Level Set Equations on Surfaces via the Closest Point Method

Macdonald, Colin B.; Ruuth, Steven J.

doi:10.1007/s10915-008-9196-6

Cited by 99 publications

(91 citation statements)

References 20 publications

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“…, using fourth-order WENO interpolation [Macdonald and Ruuth 2008] for φ(x) and ∇φ(x). Candidates are those particles with sφ(x) ∈ (−r s , +r e ), where r e is the escape radius defined in section 3.1.2.…”

Section: Particle-derived Level Setmentioning

confidence: 99%

MultiFLIP for energetic two-phase fluid simulation

Boyd

Bridson

2012

ACM Trans. Graph.

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Fig. 1:The "glugging" effect of water pouring through a spout cannot be reproduced with single-phase liquid simulation.Physically-based liquid animations often ignore the influence of air, giving up interesting behaviour. We present a new method which treats both air and liquid as incompressible, more accurately reproducing the reality observed at scales relevant to computer animation. The Fluid Implicit Particle (FLIP) method, already shown to effectively simulate incompressible fluids with low numerical dissipation, is extended to two-phase flow by associating a phase bit with each particle. The liquid surface is reproduced at each time step from the particle positions, which are adjusted to prevent mixing near the surface and to allow for accurate surface tension. The liquid surface is adjusted around small-scale features so they are represented in the grid-based pressure projection, while separate, loosely coupled velocity fields reduce unwanted influence between the phases. The resulting scheme is easy to implement, requires little parameter tuning and is shown to reproduce lively two-phase fluid phenomena.

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Section: Particle-derived Level Setmentioning

confidence: 99%

MultiFLIP for energetic two-phase fluid simulation

Boyd

Bridson

2012

ACM Trans. Graph.

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“…The Closest Point Method has been applied to a variety of time-dependent problems including in-surface advection, diffusion, reaction-diffusion, and Hamilton-Jacobi equations [30,23,22]. In these previous works, time stepping was performed explicitly and the numerical solution was propagated by alternating between two steps:…”

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confidence: 99%

The Implicit Closest Point Method for the Numerical Solution of Partial Differential Equations on Surfaces

Macdonald¹,

Ruuth²

2010

SIAM J. Sci. Comput.

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156

182

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Abstract. Many applications in the natural and applied sciences require the solutions of partial differential equations (PDEs) on surfaces or more general manifolds. The Closest Point Method is a simple and accurate embedding method for numerically approximating PDEs on rather general smooth surfaces. However, the original formulation is designed to use explicit time stepping. This may lead to a strict time-step restriction for some important PDEs such as those involving the Laplace-Beltrami operator or higher-order derivative operators. To achieve improved stability and efficiency, we introduce a new implicit Closest Point Method for surface PDEs. The method allows for large, stable time steps while retaining the principal benefits of the original method. In particular, it maintains the order of accuracy of the discretization of the underlying embedding PDE, it works on sharply defined bands without degrading the accuracy of the method, and it applies to general smooth surfaces. It also is very simple and may be applied to a rather general class of surface PDEs. Convergence studies for the in-surface heat equation and a fourth-order biharmonic problem are given to illustrate the accuracy of the method. We demonstrate the flexibility and generality of the method by treating flows involving diffusion, reaction-diffusion and fourth-order spatial derivatives on a variety of interesting surfaces including surfaces of mixed codimension.

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“…The difference is that some nonleaf nodes may have a rotation matrix R, whose purpose is to apply a coordinate transform to 14 It is often much more efficient to perform a linear search on a handful of points in the leaf nodes, compared to searching a tree whose leaf nodes contain a single point. 15 It is much faster to compute and store squared distances, rather than the distance itself, as the former avoids expensive sqrt calls. if node has a rotation matrix R then set x ← Rx.…”

Section: Discussionmentioning

confidence: 99%

High-order methods for computing distances to implicitly defined surfaces

Saye

2014

Commun. Appl. Math. Comput. Sci.

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Implicitly embedding a surface as a level set of a scalar function φ : ‫ޒ‬ d → ‫ޒ‬ is a powerful technique for computing and manipulating surface geometry. A variety of applications, e.g., level set methods for tracking evolving interfaces, require accurate approximations of minimum distances to or closest points on implicitly defined surfaces. In this paper, we present an efficient method for calculating high-order approximations of closest points on implicit surfaces, applicable to both structured and unstructured meshes in any number of spatial dimensions. In combination with a high-order approximation of φ, the algorithm uses a rapidly converging Newton's method initialised with a guess of the closest point determined by an automatically generated point cloud approximating the surface. In general, the order of accuracy of the algorithm increases with the approximation order of φ. We demonstrate orders of accuracy up to six for smooth problems, while nonsmooth problems reliably reduce to their expected order of accuracy. Accompanying this paper is C++ code that can be used to implement the algorithms in a variety of settings.

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Level Set Equations on Surfaces via the Closest Point Method

Cited by 99 publications

References 20 publications

MultiFLIP for energetic two-phase fluid simulation

MultiFLIP for energetic two-phase fluid simulation

The Implicit Closest Point Method for the Numerical Solution of Partial Differential Equations on Surfaces

High-order methods for computing distances to implicitly defined surfaces

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