Results 1  10
of
50
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 ..."
Abstract

Cited by 154 (14 self)
 Add to MetaCart
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 remark on computing distance functions
 J. Comput. Phys
"... We propose a new method for the reconstruction of the signed distance function in the context of level set methods. The new method is a modification of the algorithm which makes use of the PDE equation for the distance function introduced by M. Sussman, P. Smereka, and S. Osher (1994, J. Comput. Phy ..."
Abstract

Cited by 54 (3 self)
 Add to MetaCart
We propose a new method for the reconstruction of the signed distance function in the context of level set methods. The new method is a modification of the algorithm which makes use of the PDE equation for the distance function introduced by M. Sussman, P. Smereka, and S. Osher (1994, J. Comput. Phys. 119, 146). It is based mainly on the use of a truly upwind discretization near the interface. Comparison with the previous algorithm shows a definite improvement. When used with a firstorder upwind scheme, the method provides firstorder accuracy for the signed distance function in the whole computational domain, and secondorder accuracy in the location of the interface. A secondorder version of the method is also presented. c ○ 2000 Academic Press 1.
Spatially adaptive techniques for level set methods and incompressible flow
 Computers and Fluids
, 2005
"... Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes s ..."
Abstract

Cited by 51 (12 self)
 Add to MetaCart
Since the seminal work of [92] on coupling the level set method of [69] to the equations for twophase incompressible flow, there has been a great deal of interest in this area. That work demonstrated the most powerful aspects of the level set method, i.e. automatic handling of topological changes such as merging and pinching, as well as robust geometric information such as normals and curvature. Interestingly, this work also demonstrated the largest weakness of the level set method, i.e. mass or information loss characteristic of most Eulerian capturing techniques. In fact, [92] introduced a partial differential equation for battling this weakness, without which their work would not have been possible. In this paper, we discuss both historical and most recent works focused on improving the computational accuracy of the level set method focusing in part on applications related to incompressible flow due to both its popularity and stringent accuracy requirements. Thus, we discuss higher order accurate numerical methods such as HamiltonJacobi WENO [46], methods for maintaining a signed distance function, hybrid methods such as the particle level set method [27] and the coupled level set volume of fluid method [91], and adaptive gridding techniques such as the octree approach to free surface flows proposed in [56].
A crystalline, red green strategy for meshing highly deformable objects with tetrahedra
 In 12th Int. Meshing Roundtable
, 2003
"... Motivated by Lagrangian simulation of elastic deformation, we propose a new tetrahedral mesh generation algorithm that produces both high quality elements and a mesh that is well conditioned for subsequent large deformations. We use a signed distance function defined on a Cartesian grid in order to ..."
Abstract

Cited by 51 (13 self)
 Add to MetaCart
Motivated by Lagrangian simulation of elastic deformation, we propose a new tetrahedral mesh generation algorithm that produces both high quality elements and a mesh that is well conditioned for subsequent large deformations. We use a signed distance function defined on a Cartesian grid in order to represent the object geometry. After tiling space with a uniform lattice based on crystallography, we use the signed distance function or other user defined criteria to guide a red green mesh subdivision algorithm that results in a candidate mesh with the appropriate level of detail. Then, we carefully select the final topology so that the connectivity is suitable for large deformation and the mesh approximates the desired shape. Finally, we compress the mesh to tightly fit the object boundary using either masses and springs, the finite element method or an optimization approach to relax the positions of the nodes. The resulting mesh is well suited for simulation since it is highly structured, has robust topological connectivity in the face of large deformations, and is readily refined if deemed necessary during subsequent simulation.
ROSSIGNAC J.: An unconditionally stable MacCormack method
 SIAM J. Sci. Comput
, 2008
"... The back and forth error compensation and correction (BFECC) method advects the solution forward and then backward in time. The result is compared to the original data to estimate the error. Although inappropriate for parabolic and other nonreversible partial differential equations, it is useful for ..."
Abstract

Cited by 42 (14 self)
 Add to MetaCart
The back and forth error compensation and correction (BFECC) method advects the solution forward and then backward in time. The result is compared to the original data to estimate the error. Although inappropriate for parabolic and other nonreversible partial differential equations, it is useful for often troublesome advection terms. The error estimate is used to correct the data before advection raising the method to second order accuracy, even though each individual step is only first order accurate. In this paper, we rewrite the MacCormack method to illustrate that it estimates the error in the same exact fashion as BFECC. The difference is that the MacCormack method uses this error estimate to correct the already computed forward advected data. Thus, it does not require the third advection step in BFECC reducing the cost of the method while still obtaining second order accuracy in space and time. Recent work replaced each of the three BFECC advection steps with a simple first order accurate unconditionally stable semiLagrangian method yielding a second order accurate unconditionally stable BFECC scheme. We use a similar approach to create a second order accurate unconditionally stable MacCormack method. 1
Fast Treebased Redistancing for Level Set Computations
, 1999
"... Level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface, such as the signed distance to the interface. This paper presents ecient algorithms for this "redistancing" problem. The algorith ..."
Abstract

Cited by 37 (6 self)
 Add to MetaCart
Level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface, such as the signed distance to the interface. This paper presents ecient algorithms for this "redistancing" problem. The algorithms use quadtrees and triangulation to compute global approximate signed distance functions. A quadtree mesh is built to resolve the interface and the vertex distances are evaluated exactly with a robust search strategy to provide both continuous and discontinuous interpolants. Given a polygonal interface with N elements, our algorithms run in O(N) space and O(N log N) time. Twodimensional numerical results show they are highly efficient in practice.
Hierarchical RLE level set: A compact and versatile deformable surface representation
, 2006
"... This article introduces the Hierarchical RunLength Encoded (HRLE) Level Set data structure. This novel data structure combines the best features of the DTGrid (of Nielsen and Museth [2004]) and the RLE Sparse Level Set (of Houston et al. [2004]) to provide both optimal efficiency and extreme vers ..."
Abstract

Cited by 35 (6 self)
 Add to MetaCart
This article introduces the Hierarchical RunLength Encoded (HRLE) Level Set data structure. This novel data structure combines the best features of the DTGrid (of Nielsen and Museth [2004]) and the RLE Sparse Level Set (of Houston et al. [2004]) to provide both optimal efficiency and extreme versatility. In brief, the HRLE level set employs an RLE in a dimensionally recursive fashion. The RLE scheme allows the compact storage of sequential nonnarrowband regions while the dimensionally recursive encoding along each axis efficiently compacts nonnarrowband planes and volumes. Consequently, this new structure can store and process level sets with effective voxel resolutions exceeding 500030003000 (45 billion voxels) on commodity PCs with only 1 GB of memory. This article, besides introducing the HRLE level set data structure and its efficient core algorithms, also describes numerous applications that have benefited from our use of this structure: our unified implicit object representation, efficient and robust mesh to level set conversion, rapid ray tracing, level set metamorphosis, collision detection, and fully sparse fluid simulation (including RLE vector and matrix representations.) Our comparisons of the popular octree level set and Peng level set structures to the HRLE level set indicate that the latter is superior in both narrowband sequential access speed and overall memory usage
SemiLagrangian Methods for Level Set Equations
, 1998
"... A new numerical method for solving geometric moving interface problems is presented. The method combines a level set approach and a semiLagrangian time stepping scheme which is explicit yet unconditionally stable. The combination decouples each mesh point from the others and the time step from the ..."
Abstract

Cited by 31 (6 self)
 Add to MetaCart
A new numerical method for solving geometric moving interface problems is presented. The method combines a level set approach and a semiLagrangian time stepping scheme which is explicit yet unconditionally stable. The combination decouples each mesh point from the others and the time step from the CFL stability condition, permitting the construction of methods which are efficient, adaptive and modular. Analysis of a linear onedimensional model problem suggests a surprising convergence criterion which is supported by heuristic arguments and confirmed by an extensive collection of twodimensional numerical results. The new method computes correct viscosity solutions to problems involving geometry, anisotropy, curvature and complex topological events.
Dynamic Tubular Grid: An Efficient Data Structure and Algorithms for High Resolution Level Sets
, 2005
"... Level set methods [OS88] have proved very successful for interface tracking in many di#erent areas of computational science. However, current level set methods are limited by a poor balance between computational e#ciency and storage requirements. Treebased methods have relatively slow access ti ..."
Abstract

Cited by 29 (7 self)
 Add to MetaCart
Level set methods [OS88] have proved very successful for interface tracking in many di#erent areas of computational science. However, current level set methods are limited by a poor balance between computational e#ciency and storage requirements. Treebased methods have relatively slow access times, whereas narrow band schemes lead to very large memory footprints for high resolution interfaces.
A Fast Modular SemiLagrangian Method for Moving Interfaces
, 2000
"... A fast modular numerical method for solving general moving interface problems is presented. It simplifies code development by providing a blackbox solver which moves a given interface one step with given normal velocity. The method combines an efficiently redistanced level set approach, a problemi ..."
Abstract

Cited by 18 (4 self)
 Add to MetaCart
A fast modular numerical method for solving general moving interface problems is presented. It simplifies code development by providing a blackbox solver which moves a given interface one step with given normal velocity. The method combines an efficiently redistanced level set approach, a problemindependent velocity extension, and a secondorder semiLagrangian time stepping scheme which reduces numerical error by exact evaluation of the signed distance function. Adaptive quadtree meshes are used to concentrate computational effort on the interface, so the method moves an Nelement interface in O(N log N) work per time step. Efficiency is increased by taking large time steps even for parabolic curvature flows. Numerical results show that the method computes accurate viscosity solutions to a wide variety of difficult geometric moving interface problems involving merging, anisotropy, faceting, nonlocality and curvature.