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refine2.m
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refine2.m
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function [vert,conn,tria,tnum] = refine2(varargin)
%REFINE2 (Frontal)-Delaunay-refinement for two-dimensional,
%polygonal geometries.
% [VERT,EDGE,TRIA,TNUM] = REFINE2(NODE,EDGE) returns a co-
% nstrained Delaunay triangulation of the polygonal region
% {NODE,EDGE}. NODE is an N-by-2 array of polygonal verti-
% ces and EDGE is an E-by-2 array of edge indexing. Each
% row in EDGE represents an edge of the polygon, such that
% NODE(EDGE(JJ,1),:) and NODE(EDGE(JJ,2),:) are the coord-
% inates of the endpoints of the JJ-TH edge. If the argum-
% ent EDGE is omitted it assumed that the vertices in NODE
% are connected in ascending order.
%
% [...] = REFINE2(NODE,EDGE,PART) computes a triangulation
% for a multiply-connected geometry. PART is a cell-array
% of polygonal "parts", where each element PART{KK} is an
% array of edge indices defining a given polygonal region.
% EDGE(PART{KK}, :) is the set of edges in the KK-TH part.
%
% VERT is a V-by-2 array of XY coordinates in the triangu-
% lation, EDGE is an array of constrained edges, TRIA is a
% T-by-3 array of triangles, and TNUM is a T-by-1 array of
% part indices. Each row of TRIA and EDGE define an eleme-
% nt. VERT(TRIA(II,1),:), VERT(TRIA(II,2),:) and VERT(TRIA
% (II,3),:) are the coordinates of the II-TH triangle. The
% edges in EDGE are defined in a similar manner. NUM is an
% array of part indexing, such that TNUM(II) is the index
% of the part in which the II-TH triangle resides.
%
% [...] = REFINE2(..., OPTS) passes an additional options
% structure OPTS, containing various user-defined paramet-
% ers, including:
%
% - OPTS.KIND = {'DELFRONT'}, 'DELAUNAY' -- the type of ref-
% inement employed. The 'DELFRONT' algorithm is typically
% slower, but produces higher quality output.
%
% - OPTS.RHO2 = {1.025} -- the maximum allowable radius-edge
% ratio. Refinement proceeds until all interior triangles
% satisfy the radius-edge threshold. Smaller radius-edge
% ratios lead to improved triangle shape, with RHO2=1 req-
% uiring that all angles exceed 30 degrees. Setting RHO2<1
% may lead to non-convergence.
%
% - OPTS.REF1 = {'REFINE'}, 'PRESERVE' -- refinement 'flag'
% for 1-dimensional faces (i.e. edges). The 'PRESERVE' op-
% tion results in minimal refinement, attempting to retain
% the initial edges without further subdivision. Edges are
% split only to satisfy basic geomertical conformance.
%
% - OPTS.REF2 = {'REFINE'}, 'PRESERVE' -- refinement 'flag'
% for 2-dimensional faces (i.e. trias). The 'PRESERVE' op-
% tion results in minimal refinement, attempting to retain
% the initial trias without further subdivision. Trias are
% split only to satisfy basic geomertical conformance.
%
% - OPTS.SIZ1 = {1.333} -- the normalised rel.-length th-
% reshold for edge-elements. Each exterior edge is refined
% until LL/HH<SIZ1, where LL is the edge-length, HH is the
% edge-centred mesh-size value.
%
% - OPTS.SIZ2 = {1.300} -- the normalised rel.-length th-
% reshold for tria-elements. Each interior tria is refined
% until RE/HH<SIZ2, where RE is an effective tria length,
% based on the circumradius, HH is the tria-centred mesh-
% size value.
%
% - OPTS.DISP = { +10 } -- refinement verbosity. Set to INF
% for quiet execution.
%
% [...] = REFINE2(..., HFUN,ARGS) also passes an optional
% mesh-size function argument. Setting HFUN = HMAX, where
% HMAX is a scalar value, imposes a constant size constra-
% int over the full domain. HFUN can also be defined as a
% general function handle [HH] = HFUN(PP), where PP is an
% N-by-2 array of XY coordinates and HH is the associated
% vector of mesh-size values. User-defined HFUN must be
% fully vectorised. Additional arguments {A1,A2,...AN} for
% HFUN can be passed as trailing parameters to REFINE2. In
% such cases, HFUN must adopt a signature [HH] = HFUN(PP,
% A1,A2,...,AN). HFUN must return positive values.
%
% See also SMOOTH2, TRIDIV2, TRICOST, TRIDEMO
% This routine implements a "multi-refinement" variant of
% Delaunay-refinement type mesh-generation. Both standard
% Delaunay-refinement and Frontal-Delaunay type algorithms
% are available. The Frontal-Delaunay approach is a simpl-
% ified version of the JIGSAW algorithm, described in:
%
% * D. Engwirda, (2014): "Locally-optimal Delaunay-refineme-
% nt and optimisation-based mesh generation", Ph.D. Thesis
% School of Mathematics and Statistics, Univ. of Sydney.
% http://hdl.handle.net/2123/13148
%
% * D. Engwirda & D. Ivers, (2016): "Off-centre Steiner poi-
% nts for Delaunay-refinement on curved surfaces", Comput-
% er-Aided Design, (72), 157--171.
% http://dx.doi.org/10.1016/j.cad.2015.10.007
% This work is an extension of the "off-centre" type tech-
% niques introduced in:
%
% * H. Erten & A. Ungor, (2009): "Quality triangulation with
% locally optimal Steiner points", SIAM Journal on Scient-
% ific Comp. 31(3), 2103--2130.
% http://doi.org/10.1137/080716748
%
% * S. Rebay, (1993): "Efficient Unstructured Mesh Generati-
% on by Means of Delaunay Triangulation and Bowyer-Watson
% Algorithm, J. Comp. Physics 106(1), 125--138.
% http://dx.doi.org/10.1006/jcph.1993.1097
% Generally speaking, the Delaunay-refinement method impl-
% emented here is a variantion of the "classical" algorit-
% hm introduced in:
%
% * J. Ruppert, (1995): "A Delaunay refinement algorithm for
% quality 2-dimensional mesh generation." Journal of Algo-
% rithms 18(3), 548--585.
% http://dx.doi.org/10.1006/jagm.1995.1021
%
% See also: S. Cheng, T. Dey & J. Shewchuk, (2012): "Dela-
% unay mesh generation", CRC Press, for comprehensive cov-
% erage of Delaunay-based meshing techniques.
% A much more advanced, and fully three-dimensional imple-
% mentation is available in the JIGSAW library. For addit-
% ional information, see:
% https://github.com/dengwirda/jigsaw-matlab
%-----------------------------------------------------------
% Darren Engwirda : 2017 --
% Email : [email protected]
% Last updated : 13/02/2020
%-----------------------------------------------------------
node = []; PSLG = []; part = {}; opts = [] ;
hfun = []; harg = {};
%---------------------------------------------- extract args
if (nargin>=+1), node = varargin{1}; end
if (nargin>=+2), PSLG = varargin{2}; end
if (nargin>=+3), part = varargin{3}; end
if (nargin>=+4), opts = varargin{4}; end
if (nargin>=+5), hfun = varargin{5}; end
if (nargin>=+6), harg = varargin(6:end); end
[opts] = makeopt(opts) ;
%---------------------------------------------- default EDGE
nnod = size(node,1) ;
if (isempty(PSLG))
PSLG = [(1:nnod-1)',(2:nnod)'; nnod,1] ;
end
%---------------------------------------------- default PART
ncon = size(PSLG,1) ;
if (isempty(part)), part{1} = (1:ncon)'; end
%---------------------------------------------- basic checks
if (~isnumeric(node) || ~isnumeric(PSLG) || ...
~iscell (part) || ~isstruct (opts) )
error('refine2:incorrectInputClass' , ...
'Incorrect input class.') ;
end
%---------------------------------------------- basic checks
if (ndims(node) ~= +2 || ndims(PSLG) ~= +2)
error('refine2:incorrectDimensions' , ...
'Incorrect input dimensions.');
end
if (size(node,2) < +2 || size(PSLG,2) < +2)
error('refine2:incorrectDimensions' , ...
'Incorrect input dimensions.');
end
%---------------------------------------------- basic checks
if (min([PSLG(:)])<+1 || max([PSLG(:)])>nnod)
error('refine2:invalidInputs', ...
'Invalid EDGE input array.') ;
end
pmin = cellfun(@min,part);
pmax = cellfun(@max,part);
if (min([pmin(:)])<+1 || max([pmax(:)])>ncon)
error('refine2:invalidInputs', ...
'Invalid PART input array.') ;
end
%-------------------------------- prune any non-unique topo.
[ivec,ivec,jvec] = ...
unique(sort(PSLG,+2),'rows') ;
PSLG = PSLG(ivec,:) ;
for ppos = +1:length(part)
if ( ~isnumeric(part{ppos}) )
error ( ...
'refine2:incorrectInputClass', ...
'Incorrect input class. ') ;
end
part{ppos} = ...
unique(jvec(part{ppos})) ;
end
%-------------------------------- check part "manifold-ness"
for ppos = +1:length(part)
eloc = PSLG(part{ppos},:) ;
nadj = ...
accumarray(eloc(:),1) ;
if (any(mod(nadj,2) ~= 0) )
error('refine2:nonmanifoldInputs', ...
'Non-manifold PART detected.') ;
end
end
%---------------------------------------------- output title
if (~isinf(opts.disp))
fprintf(1,'\n') ;
fprintf(1,' Refine triangulation...\n') ;
fprintf(1,'\n') ;
fprintf(1,[...
' -------------------------------------------------------\n', ...
' |ITER.| |CDT1(X)| |CDT2(X)| \n', ...
' -------------------------------------------------------\n', ...
] ) ;
end
%-------------------------------- PASS 0: inflate box bounds
vert = node; tria = []; tnum = []; iter = 0 ;
conn = PSLG;
vmin = min(vert,[],1); % inflate bbox for stability
vmax = max(vert,[],1);
vdel = vmax - 1.*vmin;
vmin = vmin - .5*vdel;
vmax = vmax + .5*vdel;
vbox = [
vmin(1), vmin(2)
vmax(1), vmin(2)
vmax(1), vmax(2)
vmin(1), vmax(2)
] ;
vert = [vert ; vbox] ;
%-------------------------------- PASS 0: shield sharp feat.
[vert,conn,tria,tnum,iter] = ...
cdtbal0(vert,conn,tria,tnum, ...
node,PSLG,part,opts,hfun,harg,iter);
%-------------------------------- PASS 1: refine 1-simplexes
[vert,conn,tria,tnum,iter] = ...
cdtref1(vert,conn,tria,tnum, ...
node,PSLG,part,opts,hfun,harg,iter);
%-------------------------------- PASS 2: refine 2-simplexes
[vert,conn,tria,tnum,iter] = ...
cdtref2(vert,conn,tria,tnum, ...
node,PSLG,part,opts,hfun,harg,iter);
if (~isinf(opts.disp)), fprintf(1,'\n'); end
%-------------------------------- trim extra adjacency info.
tria = tria( :,1:3) ;
%-------------------------------- trim vert. - deflate bbox.
keep = false(size(vert,1),1);
keep(tria(:)) = true;
keep(conn(:)) = true;
redo = zeros(size(vert,1),1);
redo(keep) = ...
(+1:length(find(keep)))';
conn = ...
reshape(redo(conn),[],2);
tria = ...
reshape(redo(tria),[],3);
vert = vert(keep,:) ;
end
function [vert,conn,tria,tnum,iter] = ...
cdtbal0(vert,conn,tria,tnum, ...
node,PSLG,part,opts,hfun,harg,iter)
%CDTBAL0 constrained Delaunay-refinement for "sharp" 0-dim.
%features at PSLG vertices.
% [...] = CDTBAL0(...) refines the set of 1-simplex eleme-
% nts incident to "sharp" features in the PSLG. Specifica-
% lly, edges that subtend "small" angles are split about a
% set of new "collar" vertices, equi-distributed about the
% centre of "sharp" features. Collar size is computed as a
% min. of the incident edge-len. and local mesh-size cons-
% traints.
if (iter <= opts.iter)
%------------------------------------- build current CDT
[vert,conn, ...
tria,tnum] = deltri2(vert,conn, ...
node,PSLG, ...
part, ...
opts.dtri) ;
%------------------------------------- build current adj
[edge,tria] = tricon2(tria,conn) ;
[feat,ftri] = isfeat2(vert, ...
edge,tria) ;
apex = false(size(vert,1), 1) ;
apex(tria(ftri)) = true ;
%------------------------------------- eval. length-fun.
if (~isempty(hfun))
if (isnumeric(hfun))
vlen = hfun * ...
ones(size(vert,1),1) ;
else
vlen = feval( ...
hfun,vert,harg{:}) ;
vlen = vlen(:) ;
end
else
vlen = +inf * ...
ones(size(vert,1),1) ;
end
%------------------------------------- form edge vectors
evec = vert(conn(:,2),:) ...
- vert(conn(:,1),:) ;
elen = sqrt(sum(evec.^2,2));
evec = evec./[elen,elen] ;
%------------------------------------- min. adj. lengths
for epos = +1 : size(conn,1)
ivrt = conn(epos,1) ;
jvrt = conn(epos,2) ;
vlen(ivrt) = min( ...
vlen(ivrt), .67*elen(epos)) ;
vlen(jvrt) = min( ...
vlen(jvrt), .67*elen(epos)) ;
end
%------------------------------------- mark feature edge
iref = apex(conn(:,1)) ... %- refine at vert. 1
& ~apex(conn(:,2)) ;
jref = apex(conn(:,2)) ... %- refine at vert. 2
& ~apex(conn(:,1)) ;
dref = apex(conn(:,1)) ... %- refine at both!
& apex(conn(:,2)) ;
keep =~apex(conn(:,1)) ... %- refine at neither
& ~apex(conn(:,2)) ;
%------------------------------------- protecting collar
ilen = vlen(conn(iref,1)) ;
inew = vert(conn(iref,1),:) ...
+ [ilen,ilen].*evec(iref,:) ;
jlen = vlen(conn(jref,2)) ;
jnew = vert(conn(jref,2),:) ...
- [jlen,jlen].*evec(jref,:) ;
Ilen = vlen(conn(dref,1)) ;
Inew = vert(conn(dref,1),:) ...
+ [Ilen,Ilen].*evec(dref,:) ;
Jlen = vlen(conn(dref,2)) ;
Jnew = vert(conn(dref,2),:) ...
- [Jlen,Jlen].*evec(dref,:) ;
vnew = [inew; jnew; Inew; Jnew] ;
%------------------------------------- add new vert/edge
iset = (1:size(inew,1))' ...
+ size(vert,1) ;
jset = (1:size(jnew,1))' ...
+ size(inew,1) + ...
+ size(vert,1) ;
Iset = (1:size(Inew,1))' ...
+ size(inew,1) + ...
+ size(jnew,1) + ...
+ size(vert,1) ;
Jset = (1:size(Jnew,1))' ...
+ size(inew,1) + ...
+ size(jnew,1) + ...
+ size(Inew,1) + ...
+ size(vert,1) ;
vert = [vert ; vnew] ;
cnew = [conn(iref,1), iset ;
conn(iref,2), iset ;
conn(jref,2), jset ;
conn(jref,1), jset ;
conn(dref,1), Iset ;
conn(dref,2), Jset ;
Iset, Jset] ;
conn = [conn(keep,:); cnew ] ;
end
end
function [vert,conn,tria,tnum,iter] = ...
cdtref1(vert,conn,tria,tnum, ...
node,PSLG,part,opts,hfun,harg,iter)
%CDTREF1 constrained Delaunay-refinement for 1-simplex elem-
%nts embedded in R^2.
% [...] = CDTREF1(...) refines the set of 1-simplex eleme-
% nts embedded in the triangulation until all constraints
% are satisfied. Specifically, edges are refined until all
% local mesh-spacing and encroachment conditions are met.
% Refinement proceeds according to either a Delaunay-refi-
% nement or Frontal-Delaunay type approach, depending on
% user-settings. In either case, new steiner vertices are
% introduced to split "bad" edges - those that violate the
% set of prescribed constraints. In the "-DR" type process
% edges are split about their circumballs (midpoints). In
% the "-FD" approach, new vertices are positioned such th-
% at mesh-spacing constraints are satisfied in a "locally-
% optimal" fashion.
tcpu.full = +0. ;
tcpu.ball = +0. ;
tcpu.hfun = +0. ;
tcpu.encr = +0. ;
tcpu.offc = +0. ;
vidx = (1:size(vert,1))'; %- "new" vert list to test
tnow = tic ;
ntol = +1.55;
while (strcmpi(opts.ref1,'refine'))
iter = iter + 1 ;
if (iter>=opts.iter),break; end
%------------------------------------- calc. circumballs
ttic = tic ;
bal1 = cdtbal1(vert,conn) ;
tcpu.ball = ...
tcpu.ball + toc(ttic) ;
%------------------------------------- eval. length-fun.
ttic = tic ;
if (~isempty(hfun))
if (isnumeric(hfun))
fun0 = hfun * ...
ones(size(vert,1),1);
fun1 = hfun ;
else
fun0(vidx) = ...
feval(hfun, ...
vert(vidx,:), harg{:});
fun0 = fun0(:) ;
fun1 = fun0(conn(:,1))...
+ fun0(conn(:,2));
fun1 = fun1 / +2. ;
end
else
fun0 = +inf * ...
ones(size(vert,1),1);
fun1 = +inf ;
end
siz1 = ...
+4. * bal1(:,3)./(fun1.*fun1) ;
tcpu.hfun = ...
tcpu.hfun + toc(ttic) ;
%------------------------------------- test encroachment
ttic = tic ;
bal1(:,3) = ...
(1.-eps^.75) * bal1(:,3) ;
[vp,vi] = ...
findball(bal1,vert(:,1:2));
%------------------------------------- near=>[vert,edge]
next = +0;
ebad = false(size(conn,1),1) ;
near = zeros(size(conn,1),1) ;
for ii = +1 : size(vp,1)
for ip = vp(ii,1):vp(ii,2)
jj = vi(ip);
if (ii ~= conn(jj,1) ...
&& ii ~= conn(jj,2) )
next = next + 1;
near(next,1) = ii;
near(next,2) = jj;
end
end
end
near = near(1:next-0,:);
if (~isempty(near))
%-- mark edge "encroached" if there is a vert within its
%-- dia.-ball that is not joined to either of its vert's
%-- via an existing edge...
ivrt = conn(near(:,2),1);
jvrt = conn(near(:,2),2);
pair = [near(:,1), ivrt];
ivec = setset2(pair,conn) ;
pair = [near(:,1), jvrt];
jvec = setset2(pair,conn) ;
okay = ~ivec & ~jvec ;
ebad(near(okay,2))=true ;
end
tcpu.encr = ...
tcpu.encr + toc(ttic);
%------------------------------------- refinement queues
ref1 = false(size(conn,1),1);
ref1(ebad) = true ; %- edge encroachment
ref1(siz1>opts.siz1* ... %- bad equiv. length
opts.siz1) = true ;
num1 = find(ref1) ;
%------------------------------------- dump-out progess!
if (mod(iter,opts.disp)==0)
numc = size(conn,1) ;
numt = size(tria,1) ;
fprintf(+1, ...
'%11i %18i %18i\n', ...
[iter,numc,numt]) ;
end
%------------------------------------- nothing to refine
if (isempty(num1)), break; end
%------------------------------------- refine "bad" tria
switch (lower(opts.kind))
case 'delaunay'
%------------------------------------- do circ-ball pt's
new1 = bal1(ref1, 1:2) ;
vidx = (1:size(new1,1))' ...
+ size(vert,1) ;
cnew = [conn( ref1,1), vidx
conn( ref1,2), vidx];
conn = [conn(~ref1,:); cnew];
%------------------------------------- update vertex set
vert = [vert; new1(:,1:2)];
case 'delfront'
%-- symmetric off-centre scheme:- refine edges from both
%-- ends simultaneously, placing new vertices to satisfy
%-- the worst of mesh-spacing and local voronoi constra-
%-- ints.
ttic = tic ;
evec = vert(conn(ref1,2),:) ...
- vert(conn(ref1,1),:) ;
elen = sqrt(sum(evec.^2,2)) ;
evec = evec ./ [elen, elen] ;
%------------------------------------- "voro"-type dist.
vlen = sqrt(bal1(ref1,3));
%------------------------------------- "size"-type dist.
ihfn = fun0(conn(ref1,1));
jhfn = fun0(conn(ref1,2));
%------------------------------------- bind "safe" dist.
ilen = min(vlen,ihfn) ;
jlen = min(vlen,jhfn) ;
%------------------------------------- locate offcentres
inew = vert(conn(ref1,1),:) ...
+ [ilen,ilen].*evec ;
jnew = vert(conn(ref1,2),:) ...
- [jlen,jlen].*evec ;
%------------------------------------- iter. "size"-type
for ioff = +1 : +3
%------------------------------------- eval. length-fun.
if (~isempty(hfun))
if (isnumeric(hfun))
iprj = hfun * ...
ones(size(inew,1),1);
jprj = hfun * ...
ones(size(jnew,1),1);
else
iprj = feval( ...
hfun,inew,harg{:});
jprj = feval( ...
hfun,jnew,harg{:});
iprj = iprj(:);
jprj = jprj(:);
end
else
iprj = +inf * ...
ones(size(inew,1),1);
jprj = +inf * ...
ones(size(jnew,1),1);
end
iprj = 0.5*ihfn + 0.5*iprj;
jprj = 0.5*jhfn + 0.5*jprj;
%------------------------------------- bind "safe" dist.
ilen = min(vlen,iprj) ;
jlen = min(vlen,jprj) ;
%------------------------------------- locate offcentres
inew = vert(conn(ref1,1),:) ...
+ [ilen,ilen].*evec ;
jnew = vert(conn(ref1,2),:) ...
- [jlen,jlen].*evec ;
end
%------------------------------------- merge i,j if near
near = ...
ilen+jlen>=vlen*ntol ;
znew = inew(near,:) * .5 ...
+ jnew(near,:) * .5 ;
inew = inew(~near,1:2) ;
jnew = jnew(~near,1:2) ;
%------------------------------------- split constraints
zset = (1:size(znew,1))' ...
+ size(vert,1) ;
iset = (1:size(inew,1))' ...
+ size(znew,1) + ...
+ size(vert,1) ;
jset = (1:size(jnew,1))' ...
+ size(znew,1) + ...
+ size(inew,1) + ...
+ size(vert,1) ;
set1 = num1( near);
set2 = num1(~near);
cnew = [conn( set1,1), zset
conn( set1,2), zset
conn( set2,1), iset
conn( set2,2), jset
iset, jset ] ;
conn = [conn(~ref1,:); cnew];
%------------------------------------- update vertex set
vert = [vert; znew(:,1:2)];
vert = [vert; inew(:,1:2)];
vert = [vert; jnew(:,1:2)];
vidx = [zset; iset; jset] ;
tcpu.offc = ...
tcpu.offc + toc(ttic) ;
end % switch(lower(opts.kind))
end
tcpu.full = ...
tcpu.full + toc(tnow) ;
if (~isinf(opts.disp) )
%------------------------------------- print final stats
numc = size(conn,1) ;
numt = size(tria,1) ;
fprintf(+1, ...
'%11i %18i %18i\n', ...
[iter,numc,numt]) ;
end
if (opts.dbug)
%------------------------------------- print debug timer
fprintf(1,'\n') ;
fprintf(1,' 1-simplex REF. timer...\n');
fprintf(1,'\n') ;
fprintf(1, ...
' FULL: %f \n', tcpu.full);
fprintf(1, ...
' BALL: %f \n', tcpu.ball);
fprintf(1, ...
' HFUN: %f \n', tcpu.hfun);
fprintf(1, ...
' ENCR: %f \n', tcpu.encr);
fprintf(1, ...
' OFFC: %f \n', tcpu.offc);
fprintf(1,'\n') ;
end
end
function [vert,conn,tria,tnum,iter] = ...
cdtref2(vert,conn,tria,tnum, ...
node,PSLG,part,opts,hfun,harg,iter)
%CDTREF2 constrained Delaunay-refinement for 2-simplex elem-
%nts embedded in R^2.
% [...] = CDTREF2(...) refines the set of 2-simplex eleme-
% nts embedded in the triangulation until all constraints
% are satisfied. Specifically, triangles are refined until
% all local mesh-spacing and element-shape conditions are
% met. Refinement proceeds according to either a Delaunay-
% refinement or Frontal-Delaunay type approach, depending
% on user-settings. In either case, new steiner points are
% introduced to split "bad" triangles - those that violate
% the set of prescribed constraints. In the "-DR" type pr-
% ocess triangles are split about their circumballs. In
% the "-FD" approach, new vertices are positioned such th-
% at mesh-spacing and element-shape constraints are satis-
% fied in a "locally-optimal" fashion.
tcpu.full = +0. ;
tcpu.dtri = +0. ;
tcpu.tcon = +0. ;
tcpu.ball = +0. ;
tcpu.hfun = +0. ;
tcpu.offc = +0. ;
tcpu.filt = +0. ;
vidx = (1:size(vert,1))'; %- "new" vert list to test
tnow = tic ;
near = +.775;
while (strcmpi(opts.ref2,'refine'))
iter = iter + 1 ;
%------------------------------------- build current CDT
ttic = tic ;
nold = size(vert,1) ;
[vert,conn, ...
tria,tnum]= deltri2(vert,conn, ...
node,PSLG, ...
part, ....
opts.dtri) ;
nnew = size(vert,1) ;
vidx = ...
[vidx; (nold:nnew)'] ;
tcpu.dtri = ...
tcpu.dtri + toc(ttic) ;
%------------------------------------- build current adj
ttic = tic ;
[edge,tria]= tricon2(tria,conn) ;
tcpu.tcon = ...
tcpu.tcon + toc(ttic) ;
if (iter>=opts.iter),break; end
%------------------------------------- calc. circumballs
ttic = tic ;
bal1 = cdtbal1(vert,conn) ;
bal2 = cdtbal2(vert, ...
edge,tria) ;
len2 = minlen2(vert,tria) ;
rho2 = bal2(:,+3) ./ len2 ;
%------------------------------------- refinement scores
scr2 = rho2 .* bal2(:,+3) ;
tcpu.ball = ...
tcpu.ball + toc(ttic) ;
%------------------------------------- eval. length-fun.
ttic = tic ;
if (~isempty(hfun))
if (isnumeric(hfun))
fun0 = hfun * ...
ones(size(vert,1),1);
fun2 = hfun ;
else
fun0(vidx) = ...
feval(hfun, ...
vert(vidx,:), harg{:});
fun0 = fun0(:) ;
fun2 = fun0(tria(:,1))...
+ fun0(tria(:,2))...
+ fun0(tria(:,3));
fun2 = fun2 / +3. ;
end
else
fun0 = +inf * ...
ones(size(vert,1),1);
fun2 = +inf ;
end
siz2 = ...
+3. * bal2(:,3)./(fun2.*fun2) ;
tcpu.hfun = ...
tcpu.hfun + toc(ttic) ;
%------------------------------------- refinement queues
ref1 = false(size(conn,1),1);
ref2 = false(size(tria,1),1);
stri = isfeat2(vert,edge,tria) ;
ref2(rho2>opts.rho2* ... %- bad rad-edge len.
opts.rho2) = true ;
ref2(stri) = false ;
ref2(siz2>opts.siz2* ... %- bad equiv. length
opts.siz2) = true ;
num2 = find(ref2);
%------------------------------------- dump-out progess!
if (mod(iter,opts.disp)==0)
numc = size(conn,1) ;
numt = size(tria,1) ;
fprintf(+1, ...
'%11i %18i %18i\n', ...
[iter,numc,numt]) ;
end
%------------------------------------- nothing to refine
if (isempty(num2)), break; end
[scr2,idx2] = sort( ...
scr2(num2),'descend');
num2 = num2(idx2);
%------------------------------------- refine "bad" tria
switch (lower(opts.kind))
case 'delaunay'
%------------------------------------- do circ-ball pt's
new2 = zeros(length(num2),3);
new2(:,1:2) = bal2(num2,1:2);
rmin = ... %- min. insert radii
len2(num2)*(1.-eps^.75)^2 ;
new2(:, 3) = max( ...
bal2(num2,3)*near^2,rmin) ;
case 'delfront'
%-- off-centre scheme -- refine triangles by positioning
%-- new vertices along a local segment of the voronoi
%-- diagram, bounded by assoc. circmballs. New points
%-- are placed to satisfy the worst of local mesh-length
%-- and element-shape constraints.
ttic = tic ;
%------------------------------------- find frontal edge
[lmin,emin] = ...
minlen2(vert,tria(num2,:)) ;
ftri = false(length(num2),1) ;
epos = zeros(length(num2),1) ;
tadj = zeros(length(num2),1) ;
for ii = +1 : length(epos)
epos(ii) = tria( ...
num2(ii),emin(ii)+3) ;
end
%------------------------------------- find frontal tria
for enum = +1 : +3
eidx = tria(num2,enum+3) ;
ftri = ...
ftri | edge(eidx,5) > +0 ;
ione = ...
num2 ~= edge(eidx,3) ;
itwo = ~ione ;
tadj(ione) = ...
edge(eidx(ione),3);
tadj(itwo) = ...
edge(eidx(itwo),4);
okay = tadj > +0 ;
tidx = tadj(okay);
ftri(okay) = ...
ftri(okay) | ~ref2(tidx) ;
end
if (~any(ftri)) %- can this happen!?
ftri = true(length(num2),+1) ;
end
%------------------------------------- locate offcentres
emid = vert(edge(epos,+1),:) ...
+ vert(edge(epos,+2),:) ;
emid = emid * +0.50 ;
elen = sqrt(lmin(:));
%------------------------------------- "voro"-type dist.
vvec = bal2(num2,1:2)-emid ;
vlen = sqrt(sum(vvec.^2,2));
vvec = vvec ./ [vlen,vlen] ;
hmid = fun0(edge(epos,+1),:) ...
+ fun0(edge(epos,+2),:) ;
hmid = hmid * +0.50 ;
%------------------------------------- "ball"-type dist.
rtri = elen * opts.off2 ;
rfac = elen * +0.50 ;
dsqr = rtri.^2 - rfac.^2;
doff = rtri + ...
sqrt(max(+0.,dsqr)) ;
%------------------------------------- "size"-type dist.
dsiz = +sqrt(3.)/2. * hmid ;
%------------------------------------- bind "safe" dist.
[dist,ioff] = ...
min([dsiz,doff,vlen],[],2) ;
%------------------------------------- locate offcentres
off2 = ...
emid + [dist,dist] .* vvec ;
%------------------------------------- iter. "size"-type
for isub = +1 : +3
%------------------------------------- eval. length-fun.
if (~isempty(hfun))
if (isnumeric(hfun))
hprj = hfun * ...
ones(size(off2,1),1) ;
else
hprj = feval( ...
hfun,off2,harg{:}) ;
hprj = hprj(:) ;
end
else
hprj = +inf * ...
ones(size(off2,1),1) ;
end
%------------------------------------- "size"-type dist.
hprj = .33*hmid + .67*hprj ;
dsiz = +sqrt(3.)/2. * hprj ;
dsiz(dsiz<elen*.50) = +inf ; %- edge-ball limiter
dsiz(dsiz>vlen*.95) = +inf ; %- circ-ball limiter
%------------------------------------- bind "safe" dist.