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28
TopologicalderivativesandLevelsetmethodinshapeoptimization
isaharmonics.Hence,weobtainthat:
∫
(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O))+v
′(x))J′
v(x,v(x))dx=
Ω
−lim
δ→0∫
[a|x|p(x)(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O)))
aBδ
−p(x)a|x|(−
2π
1
|x|2
xT
m(ω)∇xv(O)−
2π
1
ln
|x|
E
mes2ωF(O;v(O)))]dx
−lim
δ→0
i,j,k=1∫
∑
2
[
|x|
xi
axi
ap
(O)(−
2π
1
|x|2
xk
mkj
axj
av
(O))
aBδ
−(p(O)+xi
axi
ap
(O))(−
2π
1
|x|3
xj
mjk
axk
av
(O))]dx
+lim
δ→0∫
aBδ
[2
i=1
∑
(p(O)+xi
axi
ap
(O))×(
2π
1
|x|
1
mes2ωF(O;v(O)))]dx
−lim
δ→0∫
aBδ
[2
i,j,k=1
∑
|x|
xi
axi
ap
(O)(−
1
π
|x|2
xk
mkj
axj
av
(O))
−p(O)(
2π
1
|x|
1
mes2ωF(O;v(O)))]dx
δ→0∫
lim
aBδ
[2
i,j=1
∑
|x|
xi
axi
ap
(O)(
1
π
|x|2
xi
mij
axj
av
(O))+p(O)(
2π
1
|x|
1
mes2ωF(O;v(O))
)]dx
=F(O;v(O))mes2ωp(O)+∇xp(O)Tm(ω)∇xv(O).
Thus,recalling(1.47)weconcludetherelation:
J(uE;Ω(E))=J(v;Ω)+E2[−mes2ωJ(O;v(O))+F(O;v(O))mes2ωp(O)
+∇xp(O)Tm(ω)∇xv(O)]+···.
(1.48)
Theorem50Undertheassumptions(H1),(H7)and(H8),theasymptoticexpan-
sion(1.48)isvalidwiththeremaindero(E2).
105Finiteelementapproximationsoftopological
derivatives
Ouraiminthissectionistocomputeanumericalapproximationofthetopological
derivativeoftheshapefunctional(1.14),withuEthesolutionoftheproblem(1.34)
andgiveL∞-estimatesoftheerror.
