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Page 1 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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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!
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Mode Superposition
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Modal truncation
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
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
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Modal acceleration correction
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
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 )
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Modal acceleration correction
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
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
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Modal acceleration correction (higher orders)
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Modal acceleration correction (higher orders)
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
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 )
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Modal acceleration correction (higher orders)
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
" #
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 !
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Modal truncation augmentation
1.! Modal Truncation, Mode acceleration correction, 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)
" #
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Modal truncation augmentation
1.! Modal Truncation, Mode acceleration correction, 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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Summary
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
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
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Remarks
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
•! 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)
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Remarks
1.! Modal Truncation, Mode acceleration correction, Truncation augmentation
•! 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)?
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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]
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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 ?
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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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
~
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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New feelings of a substructure
4. Craig-Bampton with MTA
! 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
~
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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New feelings of a substructure
4. Craig-Bampton with MTA
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” ?!
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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New approximation space
4. Craig-Bampton with MTA
,
( ( ( ( )
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)
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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New reduced matrices
4. Craig-Bampton with MTA
¯ 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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Example
4. Craig-Bampton with MTA
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
$ $
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Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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Example
4. Craig-Bampton with MTA
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
Autumn School on Future Developments in Model Order Reduction September 21-25, 2009, Tershelling, The Netherlands
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•! 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 …)