An Improved Jacobi Solver for Particle Simulation

Abstract

AbstractThis paper presents a new method for simulating particles for computer graphics and video games, based onan improved Jacobi solver and a hybrid approach between velocity time stepping and position based dynamics.Current constrained dynamics solvers use relaxation iterative methods like Gauss-Seidel or Jacobi. We proposea new iterative method based on a minimum residual variant of the Conjugate Gradient algorithm and show thatit can be formulated as an improvement to the Jacobi method. We also describe an adaptation of position baseddynamics to better handle contact and friction and allow tight two way coupling with velocity level methods.Categories and Subject Descriptors (according to ACM CCS) : I.3.5 [Computer Graphics]: Computational Geometryand Object Modeling—Physically Based Modeling1. IntroductionParticle systems can be used to model all kind of mechan-ical phenomena including granular matter, fluids, cloth, de-formable objects and even rigid bodies. Granular matter hasbeen used extensively in visual effects and computer gener-ated animations such as Spiderman or Rise of the Guardians[ABC07]. Cloth and soft bodies in general are now ubiqui-tous in movies and becoming more so in games. This is whythe simulation methods still need to become faster and morerobust in order to handle a growing number of objects.In this paper we present a unified approach to simulatinggranular matter and cloth using a constrained dynamics ap-proach. Our hybrid dynamics method offers tight couplingbetween all simulated objects as all constraints are treated inthe same solver loop. Also our improved Jacobi solver showsconvergence similar to the popular Gauss-Seidel method,thus allowing for more efficient parallel implementations.2. Related workGranular matter has been an area of research in computa-tional mechanics for decades. The method of choice is usu-ally the discrete element method (DEM) which treats thegranules as elastic billiard balls. The DEM method was usedin graphics too [BYM05,ATO09,Har07]. Another approachwas a continuum based one, considering the granular mattera special kind of fluid [ZB05,NGL10]. This was followed bya Lagrangian version derived from the smooth particle hy-drodynamics method for simulating fluids [AO11,IWT12].An alternative to DEM is the non-smooth constrained dy-namics approach where the particles are considered fullyrigid and this is the path we are following. In fact the methodwas developed for the more general case of rigid bodies, butthat can be turned into an advantage given the granules canhave any shape other than spherical [BYM05]. A great dealof articles have been written on the subject of multi-body dy-namics with contact and friction [ST96,AH04,AT10,Lac03],many in the computer graphics community [Bar94,Erl07,BETC14,TBV12,KSJP08], and some explicitly on the sub-ject of granular flow [TA10,RA05,LSB10].Rigid bodies can be simulated with other methods thanconstrained dynamics, e.g. the penalty method [BZX14].Other approaches consider the rigid body as a collection ofparticles and contact forces are computed using DEM or inother ways [TSIHK06,Jak01]. The particles can move underrigid transformations [Har07] or be constrained together toform a composite rigid object [Cou12,MMCK14].Cloth has been traditionally simulated as a mass-springmodel [Pro96] and great effort has been put into mak-ing the simulation stable at large time steps [BW98]. Ingames position based dynamics is usually preferred (PBD)[Jak01,MHHR07,GHF07]. The approach in [BBD09] and[How11] is similar to our hybrid algorithm, but has sepa-rate passes for velocity and position. Volumetric deformableobjects can be modeled by connecting the particles togetherin a lattice fashion, by adding internal pressure or throughshape matching [MHHR07,BMOT13]. A unified particle