geometry (version 0.1)

distmeshnd: A simple mesh generator for non-convex regions in n-D space

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

distmeshnd(fdist, fh, h, box, pfix = array(dim = c(0,
         ncol(box))), ..., ptol = 0.001, ttol = 0.1, deltat =
         0.1, geps = 0.1 * h, deps = sqrt(.Machine$double.eps)
         * h)

Arguments

fdist

Vectorized signed distance function, accepting an m-by-n matrix, where m is arbitrary, as the first argument.

fh

Vectorized function that returns desired edge length as a function of position. Accepts an m-by-n matrix, where n is arbitrary, as its first argument.

h

Initial distance between mesh nodes.

box

2-by-n matrix that specifies the bounding box. (See distmesh2d for an example.)

pfix

nfix-by-2 matrix with fixed node positions.

...

parameters that are passed to fdist and fh

ptol

Algorithm stops when all node movements are smaller than dptol

ttol

Controls how far the points can move (relatively) before a retriangulation with delaunayn.

deltat

Size of the time step in Eulers method.

geps

Tolerance in the geometry evaluations.

deps

Stepsize $\Delta x$ in numerical derivative computation for distance function.

Value

• m-by-n matrix with node positions.

Wishlist

• *

{Implement in C/Fortran} *{Translate other functions of the matlab package}

Details

This is an Rimplementation 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 n-D 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 $f(\ell, \ell_{0})$ depending on its current length $\ell$ and its unextended length $\ell_{0}$. 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 of distmesh2d.

References

http://www-math.mit.edu/~persson/mesh/ P.-O. Persson, G. Strang, A Simple Mesh Generator in MATLAB. SIAM Review, Volume 46 (2), pp. 329-345, June 2004

See Also

distmesh2d, tri.mesh, delaunayn, mesh.dsphere, mesh.hunif, mesh.diff, mesh.union, mesh.intersect

Examples

# examples distmeshnd
require(rgl)

fd = function(p, ...) sqrt((p^2)%*%c(1,1,1)) - 1
     # also predefined as `mesh.dsphere'
fh = function(p,...)  rep(1,nrow(p))
     # also predefined as `mesh.hunif'
bbox = matrix(c(-1,1),2,3)
p = distmeshnd(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