The CurvatureAugmented Closest Point method with vesicle inextensibility application
Abstract
Here, the Closest Point method, initially developed by Ruuth and Merriman, allows for the numerical solution of surface partial differential equations without the need for a parameterization of the surface itself. Surface quantities are embedded into the surrounding domain by assigning each value at a given spatial location to the corresponding value at the closest point on the surface. This embedding allows for surface derivatives to be replaced by their Cartesian counterparts (e.g. ∇ _{s}=∇). This equivalence is only valid on the surface, and thus, interpolation is used to enforce what is known as the side condition away from the surface. To improve upon the method, this work derives an operator embedding that incorporates curvature information, making it valid in a neighborhood of the surface. With this, direct enforcement of the side condition is no longer needed. Comparisons in R ^{2} and R ^{3} show that the resulting CurvatureAugmented Closest Point method has better accuracy and requires less memory, through increased matrix sparsity, than the Closest Point method, while maintaining similar matrix condition numbers. To demonstrate the utility of the method in a physical application, simulations of inextensible, bilipid vesicles evolving toward equilibrium shapes are also included.
 Authors:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Publication Date:
 Research Org.:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1410022
 Report Number(s):
 LLNLJRNL732193
Journal ID: ISSN 00219991
 Grant/Contract Number:
 AC5207NA27344
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 345; Journal Issue: C; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; Closest point method; Surface partial differential equation; Surface gradient; Laplace–Beltrami operator; Vesicle; Inextensible membrane
Citation Formats
Vogl, Christopher J. The CurvatureAugmented Closest Point method with vesicle inextensibility application. United States: N. p., 2017.
Web. doi:10.1016/j.jcp.2017.06.004.
Vogl, Christopher J. The CurvatureAugmented Closest Point method with vesicle inextensibility application. United States. doi:10.1016/j.jcp.2017.06.004.
Vogl, Christopher J. 2017.
"The CurvatureAugmented Closest Point method with vesicle inextensibility application". United States.
doi:10.1016/j.jcp.2017.06.004.
@article{osti_1410022,
title = {The CurvatureAugmented Closest Point method with vesicle inextensibility application},
author = {Vogl, Christopher J.},
abstractNote = {Here, the Closest Point method, initially developed by Ruuth and Merriman, allows for the numerical solution of surface partial differential equations without the need for a parameterization of the surface itself. Surface quantities are embedded into the surrounding domain by assigning each value at a given spatial location to the corresponding value at the closest point on the surface. This embedding allows for surface derivatives to be replaced by their Cartesian counterparts (e.g. ∇s=∇). This equivalence is only valid on the surface, and thus, interpolation is used to enforce what is known as the side condition away from the surface. To improve upon the method, this work derives an operator embedding that incorporates curvature information, making it valid in a neighborhood of the surface. With this, direct enforcement of the side condition is no longer needed. Comparisons in R2 and R3 show that the resulting CurvatureAugmented Closest Point method has better accuracy and requires less memory, through increased matrix sparsity, than the Closest Point method, while maintaining similar matrix condition numbers. To demonstrate the utility of the method in a physical application, simulations of inextensible, bilipid vesicles evolving toward equilibrium shapes are also included.},
doi = {10.1016/j.jcp.2017.06.004},
journal = {Journal of Computational Physics},
number = C,
volume = 345,
place = {United States},
year = 2017,
month = 6
}

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