Rigid Body Sampling for Rigid-fluid Coupling in SPH

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1 Journal of Computational Information Systems 10: 22 (2014) Available at Rigid Body Sampling for Rigid-fluid Coupling in SPH Xiaokun WANG, Xiaojuan BAN, Jinbiao GUAN School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing , China Abstract Rigid body sampling plays an important role on the authenticity of the simulation in the rigid-fluid coupling scenarios. The sampling method usually used is not uniform, and the effect is not ideal. This paper presents a simple and effective particle-based approach to sampling rigid body in Smoothed Particle Hydrodynamics (SPH) fluid simulation. This approach sample surface of rigid objects with boundary particles and then loose the sampling particles to get more uniform sampling results in less iterative times, so that to provide a better way for rigid body sampling based on SPH rigid-fluid coupling simulation. The method was verified correctly by experiments. Keywords: Smoothed Particles Hydrodynamics; SPH; Rigid Body Sampling; Rigid-fluid Coupling 1 Introduction Fluid simulation has drown increasing attention in computer animation in recent years.it is mainly comprised of Eulerian which is based on grid and Lagrangian methods that is based on particles. Lagrangian methods for fluid simulation have recently become more popular than Eulerian schemes. Smoothed Particle Hydrodynamics (SPH) is a particle-based method and become as one major domain for fluid animation in computer graphics. A state-of-the-art introduction of SPH fluid research and techniques has been presented by Markus Ihmsen et al. [1]. As a fluid is generally simulated in a area with fixed and moving rigid body, it is necessary to consider the interaction of the fluid with rigid body. Therefore, rigid body sampling is first issue in rigid-fluid coupling scene which have to handle. Despite the research of rigid body is a broad field and various rigid body sampling methods were addressed, only a few rigid body sampling method is suitable for use in a rigid-fluid coupling simulation directly. As a result, this paper proposes an rigid body sampling algorithm which is an extension of Poisson disk method and sampling method in [2] for rigid-fluid coupling area. Our sampling algorithm which catches boundaries precisely can also apply to other applications. Project supported by the National Nature Science Foundation of China (No , ). Corresponding author. address: banxj@ustb.edu.cn (Xiaojuan BAN) / Copyright 2014 Binary Information Press DOI: /jcis13114 November 15, 2014

2 9744 X. Wang et al. /Journal of Computational Information Systems 10: 22 (2014) Related Work SPH was introduced by Gingold and Monaghan [3] and Lucy [4] respectively, and has been used in physics research. Monaghan s Simulating free surface flows with SPH [5] serves as a basis for SPH fluid simulation. Mller et al. [6] proposed using gas state equation with surface tension and viscosity forces for interactive applications. Becker and Teschner [7] reduce compressibility with Tait Equation, and employ particle initialization with highly damped equations to reach a stable density near the surface, which is known as WCSPH. PCISPH presented by Solenthaler and Pajarola is enforced by using a prediction-correction scheme to determine the particle pressures and large time steps [8]. The coupling of SPH fluid and rigid bodies has formed a branch of SPH fluid simulation, some methods have been proposed. The fluid is represented as a group of rigid spheres exchanging impulses with surrounding rigid bodies in [9]. The pressure at the boundary is taken into account for fluid-rigid interaction in [10]. Becker et al. employed direct forcing for rigid-fluid coupling [11]. Akinci et al. presented a approach of correcting density of boundary particles [12]. Research on surface sampling includes both particle-based sampling techniques and polygonization techniques. Turk used repelling particles on surfaces to uniformly resample a static surface [13] and to simplify a polygonization by reducing the number of polygons [14]. Szeliski and Tonnesen used oriented particles to model surfaces [15]. A series of improvement based on marching cube algorithm [16] were proposed for polygonizing implicit surfaces. Physically-based approaches to the polygonization of implicit surfaces were pioneered by Figueiredoet al. [17]. Andrew and Heckbert employed local repulsion to make particles spread uniform [18]. However, only a few methods are suitable for rigid body sampling in rigid-fluid coupling scenario. This paper proposes a sampling method based on blue noise and inspired by [2]. Nehab et al. [25] presented algorithm of stratified point sampling. Cook addressed stochastic sampling of Poissondisk distributions for blue noise [19]. Turk introduced a relaxation method for surface sampling [14]. Corsini et al. sampled triangular meshes with blue noise properties [20]. Dunbar et al. [21] modified Poisson-disk sample using a spatial data structure. Bridson [22] simplified Dunbar s approach with rejection sampling and extending it to higher dimensions. 3 SPH Fluid Model SPH For the fluid simulation, it uses weakly compressible SPH formulation employing the Tait equation to estimate pressure [5, 7]. The original idea of SPH is to interpolate a function A(x) at a position x. In SPH, a field variable A at position x i is represented using a finite set of sampling points x j located within an area of x i x j < h Generally A(x) is Given as A(x) = j m j A j ρ j W (x x j, h). (1) The value m j, ρ i denotes the mass and density of particle respectively, W (x x j, h) denotes a kernel function with support radius h. For implementation, it uses Monaghan s M 4 cubic spline [23] interpolation kernel W with a support of radius 3h. Navier-Stokes equations

3 X. Wang et al. /Journal of Computational Information Systems 10: 22 (2014) The Navier-Stokes equations, is a set of partial differential equations that describe the motion of fluids. The basic form of the equations can be written as dρ = ρ v (2) dt dv dt = 1 ρ P + 1 ρ µ 2 v + g (3) where v denotes the velocity, P the pressure, g the external force density field and µ the viscosity of the fluid. The first equation assures conservation of mass while the second equation formulates conservation of momentum. Continuity equation One critical issue is how to compute density ρ i at particle i. From Eq. (1), ρ i can given as ρ i = j m j W (x i x j, h) (4) This naturally conserves mass, and forces computed from this density directly correct deficiences in the particle distribution, generally producing higher quality results. An alternative method is introduced by Monaghan [5], which results in the SPH form dρ i dt = j m j v ij W ij (5) with v ij = v i v j. Momentum equation Without regard to viscosity, the momentum equation is written as dv i dt = j m j ( P i ρ 2 i + P j ) W ρ 2 ij + g (6) j This symmetrized version conserves liner and angular momentum, viscosity will be given below. Incompressibility In order to compute the pressure gradient in Eq. (3), pressure P i is computed from the density ρ i. For fluid simulation, it uses weakly compressible SPH formulation employing the Tait equation to estimate pressure [5, 7]. Tait s equation is given as P = B(( ρ ρ 0 ) γ 1) (7) with B = ρ 0c s γ, c s denoting the speed of sound in the fluid and γ = 7. Viscosity To enhance stability and permit shock phenomena, some form of artificial viscosity is necessary. Artificial viscosity is employed [22], which given as Fi v = m i m j Π ij i W ij v T ij x ij < 0 j 0 v T ij x ij 0 where Π ij = v( vt ij x ij x 2 ij +εh2 ), v = 2αhcs ρ i +ρ j. α is viscosity constant and εh 2 is used to avoid singularities for x ij = 0. For Implementing, α = and ε = 0.01 as [7] is adopted. (8)

4 9746 X. Wang et al. /Journal of Computational Information Systems 10: 22 (2014) Rigid Objects Sampling Algorithm For rigid objects Sampling, boundary particles is used to sample the surface of rigid objects, which has several merits. For one thing, using particles permits us derive a rigid model that can handle different shapes even with complex geometry structure. For another thing, the use of boundary particles successfully alleviates sticking artifacts and makes sampling uniform. There are two components in our sampling: sampling on the surface and relaxation on the surface. In other words, it first samples the surface of rigid object image, then improves initial sampling with surface relaxation. In order to realize the first procedure, it needs fast projection of points to the surface. Hence, level set method is employed to express surface geometry with φ > 0, φ < 0 denoting exterior and interior of rigid objects respectively, while φ = 0 denoting surface of rigid objects. (a) (b) Fig. 1: Surface sampling and relaxation (a) Surface sampling. (b) Surface relaxation. Black points: newly added sample. Gray points: Surface sampling points. White points: exterior points before projected to the surface 4.1 Surface sampling After obtain the surface geometry of signed distance function, surface sampling method proposed in [2] is employed, pseudocode is provided in Algorithm 1. First, searching for seed sample points on the surface, checking every grid cell that intersection the surface (i.e. where the level set changes sign). Thus, it doesn t miss any components: in a cell take up to k attempts, projecting random points from the cell to the surface and stopping when one satisfies the Poisson disk criterion, i.e. is at least distance r from existing samples. Once obtain a seed sample, then continuing to sampling from it and taking a step of size e r from the previous sample along a random tangential direction d, again projecting to the surface and checking the Poisson disk criterion. Parameters were chosen as k = 30 and e = 1.085, but could be further tuned. 4.2 Surface relaxation Inspired by the relaxation algorithm proposed in [2] and SPH interpolation method, surface relaxation algorithm is presented. The purpose of relaxation (Algorithm 2) is to reduce noise in the SPH density and make particle distribution uniform. It starts with the initial particles seed in Algorithm 1 and attempts to reposition each sample through density gradient. Next it computes density ρ i (t) and density gradient ρ i (t) of each surface particles, emploies deviation of density ρ i (t) and average density ρ(t) as a coefficient to tune distance d. Then it emploies

5 X. Wang et al. /Journal of Computational Information Systems 10: 22 (2014) Algorithm 1 Surface Sampling Input: Level set φ, radius r, constant k, constant e Output: Sample set S 1: for each grid cells G that changes sign do 2: for each k do 3: Generate random point P in G 4: Project p to surface of φ 5: if p satisfies Poisson Disk criterion in S 6: S S {p} 7: for each point in new samples do 8: Generate random tangential direction f of surface at p 9: q p + f e r 10: Project q to surface of φ 11: if q satisfies the Poisson Disk criterion in S 12: S S {q} 13: p q d ρ i (t) to adjust particle locations. Surface sample candidates are additionally projected to the surface of the level set and merely reserved which satisfies the Poisson disk criterion. Parameter t is iterations and f is distance coefficient. Algorithm 2 Surface Relaxation Input: sample set S of Algorithm 1, Level set φ, radius r, constant t, constant f Output: relaxed sample set S 1: for each t do 2: for each p in S do 3: compute density ρ i (t), average density ρ(t) 4: compute density gradient ρ i (t) 5: d r f ρ i(t) ρ(t) ρ(t) 6: p new p + d ρ i (t) 7: if p new outside φ or came from surface sample 8: Project p new to surface of φ 9: if p new satisfies the Poisson Disk criterion in S 10: p p new 5 Implementation and Results The simulation is performed on a quad-core Intel i (8M Cache, 2.67 GHz) with 8GB memory. Bullet is used for simulating rigid objects and OpenMP is used for parallelize particle computations. The simulation and surface reconstruction actualize with C++ language and multi-threading technology. The searching process of approaching particle in simulation algorithm uses space background grid to carry out Hash lookup. Surface reconstruction uses the Anisotropic kernel function to construction color field which reads the methods from literature [24] for reference, and then uses the Stepping cubes algorithm to reconstruct surface, where in the

6 9748 X. Wang et al. /Journal of Computational Information Systems 10: 22 (2014) matrix singular value decomposition is based on the JAMA and TNT open source math library of NIST. The real time results of simulation and surface structure, is displayed by OpenGL 3D Graphics Library, while the video is recorded by OpenCV Library. In order to conduct modeling and animation for the complex container and scene, Blender software and ray tracing engine Mitsuba is used for later high quality fluid effect. Figs. 2 and 3 show the sampling of monkey-head and rabbit from our rigid sampling method, the results looks uniform and dense. Fig. 4 compares our sampling approach to possion-disk sampling method [20], stratified triangle sampling method [25] and Monte Carlo sampling method [26] through rigid-fluid coupling scene, the body are comprised of particles and the color represents the size of density. From Fig. 4 it can be found that in the first three method body changes color and the color distribution is not uniform which means the sampling is not uniform. However, the results of last row based on our sampling method is better. (a) Fig. 2: 3D monkey-head. (a): before sampling; (b): after sampling (b) (a) (b) Fig. 3: 3D rabbit. (a): before sampling; (b): after sampling 6 Conclusions This paper have addressed a rigid body sampling approach for SPH rigid-fluid coupling simulation. The method can be applied to other particle-based simulation and extended to relevant approaches like PCISPH. Ongoing work focuses on rigid body sampling of thin structure such as one line of boundary particles and boundary handling in SPH.

7 X. Wang et al. /Journal of Computational Information Systems 10: 22 (2014) (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) 9749 Fig. 4: Rigid-fluid coupling (a)-(d): possion-disk sampling; (e)-(h): stratified triangle sampling; (i)-(l): monte Carlo sampling; (m)-(p): our sampling method Acknowledgement This work is supported by National Nature Science Foundation of P.R. China (No , ), the new century personnel plan for the Ministry of Education (NCET ). References [1] Ihmsen M, Orthmann J, Solenthaler B, et al. SPH Fluids in Computer Graphics [C]. Eurographics 2014-State of the Art Reports. The Eurographics Association, 2014: [2] Schechter H, Bridson R. Ghost SPH for animating water [J]. ACM Transactions on Graphics (TOG), 2012, 31(4): 61. [3] Gingold R A, Monaghan J J. Smoothed particle hydrodynamics-theory and application to nonspherical stars [J]. Monthly notices of the royal astronomical society, 1977, 181: [4] Lucy L B. A numerical approach to the testing of the fission hypothesis [J]. The astronomical journal, 1977, 82: [5] Monaghan J J. Simulating free surface flows with SPH [J]. Journal of computational physics, 1994, 110(2):

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