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362
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 125 (8 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.
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 99 (12 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.
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 76 (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.
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 54 (3 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.
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 54 (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.
An adaptive, formally second order accurate version of the immersed boundary method
, 2006
"... ..."
An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries
 J. Comp. Phys
, 2006
"... We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular fo ..."
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Cited by 36 (3 self)
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We present an immersed interface method for the incompressible NavierStokes equations capable of handling rigid immersed boundaries. The immersed boundary is represented by a set of Lagrangian control points. In order to guarantee that the noslip condition on the boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces is determined by solving a small system of equations iteratively at each time step. The NavierStokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity. Keywords: Immersed interface method, NavierStokes equations, Cartesian grid method, finite difference, fast Poisson solvers, irregular domains.
Regularization techniques for numerical approximation of PDEs with singularities
 J. Sci. Comput
"... 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 coeÆcients. We shall discuss the technique of local regularizati ..."
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Cited by 33 (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 coeÆcients. 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 deciencies in existing regularization methods are also pointed out. 1
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 30 (3 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.
A NEW MULTISCALE FINITE ELEMENT METHOD FOR HIGHCONTRAST ELLIPTIC INTERFACE PROBLEMS
"... Abstract. We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of f ..."
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Cited by 25 (1 self)
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Abstract. We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of flow in a porous medium containing a number of inclusions of low (or high) permeability embedded in a matrix of high (respectively low) permeability. Our method is H 1 conforming, with degrees of freedom at the nodes of a triangular mesh and requires the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface, but uses standard linear approximation otherwise. A key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems. Under moderate assumptions, we prove that our methods have (optimal) convergence rate of O(h) in the energy norm and O(h 2) in the L2 norm where h is the (coarse) mesh diameter and the hidden constants in these estimates are independent of the “contrast ” (i.e. ratio of largest to smallest value) of the PDE coefficient. For standard elements the best estimate in the energy norm would be O(h 1/2−ε) with a hidden constant which in general depends on the contrast. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges. 1.