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-----------------------------------------------------------------------
-- FILE: luamesh.lua
-- DESCRIPTION: core functions of luamesh package
-- REQUIREMENTS: local scripts (luamesh-polygon and luamesh-tex)
-- AUTHOR: Maxime Chupin
-----------------------------------------------------------------------
require "luamesh-polygon"
require "luamesh-tex"
local function shallowCopy(original)
local copy = {}
for key, value in pairs(original) do
copy[key] = value
end
return copy
end
-- Bowyer and Watson algorithm
-- Delaunay meshing
function BowyerWatson (listPoints,bbox)
local triangulation = {}
lgth = #listPoints
-- add four points to listPoints to have a bounding box
listPoints = buildBoundingBox(listPoints)
-- the first triangle
triangulation[1] = {lgth+1, lgth+2, lgth+3,type="bbox"}
-- the second triangle
triangulation[2] = {lgth+1, lgth+3, lgth+4,type="bbox"}
-- add points one by one
for i=1,lgth do
-- find the triangles which the circumcircle contained the point to add
badTriangles = buildBadTriangles(listPoints[i],triangulation,listPoints)
-- build the polygon of the cavity containing the point to add
polygon = buildCavity(badTriangles, triangulation)
-- remove the bad triangles
for j=1,#badTriangles do
table.remove(triangulation,badTriangles[j]-(j-1))
end
-- build the new triangles and add them to triangulation
for j=1,#polygon do
if((polygon[j][1]>lgth) or (polygon[j][2]>lgth) or (i>lgth)) then
table.insert(triangulation,{polygon[j][1],polygon[j][2],i,type="bbox"})
else
table.insert(triangulation,{polygon[j][1],polygon[j][2],i,type="in"})
end
end
end -- end adding points of the listPoints
-- remove bounding box
if(bbox ~= "bbox") then
triangulation = removeBoundingBox(triangulation,lgth)
table.remove(listPoints,lgth+1)
table.remove(listPoints,lgth+1)
table.remove(listPoints,lgth+1)
table.remove(listPoints,lgth+1)
end
return triangulation
end
function buildBoundingBox(listPoints)
-- listPoints : list of points
-- epsV : parameter for the distance of the bounding box
local xmin, xmax, ymin, ymax, eps
xmin = 1000
ymin = 1000
xmax = -1000
ymax = -1000
for i=1,#listPoints do
if (listPoints[i].x < xmin) then
xmin = listPoints[i].x
end
if (listPoints[i].x > xmax) then
xmax = listPoints[i].x
end
if (listPoints[i].y < ymin) then
ymin = listPoints[i].y
end
if (listPoints[i].y > ymax) then
ymax = listPoints[i].y
end
end
eps = math.max(math.abs(xmax-xmin),math.abs(ymax-ymin))*0.15
xmin = xmin - eps
xmax = xmax + eps
ymin = ymin - eps
ymax = ymax + eps
-- add points of the bounding box in last positions
table.insert(listPoints,{x=xmin,y=ymin,type="bbox"})
table.insert(listPoints,{x=xmin,y=ymax,type="bbox"})
table.insert(listPoints,{x=xmax,y=ymax,type="bbox"})
table.insert(listPoints,{x=xmax,y=ymin,type="bbox"})
return listPoints
end
function BoundingBox(listPoints)
-- listPoints : list of points
-- epsV : parameter for the distance of the bounding box
local xmin, xmax, ymin, ymax, eps
xmin = 1000
ymin = 1000
xmax = -1000
ymax = -1000
for i=1,#listPoints do
if (listPoints[i].x < xmin) then
xmin = listPoints[i].x
end
if (listPoints[i].x > xmax) then
xmax = listPoints[i].x
end
if (listPoints[i].y < ymin) then
ymin = listPoints[i].y
end
if (listPoints[i].y > ymax) then
ymax = listPoints[i].y
end
end
eps = math.max(math.abs(xmax-xmin),math.abs(ymax-ymin))*0.15
xmin = xmin - eps
xmax = xmax + eps
ymin = ymin - eps
ymax = ymax + eps
return xmin, xmax, ymin, ymax
end
function removeBoundingBox(triangulation,lgth)
-- build the four bounding box edge
point1 = lgth+1
point2 = lgth+2
point3 = lgth+3
point4 = lgth+4
-- for all triangle
local newTriangulation = {}
for i=1,#triangulation do
boolE1 = pointInTriangle(point1,triangulation[i])
boolE2 = pointInTriangle(point2,triangulation[i])
boolE3 = pointInTriangle(point3,triangulation[i])
boolE4 = pointInTriangle(point4,triangulation[i])
if((not boolE1) and (not boolE2) and (not boolE3) and (not boolE4)) then
table.insert(newTriangulation,triangulation[i])
end
end
return newTriangulation
end
function buildBadTriangles(point, triangulation,listPoints)
local badTriangles = {}
for j=1,#triangulation do -- for all triangles
A = listPoints[triangulation[j][1]]
B = listPoints[triangulation[j][2]]
C = listPoints[triangulation[j][3]]
center, radius = circoncircle(A,B,C)
CP = Vector(center,point)
if(VectorNorm(CP)<radius) then -- the point belongs to the circoncirle
table.insert(badTriangles,j)
end
end
return badTriangles
end
-- construction of the cavity composed by the bad triangles around the point to add
function buildCavity(badTriangles, triangulation)
local polygon = {}
for j=1,#badTriangles do -- for all bad triangles
ind = badTriangles[j]
for k=1,3 do -- for all edges
edge = {triangulation[ind][k],triangulation[ind][k%3+1]}
edgeBord = false
for l = 1,#badTriangles do -- for all badtriangles
badInd = badTriangles[l]
if(badInd ~= ind) then -- if not the current one
edgeBord = edgeBord or edgeInTriangle(edge,triangulation[badInd])
end
end --
-- if the edge does not belong to another bad triangle
if(edgeBord == false) then
-- insert the edge to the cavity
table.insert(polygon,edge)
end
end --
end --
return polygon
end
function edgeInTriangle(e,t)
in1 = false
in2 = false
for i=1,3 do
if e[1] == t[i] then
in1 = true
end
if e[2] == t[i] then
in2 = true
end
end
out = (in1 and in2)
return out
end
function pointInTriangle(e,t)
in1 = false
for i=1,3 do
if e == t[i] then
in1 = true
end
end
return in1
end
function Vector(A,B)
local out = {x = B.x - A.x, y = B.y - A.y}
return out
end
function VectorNorm(NP)
return math.sqrt(NP.x*NP.x +NP.y*NP.y)
end
-- circoncircle
function circoncircle(M, N, P)
-- Compute center and radius of the circoncircle of the triangle M N P
-- return : (center [Point],radius [float])
local MN = Vector(M,N)
local NP = Vector(N,P)
local PM = Vector(P,M)
m = VectorNorm(NP) -- |NP|
n = VectorNorm(PM) -- |PM|
p = VectorNorm(MN) -- |MN|
d = (m + n + p) * (-m + n + p) * (m - n + p) * (m + n - p)
if d > 0 then
rad = m * n * p / math.sqrt(d)
else
rad = 0
end
d = -2 * (M.x * NP.y + N.x * PM.y + P.x * MN.y)
O = {x=0, y=0}
OM = Vector(O, M)
ON = Vector(O, N)
OP = Vector(O, P)
om2 = math.pow(VectorNorm(OM),2) -- |OM|**2
on2 = math.pow(VectorNorm(ON),2) -- |ON|**2
op2 = math.pow(VectorNorm(OP),2) -- |OP|**2
x0 = -(om2 * NP.y + on2 * PM.y + op2 * MN.y) / d
y0 = (om2 * NP.x + on2 * PM.x + op2 * MN.x) / d
if d == 0 then
Out = {nil, nil}
else
Out = {x=x0, y=y0}
end
return Out, rad -- (center [Point], R [float])
end
-- compute the list of the circumcircle of a triangulation
function listCircumCenter(listPoints,triangulation)
local list = {}
for j=1,#triangulation do
A = listPoints[triangulation[j][1]]
B = listPoints[triangulation[j][2]]
C = listPoints[triangulation[j][3]]
center, radius = circoncircle(A,B,C)
table.insert(list,{x=center.x,y=center.y,r=radius})
end
return list
end
-- find the three neighbour triangles of T
function findNeighbour(T,i,triangulation)
-- T : triangle
-- i : index of T in triangualation
-- triangulation
list = {}
-- define the three edge
e1 = {T[1],T[2]}
e2 = {T[2],T[3]}
e3 = {T[3],T[1]}
for j=1,#triangulation do
if j~= i then
if(edgeInTriangle(e1,triangulation[j])) then
table.insert(list,j)
end
if(edgeInTriangle(e2,triangulation[j])) then
table.insert(list,j)
end
if(edgeInTriangle(e3,triangulation[j])) then
table.insert(list,j)
end
end
end
return list
end
-- test if edge are the same (reverse)
function equalEdge(e1,e2)
if(((e1[1] == e2[1]) and (e1[2] == e2[2])) or ((e1[1] == e2[2]) and (e1[2] == e2[1]))) then
return true
else
return false
end
end
-- test if the edge belongs to the list
function edgeInList(e,listE)
output = false
for i=1,#listE do
if(equalEdge(e,listE[i])) then
output = true
end
end
return output
end
-- build the edges of the Voronoi diagram with a given triangulation
function buildVoronoi(listPoints, triangulation)
local listCircumCircle = listCircumCenter(listPoints, triangulation)
local listVoronoi = {}
for i=1,#listCircumCircle do
listN = findNeighbour(triangulation[i],i,triangulation)
for j=1,#listN do
edge = {i,listN[j]}
if( not edgeInList(edge, listVoronoi)) then
table.insert(listVoronoi, edge)
end
end
end
return listVoronoi
end
-- build the list of points
function buildList(chaine, mode)
-- if mode = int : the list is given in the chaine string (x1,y1);(x2,y2);...;(xn,yn)
-- if mode = ext : the list is given in a file line by line with space separation
local listPoints = {}
if mode == "int" then
local points = string.explode(chaine, ";")
local lgth=#points
for i=1,lgth do
Sx,Sy=string.match(points[i],"%((.+),(.+)%)")
listPoints[i]={x=tonumber(Sx),y=tonumber(Sy)}
end
elseif mode == "ext" then
io.input(chaine) -- open the file
text=io.read("*all")
lines=string.explode(text,"\n+") -- all the lines
tablePoints={}
for i=1,#lines do
xy=string.explode(lines[i]," +")
listPoints[i]={x=tonumber(xy[1]),y=tonumber(xy[2])}
end
else
print("Non existing mode")
end
return listPoints
end
-- function to add points on a polygon to respect
-- the size of unit mesh
function addPointsPolygon(polygon,h)
local newPolygon = shallowCopy(polygon)
k=0 -- to follow in the newPolygon
for i=1,#polygon do
k = k+1
ip = (i)%(#polygon)+1
dist = math.sqrt(math.pow(polygon[i].x-polygon[ip].x,2) + math.pow(polygon[i].y-polygon[ip].y,2))
-- if the distance between two ponits of the polygon is greater than 1.5*h
if(dist>=2*h) then
n = math.floor(dist/h)
step = dist/(n+1)
for j=1,n do
a = {x=polygon[i].x+j*step*(polygon[ip].x-polygon[i].x)/dist,y=polygon[i].y+j*step*(polygon[ip].y-polygon[i].y)/dist}
table.insert(newPolygon,k+j,a)
end
k=k+n
end
end
return newPolygon
end
-- function to build a gridpoints from the bounding box
-- with a prescribed
function buildGrid(listPoints,h,random)
-- listPoints : list of the points of the polygon, ordered
-- h : parameter for the grid
xmin, xmax, ymin, ymax = BoundingBox(listPoints)
local grid = rectangleList(xmin,xmax,ymin,ymax,h,random)
return grid
end
-- function to build the list of points in the rectangle
function rectangleList(xmin,xmax,ymin,ymax,h,random)
-- for the random
math.randomseed( os.time() )
nbrX = math.floor(math.abs(xmax-xmin)/h)
nbrY = math.floor(math.abs(ymax-ymin)/h)
local listPoints = {}
k=1
for i=1,(nbrX+1) do
for j=1,(nbrY+1) do
rd = math.random()
if(random=="perturb") then
fact = 0.3*h
--print(fact)
else
fact = 0.0
end
listPoints[k] = {x = xmin+(i-1)*h+rd*fact, y=ymin+(j-1)*h+rd*fact}
k=k+1
end
end
return listPoints
end
-- function to add points from a grid to the interior of a polygon
function addGridPoints(polygon, grid,h)
local listPoints = shallowCopy(polygon)
k = #polygon
for i=1, #grid do
--print(grid[i].x,grid[i].y)
--print(isInside(polygon,grid[i]))
if(isInside(polygon,grid[i],h)) then
k=k+1
listPoints[k] = grid[i]
end
end
return listPoints
end
-- function give a real polygon without repeting points
function cleanPoly(polygon)
local polyNew = {}
local polyCopy = shallowCopy(polygon)
e1 = polyCopy[1][1]
e2 = polyCopy[1][2]
table.insert(polyNew, e1)
table.insert(polyNew, e2)
table.remove(polyCopy,1)
j = 2
while #polyCopy>1 do
i=1
find = false
while (i<=#polyCopy and find==false) do
bool1 = (polyCopy[i][1] == polyNew[j])
bool2 = (polyCopy[i][2] == polyNew[j])
if(bool1 or bool2) then -- the edge has a common point with polyNew[j]
if(not bool1) then
table.insert(polyNew, polyCopy[i][1])
find = true
table.remove(polyCopy,i)
j = j+1
elseif(not bool2) then
table.insert(polyNew, polyCopy[i][2])
find = true
table.remove(polyCopy,i)
j = j+1
end
end
i=i+1
end
end
return polyNew
end
-- build the list of points extern and stop at nbr
function buildListExt(chaine, stop)
local listPoints = {}
io.input(chaine) -- open the file
text=io.read("*all")
lines=string.explode(text,"\n+") -- all the lines
for i=1,tonumber(stop) do
xy=string.explode(lines[i]," +")
table.insert(listPoints,{x=tonumber(xy[1]),y=tonumber(xy[2])})
end
xy=string.explode(lines[stop+1]," +")
point={x=tonumber(xy[1]),y=tonumber(xy[2])}
return point, listPoints
end
function split(pString, pPattern)
local Table = {} -- NOTE: use {n = 0} in Lua-5.0
local fpat = "(.-)" .. pPattern
local last_end = 1
local s, e, cap = pString:find(fpat, 1)
while s do
if s ~= 1 or cap ~= "" then
table.insert(Table,cap)
end
last_end = e+1
s, e, cap = pString:find(fpat, last_end)
end
if last_end <= #pString then
cap = pString:sub(last_end)
table.insert(Table, cap)
end
return Table
end
function readGmsh(file)
io.input(file) -- open the file
text=io.read("*all")
local lines = split(text,"\n+") -- all the lines
local listPoints={}
local triangulation ={}
boolNodes = false
Jnodes = 0
boolElements = false
Jelements = 0
J=0
format=split(lines[2]," +")
if((tonumber(format[1])>=2.0) and (tonumber(format[1])<3.0)) then -- Format 2
for i=4,#lines-J do
if(lines[i+J] == "$EndNodes") then
boolNodes = false
-- go to the next line
end
if(boolNodes) then -- we are in the Nodes environment
xy=split(lines[i+J]," +")
table.insert(listPoints,{x=tonumber(xy[2]),y=tonumber(xy[3])})
end
if(lines[i+J] == "$Nodes") then
boolNodes = true
-- go to the next line
J=J+1
end
if(lines[i+J] == "$EndElements") then
boolElements = false
-- go to the next line
end
if(boolElements) then -- we are in the Elements environment
xy=split(lines[i+J]," +")
if(tonumber(xy[2]) == 2) then -- if the element is a triangle
nbrTags = xy[3]+1
table.insert(triangulation,{tonumber(xy[2+nbrTags+1]),tonumber(xy[2+nbrTags+2]),tonumber(xy[2+nbrTags+3])})
end
end
if(lines[i+J] == "$Elements") then
boolElements = true
-- go to the next line
J=J+1
end
end
end
if(tonumber(format[1]) >= 4.0) then -- format 4
for i=4,#lines-J do
if(lines[i+J] == "$EndNodes") then
boolNodes = false
-- go to the next line
end
if(boolNodes) then -- we are in the Nodes environment
xy=split(lines[i+J]," +") -- bloc 4 number
nbrBloc = tonumber(xy[4])
for k=1,nbrBloc do
xyz=split(lines[i+J+nbrBloc+k]," +") -- x y z
table.insert(listPoints,{x=tonumber(xyz[1]),y=tonumber(xyz[2])})
end
J =J+ 2*tonumber(xy[4])
end
if(lines[i+J] == "$Nodes") then
boolNodes = true
-- go to the next line
J=J+1
end
if(lines[i+J] == "$EndElements") then
boolElements = false
-- go to the next line
end
if(boolElements) then -- we are in the Elements environment
xy=split(lines[i+J]," +")
if(tonumber(xy[1]) < 2) then -- if the elements block is not a triangle
J = J+tonumber(xy[4]) -- we jump to the new block
end
if(tonumber(xy[1]) == 2) then -- if the elements block is a triangle
nbrElements = tonumber(xy[4])
for k=1,nbrElements do
xyz=split(lines[i+J+k]," +") -- line tag node1 node2 node3
table.insert(triangulation,{tonumber(xyz[2]),tonumber(xyz[3]),tonumber(xyz[4])})
end
J = J+nbrElements
end
end
if(lines[i+J] == "$Elements") then
boolElements = true
-- go to the next line
J=J+1
end
end
end
return listPoints, triangulation
end
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