# GfsMetricLaplace

### From Gerris

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The syntax in parameter files is | The syntax in parameter files is | ||

- | MetricLaplace M { spherical = 0 conformal = 0 } | + | MetricLaplace NAME { spherical = 0 conformal = 0 } |

The parameter block is optional. If <code>spherical</code> is set to one, the mapping is from the plane in computational coordinates to the sphere in physical coordinates. If <code>conformal</code> is set to one, a conformal Laplace-Beltrami operator is used (the existence of such a mapping depends on the boundary conditions). | The parameter block is optional. If <code>spherical</code> is set to one, the mapping is from the plane in computational coordinates to the sphere in physical coordinates. If <code>conformal</code> is set to one, a conformal Laplace-Beltrami operator is used (the existence of such a mapping depends on the boundary conditions). | ||

+ | |||

+ | In addition to the metric scaling factors, additional variables will be defined for each of the physical coordinates i.e. <code>(NAMEx,NAMEy,NAMEz)</code>. Grid generation is controlled by imposing Dirichlet or Neumann boundary conditions on these variables. | ||

+ | |||

+ | Note that this does not work with adaptivity yet. | ||

+ | |||

+ | === References === | ||

+ | |||

+ | <bibtex> | ||

+ | @article{eca1996, | ||

+ | title = {2D orthogonal grid generation with boundary point distribution control}, | ||

+ | author = {Eca, L.}, | ||

+ | journal = {Journal of Computational Physics}, | ||

+ | volume = {125}, | ||

+ | number = {2}, | ||

+ | pages = {440-453}, | ||

+ | year = {1996}, | ||

+ | publisher = {Elsevier} | ||

+ | } | ||

+ | </bibtex> | ||

<examples/> | <examples/> |

## Current revision

GfsMetricLaplace computes a numerical orthogonal mapping of the computational space `(rx,ry)`

into the physical space

by solving
* x*=(x,y,z)

*∇*^{2}= -2**x***H**n*

where *∇ ^{2}* is the Laplace-Beltrami operator,

**is the normal to the surface and**

*n**H*is the mean curvature.

The syntax in parameter files is

MetricLaplace NAME { spherical = 0 conformal = 0 }

The parameter block is optional. If `spherical`

is set to one, the mapping is from the plane in computational coordinates to the sphere in physical coordinates. If `conformal`

is set to one, a conformal Laplace-Beltrami operator is used (the existence of such a mapping depends on the boundary conditions).

In addition to the metric scaling factors, additional variables will be defined for each of the physical coordinates i.e. `(NAMEx,NAMEy,NAMEz)`

. Grid generation is controlled by imposing Dirichlet or Neumann boundary conditions on these variables.

Note that this does not work with adaptivity yet.

### References

*Eca, L.* - **2D orthogonal grid generation with boundary point distribution control**

- Journal of Computational Physics 125(2):440-453, 1996
- Bibtex