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2014 High-order methods for computing distances to implicitly defined surfaces
Robert Saye
Commun. Appl. Math. Comput. Sci. 9(1): 107-141 (2014). DOI: 10.2140/camcos.2014.9.107

Abstract

Implicitly embedding a surface as a level set of a scalar function ϕ:d 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.

Citation

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Robert Saye. "High-order methods for computing distances to implicitly defined surfaces." Commun. Appl. Math. Comput. Sci. 9 (1) 107 - 141, 2014. https://doi.org/10.2140/camcos.2014.9.107

Information

Received: 23 January 2014; Revised: 30 April 2014; Accepted: 2 May 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1316.65030
MathSciNet: MR3212868
Digital Object Identifier: 10.2140/camcos.2014.9.107

Subjects:
Primary: 35R37 , 65D99 , 68U05
Secondary: 35R01 , 65D17 , 65D18

Keywords: closest point , high-order , implicit surfaces , level set methods , redistancing , reinitialisation

Rights: Copyright © 2014 Mathematical Sciences Publishers

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