METHODS F OR THE INTERPOLA TION OF THE REDUCED MODELS

output 1 output 2 output 3 output 4

8.2. METHODS F OR THE INTERPOLA TION OF THE REDUCED MODELS

(P1) (P2) (A1) (A2)

referene

subspae

R V =

svd([V (p 1 ), . . . . . . , V (p K )])

^, ^SVD

of the projetion

matries

R V =

svd([ω 1 V (p 1 ), . . . . . . , ω K V (p K )])

weighted SVD

of the projetion

matries

R V = V (p j 0 )

proje-tionmatrixofhosen

referenemodel

R V = V (p j 0 )

proje-tionmatrixofhosen

referenemodel

trans-formation

matries

T j = M j =

(R _V ^T V (p j )) ⁻¹

T j = M j =

(R _V ^T V (p j )) ⁻¹

T j = U j Z _j ^T

^is ^given

by the SVD of

V (p j ) ^T R V = U j Σ j Z _j ^T

and the matrix

M j

is obtained as

M j = V (p j ) ^T E(p j )V (p j ) −1

T j = U j Z _j ^T

^is ^given

by the SVD of

V (p j ) ^T R V = U j Σ j Z ^T _j

and the matrix

M j

is obtained as

M j = V (p j ) ^T E(p j )V (p j ) −1

manifolds

for

inter-polation

no manifoldis

ho-sen

nomanifoldis

ho-sen

the manifold of real

n × n

n × m

^and

p × n

^matries,^depending

onwhihmatrixto

in-terpolate

the manifold of real

n × n

n × m

^and

p × n

^matries,^depending

onwhihmatrixto

in-terpolate - for

A(p ˆ j )

the manifold of the

non-singular matries

ishosen

Table8.3. Two-stepmethodsfortheinterpolationof

re-duedordermodels.

(Af-P1) (Af-P2) (Af-A1) (Af-A2) (Af)

Firststep Calulationofreduedordermodelsfortheane

param-etersinthenewparameterpoint

p new

^see^8.2.

Seond

step

in-terpolation

method

used

(P1),

see

Table

8.2

(P2),

see

Table

8.2

(A1),

see

Table

8.2

(A2),

see

Table

8.2

inter-polation

neessary

only

ane

pa-rameters

8.3. Redutionandinterpolationusingreformulationone

Tosimplifythepresentation,wextheparameters

µ, γ, ρ

^,^so^only^one

aneparameter

θ

^remains. ^After ^thereformulation(R1) andasalingof thematries

N 1

^and

N 2

^as^explainedⁱⁿ^Setion^8.1.1,^the^following^system

isobtained:

Σ bilin (θ) :

 



 

(E 0 + θE θ ) ˙ x(t) = A 0 x (t) + X 6 k=1

gN k u ^g _k (t)x(t) + Bu ^g (t), y (t) = Cx (t ),

(8.5)

with

u ^g (t ) = ₁

g(1+θ) θ

g T 0 T _∞ θT _∞ (1 + θ)S(t) T

, A 0 = A 0 (γ, ρ) + hA h0 (γ, ρ) + 1

1 + µ A 1

1+µ (γ, ρ) + µA µ (γ, ρ), N 1 = A 1

1+θ (γ, ρ),

N 2 = A θ (γ, ρ) + hA hθ (γ, ρ),

N 3 = · · · = N 6 (γ, ρ) = 0,

B =



  0 0

. .

0 0 1 1 + µ B 1

1+µ (γ, ρ) + µB µ (γ, ρ) + B 0 (γ, ρ) hB h0 (γ, ρ) hB hθ (γ, ρ) B S (γ, ρ)

.

Nowtheresultsforthelargemodelwith

n = 71, 978

^degrees^of

free-domfrom Setion 4.3 are disussed. As noted before, the

N k

^are ^large

andneedtobesaledbeforearedutionofthesaledsystem(8.5)anbe

performed.

UsingBIRKAas givenin Algorithm3and theKronekerprodut

approxi-mation(f. Setion6.1),wereduethemodelasgiven byequation(8.5)

atvedierentsamplingpoints

θ ∈ {0, 0.5, 1, 1.5, 2}

^to^a^redued^order^of

r = 700

^. ^After^the ^redution,^stable^models^are ^obtained^by ^using ^a^one

sidedprojetion

V

ⁱⁿ^the^last^model^(f. ^BIRKA-oSⁱⁿ^Setion^7.2.1.2).

Theinterpolation between the redued models at the samplingpoints is

ondutedusingmethods(P2),(A1)and(Af)fromSetion8.2.

We examine the temperature distribution at four dierent points in the

model: At the bottom of the housing, on the oil, in the upper bearing

andatthebottomoftherotor. Resultsfortheinterpolatedmodelsattwo

dierentparameterpoints

θ new ∈ {0.45, 1.65}

^for^two

{0, 2}

^,^three

{0, 1, 2}

andve

{0, 0.5, 1, 1.5, 2}

^sampling^points^an^be^foundⁱⁿ ^the^Figures^8.1

and8.2,fortherstandthefourthoutput,respetively.

Thequalityoftheapproximationimproveswithinreasingthenumber

ofsamplingpoints. Whenusing ve samplingpoints,the interpolated

re-duedmodelsfor

θ new ∈ {0.45, 1.65}

^yield^good^results ^for^the^rst ^three

outputs.Itseemshoweverdiulttoapproximatethefourthoutput,whih

evenwithvesamplingpointsonlyleadstogoodmodelsforthe

ap-proahviaaglobalprojetionmatrix(Af),as itanbeseeninFigure8.2.

Thismight berelated to thefat that this output lieson thebottom of

therotorandisnotdiretlyattahedto thestator(asmainheatsoure).

Henetheheatanonlybetransferredviahousingandange.

250 300 350 400 450 500 550

Temperature(K)

Twosamplingpoints

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6

relativeerror

250 300 350 400 450 500 550

Temperature(K)

Threesamplingpoints

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

relativeerror

0 200 400 600

250 300 350 400 450 500 550

Time(s)

Temperature(K)

Fivesamplingpoints

Original

θ = 0.45

^Original

θ = 1.65

(P2)

θ = 0.45

^(P2)

θ = 1.65

(A1)

θ = 0.45

^(A1)

θ = 1.65

(Af)

θ = 0.45

^(Af⁾

θ = 1.65

0 200 400 600

10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

Time(s)

relativeerror

Figure8.1. Firstoutput(bottomofthehousing),

inter-polation ofreduedorder models(r=700) in adierent

numberofsamplingpoints,withresultsindierent

inter-300 350 400

Temperature(K)

Twosamplingpoints

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6

relativeerror

300 350 400

Temperature(K)

Threesamplingpoints

10− 1

10− 2

10− 3

10− 4

10− 5

relativeerror

0 200 400 600

300 350 400

Time(s)

Temperature(K)

Fivesamplingpoints

Original

θ = 0.45

^Original

θ = 1.65

(P2,2sp)

θ = 0.45

^(P2,^2sp)

θ = 1.65

(A1,2sp)

θ = 0.45

^(A1,^2sp)

θ = 1.65

(Af, 2sp)

θ = 0.45

^(Af,^2sp)

θ = 1.65

0 200 400 600

10− 1

10− 2

10− 3

10− 4

Time(s)

relativeerror

Figure 8.2. Fourthoutput (bottomoftherotor),

inter-polation of redued order modelsin a dierent number

ofsamplingpoints,withresultsindierentinterpolation

Table8.4. Costs for the redution and interpolationof

themodelwithoneaneparameter.

Method Costs

OineRedutioninoneparameterpoint 1 weekper

sam-plingpoint

OnlineInterpolationwith(A1)or(A2) 20-25min

OnlineInterpolationwith(P1)or(P2) 10-15min

OineCalulationoftheglobalprojetion

matrix

20min

OnlineAssemblingofthemodelinthenew

parameterpoint

<1min

Aswehaveonsideredamodelinoneaneparameter,itwaspossible

to usethemethod viaaglobalprojetionmatrix(Af)andno (additional)

interpolationbetweenthereduedordermodels. Thismethodalwaysleads

togoodresults,andheneitanbereommendedwhenevertheparameter

dependenyis aneand thealulationoftheSVDof allmatries

V (θ j )

does not exeed the omputational apaity. Method (A1) outperforms

(P2) in approximationof the rstoutput (ve samplingpoints), whereas

(P2)performsbetterfortheoutputstwotofour.Hene,oneannotstate

thatoneinterpolationmethodisbetterthantheother.

The redution of the large model for one sampling point required up to

oneweekon

12

^CPUs^with

3

^GB^RAM^eah. ^So^samplingⁱⁿ^more^than^one

parameterwilleasilyexeedtheavailableresouresorleadtoextremelylong

simulationtimes 2

. Hene,theinterpolationmethodswillnowbetestedon

the smaller model with

n = 2, 969

^degrees^of ^freedom^from ^Setion^4.3.

Inaddition, we willhangethe reformulationmethod, anduse theseond

reformulation(f. Setion8.1.2, (R2)),as there willbeno needto sale

themodelspriortotheredution,aswehavenotedthatasalinginthe

N k

inreasesthereduedorder(f. Remark6.3.2).

Costs for theredution and interpolationan befound in Table8.4.

Exept for the redutionthat hasbeen performed on 12CPUs with 3GB

RAM eah, the alulations have been performed on visualizationnodes

AdisussionexplainingthelongsimulationtimesanbefoundinSetion7.2.3.

thatareusedsimultaneouslybydierentusers. Dependingon thememory

demandsandtheloadsoftheotherusers,thealulationtimesandier.

8.4. Redutionandinterpolationusingtheseondreformulation

Forthepresentationoftheresultsobtainedbyusingtheseond

refor-mulation(R2)(f. Setion8.1.2),themodelwith

n = 2, 969

^will^be^used.

IthasbeenpresentedinSetion4.3andisshowninFigures4.7and4.8. For

threedierentpointsthetemperatureproleismonitored: Onthebottom

ofthehousing(output1),onthestator(output2)andontheupperpart

oftherotor(output3).

To obtain stable redued order models the one-sided approah only

V(f. Chapter7.2)ishosen. Thisleads to largerreduedordersas an

originalBIRKAhoweverstabilityispreservedautomatily,whihisruial

fortheinterpolationsteps. Foreverysamplingpoint

p j = (θ j , µ j , γ j , ρ j )

^the

originalmodelwasreduedtoanorderof

r = 100

^. ^The^parameters

(θ _j , µ j )

are ane,and theparameters

(γ j , ρ j )

^are^non-ane, ^hene^our^explained

two-stepapproahapplies. Thesamplingpointsaregivenas:

2sp:

θ j , µ j ∈ {0, 2}

^and

γ j , ρ j ∈ {1, 3}

2 ⁴

^sampling^points

3sp:

θ j , µ j ∈ {0, 1, 2}

^and

γ j , ρ j ∈ {1, 2, 3}

3 ⁴

^sampling^points

5sp:

θ j , µ j ∈ {1, 0.5, 1, 1.5, 2}

^and

γ j , ρ j ∈ {1, 1.4122, 2, 2.5878, 3}

5 ⁴

^sampling^points

where

{1.0489, 1.4122, 2, 2.5878, 2.9511}

^are^the^Chebyhev^points^within

[1, 3]

^. ^We^use

1

^and

3

^instead^of

1.0489

^and

2.9511

^as^eah^of^the

param-etersisinthelosedinterval

[1, 3]

For the interpolation of the models, we will use four dierent methods.

First,aninterpolationinallfourparameters

(θ j , µ j , γ j , ρ j )

^will^be^performed

diretly(one-step approah)by usingthetwo interpolationmethods(A1)

and (P2). In addition, a two-stepapproah will be applied by using the

methods(Af-P2)and(Af-A1)-seeSetion8.2.

0 200 400 600 300

350 400 450 500

Time(s)

Temperature(K)

Firstoutput

Original

(P2,2sp) (A1,2sp) (Af-P2,2sp)

(Af-A1,2sp) (P2,3sp) (A1,3sp) (Af-P2,3sp)

(Af-A1,3sp) (P2,5sp) (A1,5sp) (Af-P2,5sp)

(Af-A1,5sp)

0 200 400 600

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

Time(s)

relativeerror

Figure8.3. Temperatureurvesfromreduedmodels

ob-tainedbyinterpolationwithdierentmethodsand

num-bers ofsamplepointsin point

θ = 1.67, µ = 1.78, γ = 2.36, ρ = 1.22

InFigure8.3 theresultsfortwo,threeandvesamplingpointsinthe

rstoutputfortheinterpolationpoint

p new 0 = (θ = 1.67, µ = 1.78, γ = 2.36, ρ = 1.22),

and redued order

r = 100

^are ^shown. ^For ^two ^sampling^points ^(dotted

lines)thetwo-stepmethods(i.e. (Af-P2)and(Af-A1))leadto better

re-sultsthantheone-stepmethods(i.e. (P2)and(A1)). Forthreesampling

points(dashedlines), theone-stepmethodsgetbetter ingeneral,and for

ve samplingpoints(dashdotted lines), the approximation using the

one-stepmethodsissuientlyaurateespeiallyfortheapproah(A1).

Consideringthreeotherinterpolationpoints

p new 1 = (θ = 1.67, µ = 1.78, γ = 2.976, ρ = 2.73),

p new 2 = (θ = 1.56, µ = 1.2, γ = 1.47, ρ = 1.634),

and

p new 3 = (θ = 0.34, µ = 0.13, γ = 1.134, ρ = 1.22),

theresultsfortheinterpolatedmodelsanbefoundinFigures8.4 to8.6.

Oneobservesthat one obtainsgood resultsforve samplingpointsin all

fourdierentinterpolationpoints

p new _i

^. ^There^are^however^dierenesⁱⁿ^the

qualityoftheapproximation. Thepoint

p new 1

^is^for^example^not^perfetly

approximatedbytheapproahes(Af-P2)and(Af-A1). Inaddition,onean

observeosillationsin theapproximationsby (Af-P2)and(Af-A1). They

ourwheneverthereisasignianthangeinthedynamisofthemodel.

In general: For few samplingpoints, the two-step methods (Af-A1)

and(Af-P2)(i.e. usingaglobalprojetionmatrixfortheaneparameter

dependenyandtheninterpolatingthenon-aneparameters)leadto

bet-terresultsthanadiretinterpolation. However,asthenumberofsampling

pointsinreases,theapproaheswithdiretinterpolation(i.e. (A1),(P2))

performas goodastheones withaglobalprojetionmatrixfortheane

parameters,orevenbetter. Hene,iftheredutioninonesamplingpointis

timeonsuming(asit isusingBIRKAf. Setion7.2.3),itisdesirable

tosampleasfewpointsaspossible.Ifthealulationofaglobalprojetion

matrixintheaneparametersisnottootimeonsuming,usingfew

sam-plingpointsandoneofthetwo-stepmethods((Af-A1)and(Af-P2))yields

satisfatoryresults.

300 350 400

T e m p e ra tu re (K )

First Output

10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

re la ti ve e rr o r

300 350 400 450

T e m p e ra tu re (K )

Second Output

10− 1 10−2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

re la ti ve e rr o r

0 200 400 600

280 290 300 310 320

Time (s)

T e m p e ra tu re (K )

Third Output

Original θ = 1.67, µ = 1.78, γ = 2.976, ρ = 2.73 (P2, 5sp)

(A1, 5sp) (Af & P2, 5sp) (Af & A1, 5sp)

0 200 400 600

10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

Time (s)

re la ti ve e rr o r

Figure8.4. Interpolationofreduedordermodelsinve

sampling points at

p new 1 = (θ = 1.67, µ = 1.78, γ =

2.976, ρ = 2.73)

300 350 400 450

T e m p e ra tu re (K )

First Output

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

re la ti ve e rr o r

300 350 400 450

T e m p e ra tu re (K )

Second Output

10− 2

10− 3

10− 4

10− 5

10− 6

re la ti ve e rr o r

0 200 400 600

270 290 310

Time (s)

T e m p e ra tu re (K )

Third Output

Original θ = 1.56, µ = 1.2, γ = 1.47, ρ = 1.634 (P2, 5sp)

(A1, 5sp) (Af & P2, 5sp) (Af & A1, 5sp)

0 200 400 600

10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

Time (s)

re la ti ve e rr o r

Figure8.5. Interpolationofreduedordermodelsinve

sampling points at

p new 2 = (θ = 1.56, µ = 1.2, γ =

1.47, ρ = 1.634)

300 350 400

T e m p e ra tu re (K )

First Output

10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

re la ti ve e rr o r

300 350 400

T e m p e ra tu re (K )

Second Output

10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

re la ti ve e rr o r

0 200 400 600

270 290 310

Time (s)

T e m p e ra tu re (K )

Third Output

Original θ = 0.34, µ = 0.13, γ = 1.134, ρ = 1.22 (P2, 5sp)

(A1, 5sp) (Af & P2, 5sp) (Af & A1, 5sp)

0 200 400 600

10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

Time (s)

re la ti ve e rr o r

Figure8.6. Interpolationofreduedordermodelsinve

sampling points at

p new 3 = (θ = 0.34, µ = 0.13, γ =

1.134, ρ = 1.22)

Table 8.5. Costs for the redution and interpolation of

themodelwithtwoaneandtwonon-aneparameters

redutionin

sampling

points(sp)

oine redutioninoneparameterpoint:

≈ 30

^min.

for2sp:

2 ⁴ · 30

^min

= 8

^h,

for3sp:

3 ⁴ · 30

^min

≈ 1.7

^days,

for5sp:

5 ⁴ · 30

^min

≈ 13

^days.

one-stepmethod

online Interpolationwith (A1)

< 10

^min,

(A2)

< 15

^min,

(P1)

< 10

^min,

(P2)

< 5

^min.

two-stepmethod

oine one globalprojetion matrixfor xed

non-aneparameters

(θ k , µ l , γ J ˆ , ρ J ˆ )

≈ 1

^min,

for2sp:

2 ² · 1

^min

= 4

^min,

for3sp:

3 ² · 1

^min

≈ 6

^min,

for5sp:

5 ² · 1

^min

≈ 25

^min.

online interpolationofnon-aneparameterswith:

(A1)<5s,

(A2)<10s,

(P1)<20s,

(P2)<5s.

InTable8.5 approximateostsfortheredutionandinterpolationare

summarized. Againthealulationshave beenperformedon visualization

nodes used simultaneously by dierent users. The alulation times are

thereforeonlyapproximationsdependingon load andavailablememory on

thenodes. Itis notsurprising, that the interpolationusing all parameter

pointsisslower that theone, where onlythe non-aneparameters need

tobeinterpolated. Ingeneral,(A1)isfasterthan(A2)and(P2)isfaster

than(P1). Thisisduetothefollowingbehavior: Theredutionin(P2)is

performedusingaweightedSVD.Weusetheweightsthatwillbeusedfor

thelinearinterpolationofthemodelsafterwards.Asonlythenearestmodels

withrespettothenewparameterpointareusedintheinterpolation,only

theprojetionmatries

V (p j )

^from^these^models^are^used^for^the^alulation

of the referene subspae

R V

^. ^In ^ontrast, ^all ^matries

V (p j )

^are ^used

forthe SVD in (P1). This explains longeralulation times. Duringthe

exeutionof (A1)and(A2),theinterpolationisdoneontangentialspaes

ofmatrixmanifolds. Thematriesneedto bemappedto thesespaesby

usingdierentlogarithms(seeTable5.1). Whereasthemanifoldof

n × m

matriesonlyinvolvesasubtration, themanifoldofnonsingularmatries

requiresaninversionandamatrixlogarithm.Thisleadstolongeralulation

times.

8.4.0.1. Interpolationmethods(A2)and(P1). Sofar,onlyresultsfor

theinterpolationmethods(A1)and(P2)havebeenpresented. Thisisdue

to the fat that the obtainedresults for the approahes (A2) and (P1)

arein mostases notasgood asthe resultsforthe otherapproahes. A

omparisonoftheapproahes(P1)and(P2)fortheinterpolationpoint

p new 0 = (θ = 1.67, µ = 1.78, γ = 2.36, ρ = 1.22),

anbefound in Figure8.7, andresults forthe approah(A2) forthe

in-terpolationpoint

p new 3

^are^shownⁱⁿ^Figure^8.8. ^Whereas^the^method^(P1)

usuallygivesreasonableresults,themethod (A2)hassigniantproblems

intheapproximationofthethirdoutputofthemodel.

Method(A2)fails toprovideareasonableapproximation. Thismight

berelatedtotheinterpolationproedure.First,allmatriesinthesampling

points

A(p j )

^(whih^belong^to^the^manifold

M

^of^thenon-singularmatries) needtobetransferredtothetangentialspaeregardingthereferenemodel

T A(p j 0 ) M

^,^then^a^lassi"interpolationinouraselinearinterpolation isperformedontheseelementsof

T A(p _j ₀ ) M

^.^It^is^not^lear,^that^the^lassi"

interpolationstaysinthetangentialspae,andhenetheinterpolatedmatrix

A(p new )

^might^lead^to ^inaurate^results.

300 350 400 450 500

T e m p e ra tu re (K )

First Output

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7

re la ti ve e rr o r

250 300 350 400 450 500

T e m p e ra tu re (K )

Second Output

10− 2 10− 3 10− 4 10− 5 10−6 10− 7 10− 8

re la ti ve e rr o r

0 200 400 600

270 290 310 330

Time (s)

T e m p e ra tu re (K )

Third Output

Original (P2, 5sp) (P1, 5sp)

0 200 400 600

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7

Time (s)

re la ti ve e rr o r

Figure8.7. Interpolationusingtheapproahes(P1)and

(P2) in ve sampling pointsat interpolation point

θ =

1.67, µ = 1.78, γ = 2.36, ρ = 1.22

300 350 400 450

T e m p e ra tu re (K )

First Output

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

re la ti ve e rr o r

300 350 400

T e m p e ra tu re (K )

Second Output

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

re la ti ve e rr o r

0 200 400 600

270 290 310 330

Time (s)

T e m p e ra tu re (K )

Third Output

Original Two sampling points

Three sampling points Five sampling points

0 200 400 600

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

Time (s)

re la ti ve e rr o r

Figure 8.8. Interpolation of redued order models in

two,threeandvesamplingpointsatinterpolationpoint

θ = 0.34, µ = 0.13, γ = 1.134, ρ = 1.22

^for^(A2).

8.4.1. Disussionofresults. Inthishapter,resultsfortheredution

and interpolation of thermal modelswith geometri variations have been

presented. Linear parametri models have been reformulated as bilinear

modelsintwodierentways(f.Setion8.1.1and8.1.2)andthenredued

usingBIRKAwith one-sided projetions(f. Setions6.2.2 and7.2.1.2).

First, results for the rst reformulation (R1) (f. Setion 8.1.1), for a

modelwith

n = 71, 978

^and^one ^geometrial^parameter^have^been ^shown

(f. Figures8.1and 8.2). Anadditionalpreproessingstepwas neessary

to avoid problemsresulting from the fat, that the norms of

N k

^and

A

areof thesamemagnitude. A salingwas introduedandleadto alarge

reduedorder

r = 700

^. ^The^seondreformulation(R2),Setion8.1.2,does notrequire this preproessing. Due to high omputational demands (f.

Setion7.2.3),allresultsfortheseondreformulationandfourparameters

have been presented for a smaller model with

n = 2, 969

^. Interpolation ofthismodel usingdierentnumbersofsamplingpointsandinterpolation

methods(f.Setion8.2)havebeenperformed.Ingeneral,allmethodsgive

reasonableresults. However, themethod (P2) usinga weightedSVD

toobtainthereferenesubspaeusuallyoutperformsthemethod (P1)

thenon-weightedSVD.Inaddition,it wasnotpossibletoobtaingood

results for the interpolation method on tangentialspaes of non-singular

matries(A2),whereastheinterpolationontangentialspaesof

R ^k×l

leads to good results (A1). The two approahes (A1) and (P2) usually give

omparableresults,heneit isnotpossibleto favoronemethod overthe

other.Havingtwoaneandtwonon-aneparameters,itisreommended

touse atwo-stepmethod rst alulatea globalprojetion matrixfor

theaneparametersandtheninterpolatethereduedordermodelsinthe

non-aneparameters. Forfewsamplepointsthesemethodsyield usually

betterresultsthantheonestepmethods.

Conlusions and Outlook

9.1. SummaryandConlusions 171

9.2. Futureresearh 173

9.1. SummaryandConlusions

Themain objetiveofthisworkwasto investigatetheuseofbilinear

H ₂

-optimal methodsin parametriModel Order Redution. Asshownby BennerandBreiten[11℄,itispossibletoreformulateaertainlassoflinear

parametrisystems as bilinearsystems(f. Setion5.3.2). The

parame-tersanthenbeonsideredasinputsandtheredutionanbeperformed

withoutanysamplingandinterpolationintheparameterspae,as mostof

theothermethodsforpMORdo[53,3,37,13℄. Afterobtainingabilinear

model,oneanmakeuseofbilinearModelOrderRedution. Inthiswork,

we fousedon two methods for bilinear

H ₂

-optimal Model Order Redu-tion,whiharedesribedinChapter5. BIRKA(f. Algorithm3),originally

obtainedbyBennerandBreiten[12℄,isstatedandnewalgorithmsforthe

bilinear

H ₂

-optimalredutionhavebeen developed. Thesealgorithmsuse optimizationon Grassmannmanifoldsandasamainadvantagean

preservestability. Wehaveproventhestabilitypreservationforsymmetri,

bilinearsystemsandanalyzedtheonvergenebehaviorofthealgorithms.

In addition to these theoretial results, several models for the

ther-mal analysisofeletrialmotors havebeen builtusing Comsol

3.5a (f.

Chapter3). LinearparametrisystemshavebeenexportedfromComsol

R by an analysisof theunderlyingequations (f. Chapter 4). For

industri-allyrelevantproblems,bothphysialandgeometriparametersneedtobe

onsideredandtheparameterdependenyaftertheredutionmustbe

pre-served. Astheresultingmodelsareusuallylarge(inourase

n = 41, 199

n = 71, 978,

^and

n = 2, 969

^),^the^bilinear

H ₂

-optimalredutionmethods havetobeapableofdealingwiththeselargesystems.

Thenewlydevelopedmethodsfortheredutionusingoptimizationon

Grassmannmanifoldsare,however,notyetready(f. Setion7.1)forthe

usewiththeselargesystems,butresultsfortheredutionofaheat

equa-tiononasquarehavebeenstated. BIKRA(f.Setion5.5,[12℄)isapable

ofreduingthelargemodels,butseveralproblemshavebeenidentied. In

some ases,the stinessmatrix

A

^is^singular,^the^magnitude^of ^the

N k

^is

toolargeandasalingneedstobeintrodued. Alsounstablemodelshave

been obtainedafter the redution. All theseissueshaven been examined

andsolutionshavebeenproposed(f.Chapter6).

Numerial results for the redution of two dierent types of models

have been obtained. On one hand, a part of an eletrialmotor model,

inorporatingphysialparameters, hasbeen onsidered. Thesemodelsare

parametrizedwithphysialparametersandhaveastruturethat easily

al-lowstoreformulatethemasabilinearmodel. RedutionwithBIRKAyields

good results, notonly in a ertain parameter interval, butglobally in the

wholeparameterrange(f. Chapter7.2,Figure7.8). Theseondtypeof

modelsareeletrialmotormodels,thatinadditiontothephysial

parame-tersuseparametersthatdesribehangesingeometry.Thisleadstomodels

withastruturethatannoteasilyberewrittenasabilinearsystem. Hene

one anreformulatethe model as abilinearmodel for ertainparameters

andinterpolatetheotherparameters(f. Chapter8).Fortheinterpolation,

severalwellknownmethodsfrompMORhavebeenused(f. [53,3,37℄),

whihgenerallyleadtogoodresults. Thereare,however,dierenesinthe

qualityoftheapproximation. For modelswith ananeparameter

depen-deneinertainparameters,usingaglobalprojetionmatrixfortheane

parameter dependeneleads to goodresults and anoutperforma diret

interpolation,espeiallyforfewsamplingpoints.

9.2. Futureresearh

Basedon theworkthathasbeen presented inthis thesis,several

op-portunitiesforfutureresearhhavebeenidentied:

•

^The ^new ^methods ^for ^the ^bilinear

H ₂

-optimal MOR using op-timizationmethodson theGrassmannmanifoldas developedin

Setions5.5.4and7.1 stillrequiresomeinvestigation:

TheAlgorithmsbilGFA, bilFGFAand bilSQA havenotyet

beentestedonlargeproblems,duetothefatthatoneneeds

tosolvelargebilinearSylvesterequations. Inthefuture,

low-rankapproximationstothesolutionsshouldbeappliedsuh

as theADI iteration (f. [57, 14℄),to allowtreatmentof

largesystems.

ConvergeneandthestabilitypreservationfortheAlgorithms

bilGFA,bilFGFAandbilSQAhavenotyetbeenestablishedfor

bilinearsystemswithnon-symmetri

A

^and

N k

Fortheoptimization,oneneedsto orretlysetseveral

pa-rametersto ensurea desentin the objetivefuntion. It

wouldbeanadvantageto identify robustriteriabased on

whihtheseparametersanbehosen.

•

^The^redution^of^the^large^parametri ^thermal^models^has^been

doneusingBIRKA[12℄. Theredutiontimesforourlargemodels

arewithintherangeofseveralhoursto afewdaysfor12 CPUs

with 3GB RAM(see Setion for adisussion7.2.3). However,

thestrutureofBIRKAwouldallowaparallelization,whihould

signiantlyreduetheredutiontime.

•

^OneinterpolationapproahbyAmsallem[3℄shows weak perfor-mane for some models (f. Setion 8.4.0.1). This ould be

ausedby thefatthatour usedinterpolationmethod doesnot

preservethe membershipin thetangentialspae. Thisbehavior

requiresa developmentofinterpolation proedures that dostay

ontheorrespondingmanifold.

•

^Theinterpolationmethodsusedfortheredutionofthe paramet-rimodelsrequiretheredutionatseveralsamplingpoints. The

number ofsamplingpoints hasa strongimpat on the

ompu-tationaldemands,soit isworthwhiletoexploremethodsto

sys-tematiallyandoptimallysampletheparameterspae,e.g. using

sparsegrids[10℄orlatinhyperubesampling[4,20℄.

Im Dokument Bilinear H 2-optimal Model Order-Reduction with applications to thermal parametric systems (Seite 174-200)

METHODS F OR THE INTERPOLA TION OF THE REDUCED MODELS

output 1 output 2 output 3 output 4

8.2. METHODS F OR THE INTERPOLA TION OF THE REDUCED MODELS

R V =

svd([V (p 1 ), . . . . . . , V (p K )])

R V =

svd([ω 1 V (p 1 ), . . . . . . , ω K V (p K )])

R V = V (p j 0 )

R V = V (p j 0 )

T j = M j =

(R V T V (p j )) −1

T j = M j =

(R V T V (p j )) −1

T j = U j Z j T

V (p j ) T R V = U j Σ j Z j T

M j

M j = V (p j ) T E(p j )V (p j ) −1

T j = U j Z j T

V (p j ) T R V = U j Σ j Z T j

M j

M j = V (p j ) T E(p j )V (p j ) −1

n × n

n × m

p × n

n × n

n × m

p × n

A(p ˆ j )

p new

µ, γ, ρ

θ

N 1

N 2

Σ bilin (θ) :

 



 

(E 0 + θE θ ) ˙ x(t) = A 0 x (t) + X 6 k=1

gN k u g k (t)x(t) + Bu g (t), y (t) = Cx (t ),

u g (t ) = 1

g(1+θ) θ

g T 0 T ∞ θT ∞ (1 + θ)S(t) T

, A 0 = A 0 (γ, ρ) + hA h0 (γ, ρ) + 1

1 + µ A 1

1+µ (γ, ρ) + µA µ (γ, ρ), N 1 = A 1

1+θ (γ, ρ),

N 2 = A θ (γ, ρ) + hA hθ (γ, ρ),

N 3 = · · · = N 6 (γ, ρ) = 0,

B =



  0 0

0 0 1 1 + µ B 1

1+µ (γ, ρ) + µB µ (γ, ρ) + B 0 (γ, ρ) hB h0 (γ, ρ) hB hθ (γ, ρ) B S (γ, ρ)

.

n = 71, 978

N k

θ ∈ {0, 0.5, 1, 1.5, 2}

r = 700

V

θ new ∈ {0.45, 1.65}

{0, 2}

{0, 1, 2}

{0, 0.5, 1, 1.5, 2}

θ new ∈ {0.45, 1.65}

250 300 350 400 450 500 550

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6

250 300 350 400 450 500 550

10− 1 10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

0 200 400 600

250 300 350 400 450 500 550

θ = 0.45

θ = 1.65

θ = 0.45

θ = 1.65

θ = 0.45

θ = 1.65

θ = 0.45

θ = 1.65

0 200 400 600

10− 2 10− 3 10− 4 10− 5 10− 6 10− 7 10− 8

(R _V ^T V (p j )) ⁻¹

(R _V ^T V (p j )) ⁻¹

T j = U j Z _j ^T

V (p j ) ^T R V = U j Σ j Z _j ^T

M j = V (p j ) ^T E(p j )V (p j ) −1

T j = U j Z _j ^T

V (p j ) ^T R V = U j Σ j Z ^T _j

M j = V (p j ) ^T E(p j )V (p j ) −1

gN k u ^g _k (t)x(t) + Bu ^g (t), y (t) = Cx (t ),

u ^g (t ) = ₁

g T 0 T _∞ θT _∞ (1 + θ)S(t) T

(θ _j , µ j )

2 ⁴

3 ⁴

5 ⁴

p new _i