# 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 R ^{2} and R ^{3} 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 Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1410022

- Report Number(s):
- LLNL-JRNL-732193

Journal ID: ISSN 0021-9991

- Grant/Contract Number:
- AC52-07NA27344

- Resource Type:
- Journal Article: 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. doi:10.1016/j.jcp.2017.06.004.

```
Vogl, Christopher J. Tue .
"The Curvature-Augmented Closest Point method with vesicle inextensibility application". United States.
doi: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}

}