reduced basis model reduction of time-harmonic maxwell ......reduced basis model reduction of...

Reduced Basis Model Reduction ofTime-Harmonic Maxwell’s Equations

Using a Compliant Expanded FormulationPeter Benner, Martin Hess

MOR 4 MEMSNovember, 17-18, 2015

KIT, Karlsruhe

Max Planck Institute for Dynamics of Complex Technical SystemsComputational Methods in Systems and Control Theory

1. Motivation

2. Introduction to Reduced Basis Method

3. Electromagnetic Model

4. Numerical Results

5. Conclusions

Martin Hess RBM in Electromagnetics 2/19

Motivation

1. Motivation

5. Conclusions

MotivationFull wave simulations allow an accurate and reliable prediction of devicebehavior.

Figure: Printed circuit board (PCB, left) with 4 inputs and 4 outputs. Microstriplines (single and double) on the right.

Motivation

Figure: Two parallel microstrip lines in a 0.13µm CMOS process andcomputational mesh employed for solving Maxwell’s equations.

Figure: Optical lithography process (left) and phase shift mask (right).

Motivation

Figure: Two parallel microstrip lines in a 0.13µm CMOS process andcomputational mesh employed for solving Maxwell’s equations.

Figure: Optical lithography process (left) and phase shift mask (right).

Reduced Basis Concept

1. Motivation

5. Conclusions

Model Problem, [Rozza et al., 2008]For ν ∈ D evaluate

s(ν) = `(u(ν); ν),

where u(ν) ∈ X satisfies

a(u(ν), v ; ν) = f (v ; ν), ∀v ∈ X

Find linear space approximating M = {u(ν)|ν ∈ D}

Offline-Online decomposition ⇒ computational efficiency

s(ν) = `(u(ν); ν),

a(u(ν), v ; ν) = f (v ; ν), ∀v ∈ X

s(ν) = `(u(ν); ν),

a(u(ν), v ; ν) = f (v ; ν), ∀v ∈ X

Parameter-dependent linear systems

A(ν)x(ν) = b(ν)

with affine parameter dependence

A(ν) =∑Qa

q=1 Θqa(ν)Aq

b(ν) =∑Qb

q=1 Θqb(ν)bq

⇒ Offline-Online decomposition.

Parameter-dependent linear systems

A(ν)x(ν) = b(ν)

with affine parameter dependence

A(ν) =∑Qa

q=1 Θqa(ν)Aq

b(ν) =∑Qb

q=1 Θqb(ν)bq

⇒ Offline-Online decomposition.

Sampling

Greedy Sampling

Let Ξ denote a finite sample of D.

Set S1 = {ν1} and V1 = span{u(ν1)}.For N = 2, ...,Nmax , find νN = arg maxν∈Ξ∆N−1(ν),

then set SN = SN−1 ∪ νN , VN = VN−1 + span{u(νN)}.

Projection onto low order space VN (Snapshot space)

AqN = V T

N AqVN , bqN = V TN bq.

Parameter-preserving model reduction Qa∑q=1

Θqa(ν)Aq

Qb∑q=1

Θqb(ν)bqN .

Sampling

Greedy Sampling

AqN = V T

Θqa(ν)Aq

Qb∑q=1

Θqb(ν)bqN .

Sampling

Greedy Sampling

AqN = V T

Θqa(ν)Aq

Qb∑q=1

Θqb(ν)bqN .

Error EstimatorsField error estimator

∆N(ν) =‖rpr (·; ν)‖X ′

βLB(ν).

Output error estimator

∆oN(ν) =

‖rpr (·; ν)‖X ′‖rdu(·; ν)‖X ′

βLB(ν).

Output error estimator (compliant case ` = f )

∆sN(ν) =

‖rpr (·; ν)‖2X ′

βLB(ν).

∆N(ν) =‖rpr (·; ν)‖X ′

βLB(ν).

∆oN(ν) =

‖rpr (·; ν)‖X ′‖rdu(·; ν)‖X ′

βLB(ν).

∆sN(ν) =

‖rpr (·; ν)‖2X ′

βLB(ν).

∆N(ν) =‖rpr (·; ν)‖X ′

βLB(ν).

∆oN(ν) =

‖rpr (·; ν)‖X ′‖rdu(·; ν)‖X ′

βLB(ν).

∆sN(ν) =

‖rpr (·; ν)‖2X ′

βLB(ν).

Coplanar Waveguide

1. Motivation

5. Conclusions

Coplanar Waveguide

Figure: Geometry of coplanar waveguide.

Maxwell’s Equations

Time-Harmonic Maxwell’s Equations, [Hiptmair, 2002]

µ−1(∇× E ,∇× v) + iωσ(E , v)− ω2ε(E , v) = iωJ ∀v ∈ X

E × n = 0 on ΓPEC ∪ Γconductor

∇× E × n = 0 on ΓPMC

Assemble matrices

Aµ ≡ µ−1(∇× E ,∇× v)

Aσ ≡ σ(E , v)

Aε ≡ ε(E , v)

⇒ (Aµ + iωAσ − ω2Aε)(xreal + iximag ) = ib

Maxwell’s Equations

Time-Harmonic Maxwell’s Equations, [Hiptmair, 2002]

µ−1(∇× E ,∇× v) + iωσ(E , v)− ω2ε(E , v) = iωJ ∀v ∈ X

E × n = 0 on ΓPEC ∪ Γconductor

∇× E × n = 0 on ΓPMC

Assemble matrices

Aµ ≡ µ−1(∇× E ,∇× v)

Aσ ≡ σ(E , v)

Aε ≡ ε(E , v)

⇒ (Aµ + iωAσ − ω2Aε)(xreal + iximag ) = ib

Real symmetric system

Real symmetric system, [H. and Benner, 2013][Aµ − ω2Aε −ωAσ

−ωAσ −Aµ + ω2Aε

] [xrealximag

[0−b

A(ν) =Qa∑q=1

Θqa(ν)Aq = A1 + ωA2 + ω2A3

Matrices in the affine form will also be real symmetric ⇒RB computation in real arithmetics

real symmetric eigenvalue problem for β(ν), [H. et al., 2015]

BUT: system size doubles, here from 1′012 to 2′024

] [xrealximag

[0−b

A(ν) =Qa∑q=1

] [xrealximag

[0−b

A(ν) =Qa∑q=1

Quadratic OutputsOutput quantity s(ν) = |`(u)|Using the real form ⇒

s(ν) =√`1(u)2 + `2(u)2.

Define Q(·, ·) by `T1 `1 + `T2 `2 ⇒

s2(ν) = Q(u, u).

Expanded Formulation, [Sen, 2007]

A(ν) =

[2A(ν)− Q −Q−Q 2A(ν)− Q

], F =

For the parametric problem A(ν)x = F , it holds s2(ν) = Fx .BUT: system size doubles again, here from 2′024 to 4′048

s(ν) =√`1(u)2 + `2(u)2.

Define Q(·, ·) by `T1 `1 + `T2 `2 ⇒

s2(ν) = Q(u, u).

A(ν) =

[2A(ν)− Q −Q−Q 2A(ν)− Q

], F =

s(ν) =√`1(u)2 + `2(u)2.

Define Q(·, ·) by `T1 `1 + `T2 `2 ⇒

s2(ν) = Q(u, u).

A(ν) =

[2A(ν)− Q −Q−Q 2A(ν)− Q

], F =

For the parametric problem A(ν)x = F , it holds s2(ν) = Fx .

BUT: system size doubles again, here from 2′024 to 4′048

s(ν) =√`1(u)2 + `2(u)2.

Define Q(·, ·) by `T1 `1 + `T2 `2 ⇒

s2(ν) = Q(u, u).

A(ν) =

[2A(ν)− Q −Q−Q 2A(ν)− Q

], F =

Petrov-Galerkin RB

EM models contain resonances, i.e., A(ν) is singular.

A(ν) singular ⇐⇒ β(ν) = 0.

Problem: βN(ν) = 0 while β(ν) > 0.Solution: Supremizing operators T ν

T ν : X → X : (T νw , ·)X = a(w , ·; ν),

VN = span{u(ν1), u(ν2), ..., u(νN)},W ν

N = span{T νu(ν1),T νu(ν2), ...,T νu(νN)},

=⇒ βN(ν) ≥ β(ν)

Petrov-Galerkin RB

EM models contain resonances, i.e., A(ν) is singular.A(ν) singular ⇐⇒ β(ν) = 0.

T ν : X → X : (T νw , ·)X = a(w , ·; ν),

=⇒ βN(ν) ≥ β(ν)

Petrov-Galerkin RB

Problem: βN(ν) = 0 while β(ν) > 0.

Solution: Supremizing operators T ν

T ν : X → X : (T νw , ·)X = a(w , ·; ν),

=⇒ βN(ν) ≥ β(ν)

Petrov-Galerkin RB

T ν : X → X : (T νw , ·)X = a(w , ·; ν),

=⇒ βN(ν) ≥ β(ν)

PG RB - Quadratic OutputFull system

A(ν)x(ν) = b(ν), s2(ν) =(`T1 x(ν)

)2+(`T2 x(ν)

is projected as (W νT

N A(ν)VN

)x(ν) = W νT

N b(ν),

s2N(ν) =

(V TN `

T1 x(ν)

(V TN `

T2 x(ν)

A(ν) =Qa∑q=1

Θqa(ν)Aq, W ν

N =Qa∑q=1

Θqa(ν)W q

AN(ν) = W νT

N A(ν)VN =Qa∑q=1

Qa∑q′=1

Θqa(ν)Θq′

a (ν)Wq′T

N AqVN.

A(ν)x(ν) = b(ν), s2(ν) =(`T1 x(ν)

)2+(`T2 x(ν)

N A(ν)VN

)x(ν) = W νT

N b(ν),

s2N(ν) =

(V TN `

T1 x(ν)

(V TN `

T2 x(ν)

A(ν) =Qa∑q=1

Θqa(ν)Aq, W ν

N =Qa∑q=1

Θqa(ν)W q

AN(ν) = W νT

N A(ν)VN =Qa∑q=1

Qa∑q′=1

Θqa(ν)Θq′

a (ν)Wq′T

N AqVN.

A(ν)x(ν) = b(ν), s2(ν) =(`T1 x(ν)

)2+(`T2 x(ν)

N A(ν)VN

)x(ν) = W νT

N b(ν),

s2N(ν) =

(V TN `

T1 x(ν)

(V TN `

T2 x(ν)

A(ν) =Qa∑q=1

Θqa(ν)Aq, W ν

N =Qa∑q=1

Θqa(ν)W q

AN(ν) = W νT

N A(ν)VN =Qa∑q=1

Qa∑q′=1

Θqa(ν)Θq′

a (ν)Wq′T

N AqVN.

PG RB - Expanded Form

Full system

A(ν)x(ν) = F(ν), s2(ν) = FT x(ν),

N A(ν)VN

)x(ν) = W νT

N F(ν),

s2N(ν) = V T

N FT x(ν).

It is a compliant system ⇒ fast error decay expected

PG RB - Expanded Form

Full system

A(ν)x(ν) = F(ν), s2(ν) = FT x(ν),

N A(ν)VN

)x(ν) = W νT

N F(ν),

s2N(ν) = V T

N FT x(ν).

It is a compliant system ⇒ fast error decay expected

Transfer function

1. Motivation

5. Conclusions

Transfer function

2 4 6 8 102

ω in GHz

‖H(iω

)‖in

Figure: Transfer function over frequency range [0.6, 10] GHz.

Mean error in the output

50 100 150 200 250 300

10−3

10−2

10−1

Reduced order N

Figure: Mean relative error over sampled grid. Field estimator (blue), outputestimator using expanded form (green), heuristic optimum (red).

Max error in the output

50 100 150 200 250 30010−3

10−2

10−1

Reduced order N

Figure: Maximum relative error over sampled grid. Field estimator (blue), outputestimator using expanded form (green), heuristic optimum (red).

Conclusions

1. Motivation

5. Conclusions

Conclusions

expanded form improves approximation quality significantly

increased offline cost ⇒ applicable only to small or medium sizedmodels

sometimes even better than the heuristic optimum

actual optimum is infeasible to compute

Thank you for your attention!

Conclusions

References

H., M. W. and Benner, P. (2013).

Fast Evaluation of Time-Harmonic Maxwell’s Equations Using the Reduced Basis Method.IEEE Transactions on Microwave Theory and Techniques, 61:2265 – 2274.

H., M. W., Grundel, S., and Benner, P. (2015).

Estimating the Inf-Sup Constant in Reduced Basis Methods for Time-Harmonic Maxwell’s Equations.to be published: IEEE Transactions on Microwave Theory and Techniques.

Hiptmair, R. (2002).

Finite Elements in Computational Electromagnetism.Acta Numerica, pages 237 – 339.

Rozza, G., Huynh, D. B. P., and Patera, A. T. (2008).

Reduced Basis Approximation and a Posteriori Error Estimation for Affinely Parametrized Elliptic Coercive PartialDifferential Equations.Archives of Computational Methods in Engineering, 15:229 – 275.

Sen, S. (2007).

Reduced Basis Approximation and A Posteriori Error Estimation for Non-Coercive Elliptic Problems: Application toAcoustics.PhD thesis, Massachusetts Institute of Technology.

reduced basis model reduction of time-harmonic maxwell ......reduced basis model reduction of...

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