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75
Simulating Water and Smoke with an Octree Data Structure
, 2004
"... We present a method for simulating water and smoke on an unrestricted octree data structure exploiting mesh refinement techniques to capture the small scale visual detail. We propose a new technique for discretizing the Poisson equation on this octree grid. The resulting linear system is symmetric ..."
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Cited by 210 (18 self)
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We present a method for simulating water and smoke on an unrestricted octree data structure exploiting mesh refinement techniques to capture the small scale visual detail. We propose a new technique for discretizing the Poisson equation on this octree grid. The resulting linear system is symmetric positive definite enabling the use of fast solution methods such as preconditioned conjugate gradients, whereas the standard approximation to the Poisson equation on an octree grid results in a nonsymmetric linear system which is more computationally challenging to invert. The semiLagrangian characteristic tracing technique is used to advect the velocity, smoke density, and even the level set making implementation on an octree straightforward. In the case of smoke, we have multiple refinement criteria including object boundaries, optical depth, and vorticity concentration. In the case of water, we refine near the interface as determined by the zero isocontour of the level set function.
A fast variational framework for accurate solidfluid coupling
 ACM Trans. Graph
, 2007
"... Figure 1: Left: A solid stirring smoke runs at interactive rates, two orders of magnitude faster than previously. Middle: Fully coupled rigid bodies of widely varying density, with flow visualized by marker particles. Right: Interactive manipulation of immersed rigid bodies. Physical simulation has ..."
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Cited by 75 (4 self)
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Figure 1: Left: A solid stirring smoke runs at interactive rates, two orders of magnitude faster than previously. Middle: Fully coupled rigid bodies of widely varying density, with flow visualized by marker particles. Right: Interactive manipulation of immersed rigid bodies. Physical simulation has emerged as a compelling animation technique, yet current approaches to coupling simulations of fluids and solids with irregular boundary geometry are inefficient or cannot handle some relevant scenarios robustly. We propose a new variational approach which allows robust and accurate solution on relatively coarse Cartesian grids, allowing possibly orders of magnitude faster simulation. By rephrasing the classical pressure projection step as a kinetic energy minimization, broadly similar to modern approaches to rigid body contact, we permit a robust coupling between fluid and arbitrary solid simulations that always gives a wellposed symmetric positive semidefinite linear system. We provide several examples of efficient fluidsolid interaction and rigid body coupling with subgrid cell flow. In addition, we extend the framework with a new boundary condition for freesurface flow, allowing fluid to separate naturally from solids.
A second order Coupled Level Set and VolumeofFluid Method for . . .
, 2002
"... We present a coupled Level Set / VolumeofFluid (CLSVOF) method for computing growth and collapse of vapor bubbles. The liquid is assumed incompressible and the vapor is assumed to have constant pressure in space. Second order algorithms are used for nding "mass conserving" extension vel ..."
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Cited by 66 (6 self)
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We present a coupled Level Set / VolumeofFluid (CLSVOF) method for computing growth and collapse of vapor bubbles. The liquid is assumed incompressible and the vapor is assumed to have constant pressure in space. Second order algorithms are used for nding "mass conserving" extension velocities, for discretizing the local interfacial curvature and also for the discretization of the cell centered projection step. Convergence studies are given that demonstrate this second order accuracy. Examples are provided that apply to cavitating bubbles.
A Fourth Order Accurate Discretization for the Laplace and Heat Equations on Arbitrary Domains, with Applications to the Stefan Problem
 J. Comput. Phys
, 2004
"... In this paper, we first describe a fourth order accurate finite di#erence discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit time discretization to avoid the stringent tim ..."
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Cited by 52 (5 self)
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In this paper, we first describe a fourth order accurate finite di#erence discretization for both the Laplace equation and the heat equation with Dirichlet boundary conditions on irregular domains. In the case of the heat equation we use an implicit time discretization to avoid the stringent time step restrictions associated with explicit schemes. We then turn our focus to the Stefan problem and construct a third order accurate method that also includes an implicit time discretization.
A sharp interface method for incompressible twophase flows
, 2007
"... We present a sharp interface method for computing incompressible immiscible twophase flows. It couples the levelset and volumeoffluid techniques and retains their advantages while overcoming their weaknesses. It is stable and robust even for large density and viscosity ratios on the order of 100 ..."
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Cited by 45 (7 self)
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We present a sharp interface method for computing incompressible immiscible twophase flows. It couples the levelset and volumeoffluid techniques and retains their advantages while overcoming their weaknesses. It is stable and robust even for large density and viscosity ratios on the order of 1000 to 1. The numerical method is an extension of the secondorder method presented by Sussman [M. Sussman, A second order coupled levelset and volume of fluid method for computing growth and collapse of vapor bubbles, Journal of Computational Physics 187 (2003) 110–136] in which the previous method treated the gas pressure as spatially constant and the present method treats the gas as a second incompressible fluid. The new method yields solutions in the zero gas density limit which are comparable in accuracy to the method in which the gas pressure was treated as spatially constant. This improvement in accuracy allows one to compute accurate solutions on relatively coarse grids, thereby providing a speedup over continuum or "ghostfluid" methods.
A fast solver for the Stokes equations with distributed forces in complex geometries
 J. Comput. Phys
"... We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded ..."
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Cited by 41 (10 self)
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We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded Boundary Integral method, is based on Anita Mayo’s work for the Poisson’s equation: “The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions”, SIAM Journal on Numerical Analysis, 21 (1984), pp. 285–299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström’s method. The rectangular domain problem is discretized by finite elements for a velocitypressure formulation with equal order interpolation bilinear elements (£¥ ¤£¥ ¤). Stabilization is used to circumvent the ¦¨§�©������� � condition for the pressure space. For the integral equations, fast matrix vector multiplications are achieved via an ���¨���� � algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsify lowrank blocks. The regular grid solver is a Krylov method (Conjugate Residuals) combined with an optimal twolevel Schwartzpreconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates. Key Words: Stokes equations, fast solvers, integral equations, doublelayer potential, fast multipole methods, embedded domain methods, immersed interface methods, fictitious
A Level Set Approach for the Numerical Simulation of Dendritic Growth
 J. Sci. Comput
, 2002
"... In this paper, we present a level set approach for the modeling of dendritic solidification. These simulations exploit a recently developed second order accurate symmetric discretization of the Poisson equation, see [12]. Numerical results indicate that this method can be used successfully on comple ..."
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Cited by 32 (5 self)
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In this paper, we present a level set approach for the modeling of dendritic solidification. These simulations exploit a recently developed second order accurate symmetric discretization of the Poisson equation, see [12]. Numerical results indicate that this method can be used successfully on complex interfacial shapes and can simulate many of the physical features of dendritic solidification. We apply this algorithm to the simulation of the dendritic crystallization of a pure melt and find that the dendrite tip velocity and tip shapes are in excellent agreement with solvability theory. Numerical results are presented in both two and three spatial dimensions.
Incorporating diffusion in complex geometries into stochastic chemical kinetics simulations
 SIAM Journal on Scientific Computing
"... Abstract. A method is developed for incorporating diffusion of chemicals in complex geometries into stochastic chemical kinetics simulations. Systems are modeled using the reactiondiffusion master equation, with jump rates for diffusive motion between mesh cells calculated from the discretization w ..."
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Cited by 30 (3 self)
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Abstract. A method is developed for incorporating diffusion of chemicals in complex geometries into stochastic chemical kinetics simulations. Systems are modeled using the reactiondiffusion master equation, with jump rates for diffusive motion between mesh cells calculated from the discretization weights of an embedded boundary method. Since diffusive jumps between cells are treated as first order reactions, individual realizations of the stochastic process can be created by the Gillespie method. Numerical convergence results for the underlying embedded boundary method, and for the stochastic reactiondiffusion method, are presented in two dimensions. A twodimensional model of transcription, translation, and nuclear membrane transport in eukaryotic cells is presented to demonstrate the feasibility of the method in studying cellwide biological processes.
A second order accurate level set method on nongraded adaptive cartesian grids
, 2007
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Using The Particle Level Set Method And A Second Order Accurate Pressure Boundary Condition For Free Surface Flows
 In Proc. 4th ASMEJSME Joint Fluids Eng. Conf., number FEDSM2003–45144. ASME
, 2003
"... In this paper, we present an enhanced resolution capturing method for topologically complex two and three dimensional incompressible free surface flows. The method is based upon the level set method of Osher and Sethian to represent the interface combined with two recent advances in the treatment of ..."
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Cited by 28 (12 self)
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In this paper, we present an enhanced resolution capturing method for topologically complex two and three dimensional incompressible free surface flows. The method is based upon the level set method of Osher and Sethian to represent the interface combined with two recent advances in the treatment of the interface, a second order accurate discretization of the Dirichlet pressure boundary condition at the free surface (2002, J. Comput. Phys. 176, 205) and the use of massless marker particles to enhance the resolution of the interface through the use of the particle level set method (2002, J. Comput. Phys., 183, 83). Use of these methods allow for the accurate movement of the interface while at the same time preserving the mass of the the liquid, even on coarse computational grids. Also, these methods complement the level set method in its ability to handle changes in interface topology in a robust manner. Surface tension effects can be easily included in our method. The method is presented in three spatial dimensions, with numerical examples in both two and three spatial dimensions.