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# Multi-species simulation of porous sand and water mixtures

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We present a multi-species model for the simulation of gravity driven landslides and debris flows with porous sand and water interactions. We use continuum mixture theory to describe individual phases where each species individually obeys conservation of mass and momentum and they are coupled through a momentum exchange term. Water is modeled as a weakly compressible fluid and sand is modeled with an elastoplastic law whose cohesion varies with water saturation. We use a two-grid Material Point Method to discretize the governing equations. The momentum exchange term in the mixture theory is relatively stiff and we use semi-implicit time stepping to avoid associated small time steps. Our semi-implicit treatment is explicit in plasticity and preserves symmetry of force linearizations. We develop a novel regularization of the elastic part of the sand constitutive model that better mimics plasticity during the implicit solve to prevent numerical cohesion artifacts that would otherwise have occurred. Lastly, we develop an improved return mapping for sand plasticity that prevents volume gain artifacts in the traditional Drucker-Prager model.
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Multi-species simulation of porous sand and water mixtures
ANDRE PRADHANA TAMPUBOLON, University of California, Los Angeles
THEODORE GAST, University of California, Los Angeles and Jixie Eects
GERGELY KLÁR, DreamWorks Animation
CHUYUAN FU, University of California, Los Angeles
JOSEPH TERAN, University of California, Los Angeles and Jixie Eects
CHENFANFU JIANG, University of Pennsylvania and Jixie Eects
KEN MUSETH, DreamWorks Animation
Fig. 1. Levee breach. Water pours in from a reservoir and slowly erodes a dam. As water seeps into the sand, its cohesivity decreases. When it eventually
breaks, the landslide creates interesting dynamics in the debris flow.
We present a multi-species model for the simulation of gravity driven land-
slides and debris ows with porous sand and water interactions. We use
continuum mixture theory to describe individual phases where each species
individually obeys conservation of mass and momentum and they are cou-
pled through a momentum exchange term. Water is modeled as a weakly
compressible uid and sand is modeled with an elastoplastic law whose
cohesion varies with water saturation. We use a two-grid Material Point
Method to discretize the governing equations. The momentum exchange
term in the mixture theory is relatively sti and we use semi-implicit time
stepping to avoid associated small time steps. Our semi-implicit treatment is
explicit in plasticity and preserves symmetry of force linearizations. We de-
velop a novel regularization of the elastic part of the sand constitutive model
that better mimics plasticity during the implicit solve to prevent numerical
cohesion artifacts that would otherwise have occurred. Lastly, we develop
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0730-0301/2017/7-ART1 $15.00 DOI: http://dx.doi.org/10.1145/3072959.3073623 an improved return mapping for sand plasticity that prevents volume gain artifacts in the traditional Drucker-Prager model. CCS Concepts: Computing methodologies Physical simulation; Additional Key Words and Phrases: MPM, elastoplasticity, porous media ACM Reference format: Andre Pradhana Tampubolon, Theodore Gast, Gergely Klár, Chuyuan Fu, Joseph Teran, Chenfanfu Jiang, and Ken Museth. 2017. Multi-species simula- tion of porous sand and water mixtures. ACM Trans. Graph. 36, 4, Article 1 (July 2017), 11 pages. DOI: http://dx.doi.org/10.1145/3072959.3073623 1 INTRODUCTION While wet sand is both ubiquitous and literally child’s play, simulat- ing the underlying interaction of water and sand certainly is not. In fact, this type of visual eect is rarely seen in feature movie produc- tions, typically because it is considered too complex and dicult to achieve with existing particle-based simulation techniques. This is surprising given that water simulations, and to a lesser extent also sand simulations, are routinely undertaken in VFX. It is the complex nature of the changing material behaviors resulting from ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. 1:2 Pradhana Tampubolon, A. et al the mixing of sand and water that complicates simulation. However, we argue that this is precisely what makes the animation of wet sand so visually intriguing and hence desirable to master. Most practitioners in the VFX industry would agree that anima- tion of water is a mature area of research, as evident by its abundance and the fact that several excellent commercial solutions exist today, c.f. Houdini, Maya, and RealFlow just to mention a few. Conversely, animation of sand is considered challenging by most, and is still a subject of signicant research. Although the Material Point Method (MPM) [Sulsky et al . 1995] has been demonstrated to produce en- couraging results for sand simulations, commercial solutions tend to use simpler particle based techniques, like Position Based Dy- namics (PBD), which is easier to implement, but less well founded in continuum theory. However, mixtures of water and sand present a completely new set of simulation challenges for which, to the best of our knowledge, there is still no established best practice in VFX, let alone o-the-shelf commercial solutions. While there are many successful examples of wet sand animation, they tend to be based on art-directed constrained simulations or even procedural particle systems. Given the success of simulating sand with MPM it is natural to explore a similar approach for wet sand. Such a unied MPM description of the mixing of water and sand is exactly what motivated our work. In this paper, we develop a method for multi-species MPM to allow interactions between sand and water for gravity driven landslides and debris ows. We resolve wet sand transitions from cohesive rigid grains to owing slurries as water saturation increases. We use a semi-implicit time stepping scheme to deal with sti terms in the multi-species momentum exchange. The semi-implicit approach in [Stomakhin et al . 2013] produces articial cohesive eects for sand simulations. We introduce a unilateral modication to the elastic energy function that better mimics the eect of plasticity to prevent these cohesive artifacts. Furthermore, we develop an improved return mapping for Drucker-Prager plasticity that reduces the volume gain observed with lower grid resolutions. In our multi- species treatment, we use the two grid scheme proposed by Bandara et al. [2015]. Each material point is associated with a set of grid nodes and the interaction between the two species is via an interaction term proportionate to relative velocity between the phases. This term is sti for practical porosities, and it is impractical to do explicit simulation since it would require a time-step restriction on the order of t 10 6 10 5 . Our semi-implicit treatment allows us to use a time step on the order of t= 10 3 . Furthermore, using our modication to the elastic energy function and lagged plasticity approach, the system to be solved remains symmetric. We summarize our contributions as: A unilateral elastic constitutive model modication frame- work that removes numerical cohesion with semi-implicit time stepping and plasticity. An improved return mapping algorithm for Drucker-Prager plasticity with reduced volume gain artifacts. A semi-implicit two grid MPM discretization of sti inter- action terms in multi-species continuum equations. A model for wet sand cohesion based on water saturation. 2 PREVIOUS WORK Probably the earliest work on water and sand in computer graphics is by Peachey [1986]. Rungjiratananon et al. simulate sand-water interaction in real-time using a hybrid Smoothed Particles Hydro- dynamics (SPH) and Discrete-Element Method (DEM) approach [Rungjiratananon et al . 2008]. Lenaerts and Dutre [2009] also couple water with porous granular materials using SPH. Notably, these approaches capture a wider range of porous phenomena than that considered in our approach. While we focus on gravity driven land- slides and debris ows, their approaches more accurately capture surface tension driven eects like capillary action drawing water into dry sand. They can also handle landslides and debris ows, but they do so with SPH where we develop an MPM approach that naturally allows for implicit time stepping and high resolution simu- lation. Other graphics approaches have shown the ecacy of hybrid Lagrangian/Eulerian approaches like FLIP and MPM, including sand [Daviet and Bertails-Descoubes 2016; Klár et al . 2016; Narain et al . 2010; Zhu and Bridson 2005] and various other elastoplastic materi- als [Jiang et al . 2015; Ram et al . 2015; Stomakhin et al . 2013, 2014; Yue et al . 2015]. Unilateral incompressibility is an eective assump- tion for granular materials [Alduán and Otaduy 2011; Daviet and Bertails-Descoubes 2016; Ihmsen et al. 2013; Narain et al. 2010]. Our approach is the rst MPM technique in graphics that consid- ers multi-species modeling for porous sand/water. However, mixture theory and multi-species simulations have been used for a wide range of eects in computer graphics. Nielsen and Osterby [2013] simulate spray and mist with a two-continua mixture model. Takashi et al. [2003] use the Cubic Interpolation Propagation method to cou- ple spray, water and foam continua. Losasso et al. [2008] and Yang et al. [2014] also represent spray and dense water with multiple phases. Similar multi-species interaction ideas have been used for bubbles in incompressible ow [Mihalef et al . 2009; Ren et al . 2015; Song et al . 2005; Thürey et al . 2007]. Liu et al. [2008] use two continua to simulate mixtures of air and dust. Multi-species approaches have been used for miscible and immiscible uids [Bao et al . 2010; He et al. 2015; Kang et al. 2010; Ren et al. 2014; Yang et al. 2015]. Various researchers in engineering have shown the ecacy of sim- ulating water and soil interactions with the MPM. Abe et al. [2014] solve coupled hydromechanical problems of uid-saturated soil sub- jected to large deformation with a two grid MPM algorithm based on Biot’s mixture theory. Bandara et al. use a single grid MPM method for saturated and unsaturated soils that undergo large deformations in [2016] and two grid MPM to represent soil skeleton and pore water layers in [Bandara and Soga 2015]. Jassim et al. [2013] also develop a two grid MPM approach for soil mechanics problems. Mast et al. [Mast et al . 2014] use MPM to simulate large deforma- tion, gravity-driven landslides of porous soil. Mackenzie-Helnwein et al. [2010] examine the multi-species momentum exchange terms for problems with liquefaction, landslides, and sedimentation with two grid MPM. 3 MATHEMATICAL BACKGROUND We model sand and water as a multi-species continuum using mix- ture theory [Atkin and Craine 1976; Borja 2006]. With this approach, each species is given distinct material properties, and their motion is ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. Multi-species simulation of porous sand and water mixtures 1:3 derived from distinct velocity elds. This kinematic assumption al- lows sand and water to occupy the same points in space at the same time to create the mixture. We use superscript s to represent sand quantities and superscript w to represent water quantities. With this convention, the primary state is dened in terms of mass density ρα(x,t) and material velocity vα(x,t) , where α=s,w . The momen- tum density of phase α is given by ραvα . The total mass density of the mixture is the sum ρ=ρs+ρw and total momentum is the sum ρv=ρsvs+ρwvw . This denes the velocity v=1 ρ(ρsvs+ρwvw) of the mixture as the mass-averaged velocity of the constituents. Each species obeys the following conservation of mass with re- spect to its own motion Dαρα Dt +ρα∇ · vα=0.(1) Here the operator Dαf Dt =f t+vα· ∇f is the material (or total) derivative of function f with respect to the motion of either the sand ( α=s ) or water ( α=w ) phases. By summing over α in Equation 1, we obtain the standard conservation of mass Dρ Dt +ρ∇ · v=0,(2) by noting that ρDf Dt =PαραDαf Dt where Df Dt =f t+v· ∇f is the material derivative of function f with respect to the motion of the mixture. Each species also obeys conservation of linear momentum as ραDαvα Dt =∇ · σα+pα+ραg,α=s,w,(3) where pα represents the transfer of momentum due to the relative motion of the constituents, σα is the partial stress tensor associated with species α , and g is the gravitational acceleration. Because pα represent the exchange of momenta between species, the sum Pαpα=0 must be zero to not aect the total linear momentum of the mixture. Indeed with this constraint we can show by summing over αin Equation 3 ρDv Dt =∇ · σ+ρg,(4) where we have summed over α and introduced σ=Pασα , which denotes the Cauchy stress in the mixture expressed as the sum of the partial stresses in each species. In other words, with this notion of the Cauchy stress, conservation of linear momentum for the individual species implies conservation of linear momentum for the mixture. 3.1 Sand elastoplasticity We dene the constitutive behavior of sand with elastoplasticity model as in Klár et al. [2016]. We modify it slightly to include cohesive stresses. The amount of cohesion varies with the saturation level of water in the mixture. We assume that the sand partial stress σs is dened in terms of the hyperelastic potential energy density ψsas σs=1 det(Fs) ψs F(Fs,E)Fs,E.(5) Here, Fs is the deformation gradient of the sand motion, which evolves as Ds Dt Fs=vsFs . The appearance of the 1 det(Fs) and Fs,E terms arise because we write the potential energy density in terms of the deformation gradient. With this convention, its derivative gives rise to the rst Piola-Kirchho stress and these terms are need to transform it into the Cauchy stress. As typically done in nite strain elastoplasticity [Bonet and Wood 2008], it is decomposed into Fs=Fs,EFs,P to dene a plastic ow. For sand, Fs,E represents the remembered compression and shearing, while Fs,P represents the forgotten sliding and separation. We use the Drucker-Prager [Drucker and Prager 1952] plastic ow and yield condition to de- termine the evolution of the elastic ( Fs,E ) and plastic ( Fs,P ) parts of the deformation gradient. The Drucker-Prager yield condition is dened from the constraint that the shear stress should be no larger than the compressive normal stress in all directions. This expresses a mechanical interaction that is consistent with Coulomb friction. While dry sand is modeled eectively with this assumption, it precludes the eects of cohesion. However, cohesive eects can be modeled by modifying the elastic stress yield condition to be cFtr(σs)+σstr(σs) dIFcC,(6) where d= 2 , 3is the spatial dimension, cC 0increases with the amount of cohesion in the material and cF 0increases with amount of friction between grains. E.g. for dry sand in Klár et al. [2016] they used cC= 0. A positive cC shifts the yield surface along the hydrostatic axis (the line where tr(σs)= 0), which allows the material to exhibit stress under tension and thus cohere to itself. Here we model cohesivity as a function of water saturation in the sand. This is naturally measured in terms of the volume fraction of water in the mixture ϕw=ρw ρso that cC=cC(ϕw). Inequality 6 is referred to as the plastic yield condition, if it is satised, there will be no further plastic deformation. The boundary of the region in stress space dened by cFtr(σs)+σstr(σs) dIF= cC(ϕw) is called the yield surface. For states of stress on the yield surface, plastic ow will commence when a perfectly elastic assump- tion would drive the stress out of the region. The plasticity functions as a means of satisfying this inequality constraint. 3.1.1 Unilateral hyperelasticity. Here we describe the elastic part of the constitutive behavior for the sand phase. This is largely the same as in Klár et al. [2016] where the elastic potential en- ergy density is dened in terms of the logarithmic strain ϵ as ψ(Fs)=˜ ψ(ϵ)=µtr(ϵ2)+λ 2tr(ϵ) where Fs=UΣV is the singular de- composition of F and ϵ=log(Σ) . However, we provide some novel modications that allow for more ecient implicit time integra- tion. We note that with the plastic yield condition from Inequality 6, much of the energy landscape has no eect since it produces stresses outside of the yield surface. To mitigate this we modify the energy density such that it smoothly transitions to zero in these regions. This has no eect on the continuous behavior of the governing equa- tions, but it improves the performance of our semi-implicit time stepping. Essentially, our modication modies the elastic behavior outside the yield surface to better resemble the eects of plasticity. Our approach is similar to, and indeed inspired by the unilat- eral approaches in [Alduán and Otaduy 2011; Daviet and Bertails- Descoubes 2016; Ihmsen et al . 2013; Narain et al . 2010]. We thus refer to our modied energy as the unilateral energy function ˜ ψU(ϵ) . We ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. 1:4 Pradhana Tampubolon, A. et al Fig. 2. Two grids. We use separate water grid and sand grids. The blue and red dots denote water and sand particles respectively. In the overlapping region, we compute the momentum exchange term between the two species. We also compute the force based on individual constitutive model. Fig. 3. The contour plot of the multiplier in strain space. Region A is where the multiplier is the identity, region C is where the multiplier is zero, and region B is where the multiplier transitions from 1to 0in C2 manner. The green cone denotes the Drucker-Prager cone in principal strain space. dene this as the product of the original energy function ˜ ψ(ϵ) and a multiplier h(ϵ) ˜ ψU(ϵ)=˜ ψ(ϵ)h(ϵ).(7) The multiplier serves to (as smoothly as possible) transition the energy to zero in regions outside yield surface. We dene it to be symmetric about the hydrostatic axis in the strain space (axis of equal strain, see Figure 3) since the original energy ˜ ψ has this property and we wish to preserve it in ˜ ψU . To construct it, we partition strain space into the three regions labeled A, B, and C in Figure 3. In region A, the value of the multiplier is simply one and the modied constitutive model is identical to the original one. Conversely, in region C, the multiplier is set to zero. In region B, the multiplier smoothly transitions from one to zero while guaranteeing that we have a continuous second derivative. The boundary of region A and B denes two envelopes that are symmetric about the hydrostatic axis. To best preserve the behavior of sand, we want region A to cover most of the region inside of the Drucker-Prager cone, illustrated as the green cone in Figure 3. The construction of this multiplier function is done as a composi- tion of two functions: hs which is a scalar function and f which is a function of the strain ϵ. If we let oto denote the hydrostatic axis ( o=1 2(1,1) in 2D or 1 3(1,1,1) in 3D), then we can compute u=ϵ·o to be the component of the strain in the hydrostatic axis and v=ϵuo. The function fis dened as f(ϵ)=co v4 1+|v|3.(8) The coecient co controls the opening of the envelope of region A and B around the hydrostatic axis. The scalar function hs is chosen so that the multiplier function is twice continuously dierentiable, and is given by hs(z)= 1if z<0 0if z>1 110z3+15z46z5otherwise. (9) The multiplier function is then dened for some choice of param- eters a,b,and sC. h(ϵ)= 1if u+f(ϵ)<a+sC 0if u+f(ϵ)>b+sC hsu+f(ϵ)asC baotherwise. (10) The parameter a and b determines the intersection of the hydro- static axis with the boundary of region A and B respectively. The parameter sC controls a shift of this multiplier region along the hydrostatic axis. 3.2 Water We model the water as nearly incompressible [Becker and Teschner 2007] with the partial stress σw=pwI,pw=k 1 Jwγ1!.(11) We note that the water pressure is related to a potential ψw as pw= ψw Jw(Jw) , with ψw(Jw)=k(Jw)(1γ) 1γJw . This pressure pw is designed to stiy penalize the volume change of the water phase, which is characterized in terms of the determinant of the water deformation gradient Jw=det(Fw) . As with the solid phase, the uid deformation gradient evolves according to Dw Dt Fw=vwFw . Intuitively, Jw is the ratio of the current to initial local volume of ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. Multi-species simulation of porous sand and water mixtures 1:5 Fig. 4. Sand wedges. A box of sand with various values of cohesion is dropped onto a wedge. The top le simulation is done using an explicit time-stepping scheme, while all the others are done using a semi-implicit time-stepping scheme. Fig. 5. Unilateral extension. We demonstrate a 2D sand column collapse with dierent values of c0 coeicient in the unilateral modified energy density function. From top to boom, c0= 1 . 1 , 2 , 2 . 5 , 3. Here parameters a and bfrom Equation 10 are a=0.5, and b=0. material in the water phase. It evolves as DwJw Dt =∇ · vwJw.(12) Here k is the bulk modulus of the water and γ is a term that more stiy penalizes large deviations from incompressibility. 3.3 Momentum exchange The momentum exchange terms ps,pw for water and porous sand interactions can generally be viewed as a combination of dissipative and reversible interactions [Borja 2006]. We follow the formulation of Bandara and Soga [2016] because like them we are concerned pri- marily with gravity driven ows such as fast catastrophic landslides and debris ows. Their formulation assumes ps=cEvwvs+pwϕw,pw=ps(13) where cE=n2ρwд ˆ k and n is the sand porosity, ˆ k is the sand perme- ability and д is the gravitational acceleration, ϕw=ρw ρ is the water volume fraction and pw is the water pressure. The rst term repre- sents viscous forces generated by sand particles moving through the uid. Although it can be conceived from the view of an ideal- ized particle moving through a Stokes uid, it simply amounts to a Coulomb-friction-like response [Mackenzie-Helnwein et al . 2010]. The second term is often called the “buoyancy term" in mixture theories [Robert and Soga 2013]. It can be conceived from entropy equilibrium constraints or from the physical consideration that the pore uid pressure multiplying the porosity is appropriate notion of reversible pressure, however there is some debate about its ap- propriateness outside of immiscible mixtures [Drumheller 2000]. Mackenzie-Helnwein et al. [2010] omit the second term and view the momentum exchange terms as purely dissipative processes. We also follow this approach for the majority of the examples presented in the paper, however we include one example demonstrating its eect in Figure 11. Even with the inclusion of the active term from [Bandara et al . 2016], we are only capable of simulating a rather ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. 1:6 Pradhana Tampubolon, A. et al Fig. 6. The levee is going to break. Here we demonstrate the eects of our parameters on levee wall integrity. A simulation with lower drag momentum exchange coeicient and lower cohesion will fail more easily. narrow range of porous media phenomena. While this is sucient for landslides and debris ows, phenomena such as capillary action drawing water into dry sand is not achievable in our approach. 3.4 Cohesion vs. Saturation As in Robert and Soga in [2013], we assume that the sand cohesion varies as a function of water saturation. We measure saturation as the percentage of water in the mixture which we estimate as the ratio of the density of the water phase to the total density ϕw=ρw ρw+ρs. The cohesion of sand is zero when it is completely dry ( ϕw= 0). Intuitively, as dry sand becomes saturated with more water, the cohesivity of the wet sand should increase as wet sand tends to better hold its shape. Indeed this was observed in the work of Robert and Soga in [2013]. However, they also observe that this increase only continues to a maximal value cmax C when the saturation is around ϕw= 0 . 4. Beyond this point the sand becomes more compliant to owing and less cohesively elastic. In all of our multispecies examples, we model water interaction with wet sand that is capable of holding its shape. We thus set the sand cohesion to be initially maximal, even in the absence of the water phase. Based on the observations in Robert and Soga [2013], we then assume that the cohesion decreases linearly with increasing saturation beyond this point (with cohesion equal to zero at full saturation ϕw=1). 4 DISCRETIZATION First, we explain the notation used in the discretization section. There are two sets of grids: one is associated with sand material and the other is associated with water. As with the continuous equations, the superscript α=s,w indicates the corresponding species. Whenever a symbol has a subscript i or j , this denotes one degree of freedom in grid node index i or j . Subscript p denotes attributes that belong to a particle. A symbol that is not followed by a subscript refers to the whole collection of grid nodes as degrees of freedom. For example vs,n+1 refers to all of the sand grid nodes that are active at step n+1. We discretize the continuum equation using the Material Point Method (MPM). As in other approaches in the engineering literature [Bandara and Soga 2015; Jassim et al . 2013; Mackenzie-Helnwein et al . 2010], we use two sets of grids (Figure 2): one for the solid particles and the other for the water particles. This is the primary dierence between our approach and others recently used in the graphics literature. The overview of the algorithm is as follows: (1) Transfer to grids: Transfer the mass and momentum of each species to its corresponding grid (§4.1). (2) Update grids momenta: Solving for the coupled water and sand grid velocities using semi-implicit backward Euler (§4.2). (3) Update particles: Update all particle state, including the cohesion based on saturation as well as plasticity return mappings (§4.3). 4.1 Transfer to grid We transfer mass and momentum from sand and water particles to their respective grid using APIC [Jiang et al . 2015]. For each species α , the particle p interacts with the grid node α i with weight wα,n ip =N(xα,n pxα i) . The weight is computed using the quadratic B-spline interpolation kernel. Mass is computed according to mα,n i=X p wα,n ip mα p,(14) and velocity according to vα,n i=1 mα,n iX p wα,n ip mα pvα,n p+Cα,n pxα,n ixα,n p,(15) where the matrix Cα,n p is an extra matrix stored per particle which denes an ane velocity eld local to the particle [Jiang et al . 2015]. Cα,n p is initialized with Cα,0 p=0 and updated at the end of the previous time step during the grid-to-particle transfer (Equation 29). ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. Multi-species simulation of porous sand and water mixtures 1:7 4.2 Update Grids Momenta Using MPM, the forces in the sand and water phases are computed as fs i(ˆ xs)=ψs ˆ xs i =X p V0 p ψU Fs(FsE p(ˆ xs)!(Fs E,n p)ws,n ip ,(16) fw i(ˆ xw)=ψw ˆ xw i =X p V0 p ψw Jw(Jw(ˆ xw))!Jw,nww,n ip .(17) As in [Stomakhin et al . 2013] we think of ˆ xα i as the position of the grid node i corresponding to species α that has been deformed from its original position xα i by an amount of tvα i , i.e. ˆ xα i= ˆ xα i(vα,n+1 i)=xα,n i+tvα,n+1 i . The discrete momentum balance to be solved is ms,n ivs,n+1 ivs,n i=tfs iˆ xs+ms,n ig+ds i(ˆ x)(18) mw,n jvw,n+1 jvw,n j=tfs jˆ xw+mw,n jg+dw j(ˆ x),(19) where the discrete interaction term is given by ds i j (ˆ x)=cEms imw j(vs,n+1 ivw,n+1 j),(20) dw ji (ˆ x)=cEms imw j(vs,n+1 ivw,n+1 j),(21) for some drag coecient cE. Setting M= Ms,n Mw,n!,v= vs vw!,f(ˆ x(vn+1)) = fs(ˆ xs) fw(ˆ xw)!, and D to be the drag coecient matrix derived from Equations 20 and 21, we can write the coupled system as (M+tD)vn+1=Mvn+tMд+f(ˆ x(vn+1)) .(22) At each time step, we solve this nonlinear system using a few iterations of a modied Newton’s method. Since the matrix D is symmetric, and both fs and fw are derived as the negative gradient of a potential, the whole system is symmetric when linearized (as- suming the eects of plasticity are ignored in the linearization) and can be solved using MINRES. We note that we do not include the eect of plasticity when computing the derivatives of fs . Doing oth- erwise results in non-symmetric sand force derivatives which would require GMRES. Our omission of these terms in the linearization of the system is a modication to the standard Newton’s method. However, it is essential that we use implicit time stepping because of the sti momentum exchange terms and our lagged plasticity approach is the key to making this ecient. Notably, this can lead to cohesion artifacts without the unilateral modication in (§3.1.1) (see Figure 7). 4.3 Update Particles Update Jw. We do not keep track the deformation gradient Fw of water particles, but we keep track of its determinant Jw , which is updated according to the discretization of Equation 12, i.e. Jw,n+1 p=I+ttr(vw,n+1 p)Jw,n. We found that in practice, this discretization tends to oer more stability than the alternative of evolving Fw followed by computing its determinant. Update Fs. Because we ignore the eects of plasticity during the implicit solve for momenta, ˆ FsE ,n+1 evolves with the grid during the grid momentum update as in Stomakhin et al. [2013] ˆ FsE ,n+1 p=I+tvs,n+1 pFsE ,n.(23) ˆ FsE ,n+1 p is later processed for plasticity at the end of the time step to dene FsE ,n+1 p, which we discuss below. Saturation based cohesion. We dene the water saturation on sand particles based on a heuristic. First, we populate a grid whose domain is the union of the sand grid and water grid domain. We mark each node that has a non-zero mass for both sand and water species as ϕw,n+1 i= 1, otherwise ϕw,n+1 i= 0. One can think of this grid as tracking an indicator function of the overlap region between the sand and water constituents. We then compute the saturation on the sand particle by interpolating from the grid to the sand particle according to ϕw,n+1 p=X i wn s,ip ϕw,n+1 i.(24) This approximates the saturation as equal to one (maximal) deep in the interior of the overlap region with a ramp to zero exterior to the region. In all of our multi-species examples, we assume that the wet sand is already saturated and has reached its maximum cohesion level. Hence, any amount of additional water saturation will lower the cohesion level of the sand. We vary cohesion in a linear fashion as a function of water saturation as discussed in Section 3.4 cs,n+1 Cp =cs,0 Cp (1ϕw,n+1 p)(25) where we note that the saturation is always in ( 0 , 1 ) . The approxi- mation of the saturation in Equation 24 has errors biased towards full saturation in the interior. This naturally leads to more rapid fail- ure in the landslides and debris ows we consider in our examples since the cohesion decreases more rapidly than it should. This is an extreme simplication to correct behavior dened in Robert and Soga [2013], but we found that it was eective for simulating these phenomena. Projection and volume correction treatment. We now de- scribe the plastic projection of ˆ FsE ,n+1 to FsE ,n+1 . We start from the Drucker-Prager projection, P , as implemented by Klár et al. [2016]. P projects strains outside the yield surface onto the yield surface according to the ow rule. The yield region has the shape of a cone. When the material is under expansion, the strain is projected to the tip. Otherwise it is projected to the side of the cone. However, while this approach is adequate for projection directions perpendicular to the hydrostatic axis, steps which project to the tip can induce volume gain. This occurs when a particle undergoes expansion that induces a cohesive elastic stress. In this case, stress is projected to the tip, which is a stress free state. The particle is then in a new rest state and any motion that would return it to its initial volume would be penalized elastically. There are existing corrections to this behavior in the literature, e.g. Dunatunga and Kamrin [2015] model material as a disconnected stress-free medium under sucient expansion. To combat this artifact, each sand particle carries an extra scalar attribute vs cp which tracks changes in the log of the volume gained during extension. This can be naturally taken into account in the ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. 1:8 Pradhana Tampubolon, A. et al sec/frame Scheme tParticle # x Dry Sand Wedge 129.65 Explicit 1×1041×1060.003 Wet Sand Wedge 72.65 Implicit 1×1031×1060.003 Dam Breaches 268.99 Implicit 1×1023×1060.01 Piling 11.16 Implicit 1×1034.1×1060.0014 2D Piling 7.26 Explicit 5×1054.7×1040.0025 2D Piling 3.39 Implicit 1×1044.7×1040.0025 2D Piling Unilateral 0.67 Implicit 1×1024.7×1040.0025 2D Hourglass 0.84 Explicit 7.4×1053×1040.0037 2D Hourglass 0.73 Explicit 7.4×1053×1040.0074 Table 1. Performance. logarithmic-strain-based constitutive model to allow for compres- sion in the event of prior net expansion. At each time step, we update vs cp according to vs,n+1 cp =vs,n cp +log(det(Fs E,n+1) ) log(det(ˆ FsE ,n+1)),(26) where vs,0 cp = 0. Finally, when we perform the Drucker-Prager projection, we update ϵsE,n+1according to ϵsE,n+1=P* ,ˆ ϵsE,n+1+ vn cp dI+ -,(27) where Pis the projection operator described above. This can be interpreted as projecting the elastic strain plus the volume gain term. Lastly, FsE ,n+1=Uˆ ϵsE,n+1V where ˆ FsE ,n+1= Ueˆ ΣsE ,n+1 V and ˆ ϵsE,n+1=log(ˆ ΣsE,n+1) . See Figures 8 and 9 for a demonstration of this eect. We note that this projection is par- ticularly dened for constitutive models written in terms of the logarithmic strain. For a more general constitutive model, it would require non-trivial modication. Update position and velocity. Velocity is updated according to vα,n+1 p=X i wα,n ip vα,n+1 i,(28) the ane velocity matrix is updated according to Cα,n+1 p=X i wn ip vα,n+1 i(( 4 h2)(xα,n ixα,n p)) ,(29) where h is the Eulerian grid spacing. Lastly position is updated according to xα,n+1 p=xα,n p+tvα,n+1 p.(30) 5 IMPLEMENTATION AND RESULTS We use the sparse grid structure provided by OpenVDB [Museth 2013]. The table of performance for various examples is given by Table 1. The list of parameters used for these examples are listed in Table 2. 5.1 Unilateral hyperelasticity and implicit time stepping Figure 7 demonstrates the eects of our unilateral constitutive model with semi-implicit time stepping. We note that our unilateral po- tential removes articial cohesion eects in the simulation of dry sand. The cohesion in this simulation is zero so the sand should not stick together. When we use the constitutive model from [Klár et al . 2016] with a semi-implicit time integration scheme, we see articial cohesion that gets worse as we increase the time step size. Using our unilateral elastic energy function removes this articial cohe- sion. We further demonstrate that our semi-implicit scheme gives results comparable to the more accurate, but more expensive fully implicit backward Euler scheme in Figure 10. The importance of choosing the right unilateral parameters is illustrated by Figure 5. If the regions A and B in Figure 3 do not closely t the Drucker-Prager cone, then the accuracy of the simulation is compromised for large time steps. 5.2 Volume fix and wet sand To illustrate our volume x to the Drucker-Prager artifact, we run a simulation of an hourglass turned three times with explicit sym- plectic Euler, as shown by Figure 8. The left most gure depicts the initial state of the sand. For each set of three hourglass gures, we show the state of the sand after going through the neck the rst time, after the rst ip, and after the second ip. Figure 9 depicts a similar artifact in a 3D piling of sand. Without the volume correc- tion algorithm and without a suciently small grid resolution, we observe a substantial volume gain artifact. 5.3 Wet sand and dam breach In Figure 4 we demonstrate how varying cohesion gives rise to dif- ferent wet sand behaviors. In Figure 6 we demonstrate our approach with an example that is representative of the types of gravity driven ows we are interested in with our approach. As water ows into the wall of a dam, the saturation increases weakening the it. The cohesion of sand decreases with saturation and the dam eventually breaks. We demonstrate the eect of the active component of the momen- tum exchange terms in Equation 13 using a simulation of a 2D dam breach, shown in Figure 11. Again water pours in from a reservoir and slowly erodes a retaining wall. We note that the active term has only a subtle eect on the bulk dynamics of the motion for these types of ows. We discretize the active term by adding Xpw,n pws,n ip mw,n i ms,n i+mw,n i to the water drag term in Equation 21. We then dene the solid drag term to be equal and opposite to the water drag in accordance with the zero-net-sum nature of the momentum exchange. 6 LIMITATIONS AND FUTURE WORK Our approach has a number of limitations. The momentum exchange model we use in the water/sand multispecies examples is rather ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. Multi-species simulation of porous sand and water mixtures 1:9 Fig. 7. From top to boom row: initial, middle, and final configuration of a 2D sand pile. On the top le is the result of explicit time-stepping with max t= 10 4 . The two columns in the middle are the results of semi-implicit time-stepping with the regular constitutive model and max t= 10 2 and 10 3 respectively. On the boom right is semi-implicit time stepping with our unilateral energy density function with maximum time step restriction of t= 10 2 . Fig. 8. An hourglass is flipped two times. The le most hourglass shaded grey depicts the initial state of sand. Each set-of-three figures depicts the state of the sand aer going through the neck the first time, and aer flipping the hourglass for the first and second time. In the first two sets, grid dx = 7 . 4 × 10 3 , while the last two sets have grid dx = 3 . 7 × 10 3 . The first and third set are run without any volume correction fix, while the second and last set are run with our volume fix. Fig. 9. Sand piling. The top row depicts the result of an explicit simulation with a coarser grid size ( dx = 10 2) , while the boom row corresponds to a grid size of 10 3 . The first and third column correspond to a projection step without volume correction, while the second and fourth column uses our volume correction algorithm. ACM Transactions on Graphics, Vol. 36, No. 4, Article 1. Publication date: July 2017. 1:10 Pradhana Tampubolon, A. et al ρEνFriction angle h0/h1/h2/h3 Castle 2200 3.537 ×1050.3 — 35/0/0.2/10 Friction angle 2200 3.537 ×1050.3 20/25/30/35/40 Hourglass 2200 3.537 ×1050.3 — 35/9/0.3/10 Buttery 2200 3.537 ×1050.3 — 35/9/0.2/10 Buttery close 2200 3.537 ×1050.3 — 35/9/0.2/10 Raking 2200 3.537 ×1050.3 — 35/9/0.2/10 Raking close 2200 3.537 ×1050.3 — 35/9/0.2/10 Pile from spout 2200 3.537 ×1050.3 30 Splash 1582 3.537 ×1060.3 22 Shovel 2200 3.537 ×1050.3 — 35/9/0.2/10 Young’s modulus 2200 103,4,5,60.3 — 35/9/0.3/10 Table 2. Material parameters are provided for all of our 3D simulations. Friction angle ϕF and hardening parameters h0 , h1 , and h3 are listed in degrees for convenience. simplied. While adequate for gravity driven ows like landslides and levee breaches, it is inadequate for capillary driven phenom- ena like water being drawn in to dry sand. Such phenomena has been captured by prior approaches like that of Lenaerts and Dutre [2009]. Furthermore we fail to capture behavior like those in Rungji- ratananon et al. [2008] where surface tension eects in wetting are more accurately captured. Although our approximation to the dependence of sand cohesion on saturation is useful for facilitating rapid failure of water/sand mixtures, it is an extreme simplication to the correct behavior dened in Robert and Soga [2013] and this compromises its accuracy dramatically. This reduces the applicability of our approach outside of visually plausible simulation applications. Large values of the momentum exchange coecient cE can lead to ill-conditioning in the linear systems that arise during implicit time stepping. We found that these cases required many MINRES iterations to resolve and lead to excessive run times. This compli- cated the simulation of slurry materials where the water and sand remain mixed. 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The two-layer formulation provides opportunities to solve dynamic flow problems in geomechanics such as internal and external erosions as well as fluid flow failures (seepage failures). The two-layer approach was adopted to study different strategies of modeling dragging interactions; for solid-fluid interaction in animation (Gao, Tampubolon, Jiang, & Sifakis, 2017;Tampubolon et al., 2017); combine with the DDMP (Tran, Solowski, Karstunen, & Korkiala-Tanttu, 2017;Tran, Solowski, Thakur, & Karstunen, 2017;Tran & Sołowski, 2019a, 2019b to model the large consolidation; combine with the GIMP (Liu, Sun, Jin, & Zhou, 2017) and to develop the two-layer approach with a thermodynamic constitutive model (Baumgarten & Kamrin, 2019). For instance, it allows modeling of fluidization of soil particles at the water interface and simulations of internal erosion by transferring a portion of soil particle mass to fluid particle. ... 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 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. Thesis Full-text available 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. Preprint Full-text available 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. Article We present a dynamic mixture model for simulating multiphase fluids with highly dynamic relative motions. The previous mixture models assume that the multiphase fluids are under a local equilibrium condition such that the drift velocity and the phase transport can be computed analytically. By doing so, it avoids solving multiple sets of Navier-Stokes equations and improves the simulation efficiency and stability. However, due to the local equilibrium assumption, these approaches can only deal with tightly coupled multiphase systems, where the relative speed between phases are assumed stable. In this work we abandon the local equilibrium assumption, and redesign the computation workflow of the mixture model to explicitly track and decouple the velocities of all phases. The phases still share the same pressure, with which we enforce the incompressibility for the mixture. The phase transport is calculated with drift velocities, and we propose a novel correction scheme to handle the transport at fluid boundaries to ensure mass conservation. Compared with previous mixture models, the proposed approach enables the simulation of much more dynamic scenarios with negligible extra overheads. In addition, it allows fluid control techniques to be applied to individual phases to generate locally dynamic and visually interesting effects. 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. 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