Provides the Li-Lowengrub-Ratz-Voight (LLRV) domain mask characteristic functions. More...
#include <sdf_domainmask.h>
Public Member Functions | |
| LLRVDomainMask (double epsilon) | |
| Constructor for the domain mask. More... | |
| double | mask (double d) const |
| Value of characteristic function at normal distance d from the surface. More... | |
| double | dmask (double d) const |
| Derivative of characteristic function with respect to the normal distance. More... | |
| double | surface (double d) const |
| Value of surface function at distance d normal to surface. More... | |
| double | dsurface (double d) const |
| Value of d(surface)/ddistance. More... | |
| virtual | ~LLRVDomainMask () |
 Public Member Functions inherited from madness::DomainMaskInterface | |
| virtual | ~DomainMaskInterface () |
Protected Attributes | |
| const double | epsilon |
| The width of the transition region. More... | |
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,
![\[ \varphi(d) = \frac{1}{2}\left( 1 - \tanh\left( \frac{3d}{\varepsilon} \right) \right) \]](form_219.png)
where 
is the signed distance. The normalized surface function is
![\[ B(\varphi) = \frac{36}{\varepsilon} \varphi^2 (1-\varphi)^2. \]](form_221.png)
The constant 
is chosen to fulfill
![\[ \int_{-\infty}^\infty B(s) \, ds = 1 \]](form_223.png)
This class assumes the domain mask is uniformly 0 or 1 outside signed distances 
since the switching function becomes 0/1 to machine precision at these levels. Specifically, for this function the parameter 
is an effective measure of the full width of the surface layer since
![\[ \int_{-\epsilon/2}^{\epsilon/2} B(s) \, ds \doteq 0.987 \]](form_226.png)
and
![\[ \int_{-\epsilon}^{\epsilon} B(s) \, ds \doteq 0.999963 \]](form_227.png)
Constructor & Destructor Documentation
|
inline |
Constructor for the domain mask.
- Parameters
-
[in] epsilon The effective width of the surface
|
inlinevirtual |
Member Function Documentation
|
inlinevirtual |
Derivative of characteristic function with respect to the normal distance.
- Parameters
-
[in] d The signed distance
- Returns
- The derivative
Implements madness::DomainMaskInterface.
References epsilon, mpfr::fabs(), and mpfr::tanh().
Referenced by dsurface().
|
inlinevirtual |
Value of d(surface)/ddistance.
- Parameters
-
[in] d The signed distance
- Returns
- The derivative of the surface function
Implements madness::DomainMaskInterface.
|
inlinevirtual |
Value of characteristic function at normal distance d from the surface.
- Parameters
-
[in] d The signed distance. Negative is inside,'' positive isoutside.''
- Returns
- The domain mask
Implements madness::DomainMaskInterface.
References mpfr::tanh().
Referenced by dsurface(), and surface().
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inlinevirtual |
Value of surface function at distance d normal to surface.
- Parameters
-
[in] d The signed distance
- Returns
- The value of the surface function
Implements madness::DomainMaskInterface.
Member Data Documentation
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protected |
The width of the transition region.
Referenced by dmask(), dsurface(), and surface().
The documentation for this class was generated from the following file:
