Optimal Shape Design in Magnetostaticshomel.vsb.cz/~luk76/publications/PhD-talk.pdf · Optimal Shape Design in Magnetostatics Disputation of the Ph.D. thesis November 27, 2003 D

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: [email protected]

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,∞)


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


curl

(1

µB

)= J + σE +

∂(εE)

∂t

curl (E) = −∂B

∂tdiv(εE) = ρ

div(B) = 0

in Ω × T ⊂ R3 × (0,∞)


E × n = 0 and B · n = 0 on ∂Ω × T

+ initial conditions

E = E0 and B = B0 in Ω × 0


curl

(1

µB

)= J + σE +

∂(εE)

∂t

curl (E) = −∂B

∂tdiv(εE) = ρ

div(B) = 0

in Ω × T ⊂ R3 × (0,∞)


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


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))


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)


(W )


a(u,v) :=

∫

Ω

1


∫

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 )


(W )


a(u,v) :=

∫

Ω

1


∫

Ω

J · v dx ∀v ∈ H0,⊥(curl; 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)



(W )


a(u,v) :=

∫

Ω

1


∫

Ω

J · v dx ∀v ∈ H0,⊥(curl; Ω),



(Wε)



∫

Ω

uε · v dx =

∫

Ω



L. ’03: ∃!uε solution to (Wε) and |uε − u|curl,Ω → 0


(W )


a(u,v) :=

∫

Ω

1


∫

Ω

J · v dx ∀v ∈ H0,⊥(curl; Ω),



(Wε)



∫

Ω

uε · v dx =

∫

Ω



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



∣∣ 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



∣∣ 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




• 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

)







minC, ε



h(uhε) − vh,vh)

∣∣‖Xh(uh


)

• Density argument ∀τ > 0 ∀v ∈ H0(curl; Ω) ∃v ∈ H20(Ω) : ‖v − v‖curl,Ω ≤ τ







minC, ε



h(uhε) − vh,vh)

∣∣‖Xh(uh


)


• vh := Xh(πh0(uε|Ωh)) , where πh

0 is the FE–interpolation operator







minC, ε



h(uhε) − vh,vh)

∣∣‖Xh(uh


)




• Provided µh(x) → µ(x) a.e. in Ω, using Lebesgue dominated convergence theorem:

Xh(uhε) → uε in H0(curl; Ω)







minC, ε



h(uhε) − vh,vh)

∣∣‖Xh(uh


)












minC, ε



h(uhε) − vh,vh)

∣∣‖Xh(uh


)






Finite element approximation: Prof. Langer’s comment

The ellipticity constant 1/minC, ε decreases the rate of convergence to√

h only!



h only!

• Hiptmair ’96: C is common for both H0,⊥(curl; Ω) and H0,⊥(curl; Ω)h



h only!


• Schoberl ’02: uε ∈ H0,⊥(curl; Ω)



h only!



• However, H0,⊥(curl; Ω)h 6⊂ H0,⊥(curl; Ω)



h only!




• H0,⊥(curl; Ω) ∩ H20(Ω) is dense in H0,⊥(curl; Ω)



h only!





• vh := Xh(πh0(uε,⊥|Ωh))



h only!





• vh := Xh(πh0(uε,⊥|Ωh))

• The wish:

‖uε − Xh(uhε)‖curl,Ω ≤ 1

C



h(uhε) − vh,vh)

∣∣‖Xh(uh


)

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







• 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‖



U := α ∈ C(ω) | αl ≤ α(x) ≤ αu and |α(x) − α(y)| ≤ CL‖x − y‖, αn ⇒ α



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



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)

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Maltese cross electromagnet: Multistate problem

Vertical and diagonal current excitations

5 Amperes, 600 turns, relative permeability of AREMA = 5100

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Continuous cost functional

J (α) := I(α, curl(u1(α)), . . . , curl(unv(α))), where uv(α) is the solution to (W v(α))





(P )

Find α∗ ∈ U :

J(α∗) ≤ J(α) ∀α ∈ U , there exists α∗ a solution to (P )





(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 )


Existence

U compact


I : U ×[L2(Ω)



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)


⇒ ∃nk∞k=1 ⊂ n∞n=1 : αhnkεnk

∗⇒ α∗


Existence

U compact


I : U ×[L2(Ω)



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)


⇒ ∃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


(P )

minp∈R

nΥJ (p)


Structure of Jp


(P )

minp∈R

nΥJ (p)


Structure of J

pπhωF−−−→ αh


(P )

minp∈R

nΥJ (p)


Structure of J

pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh


(P )

minp∈R

nΥJ (p)


Structure of J

pπhωF−−−→ αh linear elasticity−−−−−−−−→ xh FEM−−→ An

ε , fv,n


(P )

minp∈R

nΥJ (p)


Structure of J


ε , fv,n An

ε ·uv,nε =f v,n

−−−−−−−→ uv,nε


(P )

minp∈R

nΥJ (p)


Structure of J


ε , fv,n An

ε ·uv,nε =f v,n

−−−−−−−→ uv,nε

curl−−→curl−−→ B v,n

ε


(P )

minp∈R

nΥJ (p)


Structure of J


ε , fv,n An

ε ·uv,nε =f v,n

−−−−−−−→ uv,nε


ε

Ih(αh,xh,B1,nε ,...,B

nv,nε )−−−−−−−−−−−−−→ J h

ε (p)


(P )

minp∈R

nΥJ (p)


Structure of J


ε , fv,n An

ε ·uv,nε =f v,n

−−−−−−−→ uv,nε


ε


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]


(P )

minp∈R

nΥJ (p)


Structure of J


ε , fv,n An

ε ·uv,nε =f v,n

−−−−−−−→ uv,nε


ε


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]

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PSfr

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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

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Shap

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–mes

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map

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–/pos

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cess

or

Scientific software development

Numerical methods Software (SFB F013)

- mesh generation - NETGEN (by J. Schoberl)



- mesh generation - NETGEN (by J. Schoberl)- FEM approximation using edge elements - FEPP (by J. Schoberl et al.)



- mesh generation - NETGEN (by J. Schoberl)- FEM approximation using edge elements - FEPP (by J. Schoberl et al.)- multigrid PCG solver - PEBBLES (by S. Reitzinger)



- 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)



- 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.)



- 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

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Optimized pole heads


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


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.

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General multilevel approach

optimizeddesigns

# of des.variables

# ofunknowns

SQPiters.

CPUtime

2 2777 6 44s


optimizeddesigns

# of des.variables

# ofunknowns

SQPiters.

CPUtime

2 2777 6 44s

4 12086 56 7h 27m


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




Mathematics

- analysis of optimal shape design problems governed by 3d linear magnetostatics- development of a multilevel optimization algorithm




Mathematics


Computer science

- efficient implementation of the 1st–order sensitivity analysis- development of a scientific software package NgSolve (Schoberl et al.)




Mathematics


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



– smooth hierarchical parameterization of rough interfaces

Outlook




– fixed grid approach, composite finite elements −→ overcomes the bottleneck

Outlook





– fe–adaptivity respecting the cost functional

Outlook






– simultaneous (all–at–once) approach

Outlook







– multigrid preconditioning techniques for indefinite KKT systems

Outlook








• Applications to real–life problems

Outlook









• Nonlinear problems (strongly monotone operators, . . . , hysteresis)

Outlook










• Various elliptic(–parabolic) operators: time–harmonic case, eddy currents

Outlook











• Scientific software development

Outlook











• 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.

Documents

Optimal Shape Design in Magnetostaticshomel.vsb.cz/~luk76/publications/PhD-talk.pdf · Optimal Shape Design in Magnetostatics Disputation of the Ph.D. thesis November 27, 2003 D