The Curvature-Augmented 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 R2 and R3 show that the resulting Curvature-Augmented 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, bi-lipid 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 Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1410022
- Alternate Identifier(s):
- OSTI ID: 1550442
- Report Number(s):
- LLNL-JRNL-732193
Journal ID: ISSN 0021-9991
- Grant/Contract Number:
- AC52-07NA27344
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 345; Journal Issue: C; Journal ID: ISSN 0021-9991
- 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 Curvature-Augmented 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 Curvature-Augmented Closest Point method with vesicle inextensibility application. United States. https://doi.org/10.1016/j.jcp.2017.06.004
Vogl, Christopher J. Tue .
"The Curvature-Augmented Closest Point method with vesicle inextensibility application". United States. https://doi.org/10.1016/j.jcp.2017.06.004. https://www.osti.gov/servlets/purl/1410022.
@article{osti_1410022,
title = {The Curvature-Augmented 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 Curvature-Augmented 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, bi-lipid 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 = {Tue Jun 06 00:00:00 EDT 2017},
month = {Tue Jun 06 00:00:00 EDT 2017}
}
Web of Science
Works referenced in this record:
The Closest Point Method and Multigrid Solvers for Elliptic Equations on Surfaces
journal, January 2015
- Chen, Yujia; Macdonald, Colin B.
- SIAM Journal on Scientific Computing, Vol. 37, Issue 1
Charge Diffusion on the Surface of Particles with Simple Geometries
journal, November 2015
- Eloul, Shaltiel; Compton, Richard G.
- The Journal of Physical Chemistry C, Vol. 119, Issue 49
Orientation and Dynamics of a Vesicle in Tank-Treading Motion in Shear Flow
journal, December 2005
- Kantsler, Vasiliy; Steinberg, Victor
- Physical Review Letters, Vol. 95, Issue 25
Electrohydrodynamics of Three-Dimensional Vesicles: A Numerical Approach
journal, January 2015
- Kolahdouz, Ebrahim M.; Salac, David
- SIAM Journal on Scientific Computing, Vol. 37, Issue 3
The Implicit Closest Point Method for the Numerical Solution of Partial Differential Equations on Surfaces
journal, January 2010
- Macdonald, Colin B.; Ruuth, Steven J.
- SIAM Journal on Scientific Computing, Vol. 31, Issue 6
Level Set Equations on Surfaces via the Closest Point Method
journal, March 2008
- Macdonald, Colin B.; Ruuth, Steven J.
- Journal of Scientific Computing, Vol. 35, Issue 2-3
Boundary integral method for the flow of vesicles with viscosity contrast in three dimensions
journal, October 2015
- Rahimian, Abtin; Veerapaneni, Shravan K.; Zorin, Denis
- Journal of Computational Physics, Vol. 298
A simple embedding method for solving partial differential equations on surfaces
journal, January 2008
- Ruuth, Steven J.; Merriman, Barry
- Journal of Computational Physics, Vol. 227, Issue 3
Reynolds number effects on lipid vesicles
journal, August 2012
- Salac, David; Miksis, Michael J.
- Journal of Fluid Mechanics, Vol. 711
Configurations of fluid membranes and vesicles
journal, February 1997
- Seifert, Udo
- Advances in Physics, Vol. 46, Issue 1
Fluid membranes in hydrodynamic flow fields: Formalism and an application to fluctuating quasispherical vesicles in shear flow
journal, April 1999
- Seifert, U.
- The European Physical Journal B, Vol. 8, Issue 3
A Curvature-Augmented, REA Approach to the Level Set Method
journal, January 2016
- Vogl, Chris J.
- SIAM Journal on Scientific Computing, Vol. 38, Issue 2
The effect of glass-forming sugars on vesicle morphology and water distribution during drying
journal, October 2014
- Vogl, C. J.; Miksis, M. J.; Davis, S. H.
- Journal of The Royal Society Interface, Vol. 11, Issue 99
A generalized diffusion model for growth of nanoparticles synthesized by colloidal methods
journal, April 2014
- Wen, Tianlong; Brush, Lucien N.; Krishnan, Kannan M.
- Journal of Colloid and Interface Science, Vol. 419
Charge Diffusion on the Surface of Particles with Simple Geometries
journal, November 2015
- Eloul, Shaltiel; Compton, Richard G.
- The Journal of Physical Chemistry C, Vol. 119, Issue 49
Shape transformations of vesicles: Phase diagram for spontaneous- curvature and bilayer-coupling models
journal, July 1991
- Seifert, Udo; Berndl, Karin; Lipowsky, Reinhard
- Physical Review A, Vol. 44, Issue 2
Electrohydrodynamics of Three-Dimensional Vesicles: A Numerical Approach
journal, January 2015
- Kolahdouz, Ebrahim M.; Salac, David
- SIAM Journal on Scientific Computing, Vol. 37, Issue 3