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19
Mass and Momentum Conservation for Fluid Simulation
, 2011
"... Momentum conservation has long been used as a design principle for solid simulation (e.g. collisions between rigid bodies, massspring elastic and damping forces, etc.), yet it has not been widely used for fluid simulation. In fact, semiLagrangian advection does not conserve momentum, but is still ..."
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Cited by 9 (2 self)
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Momentum conservation has long been used as a design principle for solid simulation (e.g. collisions between rigid bodies, massspring elastic and damping forces, etc.), yet it has not been widely used for fluid simulation. In fact, semiLagrangian advection does not conserve momentum, but is still regularly used as a bread and butter method for fluid simulation. In this paper, we propose a modification to the semiLagrangian method in order to make it fully conserve momentum. While methods of this type have been proposed earlier in the computational physics literature, they are not necessarily appropriate for coarse grids, large time steps or inviscid flows, all of which are common in graphics applications. In addition, we show that the commonly used vorticity confinement turbulence model can be modified to exactly conserve momentum as well. We provide a number of examples that illustrate the benefits of this new approach, both in conserving fluid momentum and passively advected scalars such as smoke density. In particular, we show that our new method is amenable to efficient smoke simulation with one time step per frame, whereas the traditional nonconservative semiLagrangian method experiences serious artifacts when run with these large time steps, especially when object interaction is considered.
A Monolithic Mass Tracking Formulation for Bubbles in Incompressible Flow
"... We devise a novel method for treating bubbles in incompressible flow that relies on the conservative advection of bubble mass and an associated equation of state in order to determine pressure boundary conditions inside each bubble. We show that executing this algorithm in a traditional manner leads ..."
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Cited by 4 (3 self)
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We devise a novel method for treating bubbles in incompressible flow that relies on the conservative advection of bubble mass and an associated equation of state in order to determine pressure boundary conditions inside each bubble. We show that executing this algorithm in a traditional manner leads to stability issues similar to those seen for partitioned methods for solidfluid coupling. Therefore, we reformulate the problem monolithically. This is accomplished by first proposing a new fully monolithic approach to coupling incompressible flow to fully nonlinear compressible flow including the effects of shocks and rarefactions, and then subsequently making a number of simplifying assumptions on the air flow removing not only the nonlinearities but also the spatial variations of both the density and the pressure. The resulting algorithm is quite robust, has been shown to converge to known solutions for test problems, and has been shown to be quite effective on more realistic problems including those with multiple bubbles, merging and pinching, etc. Notably, this approach departs from a standard twophase incompressible flow model where the air flow preserves its volume despite potentially large forces and pressure differentials in the surrounding incompressible fluid that should change its volume. Our bubbles readily change volume according to an isothermal equation of state. 1.
An adaptive discretization of incompressible flow using a multitude of moving Cartesian grids
"... We present a novel method for discretizing a multitude of moving and overlapping Cartesian grids each with an independently chosen cell size to address adaptivity. Advection is handled with first and second order accurate semiLagrangian schemes in order to alleviate any time step restriction associ ..."
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Cited by 4 (3 self)
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We present a novel method for discretizing a multitude of moving and overlapping Cartesian grids each with an independently chosen cell size to address adaptivity. Advection is handled with first and second order accurate semiLagrangian schemes in order to alleviate any time step restriction associated with small grid cell sizes. Likewise, an implicit temporal discretization is used for the parabolic terms, such as the heat equation and NavierStokes viscosity. The most intricate aspect of any such discretization is the method used in order to solve the elliptic equation for the NavierStokes pressure or that resulting from the temporal discretization of parabolic terms. We address this by first removing any degrees of freedom which duplicately cover spatial regions due to overlapping grids, and then providing a discretization for the remaining degrees of freedom adjacent to these regions. We observe that a robust second order accurate symmetric positive definite readily preconditioned discretization can be obtained by constructing a local Voronoi region on the fly for each degree of freedom in question in order to obtain both its stencil (logically connected neighbors) and stencil weights. We independently demonstrate each aspect of our approach on test problems in order to show efficacy and convergence before finally addressing a number of common test cases for incompressible flow with potentially moving solid bodies. 1.
High–Order Unstructured Lagrangian One–Step WENO Finite Volume Schemes for Non–conservative Hyperbolic Systems: Applications to Compressible Multi–Phase Flows. Computers and Fluids
"... In this article we present the first better than second order accurate unstructured Lagrangian–type one–step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non–conservative products. The method achieves high order of accuracy in space together with esse ..."
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In this article we present the first better than second order accurate unstructured Lagrangian–type one–step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non–conservative products. The method achieves high order of accuracy in space together with essentially non–oscillatory behaviour using a nonlinear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space–time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one–step finite volume scheme is derived by integration over a moving space–time control volume, where the non– conservative products are treated by a path–conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary–Lagrangian–Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity. The new scheme is applied to the full seven–equation Baer–Nunziato model of compressible multi–phase flows in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer–Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided. Key words: Arbitrary–Lagrangian–Eulerian (ALE) scheme, WENO finite volume scheme, path–conservative scheme, unstructured meshes, high order in space and time, compressible multi–phase flows, Baer–Nunziato model
Compressible, Multiphase SemiImplicit Method with Moment of Fluid Interface Representation ∗
, 2014
"... A unified method for simulating multiphase flows using an exactly mass, momentum, and energy conserving CellIntegrated SemiLagrangian advection algorithm is presented. The deforming material boundaries are represented using the momentoffluid method. The new algorithm uses a semiimplicit pressur ..."
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A unified method for simulating multiphase flows using an exactly mass, momentum, and energy conserving CellIntegrated SemiLagrangian advection algorithm is presented. The deforming material boundaries are represented using the momentoffluid method. The new algorithm uses a semiimplicit pressure update scheme that asymptotically preserves the standard incompressible pressure projection method in the limit of infinite sound speed. The asymptotically preserving attribute makes the new method applicable to compressible and incompressible flows including stiff materials; enabling large time
A Hybrid LagrangianEulerian Formulation for Bubble Generation and Dynamics
"... Figure 1: (Left) a faucet generating bubbles through air entrainment, (Center) a source seeding tiny bubbles which merge and grow as they rise, as well as interact with a moving armadillo illustrating complex object interaction, (Right) a cavitating propeller generates the characteristic helical pat ..."
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Cited by 3 (0 self)
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Figure 1: (Left) a faucet generating bubbles through air entrainment, (Center) a source seeding tiny bubbles which merge and grow as they rise, as well as interact with a moving armadillo illustrating complex object interaction, (Right) a cavitating propeller generates the characteristic helical pattern in its wake. We present a hybrid LagrangianEulerian framework for simulating both small and large scale bubble dynamics, where the bubbles can grow or shrink in volume as dictated by pressure forces in the surrounding fluid. Small underresolved bubbles are evolved using Lagrangian particles that are monolithically twoway coupled to the surrounding flow in a manner that closely approximates the analytic bubble oscillation frequency while converging to the analytic volume as predicted by the wellknown RayleighPlesset equation. We present a novel scheme for interconverting between these underresolved Lagrangian bubbles and larger wellresolved bubbles that are modeled with a traditional Eulerian level set approach. We also present a novel seeding mechanism to realistically generate bubbles when simulating fluid structure interaction with complex objects such as ship propellers. Moreover, our framework for bubble generation is general enough to be incorporated into all gridbased as well as particlebased fluid simulation methods.
Fully conservative leakproof treatment of thin solid structures immersed in compressible fluids
, 2012
"... ..."
A Mass Tracking Formulation for Bubbles in Incompressible Flow
"... We devise a novel method for treating bubbles in incompressible flow that relies on the conservative advection of bubble mass and an associated equation of state in order to determine pressure boundary conditions inside each bubble. We show that executing this algorithm in a traditional manner leads ..."
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Cited by 2 (0 self)
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We devise a novel method for treating bubbles in incompressible flow that relies on the conservative advection of bubble mass and an associated equation of state in order to determine pressure boundary conditions inside each bubble. We show that executing this algorithm in a traditional manner leads to stability issues similar to those seen for partitioned methods for solidfluid coupling. Therefore, we reformulate the problem monolithically. This is accomplished by first proposing a new fully monolithic approach to coupling incompressible flow to fully nonlinear compressible flow including the effects of shocks and rarefactions, and then subsequently making a number of simplifying assumptions on the air flow removing not only the nonlinearities but also the spatial variations of both the density and the pressure. The resulting algorithm is quite robust, has been shown to converge to known solutions for test problems, and has been shown to be quite effective on more realistic problems including those with multiple bubbles, merging and pinching, etc. Notably, this approach departs from a standard twophase incompressible flow model where the air flow preserves its volume despite potentially large forces and pressure differentials in the surrounding incompressible fluid that should change its volume. Our bubbles readily change volume according to an isothermal equation of state. 1.
On thin gaps between rigid bodies twoway coupled to incompressible flow
"... Twoway solid fluid coupling techniques typically calculate fluid pressure forces that in turn drive the solid motion. However, when solids are in close proximity (e.g. touching or in contact), the fluid in the thin gap region between the solids is difficult to resolve with a background fluid grid. ..."
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Twoway solid fluid coupling techniques typically calculate fluid pressure forces that in turn drive the solid motion. However, when solids are in close proximity (e.g. touching or in contact), the fluid in the thin gap region between the solids is difficult to resolve with a background fluid grid. Although one might attempt to address this difficulty using an adaptive, bodyfitted, or ALE fluid grid, the size of the fluid cells can shrink to zero as the bodies collide. The inability to apply pressure forces in a thin lubricated gap tends to make the solids stick together, since collision forces stop interpenetration but vanish when the solids are separating leaving the fluid pressure forces on the surface of the solid unbalanced in regards to the gap region. We address this problem by adding pressure degrees of freedom onto surfaces of rigid bodies, and subsequently using the resulting pressure forces to provide solid fluid coupling in the thin gap region. These pressure degrees of freedom readily resolve the tangential flow along the solid surface inside the gap and are twoway coupled to the pressure degrees of freedom on the grid allowing the fluid to freely flow into and out of the gap region. The twoway coupled system is formulated as a symmetric positivedefinite matrix which is solved using the preconditioned conjugate gradient method. Additionally, we provide a mechanism for advecting tangential velocities on solid surfaces in the gap region by extending semiLagrangian advection onto a curved surface mesh where a codimensionone velocity field tangential to the surface is defined. We demonstrate the convergence of our method on a number of examples, such as underwater rigid body separation and collision in both two and three spatial dimensions where typical methods do not converge. 1.