Post on 04-Oct-2020
Optimal Shape Design in Magnetostatics
Disputation of the Ph.D. thesisNovember 27, 2003
D. Lukas
SFB F013 ”Numerical and Symbolic Scientific Computing”,University Linz, AT
Department of Applied Mathematics, VSB–Technical University Ostrava, CZweb: http://lukas.am.vsb.cz, email: dalibor.lukas@vsb.cz
Supervisor: Prof. Z. Dostal, Dep. of Applied Mathematics, VSB–TU Ostrava
Referees: Prof. J. Haslinger, Charles University PragueProf. M. Krızek, Mathematical Institute of the CAS, PragueProf. U. Langer, Inst. of Computational Mathematics, University Linz
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Homogeneous electromagnets
Maltese cross electromagnet
• is used for measurements of magnetooptic ef-fects,
• produces magnetic field constant in the middle,
• is capable to rotate the magnetic field,
• is produced at Institute of Physics, VSB–TUOstrava,
• is also used at INSA Toulouse, UniversityParis VI, Simon Fraser University Vancouver,Charles University Prague
Homogeneous electromagnets
Optimization problem
Find optimal shapes of the poleheads in order to minimize inho-mogeneities of the magnetic field inthe middle area Ωm among the poleheads.
minα∈U
∫
Ωm
|Bα(x) − Bavgα |2 dx,
whereBα(x) . . . the magnetic flux density,Bavg
α (x) . . . the average mag. fluxdensity over Ωm, Bavg
α ≥ Bmin,U . . . the set of admissible shapes(bounded continuous functions)
−0.2 −0.1 0 0.1 0.2−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
x [m]
y [m
]
ferromagneticyoke
+
−
+ −
+
−
+ −
pole
pole head
magnetizationarea Ωm
magnetizationplanes
coil
Maxwell’s equations
curl
(1
µB
)= J + σE +
∂(εE)
∂t
curl (E) = −∂B
∂tdiv(εE) = ρ
div(B) = 0
in Ω × T ⊂ R3 × (0,∞)
Maxwell’s equations
curl
(1
µB
)= J + σE +
∂(εE)
∂t
curl (E) = −∂B
∂tdiv(εE) = ρ
div(B) = 0
in Ω × T ⊂ R3 × (0,∞)
+ homogeneous boundary conditions
E × n = 0 and B · n = 0 on ∂Ω × T
Maxwell’s equations
curl
(1
µB
)= J + σE +
∂(εE)
∂t
curl (E) = −∂B
∂tdiv(εE) = ρ
div(B) = 0
in Ω × T ⊂ R3 × (0,∞)
+ homogeneous boundary conditions
E × n = 0 and B · n = 0 on ∂Ω × T
+ initial conditions
E = E0 and B = B0 in Ω × 0
Maxwell’s equations
curl
(1
µB
)= J + σE +
∂(εE)
∂t
curl (E) = −∂B
∂tdiv(εE) = ρ
div(B) = 0
in Ω × T ⊂ R3 × (0,∞)
+ homogeneous boundary conditions
E × n = 0 and B · n = 0 on ∂Ω × T
+ initial conditions
E = E0 and B = B0 in Ω × 0
Maxwell’s equations for linear magnetostatics
curl
(1
µ(x)B(x)
)= J(x)
div(B(x)) = 0
in Ω
Maxwell’s equations for linear magnetostatics
curl
(1
µ(x)B(x)
)= J(x)
div(B(x)) = 0
in Ω
+ boundary condition
u(x) × n(x) = 0 on ∂Ω,
whereB(x) = curl(u(x))
Maxwell’s equations for linear magnetostatics
curl
(1
µ(x)B(x)
)= J(x)
div(B(x)) = 0
in Ω
+ boundary condition
u(x) × n(x) = 0 on ∂Ω,
whereB(x) = curl(u(x))
Maxwell’s equations for vector potential
curl
(1
µ(x)curl(u(x))
)= J(x) in Ω
u(x) × n(x) = 0 on ∂Ω
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Weak formulation of 3d linear magnetostatics
(W )
Find u ∈ H0(curl; Ω)/Ker0(curl; Ω) =: H0,⊥(curl; Ω) :
a(u,v) :=
∫
Ω
1
µcurl(u) · curl(v) dx =
∫
Ω
J · v dx ∀v ∈ H0,⊥(curl; Ω),
where 0 < µ0 ≤ µ(x) ≤ µ1 a.e. x ∈ Ω and J ∈ Ker0(div; Ω)
Weak formulation of 3d linear magnetostatics
(W )
Find u ∈ H0(curl; Ω)/Ker0(curl; Ω) =: H0,⊥(curl; Ω) :
a(u,v) :=
∫
Ω
1
µcurl(u) · curl(v) dx =
∫
Ω
J · v dx ∀v ∈ H0,⊥(curl; Ω),
where 0 < µ0 ≤ µ(x) ≤ µ1 a.e. x ∈ Ω and J ∈ Ker0(div; Ω)
Hiptmair ’96: a(v,v) ≥ C‖v‖2curl,Ω on H0,⊥(curl; Ω), i.e., ∃!u solution to (W )
Weak formulation of 3d linear magnetostatics
(W )
Find u ∈ H0(curl; Ω)/Ker0(curl; Ω) =: H0,⊥(curl; Ω) :
a(u,v) :=
∫
Ω
1
µcurl(u) · curl(v) dx =
∫
Ω
J · v dx ∀v ∈ H0,⊥(curl; Ω),
where 0 < µ0 ≤ µ(x) ≤ µ1 a.e. x ∈ Ω and J ∈ Ker0(div; Ω)
Regularization (ε > 0)
(Wε)
Find uε ∈ H0(curl; Ω) :
aε(uε,v) := a(uε,v) + ε
∫
Ω
uε · v dx =
∫
Ω
J · v dx ∀v ∈ H0(curl; Ω)
Hiptmair ’96: a(v,v) ≥ C‖v‖2curl,Ω on H0,⊥(curl; Ω), i.e., ∃!u solution to (W )
Weak formulation of 3d linear magnetostatics
(W )
Find u ∈ H0(curl; Ω)/Ker0(curl; Ω) =: H0,⊥(curl; Ω) :
a(u,v) :=
∫
Ω
1
µcurl(u) · curl(v) dx =
∫
Ω
J · v dx ∀v ∈ H0,⊥(curl; Ω),
where 0 < µ0 ≤ µ(x) ≤ µ1 a.e. x ∈ Ω and J ∈ Ker0(div; Ω)
Regularization (ε > 0)
(Wε)
Find uε ∈ H0(curl; Ω) :
aε(uε,v) := a(uε,v) + ε
∫
Ω
uε · v dx =
∫
Ω
J · v dx ∀v ∈ H0(curl; Ω)
Hiptmair ’96: a(v,v) ≥ C‖v‖2curl,Ω on H0,⊥(curl; Ω), i.e., ∃!u solution to (W )
L. ’03: ∃!uε solution to (Wε) and |uε − u|curl,Ω → 0
Weak formulation of 3d linear magnetostatics
(W )
Find u ∈ H0(curl; Ω)/Ker0(curl; Ω) =: H0,⊥(curl; Ω) :
a(u,v) :=
∫
Ω
1
µcurl(u) · curl(v) dx =
∫
Ω
J · v dx ∀v ∈ H0,⊥(curl; Ω),
where 0 < µ0 ≤ µ(x) ≤ µ1 a.e. x ∈ Ω and J ∈ Ker0(div; Ω)
Regularization (ε > 0)
(Wε)
Find uε ∈ H0(curl; Ω) :
aε(uε,v) := a(uε,v) + ε
∫
Ω
uε · v dx =
∫
Ω
J · v dx ∀v ∈ H0(curl; Ω)
Hiptmair ’96: a(v,v) ≥ C‖v‖2curl,Ω on H0,⊥(curl; Ω), i.e., ∃!u solution to (W )
L. ’03: ∃!uε solution to (Wε) and |uε − u|curl,Ω → 0
Schoberl ’02: Since J ∈ Ker0(div; Ω), then uε ∈ H0,⊥(curl; Ω) and ‖uε − u‖curl,Ω → 0
Finite element method: Nedelec’s elements
• Nedelec space Pe :=p(x) := ae × x + be
∣∣ ae,be ∈ R3, x ∈ Ke
Finite element method: Nedelec’s elements
• Nedelec space Pe :=p(x) := ae × x + be
∣∣ ae,be ∈ R3, x ∈ Ke
• Degrees of freedom σei (v) :=
∫cei
v · tei ds, i = 1, . . . , 6
PSfrag replacements
0
1
1
1
x1
x2
x3
cr1
cr2
cr3
cr4
cr5
cr6
Kr
Re
x1
x2
x3
ce1
ce2
ce3
ce4
ce5
ce6
Ke
Finite element method: Nedelec’s elements
• Nedelec space Pe :=p(x) := ae × x + be
∣∣ ae,be ∈ R3, x ∈ Ke
• Degrees of freedom σei (v) :=
∫cei
v · tei ds, i = 1, . . . , 6
PSfrag replacements
0
1
1
1
x1
x2
x3
cr1
cr2
cr3
cr4
cr5
cr6
Kr
Re
x1
x2
x3
ce1
ce2
ce3
ce4
ce5
ce6
Ke
• H(curl)–conformity condition: continuity of tangential components
Finite element approximation
• Inner approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ωh ⊂ Ω
Finite element approximation
• Inner approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ωh ⊂ Ω
• Extension (by zero) operator Xh : H0(curl; Ωh)h 7→ H0(curl; Ω)h
Finite element approximation
• Inner approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ωh ⊂ Ω
• Extension (by zero) operator Xh : H0(curl; Ωh)h 7→ H0(curl; Ω)h
• 1st Strang’s lemma: ∀vh ∈ Xh(H0(curl; Ωh)h), being a Hilbert space,
‖uε−Xh(uhε)‖curl,Ω ≤ 1
minC, ε
(‖uε − vh‖curl,Ω +
∣∣[aε − ahε ](X
h(uhε) − vh,vh)
∣∣‖Xh(uh
ε) − vh‖curl,Ω
)
Finite element approximation
• Inner approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ωh ⊂ Ω
• Extension (by zero) operator Xh : H0(curl; Ωh)h 7→ H0(curl; Ω)h
• 1st Strang’s lemma: ∀vh ∈ Xh(H0(curl; Ωh)h), being a Hilbert space,
‖uε−Xh(uhε)‖curl,Ω ≤ 1
minC, ε
(‖uε − vh‖curl,Ω +
∣∣[aε − ahε ](X
h(uhε) − vh,vh)
∣∣‖Xh(uh
ε) − vh‖curl,Ω
)
• Density argument ∀τ > 0 ∀v ∈ H0(curl; Ω) ∃v ∈ H20(Ω) : ‖v − v‖curl,Ω ≤ τ
Finite element approximation
• Inner approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ωh ⊂ Ω
• Extension (by zero) operator Xh : H0(curl; Ωh)h 7→ H0(curl; Ω)h
• 1st Strang’s lemma: ∀vh ∈ Xh(H0(curl; Ωh)h), being a Hilbert space,
‖uε−Xh(uhε)‖curl,Ω ≤ 1
minC, ε
(‖uε − vh‖curl,Ω +
∣∣[aε − ahε ](X
h(uhε) − vh,vh)
∣∣‖Xh(uh
ε) − vh‖curl,Ω
)
• Density argument ∀τ > 0 ∀v ∈ H0(curl; Ω) ∃v ∈ H20(Ω) : ‖v − v‖curl,Ω ≤ τ
• vh := Xh(πh0(uε|Ωh)) , where πh
0 is the FE–interpolation operator
Finite element approximation
• Inner approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ωh ⊂ Ω
• Extension (by zero) operator Xh : H0(curl; Ωh)h 7→ H0(curl; Ω)h
• 1st Strang’s lemma: ∀vh ∈ Xh(H0(curl; Ωh)h), being a Hilbert space,
‖uε−Xh(uhε)‖curl,Ω ≤ 1
minC, ε
(‖uε − vh‖curl,Ω +
∣∣[aε − ahε ](X
h(uhε) − vh,vh)
∣∣‖Xh(uh
ε) − vh‖curl,Ω
)
• Density argument ∀τ > 0 ∀v ∈ H0(curl; Ω) ∃v ∈ H20(Ω) : ‖v − v‖curl,Ω ≤ τ
• vh := Xh(πh0(uε|Ωh)) , where πh
0 is the FE–interpolation operator
• Provided µh(x) → µ(x) a.e. in Ω, using Lebesgue dominated convergence theorem:
Xh(uhε) → uε in H0(curl; Ω)
Finite element approximation
• Inner approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ωh ⊂ Ω
• Extension (by zero) operator Xh : H0(curl; Ωh)h 7→ H0(curl; Ω)h
• 1st Strang’s lemma: ∀vh ∈ Xh(H0(curl; Ωh)h), being a Hilbert space,
‖uε−Xh(uhε)‖curl,Ω ≤ 1
minC, ε
(‖uε − vh‖curl,Ω +
∣∣[aε − ahε ](X
h(uhε) − vh,vh)
∣∣‖Xh(uh
ε) − vh‖curl,Ω
)
• Density argument ∀τ > 0 ∀v ∈ H0(curl; Ω) ∃v ∈ H20(Ω) : ‖v − v‖curl,Ω ≤ τ
• vh := Xh(πh0(uε|Ωh)) , where πh
0 is the FE–interpolation operator
• Provided µh(x) → µ(x) a.e. in Ω, using Lebesgue dominated convergence theorem:
Xh(uhε) → uε in H0(curl; Ω)
Finite element approximation
• Inner approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ωh ⊂ Ω
• Extension (by zero) operator Xh : H0(curl; Ωh)h 7→ H0(curl; Ω)h
• 1st Strang’s lemma: ∀vh ∈ Xh(H0(curl; Ωh)h), being a Hilbert space,
‖uε−Xh(uhε)‖curl,Ω ≤ 1
minC, ε
(‖uε − vh‖curl,Ω +
∣∣[aε − ahε ](X
h(uhε) − vh,vh)
∣∣‖Xh(uh
ε) − vh‖curl,Ω
)
• Density argument ∀τ > 0 ∀v ∈ H0(curl; Ω) ∃v ∈ H20(Ω) : ‖v − v‖curl,Ω ≤ τ
• vh := Xh(πh0(uε|Ωh)) , where πh
0 is the FE–interpolation operator
• Provided µh(x) → µ(x) a.e. in Ω, using Lebesgue dominated convergence theorem:
Xh(uhε) → uε in H0(curl; Ω)
Finite element approximation: Prof. Langer’s comment
The ellipticity constant 1/minC, ε decreases the rate of convergence to√
h only!
Finite element approximation: Prof. Langer’s comment
The ellipticity constant 1/minC, ε decreases the rate of convergence to√
h only!
• Hiptmair ’96: C is common for both H0,⊥(curl; Ω) and H0,⊥(curl; Ω)h
Finite element approximation: Prof. Langer’s comment
The ellipticity constant 1/minC, ε decreases the rate of convergence to√
h only!
• Hiptmair ’96: C is common for both H0,⊥(curl; Ω) and H0,⊥(curl; Ω)h
• Schoberl ’02: uε ∈ H0,⊥(curl; Ω)
Finite element approximation: Prof. Langer’s comment
The ellipticity constant 1/minC, ε decreases the rate of convergence to√
h only!
• Hiptmair ’96: C is common for both H0,⊥(curl; Ω) and H0,⊥(curl; Ω)h
• Schoberl ’02: uε ∈ H0,⊥(curl; Ω)
• However, H0,⊥(curl; Ω)h 6⊂ H0,⊥(curl; Ω)
Finite element approximation: Prof. Langer’s comment
The ellipticity constant 1/minC, ε decreases the rate of convergence to√
h only!
• Hiptmair ’96: C is common for both H0,⊥(curl; Ω) and H0,⊥(curl; Ω)h
• Schoberl ’02: uε ∈ H0,⊥(curl; Ω)
• However, H0,⊥(curl; Ω)h 6⊂ H0,⊥(curl; Ω)
• H0,⊥(curl; Ω) ∩ H20(Ω) is dense in H0,⊥(curl; Ω)
Finite element approximation: Prof. Langer’s comment
The ellipticity constant 1/minC, ε decreases the rate of convergence to√
h only!
• Hiptmair ’96: C is common for both H0,⊥(curl; Ω) and H0,⊥(curl; Ω)h
• Schoberl ’02: uε ∈ H0,⊥(curl; Ω)
• However, H0,⊥(curl; Ω)h 6⊂ H0,⊥(curl; Ω)
• H0,⊥(curl; Ω) ∩ H20(Ω) is dense in H0,⊥(curl; Ω)
• vh := Xh(πh0(uε,⊥|Ωh))
Finite element approximation: Prof. Langer’s comment
The ellipticity constant 1/minC, ε decreases the rate of convergence to√
h only!
• Hiptmair ’96: C is common for both H0,⊥(curl; Ω) and H0,⊥(curl; Ω)h
• Schoberl ’02: uε ∈ H0,⊥(curl; Ω)
• However, H0,⊥(curl; Ω)h 6⊂ H0,⊥(curl; Ω)
• H0,⊥(curl; Ω) ∩ H20(Ω) is dense in H0,⊥(curl; Ω)
• vh := Xh(πh0(uε,⊥|Ωh))
• The wish:
‖uε − Xh(uhε)‖curl,Ω ≤ 1
C
(‖uε − vh‖curl,Ω +
∣∣[aε − ahε ](X
h(uhε) − vh,vh)
∣∣‖Xh(uh
ε) − vh‖curl,Ω
)
Finite element approximation: Prof. Haslinger’s question
Can the theory be extended to non–inner domain approximations?
Finite element approximation: Prof. Haslinger’s question
Can the theory be extended to non–inner domain approximations?
• Approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ω, Ωh ⊂ Ω
Finite element approximation: Prof. Haslinger’s question
Can the theory be extended to non–inner domain approximations?
• Approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ω, Ωh ⊂ Ω
• Extension (by zero) operators Xh,X : H0(curl; Ωh)h,H0(curl; Ω) 7→H0(curl; Ω),
• However, Xh(H0(curl; Ωh)h) 6⊂ Xh(H0(curl;Ω)) and 1st Strang’s lemma can’tbe used!
Finite element approximation: Prof. Haslinger’s question
Can the theory be extended to non–inner domain approximations?
• Approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ω, Ωh ⊂ Ω
• Extension (by zero) operators Xh,X : H0(curl; Ωh)h,H0(curl; Ω) 7→H0(curl; Ω),
• However, Xh(H0(curl; Ωh)h) 6⊂ Xh(H0(curl;Ω)) and 1st Strang’s lemma can’tbe used!
• Using 2nd Strang’s lemma, for vh := Xh(πh0(uε|Ωh)):
‖X(uε)−Xh(uhε)‖curl,Ω ≤ 1 + µ1
C‖X(uε)−vh‖curl,Ω +
1
C
|f(vh) − ahε(X(uε),v
h)|‖vh‖curl,Ω
Finite element approximation: Prof. Haslinger’s question
Can the theory be extended to non–inner domain approximations?
• Approximation of Ω by polyhedra Ωh
∀xh ∈ ∂Ωh ∃x ∈ Ω : ‖xh − x‖ ≤ h, where Ω, Ωh ⊂ Ω
• Extension (by zero) operators Xh,X : H0(curl; Ωh)h,H0(curl; Ω) 7→H0(curl; Ω),
• However, Xh(H0(curl; Ωh)h) 6⊂ Xh(H0(curl;Ω)) and 1st Strang’s lemma can’tbe used!
• Using 2nd Strang’s lemma, for vh := Xh(πh0(uε|Ωh)):
‖X(uε)−Xh(uhε)‖curl,Ω ≤ 1 + µ1
C‖X(uε)−vh‖curl,Ω +
1
C
|f(vh) − ahε(X(uε),v
h)|‖vh‖curl,Ω
• Provided µh → µ a.e. in Ω, it yields Xh(uhε) → X(uε) in H0(curl; Ω)
Optimal shape design: Continuous setting
Set of admissible shapes
U := α ∈ C(ω) | αl ≤ α(x) ≤ αu and |α(x) − α(y)| ≤ CL‖x − y‖
Optimal shape design: Continuous setting
Set of admissible shapes
U := α ∈ C(ω) | αl ≤ α(x) ≤ αu and |α(x) − α(y)| ≤ CL‖x − y‖, αn ⇒ α
Optimal shape design: Continuous setting
Set of admissible shapes
U := α ∈ C(ω) | αl ≤ α(x) ≤ αu and |α(x) − α(y)| ≤ CL‖x − y‖, αn ⇒ αPSfrag replacements
Ω0(α)Ω1(α)
α
x1
x2
x3
ω
Optimal shape design: Continuous setting
Set of admissible shapes
U := α ∈ C(ω) | αl ≤ α(x) ≤ αu and |α(x) − α(y)| ≤ CL‖x − y‖, αn ⇒ αPSfrag replacements
Ω0(α)Ω1(α)
α
x1
x2
x3
ωMultistate problem
(W v(α))
Find uv(α) ∈ H0,⊥(curl; Ω) :∫
Ω0(α)
1
µ0curl(uv(α)) · curl(v) dx +
∫
Ω1(α)
1
µ1curl(uv(α)) · curl(v) dx =
=
∫
Ω
Jv · v dx ∀v ∈ H0,⊥(curl; Ω)
Outlin
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On
the
road
PSfr
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phy
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mat
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Maltese cross electromagnet: Multistate problem
Vertical and diagonal current excitations
5 Amperes, 600 turns, relative permeability of AREMA = 5100
Outlin
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On
the
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PSfr
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Optimal shape design: Continuous setting
Continuous cost functional
J (α) := I(α, curl(u1(α)), . . . , curl(unv(α))), where uv(α) is the solution to (W v(α))
Optimal shape design: Continuous setting
Continuous cost functional
J (α) := I(α, curl(u1(α)), . . . , curl(unv(α))), where uv(α) is the solution to (W v(α))
Optimization problem
(P )
Find α∗ ∈ U :
J(α∗) ≤ J(α) ∀α ∈ U , there exists α∗ a solution to (P )
Optimal shape design: Continuous setting
Continuous cost functional
J (α) := I(α, curl(u1(α)), . . . , curl(unv(α))), where uv(α) is the solution to (W v(α))
Optimization problem
(P )
Find α∗ ∈ U :
J(α∗) ≤ J(α) ∀α ∈ U , there exists α∗ a solution to (P )
Shape parameterization
• Υ ⊂ RnΥ a compact subset
• F : Υ 7→ U a continuous mapping, e.g., Bezier parameterization
(P )
Find p∗ ∈ Υ :
[J F ](p∗) ≤ [J F ](p) ∀p ∈ Υ, there exists p∗ a solution to (P )
Optimal shape design: Existence and convergence analysis
Existence
U compact
uv : U 7→ H0,⊥(curl; Ω) continuous
I : U ×[L2(Ω)
]nv 7→ R continuous
⇒ ∃α∗ ∈ U a solution to (P )
Optimal shape design: Existence and convergence analysis
Existence
U compact
uv : U 7→ H0,⊥(curl; Ω) continuous
I : U ×[L2(Ω)
]nv 7→ R continuous
⇒ ∃α∗ ∈ U a solution to (P )
Convergence
∃α∗, αhnεn
∗solutions to (P ), (P hn
εn)
∀α ∈ U : πhnω (α) ⇒ α
uv,hnεn
: Uhn 7→ H0,⊥(curl; Ωhn)hn continuous
I : U ×[L2(Ω)
]nv 7→ R continuous
⇒ ∃nk∞k=1 ⊂ n∞n=1 : αhnkεnk
∗⇒ α∗
Optimal shape design: Existence and convergence analysis
Existence
U compact
uv : U 7→ H0,⊥(curl; Ω) continuous
I : U ×[L2(Ω)
]nv 7→ R continuous
⇒ ∃α∗ ∈ U a solution to (P )
Convergence
∃α∗, αhnεn
∗solutions to (P ), (P hn
εn)
∀α ∈ U : πhnω (α) ⇒ α
uv,hnεn
: Uhn 7→ H0,⊥(curl; Ωhn)hn continuous
I : U ×[L2(Ω)
]nv 7→ R continuous
⇒ ∃nk∞k=1 ⊂ n∞n=1 : αhnkεnk
∗⇒ α∗
Bottleneck
For fine discretizations it is hard to find a continuous shape–to–mesh mapping!
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of Jp
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of J
pπhωF−−−→ αh
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of J
pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of J
pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh FEM−−→ An
ε , fv,n
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of J
pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh FEM−−→ An
ε , fv,n An
ε ·uv,nε =f v,n
−−−−−−−→ uv,nε
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of J
pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh FEM−−→ An
ε , fv,n An
ε ·uv,nε =f v,n
−−−−−−−→ uv,nε
curl−−→curl−−→ B v,n
ε
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of J
pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh FEM−−→ An
ε , fv,n An
ε ·uv,nε =f v,n
−−−−−−−→ uv,nε
curl−−→curl−−→ B v,n
ε
Ih(αh,xh,B1,nε ,...,B
nv,nε )−−−−−−−−−−−−−→ J h
ε (p)
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of J
pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh FEM−−→ An
ε , fv,n An
ε ·uv,nε =f v,n
−−−−−−−→ uv,nε
curl−−→curl−−→ B v,n
ε
Ih(αh,xh,B1,nε ,...,B
nv,nε )−−−−−−−−−−−−−→ J h
ε (p)
Shape–to–mesh elasticity mapping: initial design
−0.2 −0.1 00
0.05
0.1
x1 [m]
x3 [
m]
−0.2 −0.1 00
0.05
0.1
x1 [m]
x3 [
m]
Finite–dimensional optimization
(P )
minp∈R
nΥJ (p)
subject to υ(p) ≤ 0
Structure of J
pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh FEM−−→ An
ε , fv,n An
ε ·uv,nε =f v,n
−−−−−−−→ uv,nε
curl−−→curl−−→ B v,n
ε
Ih(αh,xh,B1,nε ,...,B
nv,nε )−−−−−−−−−−−−−→ J h
ε (p)
Shape–to–mesh elasticity mapping: deformed design
−0.2 −0.1 00
0.05
0.1
x1 [m]
x3 [
m]
−0.2 −0.1 00
0.05
0.1
x1 [m]
x3 [
m]
Outlin
e:
On
the
road
PSfr
agre
pla
cem
ents
phy
sics
mat
hem
atic
sco
mpute
rsc
ience
PSfr
agre
pla
cem
ents
h
p0
p0
α(p
0)
x0
x0,T
Ω(α
(p0))
p∗
p
p
p
αx
T
TT
T
J,υ
J(p
),υ
(p)
gra
d( J
(p))
Gra
d(υ
(p))
gra
d(J
),G
rad(υ
)
I α,I
x,I
Bv
I Bv,δ
H0(B
;Ω),a
,fv
A,f
vu
v
Bv
Bv
Bv
τ1,τ
2,
λ
θ
θτ
γ
BFG
S–S
QP
Com
puta
tion
Com
puta
tion
ofof
Use
rin
terf
ace
Exte
rnalP
DE
tools
contr
olnodes
xω,1,.
..,x
ω,n
α
Des
ign–t
o–sh
ape
Shap
e–to
–mes
hm
appin
g
map
pin
g
Adjo
int
met
hod
Sol
ver
oflinea
rsy
stem
s
Mes
hge
ner
ator
FE
Mpre
–/pos
t–pro
cess
or
Scientific software development
Numerical methods Software (SFB F013)
- mesh generation - NETGEN (by J. Schoberl)
Scientific software development
Numerical methods Software (SFB F013)
- mesh generation - NETGEN (by J. Schoberl)- FEM approximation using edge elements - FEPP (by J. Schoberl et al.)
Scientific software development
Numerical methods Software (SFB F013)
- mesh generation - NETGEN (by J. Schoberl)- FEM approximation using edge elements - FEPP (by J. Schoberl et al.)- multigrid PCG solver - PEBBLES (by S. Reitzinger)
Scientific software development
Numerical methods Software (SFB F013)
- mesh generation - NETGEN (by J. Schoberl)- FEM approximation using edge elements - FEPP (by J. Schoberl et al.)- multigrid PCG solver - PEBBLES (by S. Reitzinger)- SQP with BFGS update of the Hessian - FEPP optimization tools (by W. Muhlhuber)
Scientific software development
Numerical methods Software (SFB F013)
- mesh generation - NETGEN (by J. Schoberl)- FEM approximation using edge elements - FEPP (by J. Schoberl et al.)- multigrid PCG solver - PEBBLES (by S. Reitzinger)- SQP with BFGS update of the Hessian - FEPP optimization tools (by W. Muhlhuber)- gradients by finite differences
or by the adjoint variable method - FEPP sensitivity analysis module (by D.L.)
Scientific software development
Numerical methods Software (SFB F013)
- mesh generation - NETGEN (by J. Schoberl)- FEM approximation using edge elements - FEPP (by J. Schoberl et al.)- multigrid PCG solver - PEBBLES (by S. Reitzinger)- SQP with BFGS update of the Hessian - FEPP optimization tools (by W. Muhlhuber)- gradients by finite differences
or by the adjoint variable method - FEPP sensitivity analysis module (by D.L.)
NETGEN + FEPP + PEBBLES = NgSolve
see http://www.hpfem.jku.at
Outlin
e:
On
the
road
PSfr
agre
pla
cem
ents
phy
sics
mat
hem
atic
sco
mpute
rsc
ience
Maltese cross electromagnet
Optimized pole heads
Maltese cross electromagnet
Optimized pole heads
Parameters
design variables 7 12deg. of freedom 12272 29541SQP iterations 72 93cost func. decrease 1.97.10−6 to 1.49.10−6 2.57.10−6 to 7.32.10−7
comput. time 2 hours 30 hours
Maltese cross electromagnet
Manufacture and measurements
The calculated cost functional has improved twice and the measured cost functionalhas improved even 4.5–times.
−8 −6 −4 −2 0 2 4 6 8600
800
1000
1200
1400
1600
1800
distance [mm]
norm
al c
ompo
nent
of m
agne
tic fl
ux d
ensi
ty [1
04 T
]
init. shape, vertical excit.init. shape, diagonal excit.opt. shape, vertical excit.opt. shape, diagonal excit.
Outlin
e:
On
the
road
PSfr
agre
pla
cem
ents
phy
sics
mat
hem
atic
sco
mpute
rsc
ience
General multilevel approach
optimizeddesigns
# of des.variables
# ofunknowns
SQPiters.
CPUtime
2 2777 6 44s
General multilevel approach
optimizeddesigns
# of des.variables
# ofunknowns
SQPiters.
CPUtime
2 2777 6 44s
4 12086 56 7h 27m
General multilevel approach
optimizeddesigns
# of des.variables
# ofunknowns
SQPiters.
CPUtime
2 2777 6 44s
4 12086 56 7h 27m
12 29541 93 29h 46m
Contributions of the thesis
Physics/electrical engineering
- development and manufacture of an electromagnet
Contributions of the thesis
Physics/electrical engineering
- development and manufacture of an electromagnet
Mathematics
- analysis of optimal shape design problems governed by 3d linear magnetostatics- development of a multilevel optimization algorithm
Contributions of the thesis
Physics/electrical engineering
- development and manufacture of an electromagnet
Mathematics
- analysis of optimal shape design problems governed by 3d linear magnetostatics- development of a multilevel optimization algorithm
Computer science
- efficient implementation of the 1st–order sensitivity analysis- development of a scientific software package NgSolve (Schoberl et al.)
Contributions of the thesis
Physics/electrical engineering
- development and manufacture of an electromagnet
Mathematics
- analysis of optimal shape design problems governed by 3d linear magnetostatics- development of a multilevel optimization algorithm
Computer science
- efficient implementation of the 1st–order sensitivity analysis- development of a scientific software package NgSolve (Schoberl et al.)
Education
- MATLAB tutorial code for 2d shape optimization (IMAMM ’03)- the thesis can serve as lecture notes
Outlook
PSfr
agre
pla
cem
ents
init
ialst
ruct
ure
(ablo
ck)
coar
seto
pol
ogy
opti
miz
atio
n
opti
miz
atio
n
coar
seop
tim
ized
stru
cture
coar
seop
tim
ized
stru
cture
wit
ha
rough
bou
ndar
y shap
e
shap
eid
enti
ficat
ion
and
hie
rarc
hic
aldes
crip
tion
wit
ha
smoo
thbou
ndar
y
shap
e
opti
miz
edst
ruct
ure
hpr
–FE
Mer
ror
anal
ysis
Isth
eer
ror
smal
len
ough
?
Yes
No
hpr
–refi
nem
ent
mult
igri
dpre
cond.
ofK
KT
syst
emre
fined
stru
cture
opti
mal
stru
cture
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
– fe–adaptivity respecting the cost functional
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
– fe–adaptivity respecting the cost functional
– simultaneous (all–at–once) approach
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
– fe–adaptivity respecting the cost functional
– simultaneous (all–at–once) approach
– multigrid preconditioning techniques for indefinite KKT systems
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
– fe–adaptivity respecting the cost functional
– simultaneous (all–at–once) approach
– multigrid preconditioning techniques for indefinite KKT systems
• Applications to real–life problems
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
– fe–adaptivity respecting the cost functional
– simultaneous (all–at–once) approach
– multigrid preconditioning techniques for indefinite KKT systems
• Applications to real–life problems
• Nonlinear problems (strongly monotone operators, . . . , hysteresis)
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
– fe–adaptivity respecting the cost functional
– simultaneous (all–at–once) approach
– multigrid preconditioning techniques for indefinite KKT systems
• Applications to real–life problems
• Nonlinear problems (strongly monotone operators, . . . , hysteresis)
• Various elliptic(–parabolic) operators: time–harmonic case, eddy currents
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
– fe–adaptivity respecting the cost functional
– simultaneous (all–at–once) approach
– multigrid preconditioning techniques for indefinite KKT systems
• Applications to real–life problems
• Nonlinear problems (strongly monotone operators, . . . , hysteresis)
• Various elliptic(–parabolic) operators: time–harmonic case, eddy currents
• Scientific software development
Outlook
• Adaptive structural optimization
– common aspects of topology and shape optimization
– smooth hierarchical parameterization of rough interfaces
– fixed grid approach, composite finite elements −→ overcomes the bottleneck
– fe–adaptivity respecting the cost functional
– simultaneous (all–at–once) approach
– multigrid preconditioning techniques for indefinite KKT systems
• Applications to real–life problems
• Nonlinear problems (strongly monotone operators, . . . , hysteresis)
• Various elliptic(–parabolic) operators: time–harmonic case, eddy currents
• Scientific software development
• Attracting some diploma students, industrial partners; cooperations abroad
Publications by Dalibor Lukas
• L. - Shape optimization of homogeneous electromagnets. Lect. Notes Comp. Sci.Eng. 18, 2001.
• L., Kopriva - Shape optimization of homogeneous electromagnets and their appli-cations for measurements of magneto–optical effects. Rec. of COMPUMAG 2001.
• L., Muhlhuber, Kuhn - An object-oriented library for the shape optimization prob-lems governed by systems of linear elliptic PDEs. Trans. of TU Ostrava 2001.
• L. - On the road – between Sobolev spaces and a manufacture of electromagnets.To appear in Trans. of TU Ostrava.
• L. - On solution to an optimal shape design problem in 3–dimensional linear mag-netostatics. To appear in Appl. Math.
• L., Ciprian, Pistora, Postava, Foldyna - Multilevel solvers for 3–dimensional optimalshape design with an application to magneto–optics. To appear in SCIE.