distmesh2d {geometry} | R Documentation |
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.
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)
fd |
Vectorized signed distance function, accepting an n -by-2 matrix,
where n is arbitrary, as the first argument. |
fh |
Vectorized function 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. |
... |
parameters to be passed to fd and/or fh |
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 Delta x in numerical derivative computation for distance function. |
maxiter |
Maximum iterations. |
This is an R 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.
n
-by-2
matrix with node positions.
n
D version as provided in the matlab packageRaoul Grasman
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
tri.mesh
,
delaunayn
,
mesh.dcircle
,
mesh.drectangle
,
mesh.diff
,
mesh.union
,
mesh.intersect
# 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