Modal Truncation Augmentation and Moment Matching · PDF fileModal Truncation Augmentation and...

Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands

Modal Truncation Augmentation and Moment Matching:

two faces of the same coin.

Daniel Rixen Delft University of Technology,

The Netherlands


Page 2

Page 2


Page 3

Content

Preliminary remarks

1.! Modal Truncation, Mode acceleration correction, Truncation augmentation

2.! Substructuring in Structural Dynamics

3.! Craig-Bampton method

4.! Craig-Bampton with Modal Truncation Augmentation


Page 4

Preliminary remarks

1. Reduction methods for i/o transfers VS. reduction for entire model

In structural dynamics typically interested in finding locations of maximum stresses or global mode shapes. So the aim is different than in control for instance.

2. A reduction method must be

•! Robust V

•! Automatic ?

Tuning free if possible, but not a black box!

If you know something about the specific problem you solve, use it!


Page 5

Page 5

Mode Superposition


M ̈ q + C ̇ q + K q = f

¨ ! s + " s ˙ ! s + # 2

s ! s = x T

s f ( t )

N decoupled oscillators

Linear dynamics

x T r M x s = $ r s

x T r K x s = $ r s #

2 s q =

N !

s = 1

x s ! s ( t )

" s = x T s C x s

Small damping

Free vibration modes


Page 6

Page 6

Modal truncation


In practice, N >> one can compute k<<N modes

Mode displacement approximation

Spatial convergence : x T

r f ( t ) ~ 0

Spectrale convergence: # r

# excit

<<

q =

!

s = 1

x s ! s ( t )

N

+

N !

r = k +1

x r ! r ( t )

k


Page 7

Page 7

Modal acceleration correction


Include contribution of higher modes to quasi-static solution

q =

!

s = 1

x s ! s ( t ) + !

r = k +1

x r ! r ( t )

N k

# 2

r ! r

= x T

r f ( t ) ¨ ! s + "

s ˙ ! s

+ # 2

s ! s

= x T

s f ( t )

Mode Acceleration approximation

[ Rayleigh (1877), Theory of Sound, chap V, § 100 ] [ Williams (1945)]

K ! 1 f ( t )


Page 8

Page 8

Modal acceleration correction


q =

k !

s = 1

x s ! s ( t ) +

N !

r = k + 1

x r

x T r f ( t )

# 2 r

=

k !

s = 1

x s ! s ( t ) +

"

K ! 1 !

k !

s = 1

x s x T s

# 2 s

#

f ( t )

static correction x c o r =

Better convergence than MDM if ~ 0 # excit


Page 9

Page 9

Modal acceleration correction (higher orders)


Other interpretation of the MAM:

M ̈ q + K q = f

Relative solution q = K ! 1 f ( t ) + y

M ̈ y + K y = ! M K ! 1 ̈ f ( t ) = f y

y =

k !

s = 1

x s % s ( t ) Modal displacement on


Page 10

Page 10



M ̈ q + K q = f

Relative solution q = K ! 1 f ( t ) + y

M ̈ y + K y = ! M K ! 1 ̈ f ( t ) = f y

Relative solution of the relative solution ... y = K

! 1 f y + z

q = K ! 1 f ( t ) ! K

! 1 + z M K

! 1 ̈ f ( t )


Page 11

Page 11



" #

n ! 1 !

j = 0 q =

!

s = 1

x s ! s ( t ) + x c o r

k

j

Generalized Mode Accelerations

[ Leung (83): Force Derivative Method] [ Akgün (93)], [Liu et al. (94)] [Rixen (00)]

Define high order corrections

K ! 1 !

k !

r = 1

x ( r ) x T ( r )

# 2 r

$ ! M K ! 1

% j ! 1 x c o r j = &

2 j ! 2 f

& t 2 j 2 !


Page 12

Page 12

Modal truncation augmentation


Generalized MAM:

n ! 1 !

j = 0

q = x s ! s ( t ) + x c o r

!

s = 1

k

j

IDEA: Corrections = Ritz vectors

Compute correction for unit

Amplitudes of are d.o.f x c o r j

& 2 j ! 2 f

& t 2 j 2 !

K!1 !

k!

r=1

x(r)xT(r)

#2r

$!MK!1%j!1

xcorj= f(1)

" #


Page 13

Page 13

Modal truncation augmentation


' j ( t ) x s ! s ( t ) +

n ! 1 !

j = 0

x c o r j

q =

Augmentation by modal truncation

[Dickens-Wilson (80 )] [ Rixen(00)]

!

s = 1

k

K!1 !

k!

r=1

x(r)xT(r)

#2r

$!MK!1%j!1

xcorj= f(1)

!! Clear similarity with moments (around 0) !! Can be obtained “for free” if eigenmodes computed by a Krylov method, starting with the static solution


Page 14

Page 14

Summary


Mode Displacements MDM q =

k !

s = 1

x s ! s ( t )

Generalized MAM

n ! 1 !

j = 0

q =

!

s = 1

x s ! s ( t ) + x c o r

k

j

Generalized MTA x s ! s ( t ) +

n ! 1 !

j = 0

x c o r j ' j ( t ) q =

!

= 1

k

s


Page 15

Page 15

Remarks


•! Modal truncation yields exact static response (and derivatives) matching moments at 0

•! Can easily be generalized for quasi-static corrections around specific frequencies matching moments at other frequencies

•! Significantly improves the truncated mode superposition (especially around expansion frequencies and in particular for the stress evaluation).

•! Orthogonalization of the corrections (to improve numerical robustness):

solve the reduced e.v.p.

then

&

X T c o r K X c o r

'

Z =

&

X T c o r M X c o r

'

Z !

!

X c o r X c o r Z

give an indication of the upper frequency range cover by the corrections

Fully decouples the reduced problem (i.e. correction modes can be handled as true modes)


Page 16

Page 16

Remarks


•! Why not replace all eigenmodes in the approximation basis by correction modes? Because we want the approximate solution to have the correct resonances!

•! Modal Truncation Augmentation is useful for “a posteriori” reduction (i.e. if one can compute the modes of the full model and use the reduced model for transient/harmonic dynamic simulations)

•! But how to reuse the same good ingredients for “a priori” reduction (i.e. to reduce the system without having to compute the eigenmodes and corrections on the full system)?


Page 17

Content

Preliminary remarks






Page 18

Page 18

Large models and how they are made

2. Substructuring

•! Many elements needed to properly represent geometry •! Many elements used because model also for static analysis •! Different groups (companies) create model of different parts •! Companies (or departments in a company) do not want to share the full model of their subpart!

o! One starts from large (sub-)models o! Not always possible to build the full model (because too big or because of confidentiality)

A priori model reductions based on reduction of sub-parts

Esa/Estec


Page 19

Page 19

Idea of substructuring

2. Substructuring

q b

q b (s)

" (s)

Create “Super-Elements” or “Macro-elements”

•!Easy to handle for FE codes •!Natural parallel procedure •!Hides the details of the subpart (confidentiality) •!Allows to localize modification during design •!Easy to reuse in other system where the same component is used

For a substructure, the interface DOFs are seen as inputs/outputs


Page 20

Content

Preliminary remarks






Page 21

Page 21

Feelings of a substructure

3. Craig-Bampton

q ( s ) i

Boundary displacements q ( s ) b

Internal displacements

M ( s ) i i ¨ q

( s ) i + K

( s ) i i q

( s ) i = ! K

( s ) i b q

( s ) b ! M

( s ) i b ¨ q

( s ) b

In each substructure (s) :

computed by modal superposition of fixed interface modes : q ( s ) i

! K ( s ) i b K ( s ) ! 1

i i q ( s ) b q

( s ) i = + # ( s ) " ( s )

static solution

[Craig-Bampton]


Page 22

Page 22

Craig-Bampton reduction space

3. Craig-Bampton

q ( ( ( )

* * * +

( ( ( ) q =

,

( ( ( )

b q ( 1 ) i

. . . q

( N s ) i

-

* * * + "

, I 0 · · · 0 $ ( 1 ) # ( 1 ) 0 . . . . . .

$ ( N s ) 0 # ( N s )

- , q b " ( 1 )

. . . " ( N s )

-

* * * +

T C B

Vibration modes (fixed interface) ( k<N )

& K

( s ) i i ! #

( s ) 2 k M

( s ) i i ' %

( s ) k = 0

Static modes (all)

= $ ( s ) ! K ( s ) i b K ( s ) ! 1

i i


Page 23

Page 23

Reduced matrices

3. Craig-Bampton

¯ K C B = T T C B K T C B =

, ( ( ( ( )

S b b 0 & ( 1 )

2

. . . 0 & ( N s )

2

- * * * * +

¯ M C B = T T C B M T C B =

, ( ( ( ( )

M # b b M ( 1 ) b ( · · · M ( N s ) b ( M ( 1 ) ( b I 0

. . . . . . M ( N s ) ( b 0 I

- * * * * +

S b b M # b b : assembly of statically condensed matrices

(Schur complement, Guyan)

Nice “diagonal” structure (Structure preserving?) 0 moment at 0 preserved, but can we do more ?


Page 24

Content

Preliminary remarks






Page 25

Page 25

New feelings of a substructure

4. Craig-Bampton with MTA

q ( s ) i

Boundary displacements q ( s ) b

Internal displacements

M ( s ) i i ¨ q

( s ) i + K

( s ) i i q

( s ) i = ! K

( s ) i b q

( s ) b ! M

( s ) i b ¨ q

( s ) b

In each substructure (s) :

! K ( s ) i b K ( s ) ! 1

i i q ( s ) b q

( s ) i = + q

( s ) i

~


Page 26

Page 26



! K ( s ) i b K

( s ) ! 1 i i q

( s ) b q

( s ) i = + q

( s ) i

~

M(s)

ii¨ +K

(s)

i i = q̈( s) bM

( s)

i i M( s)

i b!K(s)ibK

(s)! 1

ii( )

Inertia forces inside substructure associated to static modes

= Y(s)q̈

( s) b

M ( s ) i i ¨ q

( s ) i + K

( s ) i i q

( s ) i = !K

( s ) i b q

( s ) b !M

( s ) i b ¨ q

( s ) b

q ( s ) i

~ q ( s ) i

~

= K ( s ) ! 1

i i Y ( s ) ¨ q

( s )

b + z ( s )

q ( s ) i

~


Page 27

Page 27



q ( s )

i = ! ( s ) q

( s ) b + "

( s ) " ( s ) + X

( s ) c o r '

( s )

Static modes Vibration modes (fixed interface)

Projected high order static correction modes

Make M- and K- orthogonal to " ( s )

x ( s ) c o r . j =

& K

( s ) ! 1

i i ! # ( s ) &

( s ) ! 2 #

( s ) T ' &

M ( s ) i i K

( s ) ! 1

i i

' j ! 1

Y ( s )

&

X ( s ) T c o r K

( s ) X

( s ) c o r

'

Z ( s ) =

&

X ( s ) T c o r M

( s ) X

( s ) c o r

'

Z ( s ) !

( s )

Make columns of M- and K- orthogonal : solve the reduced eigenvalue problem X ( s ) c o r

X ( s ) c o r

If to many interface DOFs, compute correction modes only for interface approximation modes (e.g. [Bourquin 92]) X b

Y ( s )

Y ( s ) X b

Prof. Antoulas would call this “reduction in tangential directions for the inputs/outputs” ?!


Page 28

Page 28

New approximation space


,

( ( ( ( )

u b

u ( 1 ) i . . .

u ( N s ) i

-

* * * * + "

,

( ( ( ( )

I 0 0 · · ·

$ ( 1 ) # ( 1 ) ˜ X ( 1 ) c o r 0

. . .

$ ( N s ) 0 # ( N s ) ˜ X

( N s ) c o r

-

* * * * +

,

( ( ( ( ( ( ( ( )

u b

" ( 1 )

( ( 1 )

. . .

" ( N s )

( ( N s )

-

* * * * * * * * +

T C B . e x t

Extended Craig-Bampton reduction matrix [Dickens ’00, Rixen ‘02]

Summarize:

-! Full first order matching at 0 for the interface inputs/outputs (static modes)

-! Truncated representation of the internal dynamics -! Higher order matching on tangential directions (internal MTA assuming interface modes)


Page 29

Page 29

New reduced matrices


¯ K C B . e x t =

,

( ( ( ( ( ( ( ( )

S b b 0

& ( 1 ) 2

! ( 1 )

. . .

& ( N s ) 2

0 ! ( N s )

-

* * * * * * * * +

¯ M C B . e x t =

,

( ( ( ( ( ( ( ( ( )

M # b b M

( 1 ) b ( M

( 1 ) b X · · · M

( N s ) b ( M

( N s ) b X

M ( 1 ) ( b I

M ( 1 ) X b I 0

. . . . . .

M ( N s ) ( b I

M ( N s ) X b 0 I

-

* * * * * * * * * +

Easy to implement in existing Craig-Bampton routines


Page 30

Page 30

Example


Truss frame with beams

690 dof / substructure

Assume number of internal modes per substructure fixed

! # r = ˜ # r ! # r

# r Relative eigenfrequency error

! f = $ ( K a ! ˜ # r M a ) ̃ x ( r ) $ 2

K a ̃ x ( r ) 2 Relative force residual of modes

$ $


Page 31

Page 31

Example


50 internal modes per subdomain

10 interface modes for X ( s ) c o r

Mode number Mode number

!"# !f

10 0

10 -5

10 -10

10 -15

10 0

10 -2

10 -4

10 -6

10 -8

no X ( s ) c o r

j = 1

j = 1, 2

j = 1, 2, 3


Page 32

•! Engineers would be well inspired to look at mathematical research in model reduction

•! Mathematicians might get novel ideas for real problems from structural engineers

•! More workshops like this are needed to stimulate both communities

Conclusion

Future possibilites

•! What is the right mix of internal modes and corrections modes?

•! Define strategies to optimally choose the expansion points for correction modes (IRKA?)

•! Extend the to multi-physical problems usually non-symmetric (structure preserving ideas?)

•! Substructuring for parametric (sub-)models (sampling strategies, parametric matching …)

Modal Truncation Augmentation and Moment Matching · PDF fileModal Truncation Augmentation and...

Documents

Transcript of Modal Truncation Augmentation and Moment Matching · PDF fileModal Truncation Augmentation and...