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TopologicalderivativesandLevelsetmethodinshapeoptimization
Thenumericalapproximationvhofvisthenthesolutionoftheproblem:
{
Findvh∈Vhsuchthat:
a(vh,zh)=(F(x,vh),zh)L2(Ω),foranyzh∈Vh.
(1.51)
Theproofoftheexistenceofasolutionof(1.51)iswellknown(see[39]).Itis
enoughtoapply,inaconvenientway,Browder’sfixedpointtheoremalongwith
themonotonicityofF(x,.).
Werewriteproblem(1.46)intheform:
{−∆p(x)=F0(x,p(x)),x∈Ω,
p(x)=0,
x∈aΩ,
(1.52)
whereF0(x,p(x))=F′
v(x,v(x))p(x)+J′(x,v(x)).F0islinearwithrespecttothe
secondvariable,thenF0∈C0,1(Ω×R).
Thenthevariationalformulationofthelinearproblem(1.46)isthefollowing:
a(p,z)=(F0(x,p),z)L2(Ω),z∈H
0(Ω).
1
(1.53)
Thus,thatnumericalapproximationofpisthesolutionofthevariationalproblem:
{
Findph∈Vhsuchthat:
a(ph,zh)=(F0(x,ph),zh)L2(Ω)foranyzh∈Vh.
(1.54)
Duetothehypothesis(H1)and(H7)thelinearproblem(1.54)haveasolutionin
spaceVh.
10503Numericalexamples
Inthissectionwepresentssomenumericalexamplestoshowusthebehavior
oftopologicalderivativeapproximationw.r.t.theevolutionofdiscretizationstep
size.Wederiveerrorsandverifiesthatthecomputederrorsatisfiestheestimate
obtainedinpaper[9]ineachcase.
ForeachoftheexampleswechoosethedomainΩasasquare(0,1)×(0,1)
andthefollowingenergyfunctional:
J(v;Ω)=
1
2∫
|∇v(x)|2dx+
1
4∫
v4(x)dx−∫
f(x)v(x)dx
Ω
Ω
Ω
wherevisthesolutionofthenonlinearproblem:
{−∆xv(x)=−v(x)
v(x)=0,
3+f(x),x∈Ω,
x∈aΩ.
(1.55)
(1.56)
