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Intro H∞ Damping Examples Concl

Model reduction of large scale second-ordersystems with modal damping

Christopher Beattie and Serkan GugercinVirginia Tech

Research supported under NSF DMS0505971 and DMS0513542

Sixth International Workshopon Accurate Solution of Eigenvalue Problems

Beattie Model reduction of second-order systems w/ modal damping

Intro H∞ Damping Examples Concl Structural Dynamics Structured Model Reduction

Structural Dynamics

Second-order dynamical system:

Mx(t) + Gx(t) + Kx(t) = b u(t),y(t) = cT x(t)

symmetric positive semidefinite M, G, K ∈ Rn×n

models distributed mass, damping, and stiffnessb, c ∈ R

n models spatial distribution of input and output.Mass (M) and stiffness (K) distributions are well-modeledbut damping distribution (G) is often heuristic.Need “input-output” map u 7→ y.

Frequency domain: y(ω) = H(ıω)u(ω)

Transfer function: H(s) = cT(Ms2 + Gs + K)−1bBut n can be too large for efficient capture of i/o map u 7→ y

Structural Dynamics

Second-order dynamical system:

Mx(t) + Gx(t) + Kx(t) = b u(t),y(t) = cT x(t)

symmetric positive semidefinite M, G, K ∈ Rn×n

models distributed mass, damping, and stiffnessb, c ∈ R

n models spatial distribution of input and output.Mass (M) and stiffness (K) distributions are well-modeledbut damping distribution (G) is often heuristic.Need “input-output” map u 7→ y.

Frequency domain: y(ω) = H(ıω)u(ω)

Transfer function: H(s) = cT(Ms2 + Gs + K)−1bBut n can be too large for efficient capture of i/o map u 7→ y

Structured Model Reduction

Construct a simpler input-output map u 7→ y that maintainsthe character of the mediating system.Generate, for some r � n, an r degree of freedom“condensed structure”

Mrxr(t) + Grxr(t) + Krxr(t) = br u(t),yr(t) = cT

r xr(t)

symmetric positive semidefinite Mr, Gr, Kr ∈ Rr×r

(condensed property matrices) and br, cr ∈ Rr.

Can maintain structure with a Ritz approximation onto asubspace Wr = Ran(W) with WTW = Ir and

Mr = WTMW, Gr =WTGW, Kr = WTKW,

br = WTb, and cTr = cT W

Choose Wr so yr ≈ y over a wide range of u(t).

r xr(t)

Intro H∞ Damping Examples Concl PerfMeas Interpolation RatKry

H∞ Performance Measure

‖yr − y‖L2 ≤ (small number) ‖u‖L2 ;Choose Wr to make (small number) small

Frequency domain:

Full response: y(ω) = H(ıω)u(ω)

Reduced order response: yr(ω) = Hr(ıω)u(ω)

with transfer functions:

H(s) = cT(Ms2 + Gs + K)−1b

Hr(s) = cTr (Mrs2 + Grs + Kr)

−1br

Uniformly small L2 error guaranteed by:

‖yr − y‖L2 ≤(

maxω∈R

|Hr(ıω) − H(ıω)|)

︸︷︷︸

small H∞ error

‖u‖L2

Approximation by Interpolation

Want a reduced order model leading to small H∞ error:maxω∈R |Hr(ıω) − H(ıω)|.

Suppose H(opt)r is a stable transfer function producing the

optimal H∞ error

maxω∈R

|H(opt)r (ıω) − H(ıω)| ≤ max

ω∈R

|Hr(ıω) − H(ıω)|

and Hr is a stable transfer function that interpolates H at2r + 1 points in the RHP.Then

minω∈R

|Hr(ıω) − H(ıω)| ≤ maxω∈R

|H(opt)r (ıω) − H(ıω)|

optimal H∞ error

maxω∈R

ω∈R

minω∈R

optimal H∞ error

maxω∈R

ω∈R

minω∈R

Related to ”near-circularity” in error of best uniform rationalapproximation on unit disk (Trefethen, 1981)complex analog of de la Valee-Poussin theorem.Hr(s) is close to the best H∞ approx to H(s) when

minω∈R

|Hr(ıω) − H(ıω)| ≈ maxω∈R

(that is, |H(ω) − Hr(ω)| ≈ constant)and

Hr(s) interpolates H(s) on at least 2r + 1 points in RHP

Best uniform approximations are hard to calculate.Interpolants are (comparatively) easy to calculate.

minω∈R

How to choose interpolation points ?

|H(ω) − Hr(ω)| ≈ constant ?Φ(s) = log |H(s) − Hr(s)|· has positive singularities at system eigenvalues.· has negative singularities at interpolation points.· is harmonic everywhere else

Φ is a potential function - electrostatic analogyLocate interpolation points (negative point charges) tobalance equipotentials of log |H(s) − Hr(s)| (makeslog |H(s) − Hr(s)| nearly constant along the imaginary axis)Interpolate at points that mirror singularities across theimaginary axis (but there are too many !)So mirror “equivalent charges” instead; e.g., Ritz values.Mirrored Ritz values are optimal choice for H2 minimizationas well (Gugercin/Antoulas/B [2004])

Rational Krylov-based model reduction

Examine the pointwise error: H(s) − Hr(s)

Define Kσ = Mσ2 + Gσ + K

H(σ) − Hr(σ) = cT [K−1

σ − Wr(Mrs2 + Grs + Kr)−1WT

So K−1σ b ∈ Wr implies H(σ) = Hr(σ).

Matching H′(σ) = H′r(σ) and higher moments can be done

similarly.

H(σ) − Hr(σ) = cT [K−1

= ct[I − Wr(Mrs2 + Grs + Kr)−1WT

r Kσ]K−1σ b

(factor out K−1σ )

similarly.

H(σ) − Hr(σ) = cT [K−1

= cT [I − Wr(Mrs2 + Grs + Kr)−1WT

r Kσ︸︷︷︸

projection onto Wr∆= Pr(σ)

]K−1σ b

similarly.

H(σ) − Hr(σ) = cT [K−1

r Kσ︸︷︷︸

]K−1σ b

= cT [I − Pr(σ)]K−1σ b

similarly.

H(σ) − Hr(σ) = cT [K−1

r Kσ︸︷︷︸

]K−1σ b

= 0 if K−1σ b ∈ Wr

similarly.

H(σ) − Hr(σ) = cT [K−1

r Kσ︸︷︷︸

]K−1σ b

= 0 if K−1σ b ∈ Wr

similarly.

K−1σ b ∈ Wr implies H(σ) = Hr(σ)

In particular, if for {σ1, σ2, . . . , σr, } ⊂ C,

span{K−1σ1

b, K−1σ2

b, . . . , K−1σr

b} = Wr

then H(σ) = Hr(σ) for σ = σ1, σ2, . . . , σr.Wr is not a rational Krylov subspace in the usual senseexcept in special circumstances.(Note there is no useful commutation property in general:KσiKσj 6= KσjKσi , KσiM−1Kσj 6= KσjM−1Kσi , orKσiK−1Kσj 6= KσjK−1Kσi)Bai[2003] discovered these interpolation conditions andcalled the associated spaces “second-order Krylovsubspaces”

span{K−1σ1

b, K−1σ2

b, . . . , K−1σr

b} = Wr

span{K−1σ1

b, K−1σ2

b, . . . , K−1σr

b} = Wr

Intro H∞ Damping Examples Concl Modal ShftSel PropDmp

Second-order systems with modal damping

Assume that KσiM−1Kσj = Kσj M−1Kσi independent of {σ`}.Then M, G, and K can be simultaneously diagonalized:

XTMX = I, XTKX = diag(ω2i ), XTGX = diag(γ2

Simple heuristic model of damping: γ2i = 2ξiωi

Modal damping can be characterized by a function g(z)that is real analytic on R+ that interpolates the values

g(ω2i ) = γ2

i and then G = Mg(M−1K)

More generally G = Mg1(M−1K)P1 + Mg2(M−1K)P2where P1 and P2 are complementary spectral projectors.Common choices:

g(z) = α + βz with α, β > 0 (“proportional damping”)g(z) = 2ξ

√z with ξ > 0 (“fractional damping”)

g(ω2i ) = γ2

The damped eigenvalue λ(ω) associated with ω satisfiesλ2 + λ g(ω2) + ω2 = 0.

Damped eigenvalues are constrained to curves in LHPRe(λ) ≤ g(ω2)

2 ; |λ| = ω

-6 -4 -2 2 4 6

Shift Selection for Modal Damping

Damped eigenvalues are constrained to lie on curves inLHP (change in stiffness or mass properties only changeseigenvalue distribution on the curve).Effective shift strategies (mirroring eigenvalue distribution)could be constructed on the basis of g(z).

-6 -4 -2 2 4 6

(largely independent of mass and stiffness)

-6 -4 -2 2 4 6

Replace w/ equivalent charges. (Balyage)

-6 -4 -2 2 4 6

Interesting special case: proportional damping

Mx + (αM + βK)x + Kx = b u(t).

All damped eigenvalues (system poles) are on circle withcenter:− 1

β, radius:

√1−αββ

and on ray (∞, − 1β].

Distribution depends on undamped natural frequencies,but usual elastic vibration models lead to distributions thatare close to “equilibrium condenser distributions”

β, radius:

√1−αββ

-6 -4 -2 2 4 6

β, radius:

√1−αββ

-6 -4 -2 2 4 6

Shift Selection for Proportional Damping

Only ONE shift is necessary - replace aggregate ofinterpolation points (negative charge distribution) with

single shift (an equivalent lumped charge) at σ∗ =

Optimal choice for condenser distribution of system poles;pretty good choice for most K and M.Wr is a true rational Krylov space in this case.

Intro H∞ Damping Examples Concl Condenser Beam1 Beam2

Exact Condenser Distribution• Pick α, β ∈ (0, 1)

2−√

1−αβ√

1−αβ−1 0 . . .

−1 2√1−αβ

0. . . 0

. 2√1−αβ

0 −1 2−√

1−αβ√

1−αβ

1−αβ√

1−αβ1 0 . . .

1 2√1−αβ

0. . . 0

. 2√1−αβ

0 1 2+√

1−αβ√

1−αβ

• G = α M + β KSingle shift is exactly equivalentto mirrored eigenvalue distribution !!

• Reduction from n = 2000 to r = 30 using a single shift• α = β = 0.05, b = c = [ 1 0 0 · · · 0 ]T .

−40 −35 −30 −25 −20 −15 −10 −5 0−20

Pole locations for exact condenser distribution

10−3

10−2

10−1

10−2

10−1

H∞ error vs interpolation point

H∞ e

• σ∗ =√

= 1 is the optimal shift.

A 1-D Beam Model

• n = 2000. α = 1/10, β = 1/500, b = e1, c = e200.

−1000 −900 −800 −700 −600 −500 −400 −300 −200 −100 0−500

−400

−300

−200

−100

Distributiion of Observed and Condenser Poles

Observed Poles

Condenser Poles

2 4 6 8 10 12 14

H∞ error vs interpolation point

• σ∗ =√

= 7.0711: Very close to being optimal.

Another 1-D Beam Model

• n = 200 and α = β = 1/300, b = c = e1.• Compare with balanced truncation and other shift selections• Balanced reduction done on ‘linearized’ system to r(1) = 40(r = 20).

10−3 10−2 10−1 100 101 102 10310−7

10−6

10−5

10−4

10−3

freq (rad/sec)

Amplitude bode plots for the beam model

10−3 10−2 10−1 100 101 102 10310−10

10−9

10−8

10−7

10−6

10−5

10−4

freq (rad/sec)

Amplitude bode plots of the error systems for the beam model

H − Hσ*

H − HBT

10−3 10−2 10−1 100 101 102 103

10−6

10−5

10−4

freq (rad/sec)

Amplitude bode plots for the beam model

σ = 1

σi, i=1:40

0 10 20 30 40 50 60 70 80 90 100−5.6

−5.4

−5.2

−4.8

−4.6

−4.4

−4.2

−3.8

−3.6

Beam with n=200, α = β = 1/300

σ = 1

σ = 5

σ = 10

Observed Convergence rate: 0.9428 Expected Convergence rate: 0.9867

• Convergence rates are very close to what is predicted by“condenser capacity” estimates.

Intro H∞ Damping Examples Concl

Conclusions

Considered strategies for structure-maintaining modelreduction of large scale second-order systems.

Mx + Gx + Kx = b u(t)

y(t) = cTx(t)

The use of modal damping models G = Mg(M−1K) canfacilitate model reduction through well chosen interpolationpoints.Proportional damping provides special computationaladvantages: an equivalent single shift (lumped charge) toaggregate interpolation points produced by mirroring poles.This single shift is exactly optimal for a class of mass andstiffness matrices (condenser distribution)Close to optimal in general.Future work? Extend to other damping models.

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