David Lambert/mesh

This work contributes to the ultimate goal to produce meshes suitable for wire frame graphics and finite element programs, such as CalculiX. The verbs mesh2, mesh3 and plot_mesh provide the main functionality to produce and display uniform rectilinear grids which can later be warped, repositioned, and merged. The meshes are scaled from _1 to 1 on all axes. Index position serves as node or element number, and otherwise, with so many good meshing programs available, this is an yet another effort---in j.

Ultimately---a nice mesher might take a bounded 3D figure, use Schwarz–Christoffel transform to mesh the 2D (possibly warped) surfaces, then mesh the interiors with Delaunay simplexes. Missing, a method to choose interior nodes. I've use such a 2d remesh algorithm in which uniformly spaced points on complex circle worked out nicely on the inverse transformation.

require'plot'

NB. The mesh data structure is elements ; nodes
NB. elements are the connectivity of nodes by index

NB. extraction, although most common is
NB. 'nodes elements'=. MESH
elements =: 0&{::  NB. elements MESH
nodes =: 1&{::     NB. nodes MESH

Note 'element node order'
 Node order is non-standard,
 The n-dimensional linear element node order given by
 coordinates
    ('x';'x y';'x y z'),: |."1@#:@:i.&.>2^1 2 3
 +-+---+-----+
 |x|x y|x y z|
 +-+---+-----+
 |0|0 0|0 0 0|
 |1|1 0|1 0 0|
 | |0 1|0 1 0|
 | |1 1|1 1 0|
 | |   |0 0 1|
 | |   |1 0 1|
 | |   |0 1 1|
 | |   |1 1 1|
 +-+---+-----+
 These are the corner positions of linear elements 0 1, bilinear 0 1 2 3, and trilinear i.8 .
)

Monad=: :[:
Dyad=: [: :

cross =: 1 |. ([ * 1 |. ]) - ] * 1 |. [ NB. j forum

odometer6=: (4 $. $.)@:($&1)  NB. contributed by David Ward Lambert
grid=: ([: <: odometer6 (*"1) 2&%@:<:)&.:(|."1) Monad NB.  grid Nx [Ny [Nz]]
NB.grid =: ([: <: odometer6 (*"1) 2&%@:<:) Monad
square_grid =: ([: grid 2 # ]) Monad

e =: ,: NB. itemize a vector
egen =: ([: ,/ (+/~ ({: * [: i. {.)))~ Dyad NB.  (repetition, nodal increment) egen elements
egrid2 =: (<:@:(1&{) , {.) egen ((1 ,~ <:@:{.) egen [: e 0 1 , 0 1 + {.) Monad
egrid3=: ((<:@:(2&{) , */@:(2&{.)) egen (,. (+ >:@:((<_1 _1)&{)))@:egrid2) Monad

NB.egrid3 =: (] ,. */@:(2&{.)@:[ + ]) egrid2

mesh2=: (egrid2 ; grid) Monad  NB. mesh Nx, Ny
mesh3=: (egrid3 ; grid) Monad  NB. mesh Nx, Ny, Nz

NB. segments ELEMENTS  NB. constructs a list of segments to draw
NB. (>0 1;1 3;3 2;2 0) are corners of mesh
segments =: (>0 1;1 3;3 2;2 0)&{"1  NB. per element arrays of faces
unique_segments =: [: ~. [: /:"1~ [: ,/ segments NB. nub of sorted faces 
plot_mesh =: 'color black' plot j./"1@:(({~ unique_segments)&>~/)

echo 'example: plot_mesh mesh2 4 9'