R: A simple mesh generator for non-convex regions

distmesh2d {geometry} | R Documentation

## A simple mesh generator for non-convex regions

### Description

An unstructured simplex requires a choice of meshpoints (vertex nodes) and a triangulation. This is a simple and short algorithm that improves the quality of a mesh by relocating the meshpoints according to a relaxation scheme of forces in a truss structure. The topology of the truss is reset using Delaunay triangulation. A (sufficiently smooth) user supplied signed distance function (`fd`) indicates if a given node is inside or outside the region. Points outside the region are projected back to the boundary.

### Usage

    distmesh2d(fd, fh, h0, bbox, p = NULL, pfix = array(0, dim = c(0, 2)), ...,
      dptol = 0.001, ttol = 0.1, Fscale = 1.2, deltat = 0.2, geps = 0.001
      * h0, deps = sqrt(.Machine$double.eps) * h0, maxiter = 1000)

### Arguments

`fd` | Vectorized signed distance function, for example `mesh.dcircle` or `mesh.diff`, accepting an `n`-by-`2` matrix, where `n` is arbitrary, as the first argument.
`fh` | Vectorized function, for example `mesh.hunif`, that returns desired edge length as a function of position. Accepts an `n`-by-`2` matrix, where `n` is arbitrary, as its first argument.
`h0` | Initial distance between mesh nodes. (Ignored of `p` is supplied)
`bbox` | Bounding box cbind(c(xmin,xmax), c(ymin,ymax))
`p` | An `n`-by-`2` matrix. The rows of `p` represent locations of starting mesh nodes.
`pfix` | `nfix`-by-2 matrix with fixed node positions.
`dptol` | Algorithm stops when all node movements are smaller than `dptol`
`ttol` | Controls how far the points can move (relatively) before a retriangulation with `delaunayn`.
`Fscale` | “Internal pressure” in the edges.
`deltat` | Size of the time step in Eulers method.
`geps` | Tolerance in the geometry evaluations.
`deps` | Stepsize Δ x in numerical derivative computation for distance function.
`maxiter` | Maximum iterations.
`...` | parameters to be passed to `fd` and/or `fh`

### Details

This is an implementation of original Matlab software of Per-Olof Persson.

Excerpt (modified) from the reference below:

‘The algorithm is based on a mechanical analogy between a triangular mesh and a 2D truss structure. In the physical model, the edges of the Delaunay triangles of a set of points correspond to bars of a truss. Each bar has a force-displacement relationship F(L,L0) depending on its current length L and its unextended length L0.’

‘External forces on the structure come at the boundaries, on which external forces have normal orientations. These external forces are just large enough to prevent nodes from moving outside the boundary. The position of the nodes are the unknowns, and are found by solving for a static force equilibrium. The hope is that (when `fh = function(p) return(rep(1,nrow(p)))`), the lengths of all the bars at equilibrium will be nearly equal, giving a well-shaped triangular mesh.’

See the references below for all details. Also, see the comments in the source file.

### Value

`n`-by-`2` matrix with node positions.

### Wishlist

- *Implement in C/Fortran

- *Implement an `n`D version as provided in the matlab package

- *Translate other functions of the matlab package

### Author(s)

Raoul Grasman

### References

http://persson.berkeley.edu/distmesh/

P.-O. Persson, G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004

### See Also

`tri.mesh`, `delaunayn`, `mesh.dcircle`, `mesh.drectangle`,
`mesh.diff`, `mesh.union`, `mesh.intersect`

### Examples

    # examples distmesh2d
    fd <- function(p, ...) sqrt((p^2)%*%c(1,1)) - 1
         # also predefined as `mesh.dcircle'
    fh <- function(p,...)  rep(1,nrow(p))
    bbox <- matrix(c(-1,1,-1,1),2,2)
    p <- distmesh2d(fd,fh,0.2,bbox, maxiter=100)
        # this may take a while:
        # press Esc to get result of current iteration

    # example with non-convex region
    fd <- function(p, ...) mesh.diff(p , mesh.drectangle, mesh.dcircle, radius=.3)
         # fd defines difference of square and circle

    p <- distmesh2d(fd,fh,0.05,bbox,radius=0.3,maxiter=4)
    p <- distmesh2d(fd,fh,0.05,bbox,radius=0.3, maxiter=10)
         # continue on previous mesh

[Package geometry version 0.3-6 Index]