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5.5. ADAPTMESH
121
otherwise,themetricisevaluatedusingthecriteriumofequi-distributionoferrors.In
thiscasethemetricisdenedby
M=
1
errcoef
2
jHj
sup() inf()
p
:
(5.2)
cutoff= lowerlimitfortherelativeerrorevaluation(1.0e-6isthedefault).
verbosity= informationalmessageslevel(canbechosenbetween0and1).Alsochanges
thevalueoftheglobalvariableverbosity(obsolete).
inquire= Toinquiregraphicallyaboutthemesh(falseisthedefault).
splitpbedge= Iftrue,splitsallinternaledgesinhalfwithtwoboundaryvertices(trueis
thedefault).
maxsubdiv= Changesthemetricsuchthatthemaximumsubdivisionofabackgroundedge
isboundbyval(alwayslimitedby10,and10isalsothedefault).
rescaling= iftrue,thefunctionwithrespecttowhichthemeshisadaptedisrescaledto
bebetween0and1(trueisthedefault).
keepbackvertices= iftrue, tries s to keep as many vertices from the originalmeshas
possible(trueisthedefault).
isMetric= iftrue,themetricisdenedexplicitly(falseisthedefault).Ifthe3functions
m
11
;m
12
;m
22
aregiven,they directlydeneasymmetricmatrixeldwhoseHessian
is computedtodeneametric. . Ifonly y one functionis given,thenitrepresents the
isotropicmeshsizeateverypoint.
For example,ifthe partialderivatives fxx (= @
2
f=@x
2
), fxy (= = @
2
f=@x@y), fyy
(=@
2
f=@y
2
)aregiven,wecanset
Th=
adaptmesh
(Th,fxx,fxy,fyy,IsMetric=1,nbvx=10000,hmin=hmin);
power= exponentpoweroftheHessianusedtocomputethemetric(1isthedefault).
thetamax= minimumcornerangleofindegrees(defaultis10
)wherethecornerisABC
andtheangleistheangleofthetwovectorsAB;BC,(0imply nocorner,90imply
perp.corner,...).
splitin2= booleanvalue.Iftrue,splitsalltrianglesofthenalmeshinto4sub-triangles.
metric= anarrayof3realarraystosetorgetmetricdatainformation. Thesizeofthese
threearraysmustbethenumberofvertices.Soifm11,m12,m22arethreeP1niteel-
ementsrelatedtothemeshtoadapt,youcanwrite:metric=[m11[],m12[],m22[]]
(seeleconvect-apt.edpforafullexample)
nomeshgeneration= Iftrue, no adaptedmesh is s generated(usefultocompute only a
metric).
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122
CHAPTER5. MESHGENERATION
periodic= Writing periodic=[[4,y],[2,y],[1,x],[3,x]]; builds s an adapted
periodicmesh. Thesamplebuildabiperiodicmeshofa a square. . (seeperiodicnite
elementspaces6,andseesphere.edpforafullexample)
Wecanusethecommand adaptmeshtobuilduniformmeshwithacontantmeshsize.
Sotobuildameshwithaconstantmeshsizeequalto
1
30
try:
Example5.6 uniformmesh.edp
mesh Th=square(2,2);
//
to have initial mesh
plot(Th,wait=1,ps="square-0.eps");
Th= adaptmesh(Th,1./3As writing
0.,IsMetric=1,nbvx=10000);
//
plot(Th,wait=1,ps="square-1.eps");
Th= adaptmesh(Th,1./30.,IsMetric=1,nbvx=10000);
//
more the one time du to
Th= adaptmesh(Th,1./30.,IsMetric=1,nbvx=10000);
//
adaptation bound
maxsubdiv=
plot(Th,wait=1,ps="square-2.eps");
Figure5.17:Initialmesh
Figure5.18: rstiteration
Figure5.19:lastiteration
5.6 Trunc
Two operators have been introduce to remove e triangles s from a mesh or to divide them.
Operatortrunc hastwoparameters
label= setsthelabelnumberofnewboundaryitem(onebydefault)
split= sets thelevelnoftrianglesplitting. . eachtriangleis s splittedin nn( oneby
default).
TocreatethemeshTh3whereallstrianglesofameshTharesplittedin33,justwrite:
mesh Th3 = trunc(Th,1,split=3);
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5.7. SPLITMESH
123
The truncmesh.edp example construct all "trunc" mesh h to o the e support t of the e basic
functionofthe space Vh(cf. . abs(u)>0),split t allthetriangles in55,andput alabel
numberto2onnewboundary.
mesh Th=square(3,3);
fespace Vh(Th,P1);
Vh u;
int i,n=u.n;
u=0;
for (i=0;i<n;i++)
//
all degree of freedom
{
u[][i]=1;
//
the basic function i
plot(u,wait=1);
mesh Sh1=trunc(Th,abs(u)>1.e-10,split=5,label=2);
plot(Th,Sh1,wait=1,ps="trunc"+i+".eps");
//
plot the mesh of
//
the function’s support
u[][i]=0;
//
reset
}
Figure5.20: meshofsupportthefunction
P1number0,splittedin55
Figure 5.21: : meshofsupportthefunction
P1number6,splittedin55
5.7 Splitmesh
Anotherwaytosplitmeshtrianglesistousesplitmesh,forexample:
{
//
new stuff f 2004 4 splitmesh (version 1.37)
assert(version>=1.37);
border a(t=0,2
*
pi){ x=cos(t); y=sin(t);label=1;}
mesh Th=buildmesh(a(20));
plot(Th,wait=1,ps="nosplitmesh.eps");
//
see figure 5.22
Th=splitmesh(Th,1+5
*
(square(x-0.5)+y
*
y));
plot(Th,wait=1,ps="splitmesh.eps");
//
see figure 5.23
}
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124
CHAPTER5. MESHGENERATION
Figure5.22: initialmesh
Figure 5.23:
all left t mesh
tri-
angle
is
split
conformaly
in
int(1+5
*
(square(x-0.5)+y
*
y)
ˆ
2
triangles.
5.8 MeshingExamples
Example5.7(Tworectanglestouchingbyaside)
border a(t=0,1){x=t;y=0;};
border b(t=0,1){x=1;y=t;};
border c(t=1,0){x=t ;y=1;};
border d(t=1,0){x = = 0; y=t;};
border c1(t=0,1){x=t ;y=1;};
border e(t=0,0.2){x=1;y=1+t;};
border f(t=1,0){x=t ;y=1.2;};
border g(t=0.2,0){x=0;y=1+t;};
int n=1;
mesh th = buildmesh(a(10
*
n)+b(10
*
n)+c(10
*
n)+d(10
*
n));
mesh TH = buildmesh ( c1(10
*
n) + e(5
*
n) + f(10
*
n) + + g(5
*
n) );
plot(th,TH,ps="TouchSide.esp");
//
Fig.
5.24
Example5.8(NACA0012Airfoil)
border upper(t=0,1) { x = t;
y = 0.17735
*
sqrt(t)-0.075597
*
t
- 0.212836
*
(tˆ2)+0.17363
*
(tˆ3)-0.06254
*
(tˆ4); }
border lower(t=1,0) { x = t;
y= -(0.17735
*
sqrt(t)-0.075597
*
t
-0.212836
*
(tˆ2)+0.17363
*
(tˆ3)-0.06254
*
(tˆ4)); }
border c(t=0,2
*
pi) { x=0.8
*
cos(t)+0.5;
y=0.8
*
sin(t); }
mesh Th = buildmesh(c(30)+upper(35)+lower(35));
plot(Th,ps="NACA0012.eps",bw=1);
//
Fig.
5.25
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5.8. MESHINGEXAMPLES
125
     
  
  
      
 

Figure5.24: Tworectanglestouchingby y a
side
Figure5.25: NACA0012Airfoil
Example5.9(Cardioid)
real b = 1, a = b;
border C(t=0,2
*
pi) { x=(a+b)
*
cos(t)-b
*
cos((a+b)
*
t/b);
y=(a+b)
*
sin(t)-b
*
sin((a+b)
*
t/b); }
mesh Th = buildmesh(C(50));
plot(Th,ps="Cardioid.eps",bw=1);
//
Fig.
5.26
Example5.10(CassiniEgg)
border C(t=0,2
*
pi) { x=(2
*
cos(2
*
t)+3)
*
cos(t);
y=(2
*
cos(2
*
t)+3)
*
sin(t); }
mesh Th = buildmesh(C(50));
plot(Th,ps="Cassini.eps",bw=1);
//
Fig.
5.27
Figure 5.26: : Domain n with Cardioid d curve
boundary
Figure5.27: DomainwithCassiniEggcurve
boundary
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126
CHAPTER5. MESHGENERATION
Example5.11(BycubicBeziercurve)
//
A cubic Bezier curve connecting two points with two control points
func real bzi(real p0,real p1,real q1,real q2,real t)
{
return p0
*
(1-t)ˆ3+q1
*
3
*
(1-t)ˆ2
*
t+q2
*
3
*
(1-t)
*
tˆ2+p1
*
tˆ3;
}
real[int] p00=[0,1], p01=[0,-1], q00=[-2,0.1], q01=[-2,-0.5];
real[int] p11=[1,-0.9], q10=[0.1,-0.95], q11=[0.5,-1];
real[int] p21=[2,0.7], q20=[3,-0.4], q21=[4,0.5];
real[int] q30=[0.5,1.1], q31=[1.5,1.2];
border G1(t=0,1) { { x=bzi(p00[0],p01[0],q00[0],q01[0],t);
y=bzi(p00[1],p01[1],q00[1],q01[1],t); }
border G2(t=0,1) { { x=bzi(p01[0],p11[0],q10[0],q11[0],t);
y=bzi(p01[1],p11[1],q10[1],q11[1],t); }
border G3(t=0,1) { { x=bzi(p11[0],p21[0],q20[0],q21[0],t);
y=bzi(p11[1],p21[1],q20[1],q21[1],t); }
border G4(t=0,1) { { x=bzi(p21[0],p00[0],q30[0],q31[0],t);
y=bzi(p21[1],p00[1],q30[1],q31[1],t); }
int m=5;
mesh Th = buildmesh(G1(2
*
m)+G2(m)+G3(3
*
m)+G4(m));
plot(Th,ps="Bezier.eps",bw=1);
//
Fig 5.28
Example5.12(SectionofEngine)
real a= 6., b= 1., c=0.5;
border L1(t=0,1) ) { { x= -a; y= 1+b - 2
*
(1+b)
*
t; }
border L2(t=0,1) ) { { x= -a+2
*
a
*
t; y= -1-b
*
(x/a)
*
(x/a)
*
(3-2
*
abs(x)/a );}
border L3(t=0,1) ) { { x= a; y=-1-b + (1+ b )
*
t; }
border L4(t=0,1) ) { { x= a - a
*
t;
y=0; }
border L5(t=0,pi) ) { { x= -c
*
sin(t)/2; y=c/2-c
*
cos(t)/2; }
border L6(t=0,1) ) { { x= a
*
t;
y=c; }
border L7(t=0,1) ) { { x= a;
y=c + (1+ b-c )
*
t; }
border L8(t=0,1) ) { { x= a-2
*
a
*
t; y= 1+b
*
(x/a)
*
(x/a)
*
(3-2
*
abs(x)/a); }
mesh Th = buildmesh(L1(8)+L2(26)+L3(8)+L4(20)+L5(8)+L6(30)+L7(8)+L8(30));
plot(Th,ps="Engine.eps",bw=1);
//
Fig.
5.29
Example5.13(DomainwithU-shapechannel)
real d = 0.1;
//
width of U-shape
border L1(t=0,1-d) { x=-1; y=-d-t; }
border L2(t=0,1-d) { x=-1; y=1-t; }
border B(t=0,2) { { x=-1+t; ; y=-1; }
border C1(t=0,1) { { x=t-1; y=d; }
border C2(t=0,2
*
d) { x=0; y=d-t; }
border C3(t=0,1) { { x=-t; y=-d; }
border R(t=0,2) { { x=1; ; y=-1+t; }
border T(t=0,2) { { x=1-t; ; y=1; }
int n = 5;
mesh Th = buildmesh (L1(n/2)+L2(n/2)+B(n)+C1(n)+C2(3)+C3(n)+R(n)+T(n));
plot(Th,ps="U-shape.eps",bw=1);
//
Fig 5.30
5.8. MESHINGEXAMPLES
127
 
 
 
 
Figure 5.28: : Boundary y drawed d by Bezier
curves
 
 
 
 
 
 
 
 
Figure5.29: SectionofEngine
Example5.14(DomainwithV-shapecut)
real dAg = 0.01;
//
angle of V-shape
border C(t=dAg,2
*
pi-dAg) { x=cos(t); y=sin(t); };
real[int] pa(2), , pb(2), , pc(2);
pa[0] = cos(dAg); ; pa[1] ] = sin(dAg);
pb[0] = cos(2
*
pi-dAg); pb[1] = sin(2
*
pi-dAg);
pc[0] = 0; pc[1] ] = = 0;
border seg1(t=0,1) { x=(1-t)
*
pb[0]+t
*
pc[0]; y=(1-t)
*
pb[1]+t
*
pc[1]; };
border seg2(t=0,1) { x=(1-t)
*
pc[0]+t
*
pa[0]; y=(1-t)
*
pc[1]+t
*
pa[1]; };
mesh Th = buildmesh(seg1(20)+C(40)+seg2(20));
plot(Th,ps="V-shape.eps",bw=1);
//
Fig.
5.31
 
 
 
 
 
       
     
  
   
Figure5.30: DomainwithU-shapechannel
changedbyd
    
    
Figure 5.31:
Domain with V-shape cut
changedbydAg
Example5.15(Smilingface)
real d=0.1;
128
CHAPTER5. MESHGENERATION
int m=5;
real a=1.5, b=2, , c=0.7, , e=0.01;
border F(t=0,2
*
pi) { x=a
*
cos(t); y=b
*
sin(t); }
border E1(t=0,2
*
pi) { x=0.2
*
cos(t)-0.5; y=0.2
*
sin(t)+0.5; }
border E2(t=0,2
*
pi) { x=0.2
*
cos(t)+0.5; y=0.2
*
sin(t)+0.5; }
func real st(real t) ) {
return sin(pi
*
t)-pi/2;
}
border C1(t=-0.5,0.5) { x=(1-d)
*
c
*
cos(st(t)); y=(1-d)
*
c
*
sin(st(t)); }
border C2(t=0,1){x=((1-d)+d
*
t)
*
c
*
cos(st(0.5));y=((1-d)+d
*
t)
*
c
*
sin(st(0.5));}
border C3(t=0.5,-0.5) { x=c
*
cos(st(t)); y=c
*
sin(st(t)); }
border C4(t=0,1) { { x=(1-d
*
t)
*
c
*
cos(st(-0.5)); y=(1-d
*
t)
*
c
*
sin(st(-0.5));}
border C0(t=0,2
*
pi) { x=0.1
*
cos(t); y=0.1
*
sin(t); }
mesh Th=buildmesh(F(10
*
m)+C1(2
*
m)+C2(3)+C3(2
*
m)+C4(3)
+C0(m)+E1(-2
*
m)+E2(-2
*
m));
plot(Th,ps="SmileFace.eps",bw=1);
//
see Fig.
5.32
}
Example5.16(3pointbending)
//
Square for Three-Point Bend Specimens fixed on
Fix1, Fix2
//
It will be loaded on
Load.
real a=1, b=5, c=0.1;
int n=5, m=b
*
n;
border Left(t=0,2
*
a) { x=-b; y=a-t; }
border Bot1(t=0,b/2-c) { x=-b+t; y=-a; }
border Fix1(t=0,2
*
c) { x=-b/2-c+t; y=-a; }
border Bot2(t=0,b-2
*
c) { x=-b/2+c+t; y=-a; }
border Fix2(t=0,2
*
c) { x=b/2-c+t; y=-a; }
border Bot3(t=0,b/2-c) { x=b/2+c+t; y=-a; }
border Right(t=0,2
*
a) { x=b; y=-a+t; }
border Top1(t=0,b-c) { x=b-t; y=a; }
border Load(t=0,2
*
c) { x=c-t; y=a; }
border Top2(t=0,b-c) { x=-c-t; y=a; }
mesh Th = buildmesh(Left(n)+Bot1(m/4)+Fix1(5)+Bot2(m/2)+Fix2(5)+Bot3(m/4)
+Right(n)+Top1(m/2)+Load(10)+Top2(m/2));
plot(Th,ps="ThreePoint.eps",bw=1);
//
Fig.
5.33
5.9 Howtochangethelabelofelementsandborderelements
ofamesh
Changingthelabelofelementsandborderelementswillbedoneusingthekeywordchange.
Theparametersforthiscommandlineareforatwodimensionalanddimensionalcase:
label = isavectorofintegerthatcontainssuccessivepairoftheoldlabelnumbertothe
newlabelnumber.
region = isavectorofintegerthatcontainssuccessivepairoftheoldregionnumberto
newregionnumber.
5.9. HOWTOCHANGETHELABELOFELEMENTSANDBORDERELEMENTSOFAMESH129
 
 
 
 
 
 
Figure5.32: Smilingface(Mouthischange-
able)
  
    
   
   
   
    
    
   
   
   
  
  
Figure5.33: Domainforthree-pointbending
test
flabel = isaintegerfunctionwithgiventhenewvalueofthelabel(version3.21).
fregion = isaintegerfunctionwithgiventhenewvalueoftheregion.
Thesevectorsarecomposedofn
l
successivepairofnumber O;N N where n
l
isthenumber
(labelorregion)thatwewanttochange.Forexample,wehave
label = = [O
1
;N
1
;:::;O
n
l
;N
n
l
]
(5.3)
region = = [O
1
;N
1
;:::;O
n
l
;N
n
l
]
(5.4)
Anexampleofusingthisfunctionisgivenin"glumesh2D.edp":
Example5.17(glumesh2D.edp)
1:
2: mesh Th1=square(10,10);
3: mesh Th2=square(20,10,[x+1,y]);
4: verbosity=3;
5: int[int] r1=[2,0],
r2=[4,0];
6: plot(Th1,wait=1);
7: Th1=change(Th1,label=r1);
//
Change the label of Edges 2 in 0.
8: plot(Th1,wait=1);
9: Th2=change(Th2,label=r2);
//
Change the label of Edges 4 in 0.
10: mesh Th=Th1+Th2;
//
‘‘gluing together’’ of meshes Th1 and Th2
11: cout << " nb b lab b = " << int1d(Th1,1,3,4)(1./lenEdge)+int1d(Th2,1,2,3)(1./lenEdge)
12:
<< " " == = " << int1d(Th,1,2,3,4)(1./lenEdge) <<" == " << ((10+20)+10)
*
2
<< endl;
13: plot(Th,wait=1);
14: fespace Vh(Th,P1);
15: macro Grad(u) ) [dx(u),dy(u)];
//
definition of a macro
16: Vh u,v;
17: solve P(u,v)=int2d(Th)(Grad(u)’
*
Grad(v))-int2d(Th)(v)+on(1,3,u=0);
18: plot(u,wait=1);
130
CHAPTER5. MESHGENERATION
\gluing"dierentmesh Inline10ofpreviousle,themethodto\gluing"dierentmesh
ofthesamedimensioninFreeFem++ isusing. . Thisfunctionistheoperator"+"between
meshes. Themethodimplementedneedthatthepointinadjacentmesharethesame.
5.10 Meshinthreedimensions
5.10.1 cube
Fromversion(3.38-2),anewfunctioncubelikethefunctionsquarein2disthesimple
waytobuildcubicobject,inpluginmsh3(needload "msh3").
Thefollowingcode
mesh3 Th=cube(3,4,5);
generatesa345gridintheunitcube[0;1]
3
.
Bydefaultsthelabelare:
1. facey=0;
2. facex=1,
3. facey=1,
4. facex=0,
5. facez=0,
6. facez=1,
andtheregionnumberis0.
Afullexamplesofthethisfunctiontobuildameshofcube] 1;1[
3
withfacelabelgiven
by(ix+4(iy+1)+16(iz+1))where(ix;iy;iz)iscoordinateofthebarycenterofthe
currentface.
load "msh3"
int[int] l6=[21,42,45,40,25,53];
int r11=11;
mesh3 Th=cube(4,5,6,[x
*
2-1,y
*
2-1,z
*
2-1],label=l6,flags =3,region=r11);
//
Check label dans region numbering
int err =0;
for(int i=0; i<100; ++i)
{
real s =int2d(Th,i)(1.);
real sx=int2d(Th,i)(x);
real sy=int2d(Th,i)(y);
real sz=int2d(Th,i)(z);
if( s )
{
int ix = sx/s+1, iy=sy/s+1, iz=sz/s+1, ii=(ix x + + 4
*
(iy+1) + 16
*
(iz+1) ) ;
//
value of ix,iy,iz => face min 0 , face max 2 , no face 1
cout <<" label="<< i << " s " << s << " " << < ix x << iy << iz << " : " << ii
<< endl;
if( i != ii i ) ) err++;
}
}
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