convert byte array to pdf mvc : Convert pdf to html online software application dll winforms html windows web forms freefem++doc13-part1953

5.10. MESHINTHREEDIMENSIONS
131
real volr11 = int3d(Th,r11)(1.) ;
cout << " vol region " << 11 << ": " << volr11 << endl;
if( (volr11 - Th.mesure )>1e-8) err++;
plot(Th,fill=0);
cout << " nb err= = " " << err <<endl;
assert(err==0);
theoutputofthisscriptis:
Transfo TH3 3 : Vertex, Tetrahedra, Border : nv_t=210 nt_t=720 nbe_t=296
label=21 s 4 100 : 21
label=25 s 4 110 : 25
label=40 s 4 011 : 40
label=42 s 4 211 : 42
label=45 s 4 121 : 45
label=53 s 4 102 : 53
vol region 11: 8
nb err= 0
times: compile 0.014094s, execution 0.023072s,
mpirank:0
CodeAlloc : nb ptr
2710,
size :327200 0 mpirank: 0
Ok: Normal End
Figure5.34:Themesh3doffunctioncube(4,5,6,flags =3)
5.10.2 Read/WriteStatementsforaMeshin3D
Inthreedimensions,thelemeshformatsupportedforinputandoutputlesbyFreeFem++
aretheextension.mshand.mesh.TheseformatsaredescribedinthechapteronMeshFiles
intwodimensions.
extensionle.msh Thestructureoftheleswithextension.mshin3DisgiveninTable
5.2.Inthisstructure,n
v
denotesthenumberofvertices,n
tet
thenumberoftetrahedraand
n
tri
thenumberoftrianglesForeachvertex q
i
;i=1;;n
v
,wedenoteby(q
i
x
;q
i
y
;q
i
z
)the
x-coordinate,they-coordinateandthez-coordinate.EachtetrahedraT
k
;k=1;;n
tet
has
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132
CHAPTER5. MESHGENERATION
fourverticesq
k
1
;q
k
2
;q
k
3
;q
k
4
.Theboundaryconsistsofanunionoftriangles.Eachtriangle
be
j
;j=1;;n
tri
hasthreeverticesq
j
1
;q
j
2
;q
j
3
.
n
v
n
tet
n
tri
q
1
x
q
1
y
q
1
z
Vertexlabel
q
2
x
q
2
y
q
2
z
Vertexlabel
.
.
.
.
.
.
.
.
.
.
.
.
q
n
v
x
q
n
v
y
q
n
v
z
Vertexlabel
1
1
1
2
1
3
1
4
regionlabel
2
1
2
2
2
3
2
4
regionlabel
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(n
tet
)
1
(n
tet
)
2
(n
tet
)
3
(n
tet
)
4
regionlabel
1
1
1
2
1
3
boundarylabel
2
1
2
2
2
3
boundarylabel
.
.
.
.
.
.
.
.
.
.
.
.
(n
t
ri)
1
(n
tri
)
2
(n
tri
)
3
boundarylabel
Table5.2:Thestructureofmeshleformat\.msh"inthreedimensions.
extensionle.mesh Thedatastructureforathreedimensionalmeshiscomposedofthe
datastructurepresentedinSection12.1andadatastructurefortetrahedra.Thetetrahedra
ofathreedimensionalmesharerefereedusingthefollowingeld:
• Tetrahedra
(I)NbOfTetrahedrons
((
@@Vertex
j
i
,
j=1,4
)
, (I)Ref
tet
i
,
i=1,NbOfTetrahedrons
)
ThiseldisexpresswiththenotationofSection12.1.
5.10.3 TeGen: : Atetrahedralmeshgenerator
TetGen
TetGenisasoftwaredevelopedbyDr.HangSiofWeierstrassInstituteforAppliedAnal-
ysisandStochasticsofBerlininGermany[41]. TetGenisafreeforresearchandnon-commercial
uses.Foranycommerciallicenceutilization,acommerciallicenceisavailableuponrequesttoHang
Si.
Thissoftwareisatetrahedralmeshgeneratorofathreedimensionaldomaindenedbyitsboundary.
Theinputdomaintakeintoaccountapolyhedralorapiecewiselinearcomplex. Thistetrahedral-
izationisaconstrainedDelaunaytetrahedralization.
ThemethodusedinTetGentocontrolthequality ofthemeshisaDelaunayrenement dueto
Shewchuk[42]. ThequalitymeasureofthisalgorithmistheRadius-EdgeRatio(seeSection1.3.1
[41]formoredetails). AtheoreticalboundsofthisratioofthealgorithmofShewchukisobtained
fora givencomplex of vertices, constrainedsegments andfacets of surface mesh, withnoinput
anglelessthan90degree. Thistheoreticalboundsis2.0.
ThelaunchofTetgenisdonewiththekeywordtetg.Theparametersofthiscommandlineis:
label = isavectorofintegerthatcontainstheoldlabelsnumberatindex2iandthenewlabels
numberatindex2i+1ofTriangles. Thisparametersisinitializedaslabelforthekeyword
change(5.3).
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5.10. MESHINTHREEDIMENSIONS
133
switch = Astringexpression.ThisstringcorrespondstothecommandlineswitchofTetgensee
Section3.2of[41].
nbofholes= Numberofholes(defaultvaluesize of holelist/3(version3.11)).
holelist = Thisarraycorrespondtoholelist oftetgeniodatastructure[41]. . Arealvector r of
size 3nbofholes. . InTetGen,eachholeisassociatedwithapoint t insidethisdomain.
Thisvectorisx
h
1
;y
h
1
;z
h
1
;x
h
2
;y
h
2
;z
h
2
;;wherex
h
i
;y
h
i
;z
h
i
is theassociatedpointwiththei
th
hole.
nbofregions = Numberofregions(size of regionlist/5(version3.11)).
regionlist = Thisarraycorrespondstoregionlistoftetgeniodatastructure[41].Theattribute
andthevolumeconstraintofregionaregiveninthisrealvectorofsize5nbofregions.
Thei
th
regionisdescribedbyveelements:x coordinate,y coordinateandz coordinateof
apointinsidethisdomain(x
i
;y
i
;z
i
);theattribute(at
i
)andthemaximumvolumefortetrahe-
dra(mvol
i
)forthisregion.Theregionlistvectoris:x
1
;y
1
;z
1
;at
1
;mvol
1
;x
2
;y
2
;z
2
;at
2
;mvol
2
;.
nboffacetcl= Numberoffacetsconstraintssize of facetcl/2(version3.11)).
facetcl= This array correspondstofacetconstraintlist oftetgeniodatastructure[41]. . Thei
th
facet constraint is dened d by y the facet marker r Ref
fc
i
and the maximum area for r faces
marea
fc
i
. The e facetcl array is: : Ref
fc
1
;marea
fc
1
;Ref
fc
2
;marea
fc
2
;. This s parameters
hasnoeectifswitchqisnotselected.
PrincipalswitchparametersinTetGen:
p Tetrahedralizationofboundary.
q Qualitymeshgeneration.TheboundofRadius-EdgeRatiowillbegivenaftertheoptionq.
Bydefault,thisvalueis2.0.
a Constructwiththevolumesconstraintsontetrahedra. . Thesevolumesconstraintsaredened
withtheboundofthepreviousswitchqorintheparameterregionlist.
A Attributesreference e toregiongiveninthe regionlist. . The e other regionshavelabel0.
TheoptionAAgivesadierentlabelateachregion. This s switchworkwiththeoption’p’.
Ifoption’r’isused,thisswitchhasnoeect.
r ReconstructsandRenesapreviouslygeneratedmesh.Thischaracterisonlyusedwiththe
commandlinetetgreconstruction.
Y Thisswitchallowtopreservethemeshontheexteriorboundary. . Thisswitchmustbeused
toensureconformalmeshbetweentwoadjacentsmesh.
YY Thisswitchallowtopreservethemeshontheexteriorandinteriorboundary.
C Theconsistencyoftheresult’smeshistestingbyTetGen.
CC Theconsistencyoftheresult’smeshistestingbyTetGenandalsochecksconstraineddelaunay
mesh (if ’p’ switch is s selected) or the consistency of Conformal l Delaunay y (if ’q’switchis
selected).
V GiveinformationoftheworkofTetGen.Moreinformationcanbeobtainedinspecied’VV’
or’VVV’.
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134
CHAPTER5. MESHGENERATION
Q Quiet:Noterminaloutputexcepterrors
M Thecoplanarfacetsarenotmerging.
T Setatoleranceforcoplanartest. . Thedefaultvalueis1e 8.
d Itersectionsoffacetsaredetected.
Toobtainatetrahedralmeshgeneratorwithtetgen,weneedthesurfacemeshofthreedimensional
domain.WegivenowthecommandlineinFreeFem++ toconstructthesemeshes.
keyword: \movemesh23"
Asimplemethodtoconstructasurfaceistoplaceatwodimensional
domaininathreedimensionalspace. Thiscorrespondingtomovethedomainbyadisplacement
vectorofthisform(x;y)=(1(x;y);2(x;y);3(x;y)). Theresultofmovingatwodimensional
meshTh2bythisthreedimensionaldisplacementisobtainedusing:
mesh3 Th3 = movemesh23(Th2,transfo=[1,2,3]);
Theparametersofthiscommandlineare:
transfo = [1, 2, , 3]setthedisplacementvectoroftransformation(x;y)=[1(x;y);2(x;y);3(x;y)].
label = setintegerlabeloftriangles
orientation= setintegerorientationofmesh.
ptmerge = Arealexpression. Whenyoutransformamesh,somepointscanbe e merged. . This
parametersisthecriteriatodenetwomergingpoints.Bydefault,weuse
ptmerge = 1e 7Vol(B);
whereBisthesmallestaxisparallelboxescontainingthediscretizeddomainof andVol(B)
isthevolumeofthisbox.
Wecan‘do a‘gluing"of surface meshes using theprocess giveninSection5.9. . Anexampleto
obtainathreedimensionalmeshusingthecommandlinetetgandmovemesh23isgiveninthe
letetgencube.edp.
Example5.18(tetgencube.edp)
//
file tetgencube.edp
load "msh3"
load "tetgen"
real x0,x1,y0,y1;
x0=1.; x1=2.; y0=0.; y1=2
*
pi;
mesh Thsq1 = square(5,35,[x0+(x1-x0)
*
x,y0+(y1-y0)
*
y]);
func ZZ1min = 0;
func ZZ1max = 1.5;
func XX1 = x;
func YY1 = y;
mesh3 Th31h = movemesh23(Thsq1,transfo=[XX1,YY1,ZZ1max]);
mesh3 Th31b = movemesh23(Thsq1,transfo=[XX1,YY1,ZZ1min]);
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5.10. MESHINTHREEDIMENSIONS
135
//
///////////////////////////////
x0=1.; x1=2.; y0=0.; y1=1.5;
mesh Thsq2 = square(5,8,[x0+(x1-x0)
*
x,y0+(y1-y0)
*
y]);
func ZZ2 = y;
func XX2 = x;
func YY2min = 0.;
func YY2max = 2
*
pi;
mesh3 Th32h = movemesh23(Thsq2,transfo=[XX2,YY2max,ZZ2]);
mesh3 Th32b = movemesh23(Thsq2,transfo=[XX2,YY2min,ZZ2]);
//
///////////////////////////////
x0=0.; x1=2
*
pi; y0=0.; y1=1.5;
mesh Thsq3 = square(35,8,[x0+(x1-x0)
*
x,y0+(y1-y0)
*
y]);
func XX3min = 1.;
func XX3max = 2.;
func YY3 = x;
func ZZ3 = y;
mesh3 Th33h = movemesh23(Thsq3,transfo=[XX3max,YY3,ZZ3]);
mesh3 Th33b = movemesh23(Thsq3,transfo=[XX3min,YY3,ZZ3]);
//
//////////////////////////////
mesh3 Th33 = Th31h+Th31b+Th32h+Th32b+Th33h+Th33b; //
"gluing" surface meshs
to obtain the surface of cube
savemesh(Th33,"Th33.mesh");
//
build a mesh of a a axis s parallel box with TetGen
real[int] domain n =[1.5,pi,0.75,145,0.0025];
mesh3 Thfinal = tetg(Th33,switch="paAAQY",regionlist=domain);
//
Tetrahelize the interior of the cube with tetgen
savemesh(Thfinal,"Thfinal.mesh");
//
build a a mesh of a half cylindrical shell of interior radius 1.
and
exterior radius 2 2 and d heigh 1.5
func mv2x = x
*
cos(y);
func mv2y = x
*
sin(y);
func mv2z = z;
mesh3 Thmv2 = movemesh3(Thfinal, transfo=[mv2x,mv2y,mv2z]);
savemesh(Thmv2,"halfcylindricalshell.mesh")
Thecommandmovemeshisdescribeinthefollowingsection.
Thekeyword\tetgtransfo"
Thiskeywordcorrespondtoacompositionofcommandlinetetg
andmovemesh23:
tetgtransfo( Th2, , transfo= = [1, 2, 3] ),   ) ) = tetg( Th3surf,   ),
whereTh3surf=movemesh23(Th2,tranfo=[1,2,3])andTh2istheinputtwodimensional
meshof tetgtransfo.
Theparametersofthiscommandlineareontheonehandtheparameters:
label,switch,regionlistnboffacetclfacetcl
ofkeywordtetgandontheotherhandtheparameterptmergeofkeywordmovemesh23.
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136
CHAPTER5. MESHGENERATION
Remark:
Tousetetgtransfo,theresult’s meshof f movemesh23 3 must beanclosedsurface
anddeneoneregiononly. Therefore,theparameterregionlistisdenedforoneregion.
Anexampleofthiskeywordcanbefoundinlineofle\buildlayers.edp"
Thekeyword"tetgconvexhull"
FreeFem++ ,usingtetgen,isabletobuildatetrahedralization
fromasetofpoints. ThistetrahedralizationisaDelaunaymeshoftheconvexhullofthesetof
points.
Thecoordinatesofthepoints canbeinitializedintwoways. . Therstisalethatcontainsthe
coordinateofpointsX
i
=(x
i
;y
i
;z
i
).Thislesisorganizedasfollows:
n
v
x
1
y
1
z
1
x
2
y
2
z
2
.
.
.
.
.
.
.
.
.
x
n
v
y
n
v
z
n
v
Thesecondwayistogivethreearraysthatcorrespondrespectivelytothex coordinates,y coordinates
andz coordinates.
Theparametersofthiscommandlineare
switch = Astringexpression. This s stringcorresponds to the commandline switch ofTetGen
seeSection3.2of[41].
reftet = Anintegerexpression.setthelabeloftetrahedra.
label = Anintegerexpression.setthelabeloftriangles.
Inthestringswitch,wecan’tusedtheoption’p’and’q’oftetgen.
5.10.4 Reconstruct/ReneathreedimensionalmeshwithTetGen
MeshesinthreedimensioncanberenedusingTetGenwiththecommandlinetetgreconstruction.
Theparameterofthiskeywordare
region= anintegerarraythat allow tochange the regionnumberoftetrahedra. . Thisarrayis
denedastheparameterreftetinthekeywordchange.
label= anintegerarraythatallowtochangethelabelofboundarytriangles.Thisarrayisdened
astheparameterlabelinthekeywordchange.
sizevolume= areelfunction.Thisfunctionallowstoconstraintvolumesizeoftetrahedrainthe
domain.(seeexample5.31tobuild3dadaptmesh)
Theparameterswitchnbofregions,regionlist,nboffacetclandfacetclofthecom-
mandlinewhichcallTetGen(tetg)isusedfortetgrefine.
Intheparameterswitch=,thecharacter’r’shouldbeusedwithoutthecharacter’p’.Forinstance,
seethemanualofTetGen[41]foreectof’r’toothercharacter.
Theparameterregionlistallowstodeneanewvolumeconstraintintheregion.Thelabelin
theregionlist willbe thepreviouslabelofregion. . This s parameterandnbofregionscan’t
beusedwithparametersizevolume.
Example:
5.10. MESHINTHREEDIMENSIONS
137
Example5.19(renesphere.edp)
//
file refinesphere.edp
load "msh3"
load "tetgen"
load "medit"
mesh Th=square(10,20,[x
*
pi-pi/2,2
*
y
*
pi]);
//
]
pi
2
;frac pi2[]0;2[
//
a parametrization of a sphere
func f1 =cos(x)
*
cos(y);
func f2 =cos(x)
*
sin(y);
func f3 = sin(x);
//
partiel derivative of the parametrization DF
func f1x=sin(x)
*
cos(y);
func f1y=-cos(x)
*
sin(y);
func f2x=-sin(x)
*
sin(y);
func f2y=cos(x)
*
cos(y);
func f3x=cos(x);
func f3y=0;
//
M=DF
t
DF
func m11=f1xˆ2+f2xˆ2+f3xˆ2;
func m21=f1x
*
f1y+f2x
*
f2y+f3x
*
f3y;
func m22=f1yˆ2+f2yˆ2+f3yˆ2;
func perio=[[4,y],[2,y],[1,x],[3,x]];
real hh=0.1;
real vv= 1/square(hh);
verbosity=2;
Th=adaptmesh(Th,m11
*
vv,m21
*
vv,m22
*
vv,IsMetric=1,periodic=perio);
Th=adaptmesh(Th,m11
*
vv,m21
*
vv,m22
*
vv,IsMetric=1,periodic=perio);
plot(Th,wait=1);
verbosity=2;
//
construction of the surface of spheres
real Rmin
= 1.;
func f1min = Rmin
*
f1;
func f2min = Rmin
*
f2;
func f3min = Rmin
*
f3;
mesh3 Th3=movemesh23(Th,transfo=[f1min,f2min,f3min]);
real[int] domain n = = [0.,0.,0.,145,0.01];
mesh3 Th3sph=tetg(Th3,switch="paAAQYY",nbofregions=1,regionlist=domain);
int[int] newlabel l = = [145,18];
real[int] domainrefine = [0.,0.,0.,145,0.0001];
mesh3 Th3sphrefine=tetgreconstruction(Th3sph,switch="raAQ",reftet=newlabel,
nbofregions=1,regionlist=domain,refinesizeofvolume=0.0001);
int[int] newlabel2 = [145,53];
func fsize = 0.01/(( 1 + 5
*
sqrt( (x-0.5)ˆ2+(y-0.5)ˆ2+(z-0.5)ˆ2) )ˆ3);
mesh3 Th3sphrefine2=tetgreconstruction(Th3sph,switch="raAQ",reftet=newlabel2,
sizeofvolume=fsize);
medit(‘‘sphere’’,Th3sph);
138
CHAPTER5. MESHGENERATION
medit(‘‘isotroperefine’’ ,Th3sphrefine);
medit(‘‘anisotroperefine’’,Th3sphrefine2);
5.10.5 Movingmeshinthreedimensions
Meshes in n three dimensions s can n be translated rotated d and deformed using g the command d line
movemeshasinthe2Dcase(seesectionmovemeshinchapiter5). If istetrahedrizedasT
h
( ),
and(x;y)=(1(x;y;z);1(x;y;z);3(x;y;z))isadisplacementvectorthen(T
h
)isobtained
by
mesh3 Th = movemesh( Th, [1, 2, 3], ... );
Theparametersofmovemeshinthreedimensionsare
region = setintegerlabeloftetrahedra. 0bydefault.
label = setthelabeloffacesofborder. Thisparametersisinitializedaslabelforthekeyword
change(5.3).
facemerge = Aninteger expression. . Whenyoutransformamesh,some e faces canbemerged.
Thisparameters equalstooneifmerge’sfacesis considered. . Otherwiseequalstozero. By
default,thisparameterisequalsto1.
ptmerge = Arealexpression. Whenyoutransformamesh,somepointscanbe e merged. . This
parametersisthecriteriatodenetwomergingpoints.Bydefault,weuse
ptmerge = 1e 7Vol(B);
whereBisthesmallestaxisparallelboxescontainingthediscretiondomainof andVol(B)
isthevolumeofthisbox.
orientation= Anintegerexpression(1bybydefault),toreverseornottheorientationoftet
ifnotpositive.
Anexampleofthis command d can be foundinthe le"Poisson3d.edp" locatedin the directory
examples++-3d.
5.10.6 Layermesh
Inthissection,wepresentthecommandline toobtainaLayermesh: : buildlayermesh. . This
meshisobtainedbyextendingatwodimensionalmeshinthez-axis.
Thedomain
3d
denedbythelayermeshisequalto
3d
=
2d
[zmin;zmax]where
2d
isthe
domaindenebythetwodimensionalmesh,zminandzmaxarefunctionof
2d
inRthatdenes
respectivelythelowersurfaceanduppersurfaceof
3d
.
For avertex of a two dimensional l mesh V
2d
i
= (x
i
;y
i
),weintroduce the number of associated
verticesinthez axisM
i
+1. WedenotebyM themaximumofM
i
overtheverticesofthetwo
dimensionalmesh. Thisvaluearecalledthenumberoflayers(if8i; ; M
i
=M thenthereareM
layersinthemeshof
3d
). V
2d
i
generatedM+1verticeswhicharedenedby
8j=0;:::;M;
V
3d
i;j
=(x
i
;y
i
;
i
(z
i;j
));
5.10. MESHINTHREEDIMENSIONS
139
Middle surface
Lower surface
upper surface
Figure5.35: ExampleofLayermeshinthreedimension.
where(z
i;j
)
j=0;:::;M
aretheM+1equidistantpointsontheinterval[zmin(V
2d
i
);zmax(V
2d
i
)]:
z
i;j
=j+zmin(V
2d
i
);
=
zmax(V
2d
i
) zmin(V
2d
i
)
M
:
Thefunction
i
,denedon[zmin(V
2d
i
);zmax(V
2d
i
)],isgivenby
i
(z)=
i;0
ifz=zmin(V
2d
i
);
i;j
ifz2]
i;j 1
;
i;j
];
with(
i;j
)
j=0;:::;M
i
aretheM
i
+1equidistantpointsontheinterval[zmin(V
2d
i
);zmax(V
2d
i
)].
SetatriangleK =(V
2d
i1
,V
2d
i2
,V
2d
i3
) ofthetwodimensionalmesh. . K K is associatedwithatrian-
gle on the upper surface (resp. . on n the lower r surface)oflayer mesh: : (V
3d
i1;M
;V
3d
i2;M
;V
3d
i3;M
) (resp.
(V
3d
i1;0
;V
3d
i2;0
;V
3d
i3;0
)).
AlsoKisassociatedwithM volumeprismaticelementswhicharedenedby
8j=0;:::;M; H
j
=(V
3d
i1;j
;V
3d
i2;j
;V
3d
i3;j
;V
3d
i1;j+1
;V
3d
i2;j+1
;V
3d
i3;j+1
):
Thesesvolumeelementscanhavesomemergedpoint:
• 0mergedpoint:prism
• 1mergedpoints:pyramid
• 2mergedpoints:tetrahedra
• 3mergedpoints:noelements
Theelementswithmergedpointsarecalleddegenerateelements. Toobtainameshwithtetrahe-
dra,wedecomposethepyramidintotwotetrahedraandtheprismintothree tetrahedra. . These
tetrahedraareobtainedbycuttingthequadrilateralfaceofpyramidandprismwiththediagonal
whichhavethevertexwiththemaximumindex(see[12]forthereaspnofthischoice).
Thetrianglesonthemiddlesurfaceobtainedwiththedecompositionofthevolumeprismaticel-
ementsarethetrianglesgeneratedbytheedgesontheborderofthetwodimensionalmesh. The
labeloftrianglesontheborderelementsandtetrahedraaredenedwiththelabeloftheseassoci-
atedelements.
TheargumentsofbuildlayermeshisatwodimensionalmeshandthenumberoflayersM.
Theparametersofthiscommandare:
140
CHAPTER5. MESHGENERATION
zbound = [zmin,zmax]wherezminandzmaxarefunctions expression. . Theses s functionsdene
thelowersurfacemeshanduppermeshofsurfacemesh.
coef = A function expression between [0,1]. . This s parameter r is used to introduce e degenerate
element in mesh. . The e number r of f associated points or vertex x V
2d
i
is the integer part of
coef(V
2d
i
)M.
region = Thisvectorisusedtoinitializedtheregionoftetrahedra. Thisvectorcontainsuccessive
pairofthe2dregionnumberatindex2iandthecorresponding3dregionnumberatindex
2i+1,like(5.3).becomethe
labelmid = Thisvector isusedtoinitializedthe3dlabelsnumberoftheverticalfaceor mid
faceformthe2dlabelnumber. Thisvectorcontainssuccessivepairofthe2dlabelnumber
atindex2iandthecorresponding3dlabelnumberatindex2i+1,like(5.3).
labelup = Thisvectorisusedtoinitializedthe3dlabelnumbersoftheupper/topfaceformthe
2dregionnumber. Thisvectorcontainssuccessivepairofthe2dregionnumberatindex2i
andthecorresponding3dlabelnumberatindex2i+1,like(5.3).
labeldown = Sameasthepreviouscasebutforthelower/downfacelabel.
Moreover,wealsoaddpostprocessingparametersthatallowtomovingthemesh.Theseparameters
correspondtoparameterstransfo,facemergeandptmergeofthecommandlinemovemesh.
The vector region, labelmid, labelup andlabeldown These vectors are composedof n
l
successivepairsofnumberO
i
;N
l
wheren
l
isthenumber(labelorregion)thatwewanttoget.
Anexampleofthiscommandlineisgiveninbuildlayermesh.edp.
Example5.20(cube.idp)
load "medit"
load "msh3"
func mesh3 Cube(int[int] & NN,real[int,int] &BB ,int[int,int] & L)
{
//
first build d the 6 faces of the hex.
real x0=BB(0,0),x1=BB(0,1);
real y0=BB(1,0),y1=BB(1,1);
real z0=BB(2,0),z1=BB(2,1);
int nx=NN[0],ny=NN[1],nz=NN[2];
mesh Thx = square(nx,ny,[x0+(x1-x0)
*
x,y0+(y1-y0)
*
y]);
int[int] rup=[0,L(2,1)],
rdown=[0,L(2,0)],
rmid=[1,L(1,0),
2,L(0,1),
3, L(1,1),
4, L(0,0) ];
mesh3 Th=buildlayers(Thx,nz,
zbound=[z0,z1],
labelmid=rmid,
labelup = rup,
labeldown = rdown);
return Th;
}
Theunitcubeexample:
include "Cube.idp"
int[int]
NN=[10,10,10];
//
the number of step in each direction
real [int,int] BB=[[0,1],[0,1],[0,1]];
//
bounding box
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