Provides the Li-Lowengrub-Ratz-Voight (LLRV) domain mask characteristic functions. More...

#include <sdf_domainmask.h>

Inheritance diagram for madness::LLRVDomainMask:

Collaboration diagram for madness::LLRVDomainMask:

Public Member Functions
	LLRVDomainMask (double epsilon)
	Constructor for the domain mask.

virtual	~LLRVDomainMask ()

double	dmask (double d) const
	Derivative of characteristic function with respect to the normal distance.

double	dsurface (double d) const
	Value of d(surface)/ddistance.

double	mask (double d) const
	Value of characteristic function at normal distance d from the surface.

double	surface (double d) const
	Value of surface function at distance d normal to surface.

Public Member Functions inherited from madness::DomainMaskInterface
virtual	~DomainMaskInterface ()

Protected Attributes
const double	epsilon
	The width of the transition region.

Private Member Functions
	LLRVDomainMask ()
	Forbidden.

Detailed Description

Provides the Li-Lowengrub-Ratz-Voight (LLRV) domain mask characteristic functions.

See X. Li, J. Lowengrub, A. Rätz, and A. Voight, `‘Solving PDEs in Complex Geometries: A Diffuse Domain Approach,’' Commun. Math. Sci., 7, p81-107, 2009.

Given a signed distance, this class implements in the domain mask and surface functions from the above reference. For the domain mask,

$\varphi(d) = \frac{1}{2}\left( 1 - \tanh\left( \frac{3d}{\varepsilon} \right) \right)$

where is the signed distance. The normalized surface function is

$B(\varphi) = \frac{36}{\varepsilon} \varphi^2 (1-\varphi)^2.$

The constant $36/\varepsilon$ is chosen to fulfill

$\int_{-\infty}^\infty B(s) \, ds = 1$

This class assumes the domain mask is uniformly 0 or 1 outside signed distances $|8 \epsilon|$ since the switching function becomes 0/1 to machine precision at these levels. Specifically, for this function the parameter $\epsilon$ is an effective measure of the full width of the surface layer since