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124
Rigid fluid: Animating the interplay between rigid bodies and fluid
 ACM Trans. Graph
, 2004
"... Figure 1: A silver block catapulting some wooden blocks into an oncoming wall of water. We present the Rigid Fluid method, a technique for animating the interplay between rigid bodies and viscous incompressible fluid with free surfaces. We use distributed Lagrange multipliers to ensure twoway coupl ..."
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Cited by 97 (7 self)
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Figure 1: A silver block catapulting some wooden blocks into an oncoming wall of water. We present the Rigid Fluid method, a technique for animating the interplay between rigid bodies and viscous incompressible fluid with free surfaces. We use distributed Lagrange multipliers to ensure twoway coupling that generates realistic motion for both the solid objects and the fluid as they interact with one another. We call our method the rigid fluid method because the simulator treats the rigid objects as if they were made of fluid. The rigidity of such an object is maintained by identifying the region of the velocity field that is inside the object and constraining those velocities to be rigid body motion. The rigid fluid method is straightforward to implement, incurs very little computational overhead, and can be added as a bridge between current fluid simulators and rigid body solvers. Many solid objects of different densities (e.g., wood or lead) can be combined in the same animation.
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 49 (3 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.
Numerical mathematics
, 2000
"... Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a ..."
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Cited by 40 (5 self)
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Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physiopathology. Mathematics Subject Classification (2000). 92C50,96C10,76Z05,74F10,65N30,65M60. Keywords. Cardiovascular mathematics; mathematical modeling; fluid dynamics; Navier– Stokes equations; numerical approximation; finite element method; differential equations. 1.
Discretization of Dirac Delta Functions in Level Set Methods
 J. Comput. Phys
, 2004
"... Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to ..."
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Cited by 27 (2 self)
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Discretization of singular functions is an important component in many problems to which level set methods have been applied. We present two methods for constructing consistent approximations to Dirac delta measures concentrated on piecewise smooth curves or surfaces. Both methods are designed to be convenient for level set simulations on Cartesian grids and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on the distance to the singularity with a regularization parameter proportional to the mesh size. The first algorithm is based on a tensor product of regularized onedimensional delta functions.
Numerical Approximations of Singular Source Terms in Differential Equations
 J. Comput. Phys
, 2003
"... Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the ..."
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Cited by 25 (1 self)
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Singular terms in di#erential equations pose severe challenges for numerical approximations on regular grids. Regularization of the singularities is a very useful technique for their representation on the grid. We analyze such techniques for the practically preferred case of narrow support of the regularizations, extending our earlier results for wider support. The analysis also generalizes existing theory for one dimensional problems to multi dimensions. New high order multi dimensional techniques for di#erential equations and numerical quadrature are introduced based on the analysis and numerical results are presented. We also show that the common use of distance functions to extend one dimensional regularization to higher dimensions may produce O(1) errors.
Regularization Techniques for Numerical Approximation of PDEs with Singularities
 J. of Sci. Comput
, 2002
"... The rate of convergence for numerical methods approximating dierential equations are often drastically reduced from lack of regularity in the solution. Typical examples are problems with singular source terms or discontinuous material coecients. We shall discuss the technique of local regulariza ..."
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Cited by 23 (2 self)
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The rate of convergence for numerical methods approximating dierential equations are often drastically reduced from lack of regularity in the solution. Typical examples are problems with singular source terms or discontinuous material coecients. We shall discuss the technique of local regularization for handling these problems. New numerical methods are presented and analyzed and numerical examples are given. Some serious de ciencies in existing methods are also pointed out.
Wave particles
 ACM Transactions on Graphics (Proceedings of SIGGRAPH
, 2007
"... Figure 1: Sample frames captured from our realtime simulation system (approximately 100,000 wave particles) We present a new method for the realtime simulation of fluid surface waves and their interactions with floating objects. The method is based on the new concept of wave particles, which offer ..."
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Cited by 19 (1 self)
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Figure 1: Sample frames captured from our realtime simulation system (approximately 100,000 wave particles) We present a new method for the realtime simulation of fluid surface waves and their interactions with floating objects. The method is based on the new concept of wave particles, which offers a simple, fast, and unconditionally stable approach to wave simulation. We show how graphics hardware can be used to convert wave particles to a height field surface, which is warped horizontally to account for local waveinduced flow. The method is appropriate for most fluid simulation situations that do not involve significant global flow. It is demonstrated to work well in constrained areas, including wave reflections off of boundaries, and in unconstrained areas, such as an ocean surface. Interactions with floating objects are easily integrated by including wave forces on the objects and wave generation due to object motion. Theoretical foundations and implementation details are provided, and experiments demonstrate that we achieve plausible realism. Timing studies show that the method is scalable to allow simulation of wave interaction with several hundreds of objects at realtime rates.
An adaptive, formally second order accurate version of the immersed boundary method
, 2006
"... ..."
When vortices stick: an aerodynamic transition in tiny insects
 J. Exp. Biol
"... We have used computational fluid dynamics to study changes in lift generation and vortex dynamics for Reynolds numbers (Re) between 8 and 128. The immersed boundary method was used to model a twodimensional wing through one stroke cycle. We calculated lift and drag coefficients as a function of tim ..."
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Cited by 17 (2 self)
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We have used computational fluid dynamics to study changes in lift generation and vortex dynamics for Reynolds numbers (Re) between 8 and 128. The immersed boundary method was used to model a twodimensional wing through one stroke cycle. We calculated lift and drag coefficients as a function of time and related changes in lift to the shedding or attachment of the leading and trailing edge vortices. We find that the fluid dynamics around the wing fall into two distinct patterns. For Re�64, leading and trailing edge vortices are alternately shed behind the wing, forming the von Karman vortex street. For Re�32, the leading and trailing edge vortices remain attached to the Summary wing during each ‘half stroke’. In threedimensional studies, large lift forces are produced by ‘vortical asymmetry ’ when the leading edge vortex remains attached to the wing for the duration of each half stroke and the trailing edge vortex is shed. Our twodimensional study suggests that this asymmetry is lost for Re below some critical value (between 32 and 64), resulting in lower lift forces. We suggest that this transition in fluid dynamics is significant for lift generation in tiny insects. Key words: insect flight, Reynolds number, aerodynamics, computational fluid dynamics.
On the accuracy of finite difference methods for elliptic problems with interfaces
 Commun. Appl. Math. Comput. Sci
"... In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator a ..."
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Cited by 16 (9 self)
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In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and modifying the operator if necessary. It has long been observed that the solution can be computed with uniform O(h 2) accuracy even if the truncation error is O(h) at the interface, while O(h 2) in the interior. We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to O(h 2 log (1/h)). Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Twofluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is O(h 2).