggle to solve most of the tasks efficiently; 2) gradient-based approaches, by optimizing open-loop control sequences with the built-in differentiable physics engine, can rapidly find a solution within tens of iterations, but still fall short on multi-stage tasks that require long-term planning. We expect that PlasticineLab will encourage the development of novel algorithms that combine differentiable physics and RL for more complex physics-based skill learning tasks.
... When the plastic flow of the Drucker-Prager model is assumed to be isochoric, the two models also use the same flow rule whereby the the plastic strain is purely deviatoric. Therefore, the lðIÞ rheology model can be utilized without significant change in the existing algorithm for the Drucker-Prager plasticity model [5,29]. ...
Article
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Granular impact—the dynamic intrusion of solid objects into granular media—is widespread across scientific and engineering applications including geotechnics. Existing approaches to the simulation of granular impact dynamics have relied on either a purely discrete method or a purely continuum method. Neither of these methods, however, is deemed optimal from the computational perspective. Here, we introduce a hybrid continuum–discrete approach, built on the coupled material-point and discrete-element method (MP–DEM), for simulation of granular impact dynamics with unparalleled efficiency. To accommodate highly complex solid–granular interactions, we enhance the existing MP–DEM formulation with three new ingredients: (i) a robust contact algorithm that couples the continuum and discrete parts without any interpenetration under extreme impact loads, (ii) large deformation kinematics employing multiplicative elastoplasticity, and (iii) a trans-phase constitutive relation capturing gasification of granular media. For validation, we also generate experimental data through laboratory measurement of the impact dynamics of solid spheres dropped onto dry sand. Simulation of the experiments shows that the proposed approach can well reproduce granular impact dynamics in terms of impact forces, intrusion depths, and splash patterns. Furthermore, through parameter studies on material properties, model formulations, and numerical schemes, we identify key factors for successful continuum–discrete simulation of granular impact dynamics.
... MPM is a hybrid Lagrangian-Eulerian method widely used in different fields, e.g., computer graphics [28,29], civil engineering [30,31], mechanical engineering [15,32,33,34]. With the capability of handling large deformation [35,36,37,38,39,40], topology changes, and coupled materials, MPM has been considered as one of the top choices in various physics-based simulations, including fracture [41,29,42,43,44,45], viscoelastic and elastoplastic solids [46,47], incompressible materials [48,49], high explosive explosion [50], snow [28,51,52], granular material [53,54,55,56] and mixtures [57,58,59]. In MPM, Lagrangian particles, which are also known as material points, are used to track quantities like mass, momentum, and deformation. ...
Article
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In this paper, a hybrid Lagrangian‐Eulerian topology optimization (LETO) method is proposed to solve the elastic force equilibrium with the Material Point Method (MPM). LETO transfers density information from freely movable Lagrangian carrier particles to a fixed set of Eulerian quadrature points. The transfer is based on a smooth radial kernel involved in the compliance objective to avoid the artificial checkerboard pattern. The quadrature points act as MPM particles embedded in a lower‐resolution grid and enable a sub‐cell multi‐density resolution of intricate structures with a reduced computational cost. A quadrature‐level connectivity graph‐based method is adopted to avoid the artificial checkerboard issues commonly existing in multi‐resolution topology optimization methods. Numerical experiments are provided to demonstrate the efficacy of the proposed approach.
... Hyde et al. [2020] provide a thorough discussion of the state of the art. Our approach utilizes the particle-based MPM [de Vaucorbeil et al. 2020;Sulsky et al. 1994] PIC technique, largely due to its natural ability to handle self collision [Fei et al. 2018[Fei et al. , 2017Guo et al. 2018;], topology change Wolper et al. 2020Wolper et al. , 2019, diverse materials [Daviet and Bertails-Descoubes 2016;Klár et al. 2016;Ram et al. 2015;Schreck and Wojtan 2020;Stomakhin et al. 2013;Wang et al. 2020c;Yue et al. 2015] as well as implicit time stepping with elasticity Fei et al. 2018;Stomakhin et al. 2013;Wang et al. 2020b]. We additionally use the APIC method [Fu et al. 2017;Jiang et al. 2015 for its conservation properties and beneficial suppression of noise. ...
Preprint
We present a novel Material Point Method (MPM) discretization of surface tension forces that arise from spatially varying surface energies. These variations typically arise from surface energy dependence on temperature and/or concentration. Furthermore, since the surface energy is an interfacial property depending on the types of materials on either side of an interface, spatial variation is required for modeling the contact angle at the triple junction between a liquid, solid and surrounding air. Our discretization is based on the surface energy itself, rather than on the associated traction condition most commonly used for discretization with particle methods. Our energy based approach automatically captures surface gradients without the explicit need to resolve them as in traction condition based approaches. We include an implicit discretization of thermomechanical material coupling with a novel particle-based enforcement of Robin boundary conditions associated with convective heating. Lastly, we design a particle resampling approach needed to achieve perfect conservation of linear and angular momentum with AffineParticle-In-Cell (APIC) [Jiang et al. 2015]. We show that our approach enables implicit time stepping for complex behaviors like the Marangoni effect and hydrophobicity/hydrophilicity. We demonstrate the robustness and utility of our method by simulating materials that exhibit highly diverse degrees of surface tension and thermomechanical effects, such as water, wine and wax.
... For example, fluid simulations using FLIP have been coupled with hair [60] and cloth [61] simulations. The Material Point Method [62] is another good example, providing simulations of various material types, such as snow [62], multi-species [63], [64] with phase transition [63], sand [65], [66], elastoplastic solids, viscoelastic fluids, foams and sponges [67]- [69], anisotropic elastoplastic materials [70], [71], fluid-sediment mixture [72]. MPM can also achieve solid-fluid coupling simulation [64], [72], [73], dynamic fracture [71], ductile fracture [74] and frictional contact [75]. ...
Preprint
Robustly handling collisions between individual particles in a large particle-based simulation has been a challenging problem. We introduce particle merging-and-splitting, a simple scheme for robustly handling collisions between particles that prevents inter-penetrations of separate objects without introducing numerical instabilities. This scheme merges colliding particles at the beginning of the time-step and then splits them at the end of the time-step. Thus, collisions last for the duration of a time-step, allowing neighboring particles of the colliding particles to influence each other. We show that our merging-and-splitting method is effective in robustly handling collisions and avoiding penetrations in particle-based simulations. We also show how our merging-and-splitting approach can be used for coupling different simulation systems using different and otherwise incompatible integrators. We present simulation tests involving complex solid-fluid interactions, including solid fractures generated by fluid interactions.
... However, the middle regime of (visco) plastic deformation is more challenging. Plastic models can suffer from rate-independency [6] and in some cases, they may have issues with modeling strain hardening [7]. Whereas, viscoplastic models eliminate some numerical difficulties associated with plastic models, such as hardening [8]. ...
Conference Paper
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This research investigates the development and validation of state-of-the-art high-fidelity models of soil cutting operations. The accurate and efficient modeling of complex tool-soil interactions is an open problem in the literature. Modeling options that provide more flexibility in trading off accuracy and computational efficiency than current state-of-the-art continuum or discrete element methods are sought. In this work, two modern numerical methods, the material point method (MPM) and a hybrid approach, are presented with the goal to simulate excavation maneuvers efficiently and with high accuracy. MPM, as an accurate, continuum-based and meshfree method, uses a constitutive model (here, non-local granular fluidity model) for computing internal forces to update particle velocities and positions. The hybrid approach, a combination of particle and grid-based methods, avoids explicit integration scheme difficulties and unnecessary computations in the static regime. Visual and quantitative data, including forces on the excavation tool, are collected experimentally to evaluate these two simulation methods with respect to geometry of the soil deformation as well as interaction forces, both as a function of time.
... Note that various constitutive models can be implemented into the framework of the MPM to capture different materials and their distinct behaviors. For example, a nonassociated Mohr-Coulomb model was applied to model landslide and dam failure (Zabala and Alonso, 2011;Soga et al., 2016) and a nonassociated Drucker-Prager model was used to simulate sand (Klár et al., 2016). In this study, we use the associated modified cam clay model developed for snow (Gaume et al., 2018), which reproduces mixed-mode snow fracture and compaction hardening. ...
Article
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Snow avalanches cause fatalities and economic damage. Key to their mitigation is the understanding of snow avalanche dynamics. This study investigates the dynamic behavior of snow avalanches, using the material point method (MPM) and an elastoplastic constitutive law for porous cohesive materials. By virtue of the hybrid Eulerian–Lagrangian nature of the MPM, we can handle processes involving large deformations, collisions and fractures. Meanwhile, the elastoplastic model enables us to capture the mixed-mode failure of snow, including tensile, shear and compressive failure. Using the proposed numerical approach, distinct behaviors of snow avalanches, from fluid-like to solid-like, are examined with varied snow mechanical properties. In particular, four flow regimes reported from real observations are identified, namely, cold dense, warm shear, warm plug and sliding slab regimes. Moreover, notable surges and roll waves are observed peculiarly for flows in transition from cold dense to warm shear regimes. Each of the flow regimes shows unique flow characteristics in terms of the evolution of the avalanche front, the free-surface shape, and the vertical velocity profile. We further explore the influence of slope geometry on the behavior of snow avalanches, including the effect of slope angle and path length on the maximum flow velocity, the runout angle and the deposit height. Unified trends are obtained between the normalized maximum flow velocity and the scaled runout angle as well as the scaled deposit height, reflecting analogous rules with different geometry conditions of the slope. It is found that the maximum flow velocity is mainly controlled by the friction between the bed and the flow, the geometry of the slope, and the snow properties. We reveal the crucial effect of both flow and deposition behaviors on the runout angle. Furthermore, our MPM modeling is calibrated and tested with simulations of real snow avalanches. The evolution of the avalanche front position and velocity from the MPM modeling shows reasonable agreement with the measurement data from the literature. The MPM approach serves as a novel and promising tool to offer systematic and quantitative analysis for mitigation of gravitational hazards like snow avalanches.
... Soil flush. IQ-MPM can also animate the detailed interaction between fluid and granular media (modeled as an elastoplastic solid as in [Klár et al. 2016;). In Fig. 21 we show the runtime breakdown for a representative time step of the "dam jello" example ( Fig. 9). ...
... MPM is widely used in various graphics applications such as sand [37], [39], [40], foam and complex fluids [41], [42], large strain elasticity [43], cloth and fiber collision [44], and phase change [31], as well as snow simulations [31]. As mentioned in the aforementioned studies, the standard FLIP technique is being improved and used for various materials. ...
Article
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Physically-based simulation is being expanded to simulate various materials such as deformable bodies, sand, and lava as well as liquid, and has been widely used as plug-in and software in actual related industries. Contrary to this trend, however, most of the simulation software products are built as an in-house solution, making it difficult for users to add new features not provided by the software. In this paper, we propose a post-processing framework that can improve the visual quality of liquid surfaces by receiving only the positions of liquid particles created in the existing simulation tools. Our framework includes the following methods: 1) analyzing liquid flow from particle position, 2) sampling technique for creating liquid sheets. Our framework improves the detail of liquid surfaces by temporarily adding new liquid particles through sampling of particles from existing simulation techniques. Our technique does not depend on any particular simulation tool. Experiments with various scenarios have also shown that liquid sheet detail, which was difficult to express in traditional software, has been improved, and problems such as break-up fluid surfaces and noisy surfaces have been mitigated. In addition, our research is a post-processing framework that uses only particle position as input data, making it easy to integrate with existing simulation tools that provide particle position.
... The MPM has a rich developmental history from both a solid and fluid mechanics [2] perspective and a computer graphics [15] perspective. MPM is a hybrid Eulerian-Lagrangian simulation method which has proven success and versatility in faithfully simulating snow [16], sand [17], non-Newtonian fluids, silicones [18], and cloth [19], as well as multiphysics applications in elastic-fluid and soft-rigid body coupling [20]. Further, MPM can easily be extended to extreme plastic phenomena such as fracture [21]. ...
Article
Full-text available
We present extensions to ChainQueen, an open source, fully differentiable material point method simulator for soft robotics. Previous work established ChainQueen as a powerful tool for inference, control, and co-design for soft robotics. We detail enhancements to ChainQueen, allowing for more efficient simulation and optimization and expressive co-optimization over material properties and geometric parameters. We package our simulator extensions in an easy-to-use, modular application programming interface (API) with predefined observation models, controllers, actuators, optimizers, and geometric processing tools, making it simple to prototype complex experiments in 50 lines or fewer. We demonstrate the power of our simulator extensions in over nine simulated experiments.
... We focus on the performance analysis and optimization for explicit integration of traditional MPM. Other forms or variants of MPM, e.g., semi-implicit MPM [Klár et al. 2016], augmented MPM [Stomakhin et al. 2014], IQ-MPM [Fang et al. 2020], and MPM with codimensional objects [Fei et al. 2018[Fei et al. , 2021Guo et al. 2018; involve additional components that are not studied in this work. And we leave the analysis and/or optimization of them as future work. ...
Preprint
Physics-based simulation has been actively employed in generating offline visual effects in the film and animation industry. However, the computations required for high-quality scenarios are generally immense, deterring its adoption in real-time applications, e.g., virtual production, avatar live-streaming, and cloud gaming. We summarize the principles that can accelerate the computation pipeline on single-GPU and multi-GPU platforms through extensive investigation and comprehension of modern GPU architecture. We further demonstrate the effectiveness of these principles by applying them to the material point method to build up our framework, which achieves $1.7\times$--$8.6\times$ speedup on a single GPU and $2.5\times$--$14.8\times$ on four GPUs compared to the state-of-the-art. Our pipeline is specifically designed for real-time applications (i.e., scenarios with small to medium particles) and achieves significant multi-GPU efficiency. We demonstrate our pipeline by simulating a snow scenario with 1.33M particles and a fountain scenario with 143K particles in real-time (on average, 68.5 and 55.9 frame-per-second, respectively) on four NVIDIA Tesla V100 GPUs interconnected with NVLinks.
... When the plastic flow of the Drucker-Prager model is assumed to be isochoric, the two models also use the same flow rule whereby the the plastic strain is purely deviatoric. Therefore, the µ(I) rheology model can be utilized without significant change in the existing algorithm for the Drucker-Prager plasticity model [41,42]. The Drucker-Prager and µ(I) models are distinguished according to the specific forms and rate dependence ofμ, asμ ...
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Granular impact – the dynamic intrusion of solid objects into granular media – is widespread across scientific and engineering applications including geotechnics. Existing approaches for simulating granular impact dynamics have relied on either a pure discrete method or a pure continuum method. Neither of these methods, however, is deemed optimal from the computational perspective. Here, we introduce a hybrid continuum–discrete approach, built on the coupled material-point and discrete-element method (MP-DEM), for simulating granular impact dynamics with unparalleled efficiency. To accommodate highly complex solid–granular interactions, we enhance the existing MP-DEM formulation with three new ingredients: (i) a robust contact algorithm that couples the continuum and discrete parts without any interpenetration under extreme impact loads, (ii) large deformation kinematics employing multiplicative elastoplasticity, and (iii) a trans-phase constitutive relation capturing gasification of granular media. For validation, we also generate experimental data through laboratory measurement of the impact dynamics of solid spheres dropped onto dry sand. Simulation of the experiments shows that the proposed approach can well reproduce granular impact dynamics in terms of impact forces, intrusion depths, and splash patterns. Further, through parameter studies on material properties, model formulations, and numerical schemes, we identify key factors for successful continuum–discrete simulation of granular impact dynamics.
... It is a hybrid Eulerian-Lagrangian discretization method widely employed in solid, fluid, and multiphase simulations. Due to its dual Eulerian and Lagrangian representations, MPM offers several advantages over the Finite Element Method: large deformation, fracture, as well as automatic contact and collision [7,108,75,80,113,63,99,54,84,60,39,100,55,114,115,22,25,70,24,91,31,111,44,104,32,57,64,33,101]. ...
Preprint
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This work proposes a model-reduction approach for the material point method on nonlinear manifolds. The technique approximates the $\textit{kinematics}$ by approximating the deformation map in a manner that restricts deformation trajectories to reside on a low-dimensional manifold expressed from the extrinsic view via a parameterization function. By explicitly approximating the deformation map and its spatial-temporal gradients, the deformation gradient and the velocity can be computed simply by differentiating the associated parameterization function. Unlike classical model reduction techniques that build a subspace for a finite number of degrees of freedom, the proposed method approximates the entire deformation map with infinite degrees of freedom. Therefore, the technique supports resolution changes in the reduced simulation, attaining the challenging task of zero-shot super-resolution by generating material points unseen in the training data. The ability to generate material points also allows for adaptive quadrature rules for stress update. A family of projection methods is devised to generate $\textit{dynamics}$, i.e., at every time step, the methods perform three steps: (1) generate quadratures in the full space from the reduced space, (2) compute position and velocity updates in the full space, and (3) perform a least-squares projection of the updated position and velocity onto the low-dimensional manifold and its tangent space. Computational speedup is achieved via hyper-reduction, i.e., only a subset of the original material points are needed for dynamics update. Large-scale numerical examples with millions of material points illustrate the method's ability to gain an order-of-magnitude computational-cost saving -- indeed $\textit{real-time simulations}$ in some cases -- with negligible errors.
... From the continuum point of view, the (MPM) particles represent discrete samples of the continuous material and the grid is just a helper for computing their physical interactions. In the work of [60][61][62], on a sandy soil specimen, the particles were rendered as individual grains. Herein, for the proposed numerical approach we propose the following assumption: porosity will be considered as the particle representation of the continuum. ...
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Desiccation cracking is a critical phenomenon soliciting the soil hydro-mechanical behavior, and significantly affects the performance of soil in geotechnical engineering. For this reason, an increasing interest toward studying and simulating the soil crack propagation, after a severe exposure to dry conditions (induced by desiccation), has been noticed during the recent years. However, major gaps remain in the previously developed models to properly provide a realistic prediction of the cracks pattern scheme especially when using the classical Finite Element Method (FEM), widely used in the geotechnical application. In this study, owing to the limitation of this method in re-meshing and dealing with large deformation, the authors were prompted to couple FEM with a mesh-free method: The Material Point Method (MPM) to overcome the individual drawbacks of each method. The dominant influencing factors on soil desiccation cracking have been assessed through a desiccation test performed in climatic chamber and using a digital image processing technique (image analysis) for a quantitative description of the studied sample. A model that relates porosity with suction and tensile strength was used to study the effect of the shrinkage phenomena in desiccation term, and to simulate the crack propagation in a thin clayey soil layer using the Code_Bright software. Consequently, a clear and connected crack pattern was observed, the problem of mesh dependency was clearly overcome proving the validity of the approach and providing a further insight into the behavior of clayey soil exposed to desiccation factors.
Article
The paper gives an overview of Material Point Method and shows its evolution over the last 25 years. The Material Point Method developments followed a logical order. The article aims at identifying this order and show not only the current state of the art, but explain the drivers behind the developments and identify what is currently still missing. The paper explores modern implementations of both explicit and implicit Material Point Method. It concentrates mainly on uses of the method in engineering, but also gives a short overview of Material Point Method application in computer graphics and animation. Furthermore, the article gives overview of errors in the material point method algorithms, as well as identify gaps in knowledge, filling which would hopefully lead to a much more efficient and accurate Material Point Method. The paper also briefly discusses algorithms related to contact and boundaries, coupling the Material Point Method with other numerical methods and modeling of fractures. It also gives an overview of modeling of multi-phase continua with Material Point Method. The paper closes with numerical examples, aiming at showing the capabilities of Material Point Method in advanced simulations. Those include landslide modeling, multiphysics simulation of shaped charge explosion and simulations of granular material flow out of a silo undergoing changes from continuous to discontinuous and back to continuous behavior. The paper uniquely illustrates many of the developments not only with figures but also with videos, giving the whole extend of simulation instead of just a timestamped image.
Article
The accurate and efficient modeling of granular flows and their interactions with external bodies is an open research problem. Continuum methods can be used to capture complexities neglected by terramechanics models without the computational expense of discrete element methods. Constitutive models and numerical solvers are the two primary aspects of the continuum methods. The viscoplastic size-dependent nonlocal granular fluidity (NGF) constitutive model has successfully provided a quantitative description of experimental flows in many different configurations in literature. This research develops a numerical approach, within a hyperelasticity framework, for implementing the dynamical form of NGF in three-dimensional material point method (3D MPM, an appropriate numerical solver for granular flow modeling). This approach is thermodynamically consistent to conserve energy, and the dynamical form includes the nonlocal effect of flow cessation. Excavation data, both quantitative measurements and qualitative visualization, are collected experimentally via our robotic equipment to evaluate the model with respect to the flow geometry as well as interaction forces. The results are further compared with the results from a recent modified plastic Drucker–Prager constitutive model, and in other configurations including wheel–soil interactions, a gravity-driven silo, and Taylor–Couette flow.
Article
Large-scale topological changes play a key role in capturing the fine debris of fracturing virtual brittle material. Real-world, tough brittle fractures have dynamic branching behaviour but numerical simulation of this phenomena is notoriously challenging. In order to robustly capture these visual characteristics, we simulate brittle fracture by combining elastodynamic continuum mechanical models with rigid-body methods: A continuum damage mechanics problem is solved, following rigid-body impact, to simulate crack propagation by tracking a damage field. We combine the result of this elastostatic continuum model with a novel technique to approximate cracks as a non-manifold mid-surface, which enables accurate and robust modelling of material fragment volumes to compliment fast-and-rigid shatter effects. For enhanced realism, we add fracture detail, incorporating particle damage-time to inform localised perturbation of the crack surface with artistic control. We evaluate our method with numerous examples and comparisons, showing that it produces a breadth of brittle material fracture effects and with low simulation resolution to require much less time compared to fully elastodynamic simulations.
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When we move on snow, sand, or mud, the ground deforms under our feet, immediately affecting our gait. We propose a physically based model for computing such interactions in real time, from only the kinematic motion of a virtual character. The force applied by each foot on the ground during contact is estimated from the weight of the character, its current balance, the foot speed at the time of contact, and the nature of the ground. We rely on a standard stress-strain relationship to compute the dynamic deformation of the soil under this force, where the amount of compression and lateral displacement of material are, respectively, parameterized by the soil’s Young modulus and Poisson ratio. The resulting footprint is efficiently applied to the terrain through procedural deformations of refined terrain patches, while the addition of a simple controller on top of a kinematic character enables capturing the effect of ground deformation on the character’s gait. As our results show, the resulting footprints greatly improve visual realism, while ground compression results in consistent changes in the character’s motion. Readily applicable to any locomotion gait and soft soil material, our real-time model is ideal for enhancing the visual realism of outdoor scenes in video games and virtual reality applications.
Article
In this paper, we articulate a novel plastic phase‐field (PPF) method that can tightly couple the phase‐field with plastic treatment to efficiently simulate ductile fracture with GPU optimization. At the theoretical level of physically‐based modeling and simulation, our PPF approach assumes the fracture sensitivity of the material increases with the plastic strain accumulation. As a result, we first develop a hardening‐related fracture toughness function towards phase‐field evolution. Second, we follow the associative flow rule and adopt a novel degraded von Mises yield criterion. In this way, we establish the tight coupling of the phase‐field and plastic treatment, with which our PPF method can present distinct elastoplasticity, necking, and fracture characteristics during ductile fracture simulation. At the numerical level towards GPU optimization, we further devise an advanced parallel framework, which takes the full advantages of hierarchical architecture. Our strategy dramatically enhances the computational efficiency of preprocessing and phase‐field evolution for our PPF with the material point method (MPM). Based on our extensive experiments on a variety of benchmarks, our novel method's performance gain can reach 1.56× speedup of the primary GPU MPM. Finally, our comprehensive simulation results have confirmed that this new PPF method can efficiently and realistically simulate complex ductile fracture phenomena in 3D interactive graphics and animation.
Article
We present a novel divergence free mixture model for multiphase flows and the related fluid‐solid coupling. The new mixture model is built upon a volume‐weighted mixture velocity so that the divergence free condition is satisfied for miscible and immiscible multiphase fluids. The proposed mixture velocity can be solved efficiently by adapted single phase incompressible solvers, allowing for larger time steps and smaller volume deviations. Besides, the drift velocity formulation is corrected to ensure mass conservation during the simulation. The new approach increases the accuracy of multiphase fluid simulation by several orders. The capability of the new divergence‐free mixture model is demonstrated by simulating different multiphase flow phenomena including mixing and unmixing of multiple fluids, fluid‐solid coupling involving deformable solids and granular materials.
Article
Physically plausible fracture animation is a challenging topic in computer graphics. Most of the existing approaches focus on the fracture of isotropic materials. We proposed a frame‐field method for the design of anisotropic brittle fracture patterns. In this case, the material anisotropy is determined by two parts: anisotropic elastic deformation and anisotropic damage mechanics. For the elastic deformation, we reformulate the constitutive model of hyperelastic materials to achieve anisotropy by adding additional energy density functions in particular directions. For the damage evolution, we propose an improved phase‐field fracture method to simulate the anisotropy by designing a deformation‐aware second‐order structural tensor. These two parts can present elastic anisotropy and fractured anisotropy independently, or they can be well coupled together to exhibit rich crack effects. To ensure the flexibility of simulation, we further introduce a frame‐field concept to assist in setting local anisotropy, similar to the fiber orientation of textiles. For the discretization of the deformable object, we adopt a novel Material Point Method(MPM) according to its fracture‐friendly nature. We also give some design criteria for anisotropic models through comparative analysis. Experiments show that our anisotropic method is able to be well integrated with the MPM scheme for simulating the dynamic fracture behavior of anisotropic materials.
Article
The material point method (MPM) recently demonstrated its efficacy at simulating many materials and the coupling between them on a massive scale. However, in scenarios containing debris, MPM manifests more dissipation and numerical viscosity than traditional Lagrangian methods. We have two observations from carefully revisiting existing integration methods used in MPM. First, nearby particles would end up with smoothed velocities without recovering momentum for each particle during the particle-grid-particle transfers. Second, most existing integrators assume continuity in the entire domain and advect particles by directly interpolating the positions from deformed nodal positions, which would trap the particles and make them harder to separate. We propose an integration scheme that corrects particle positions at each time step. We demonstrate our method's effectiveness with several large-scale simulations involving brittle materials. Our approach effectively reduces diffusion and unphysical viscosity compared to traditional integrators.
Article
Shear localization is a frequent feature of granular materials. While the discrete element method can properly simulate such a phenomenon as long as the grain representation is accurate, it is computationally intractable when there are a large number of grains. The general continuum-based finite element method is computationally tractable, yet struggles to capture many grain-scale effects, e.g., shear band thickness, because of mesh dependence and a lack of proper length scale in the model. We propose a hybrid discrete-continuum technique that combines the speed of the continuum method with the grain-scale accuracy of the discrete method. In the case of shear localization problems, we start the simulation using the continuum-based material point method. As the simulation evolves, we monitor an adaptation oracle to identify the onset of shear bands and faithfully enrich the macroscopic continuum shear bands into the microscopic-scale grains using the discrete element method. Our algorithm then simulates the shear band region with the discrete method while continuing to simulate the rest of the domain with the continuum method so that the computational cost remains significantly cheaper than a purely discrete solution. We validate our technique in simple shear, triaxial compression, and plate indentation tests for both dry and cohesive granular media. Our method is as accurate as a purely discrete simulation but shown to be over 100 times faster than a discrete simulation that would require tens of millions of grains.
Article
We present a novel Material Point Method (MPM) discretization of surface tension forces that arise from spatially varying surface energies. These variations typically arise from surface energy dependence on temperature and/or concentration. Furthermore, since the surface energy is an interfacial property depending on the types of materials on either side of an interface, spatial variation is required for modeling the contact angle at the triple junction between a liquid, solid and surrounding air. Our discretization is based on the surface energy itself, rather than on the associated traction condition most commonly used for discretization with particle methods. Our energy based approach automatically captures surface gradients without the explicit need to resolve them as in traction condition based approaches. We include an implicit discretization of thermomechanical material coupling with a novel particle-based enforcement of Robin boundary conditions associated with convective heating. Lastly, we design a particle resampling approach needed to achieve perfect conservation of linear and angular momentum with Affine-Particle-In-Cell (APIC) [Jiang et al. 2015]. We show that our approach enables implicit time stepping for complex behaviors like the Marangoni effect and hydrophobicity/hydrophilicity. We demonstrate the robustness and utility of our method by simulating materials that exhibit highly diverse degrees of surface tension and thermomechanical effects, such as water, wine and wax.
Article
Background and Objective Thrombus simulation plays an important role in many specialist areas in the field of medicine such as surgical education and training, clinical diagnosis and prediction, treatment planning, etc. Although a considerable number of methods have been developed to simulate various kinds of fluid flows, it remains a non-trivial task to effectively simulate thrombus because of its unique physiological properties in contrast to other types of fluids. To tackle this issue, this study introduces a novel method to model the formation mechanism of thrombus and its interaction with blood flow. Methods The proposed method for thrombus formation simulation mainly consists of three steps. First, we formulate the formation of thrombus as a particle-based model and obtain the fibrin concentration of the particles with a discretized form of the convection-diffusion-reaction equation; then, we calculate the velocity decay factor using the obtained fibrin concentration. Finally, the formation of thrombus can be simulated by applying the velocity decay factor on particles. Results We carried out extensive experiments under different settings to verify the efficacy of the proposed method. The experimental results demonstrate that our method can yield more realistic simulation of thrombus and is superior to peer method in terms of computational efficiency, maintaining the stability of the dynamic particle motion, and preventing particle penetration at the boundary. Conclusion The proposed method can simulate the formation mechanism of thrombus and the interaction between blood flow and thrombus both efficiently and effectively.
Conference Paper
One of the major challenges in space exploration robotics is understanding the interactions between robot wheels and planetary terrains that consist of granular regolith under reduced gravity conditions. A deeper understanding can contribute to robot wheel design and mobility control. A key factor, and the focus of this work, is the eﬀect of gravity. A candidate theoretical framework for wheel-soil interactions is Taylor Couette (TC) ﬂow, which models the ﬂow between a rotating inner and a stationary outer concentric cylinder. This research models TC continuum granular ﬂow in quasi-static and intermediate regimes to capture complexities neglected by terramechanics models without the computational expense of discrete element method (DEM). This research uses the experimental results produced by ﬂying a TC cell aboard a reduced-gravity aircraft presented in literature to investigate the three following methods: (1) an analytical model which captures the relative trends of velocity proﬁles at various vertical positions and gravities, but not the absolute values. (2) Finite element method (FEM) accompanied by a nonlocal constitutive model, which on average is consistent with the experimental results, however, does not capture the diﬀerences at various vertical positions and gravities. (3) Material point method (MPM) accompanied by a Drucker-Prager plasticity model, which results in both absolute velocity values and trends consistent with the experimental results, at diﬀerent gravity conditions. This research identiﬁes MPM as an appropriate continuum solver to model granular ﬂows under the inﬂuence of gravity.
Article
Diagenetic natural gas hydrate (DNGH) is a metastable material that is widely distributed in the frozen formations of Qinghai–Tibet Plateau in China. Drilling boreholes in such frozen formations can lead to the decomposition of DNGH, which is deleterious for wellbore stability as this process can lead to a complex and uncontrollable changes in the structure of the frozen formation. In this study, a fluid–solid–heat coupling mathematical model was developed for evaluating the wellbore deformation of a DNGH reservoir while considering the effect of DNGH decomposition. This mathematical model includes the kinetic equations describing DNGH decomposition, rock skeleton deformation equations, seepage field equations, temperature field equations, dynamic porosity equations, and dynamic permeability equations. COMSOL Multiphysics software was used to solve this new mathematical model and the deformation of a real wellbore in the permafrost of Qilian Mountains was analyzed. Two drilling methods were considered in the numerical simulation. The results indicate that: 1) the stress redistribution that occurs immediately after drilling the DNGH reservoir stabilizes rapidly. The maximum stress observed was at an angle of 135° with respect to the minimum horizontal principal stress. 2) The stress in the wellbore surrounding rock under the condition of micro-overbalanced drilling stabilizes more rapidly, and the values are higher compared with the results of micro-underbalanced drilling. 3) The most significant compression deformation occurs in the direction of minimum stress and the displacement in the wellbore surrounding rock simulated by micro-overbalanced drilling increases more rapidly and produces lower stress compared to that simulated with micro-underbalanced drilling.
Article
We present an adaptively updated Lagrangian Material Point Method (A‐ULMPM) to alleviate non‐physical artifacts, such as the cell‐crossing instability and numerical fracture, that plague state‐of‐the‐art Eulerian formulations of MPM, while still allowing for large deformations that arise in fluid simulations. A‐ULMPM spans MPM discretizations from total Lagrangian formulations to Eulerian formulations. We design an easy‐to‐implement physics‐based criterion that allows A‐ULMPM to update the reference configuration adaptively for measuring physical states, including stress, strain, interpolation kernels and their derivatives. For better efficiency and conservation of angular momentum, we further integrate the APIC [JSS*15] and MLS‐MPM [HFG*18] formulations in A‐ULMPM by augmenting the accuracy of velocity rasterization using both the local velocity and its first‐order derivatives. Our theoretical derivations use a nodal discretized Lagrangian, instead of the weak form discretization in MLS‐MPM [HFG*!!18], and naturally lead to a “modified” MLS‐MPM in A‐ULMPM, which can recover MLS‐MPM using a completely Eulerian formulation. A‐ULMPM does not require significant changes to traditional Eulerian formulations of MPM, and is computationally more efficient since it only updates interpolation kernels and their derivatives during large topology changes. We present end‐to‐end 3D simulations of stretching and twisting hyperelastic solids, viscous flows, splashing liquids, and multi‐material interactions with large deformations to demonstrate the efficacy of our new method.
Article
This paper introduces a simple method for simulating highly anisotropic elastoplastic material behaviors like the dissolution of fibrous phenomena (splintering wood, shredding bales of hay) and materials composed of large numbers of irregularly‐shaped bodies (piles of twigs, pencils, or cards). We introduce a simple transformation of the anisotropic problem into an equivalent isotropic one, and we solve this new “fictitious” isotropic problem using an existing simulator based on the material point method. Our approach results in minimal changes to existing simulators, and it allows us to re‐use popular isotropic plasticity models like the Drucker‐Prager yield criterion instead of inventing new anisotropic plasticity models for every phenomenon we wish to simulate.
Conference Paper
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Hybrid Lagrangian/Eulerian simulation is commonplace in computer graphics for fluids and other materials undergoing large deformation. In these methods, particles are used to resolve transport and topological change, while a background Eulerian grid is used for computing mechanical forces and collision responses. Particle-in-Cell (PIC) techniques, particularly the Fluid Implicit Particle (FLIP) variants have become the norm in computer graphics calculations. While these approaches have proven very powerful, they do suffer from some well known limitations. The original PIC is stable, but highly dissipative, while FLIP, designed to remove this dissipation, is more noisy and at times, unstable. We present a novel technique designed to retain the stability of the original PIC, without suffering from the noise and instability of FLIP. Our primary observation is that the dissipation in the original PIC results from a loss of information when transferring between grid and particle representations. We prevent this loss of information by augmenting each particle with a locally affine, rather than locally constant, description of the velocity. We show that this not only stably removes the dissipation of PIC, but that it also allows for exact conservation of angular momentum across the transfers between particles and grid.
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We present a solution method that, compared to the traditional Gauss-Seidel approach, reduces the time required to simulate the dynamics of large systems of rigid bodies interacting through frictional contact by one to two orders of magnitude. Unlike Gauss-Seidel, it can be easily parallelized, which allows for the physics-based simulation of systems with millions of bodies. The proposed accelerated projected gradient descent (APGD) method relies on an approach by Nesterov in which a quadratic optimization problem with conic constraints is solved at each simulation time step to recover the normal and friction forces present in the system. The APGD method is validated against experimental data, compared in terms of speed of convergence and solution time with the Gauss-Seidel and Jacobi methods, and demonstrated in conjunction with snow modeling, bulldozer dynamics, and several benchmark tests that highlight the interplay between the friction and cohesion forces.
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We propose a new method for auto stylized sand art drawing. Our system is built on top of our sand drawing system which is mixed with a particle system and a height map. An image is input and the patterns of the image is analyzed. The contours of the image are computed and they are converted into strokes. In sand art drawing, artists can use multiple fingers to draw the strokes which are symmetric or parallel at the same time. Our system detects such strokes and groups them according to geometric rules. After analyzing the image and collecting a set of strokes, we enter the drawing stage. In the drawing stage, our system automatically performs sand spilling, drawing strokes, leaking, erosion, pinch spilling and pinch erosion.
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We present an efficient Lagrangian framework for simulating granular material with high visual detail. Our model solves the computationally and numerically critical forces on a coarsely sampled particle simulation. Pressure and friction forces are expressed as constraint forces which are iteratively computed. We realize stable and realistic interactions with rigid bodies by employing pressure and friction-based boundary forces. Stable formations of sand piles are realized by employing the concept of rigid-body sleeping. Furthermore, material transitions from dry to wet can be modeled. Visual realism is achieved by coupling a set of highly resolved particles with the base simulation at low computational costs. Thereby, detail is added which is not resolved by the base simulation. The practicability of the approach is demonstrated by showing various high-resolution simulations with up to 20 million particles.
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We present a new particle-based method for granular flow simulation. In the method, a new elastic stress term, which is derived from a modified form of the Hooke’s law, is included in the momentum governing equation to handle the friction of granular materials. Viscosity force is also added to simulate the dynamic friction for the purpose of smoothing the velocity field and further maintaining the simulation stability. Benefiting from the Lagrangian nature of the SPH method, large flow deformation can be well handled easily and naturally. In addition, a signed distance field is also employed to enforce the solid boundary condition. The experimental results show that the proposed method is effective and efficient for handling the flow of granular materials, and different kinds of granular behaviors can be well simulated by adjusting just one parameter.
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We have performed an experiment in which a conical sandpile was built by slowly dropping sand onto a circular disk through a funnel with a small outlet. Avalanches (sand dropping off the disk) occurred, the size and the number of which were observed. It was seen that the behavior of avalanches (frequency-size distribution) was determined solely by the ratio of grain size to disk size, which is consistent with earlier studies. We categorize the behavior into three types: (1) the self-organized criticality (SOC) type (obeying Gutenberg-Richter's law), (2) the characteristic earthquake (CE) type where only large avalanches are almost periodically generated, and (3) the transition type. The transition from SOC to CE type drastically occurs when the ratio of grain diameter to disk radius is reduced to about 0.02. The underlying mechanism to cause the transition is considered. Results of simulation by cellular automaton models, an experimental result showing that a conical pile has a stress dip near its center, and a two-dimensional simulation building up a conical pile, all suggest that the transition occurs due to a change in stress profile inside and near the surface of the pile. Although we are unfortunately not able to understand the detailed mechanism at the present stage, it seems very important to further investigate the underlying physics of the transition because it presumably provides us a clue to understand the mechanism of the periodicity of large characteristic earthquakes and may open a way for earthquake prediction.
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Granular materials are ubiquitous in the world around us. They have properties that are different from those commonly associated with either solids, liquids, or gases. In this review the authors select some of the special properties of granular materials and describe recent research developments.[S0034-6861(96)00204-8]
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Nonlinear continuum mechanics of solids is a fascinating subject. All the assumptions inherited from an overexposure to linear behaviour and analysis must be re-examined. The standard definitions of strain designed for small deformation linear problems may be totally misleading when finite motion or large deformations are considered. Nonlinear behaviour includes phenomena like snap-through', where bifurcation theory is applied to engineering design. Capabilities in this field are growing at a fantastic speed; for example, modern automobiles are presently being designed to crumple in the most energy absorbing manner in order to protect the occupants. The combination of nonlinear mechanics and the finite element method is a very important field.Most engineering designs encountered in the fusion effort are strictly limited to small deformation linear theory. In fact, fusion devices are usually kept in the low stress, long life regime that avoids large deformations, nonlinearity and any plastic behaviour. The only aspect of nonlinear continuum solid mechanics about which the fusion community now worries is that rare case where details of the metal forming process must be considered.This text is divided into nine sections: introduction, mathematical preliminaries, kinematics, stress and equilibrium, hyperelasticity, linearized equilibrium equations, discretization and solution, computer implementation and an appendix covering an introduction to large inelastic deformations. The authors have decided to use vector and tensor notation almost exclusively. This means that the usual maze of indicial equations is avoided, but most readers will therefore be stretched considerably to follow the presentation, which quickly proceeds to the heart of nonlinear behaviour in solids. With great speed the reader is led through the material (Lagrangian) and spatial (Eulerian) co-ordinates, the deformation gradient tensor (an example of a two point tensor), the right and left Cauchy-Green tensors, the Eulerian or Almansi strain tensor, distortional components, strain rate tensors, rate of deformation tensors, spin tensors and objectivity. The standard Cauchy stress tensor is mentioned in passing, and then virtual work and work conjugacy lead to alternative stress representations such as the Piola-Kirchoff representation. Chapter 5 concentrates on hyperelasticity (where stresses are derived from a stored energy function) and its subvarieties. Chapter 6 proceeds by linearizing the virtual work statement prior to discretization and Chapter 7 deals with approaches to solving the formulation. In Chapter 8 the FORTRAN finite element code written by Bonet (available via the world wide web) is described.In summary this book is written by experts, for future experts, and provides a very fast review of the field for people who already know the topic. The authors assume the reader is familiar with elementary stress analysis' and has had some exposure to the principle of the finite element method'. Their goals are summarized by the statement, If the reader is prepared not to get too hung up on details, it is possible to use the book to obtain a reasonable overview of the subject'. This is a very nice summary of what is going on in the field but as a stand-alone text it is much too terse. The total bibliography is a page and a half. It would be an improvement if there were that much reference material for each chapter.
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A model for granular materials is presented that describes both the internal deformation of each granule and the interactions between grains. The model, which is based on the FLIP-material point, particle-in-cell method, solves continuum constutitive models for each grain. Interactions between grains are calculated with a contact algorithm that forbids interpenetration, but allows separation and sliding and rolling with friction. The particle-in-cell method eliminates the need for a separate contact detection step. The use of a common rest frame in the contact model yields a linear scaling of the computational cost with the number of grains. The properties of the model are illustrated by numerical solutions of sliding and rolling contacts, and for granular materials by a shear calculation. The results of numerical calculations demonstrate that contacts are modeled accurately for smooth granules whose shape is resolved by the computations mesh.
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An implicit-in-time method for granular materials is described. The method combines the material point method, a first-order contact algorithm, and a Newton–Krylov equation solver to give improved energy conservation, stabilization of the finite-grid-instability, and the correct description of collisions between grains.
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A broad class of engineering problems including penetration, impact and large rotations of solid bodies causes severe numerical problems. For these problems, the constitutive equations are history dependent so material points must be followed; this is difficult to implement in a Eulerian scheme. On the other hand, purely Lagrangian methods typically result in severe mesh distortion and the consequence is ill conditioning of the element stiffness matrix leading to mesh lockup or entanglement. Remeshing prevents the lockup and tangling but then interpolation must be performed for history dependent variables, a process which can introduce errors. Proposed here is an extension of the particle-in-cell method in which particles are interpreted to be material points that are followed through the complete loading process. A fixed Eulerian grid provides the means for determining a spatial gradient. Because the grid can also be interpreted as an updated Lagrangian frame, the usual convection term in the acceleration associated with Eulerian formulations does not appear. With the use of maps between material points and the grid, the advantages of both Eulerian and Lagrangian schemes are utilized so that mesh tangling is avoided while material variables are tracked through the complete deformation history. Example solutions in two dimensions are given to illustrate the robustness of the proposed convection algorithm and to show that typical elastic behavior can be reproduced. Also, it is shown that impact with no slip is handled without any special algorithm for bodies governed by elasticity and strain hardening plasticity.
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Connected particle systems can depict many objects difficult to model in any other fashion. We present a method for animating viscous fluids by simulating the forces of such particles interacting with each other. This method allows for collision detection between the particles and obstacles, both stationary and mobile, and it allows solid objects to break and melt. An approximate method for covering the particles with an isosurface for efficient rendering is also presented.
Conference Paper
A physically based model of an object is a mathematical representation of its behavior, which incorporates principles of Newtonian physics. Dynamic soil models are required in animations and realtime interactive simulations in which changes of natural terrain are involved. Analytic methods, based on soil properties and Newtonian physics, are presented in the paper to model soil slippage and soil manipulations. These methods can be used to calculate the evolution of a given soil configuration under the constraint of volume conservation and to simulate excavating activities such as digging, cutting, piling, carrying or dumping soil. Numerical algorithms with linear time complexity are also developed to meet the requirement of realtime computer simulation.
Article
Abstract Fluid animations in computer graphics show interactions with various kinds of objects. However, fluid flowing through a granular material such as sand is still not possible within current frameworks. In this paper, we present the simulation of fine granular materials interacting with fluids. We propose a unified Smoothed Particle Hydrodynamics framework for the simulation of both fluid and granular material. The granular volume is simulated as a continuous material sampled by particles. By incorporating previous work on porous flow in this simulation framework we are able to fully couple fluid and sand. Fluid can now percolate between sand grains and influence the physical properties of the sand volume. Our method demonstrates various new effects such as dry soil transforming into mud pools by rain or rigid sand structures being eroded by waves.
Article
We present a novel continuum-based model that enables efficient simulation of granular materials. Our approach fully solves the internal pressure and frictional stresses in a granular material, thereby allows visually noticeable behaviors of granular materials to be reproduced, including freely dispersing splashes without cohesion, and a global coupling between friction and pressure. The full treatment of internal forces in the material also enables two-way interaction with solid bodies. Our method achieves these results at only a very small fraction of computational costs of the comparable particle-based models for granular flows.
Article
We present a physics-based simulation method for animating sand. To allow for efficiently scaling up to large volumes of sand, we abstract away the individual grains and think of the sand as a continuum. In particular we show that an existing water simulator can be turned into a sand simulator with only a few small additions to account for inter-grain and boundary friction.We also propose an alternative method for simulating fluids. Our core representation is a cloud of particles, which allows for accurate and flexible surface tracking and advection, but we use an auxiliary grid to efficiently enforce boundary conditions and incompressibility. We further address the issue of reconstructing a surface from particle data to render each frame.
Article
Computer animations often lack the subtle environmental changes that should occur due to the actions of the characters. Squealing car tires usually leave no skid marks, airplanes rarely leave jet trails in the sky, and most runners leave no footprints. In this paper, we describe a simulation model of ground surfaces that can be deformed by the impact of rigid body models of animated characters. To demonstrate the algorithms, we show footprints made by a simulated runner in sand, mud, and snow as well as bicycle tire tracks, a bicycle crash, and a falling runner. The shapes of the footprints in the three surfaces are quite different, but the effects were controlled through six essentially orthogonal parameters. To assess the realism of the resulting motion, we compare the simulated footprints to video footage of human footprints in sand.
Conference Paper
We present a method to compute friction in a particle-based simulation of granular materials on GPUs and its data structure. We use Distinct Element Method to compute the force between particles. There has been a method to accelerate Distinct Element Method using GPUs, but the method does not compute friction. We implemented friction into the DEM simulation on GPUs and this leads to the real-time simulation of granular materials.
Conference Paper
Interactive applications such as virtual reality systems have become popular in recent years. A ground surface composed of a granular material can be deformed when it comes into contact with an object, and, in this paper, we propose a deformation algorithm for the ground surface which is useful for such applications. The deformation algorithm is divided into three steps: (1) detection of the collision between an object and the ground surface, (2) displacement of the granular material, and (3) erosion of the material at steep slopes. The proposed algorithm can handle objects of various shapes, including a concave polyhedron, and a texture sliding technique is proposed to represent the motion of the granular materials. In addition, the proposed algorithm can be used at an interactive frame rate.
Conference Paper
The paper deals with the modeling of loose soil (sandy, muddy, etc.). When an object moves on such grounds, the object's and the soil's movement both depend on the mutual physical interactions, and therefore are very difficult to achieve with kinematic or geometric models. We use a particle based dynamic modeler and achieve a discrete model of plasticity, which accounts for the influence of the soil on objects moving on this soil, but also for the influence of the object on the movement and the shape of the soil. Thus we have simulated soil compression and piling, vehicles leaving tyre traces spinning, skidding and even sinking. This first step is the simulation of the soil object system at a discretization scale that can be termed “intermediate”. A subsequent step consists of the simulation of a finer physical soil model in order to account for smaller scale dynamic phenomena
Article
This paper proposes a simplified position-based physics that allows us to rapidly generate "piles" or "clumps" of many objects: local energy minima under a variety of potential energy functions. We can also generate plausible motions for many highly interacting objects from arbitrary starting positions to a local energy minimum. We present an efficient and numerically stable algorithm for carrying out position-based physics on spheres and non-rotating polyhedra through the use of linear programming. This algorithm is a generalization of an algorithm for finding tight packings of (nonrotating) polygons in two dimensions. This work introduces linear programming as a useful tool for graphics animation. As its name implies, position-based physics does not contain a notion of velocity, and thus it is not suitable for simulating the motion of free-flying, unencumbered objects. However, it generates realistic motions of "crowded" sets of objects in confined spaces, and it does so at least two...
The material point method for the physics-based simulation of solids and fluids
• C Jiang
JIANG, C. 2015. The material point method for the physics-based simulation of solids and fluids. PhD thesis, University of California, Los Angeles.
Modeling landslide-induced flow interactions with structures using the Material Point Method. PhD thesis. Mast C. 2013. Modeling landslide-induced flow interactions with structures using the Material Point Method
• C Mast
MAST, C. 2013. Modeling landslide-induced flow interactions with structures using the Material Point Method. PhD thesis.
Continuum foam: a material point method for sheardependent flows
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YUE, Y., SMITH, B., BATTY, C., ZHENG, C., AND GRINSPUN, E. 2015. Continuum foam: a material point method for sheardependent flows. ACM Trans Graph 34, 5, 160:1-160:20.
Numerical Optimization. Springer series in operations research and financial engineering
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NOCEDAL, J., AND WRIGHT, S. 2006. Numerical Optimization. Springer series in operations research and financial engineering. Springer.