progress in parametric model reduction · 2012-06-27 · matlab ® toolbox mrt (fraunhofer model...
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Progress in Parametric Model Reduction
J. Mohring, Paris, 15.12.2009
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Slide 2
Overview
• Need for parametric model reduction
• Optimal H2 controller for LMS concrete car benchmark
• The LBF acoustic aquarium
• Numerical improvements in Fraunhofer model reduction toolbox
• New theoretical results
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Slide 3
Need for parametric model reduction
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Slide 4
Need for parametric model reduction
• Fitting FE-models to measurements
• Treatment of non-linear boundary conditions
• Sensitivity analyses
• Model based controller design
• Simulation based design optimization
Whenever evaluating large FE-models
• must be repeated many times (e.g. optimization)
• is very expensive (e.g. frequency response)
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Slide 5
Optimal H 2 controller for
LMS concrete Car benchmark
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Slide 6
Optimal model based H 2 controller of LMS concrete car
Optimization parameters:Controller gains + Patch coordinates
• (x,y) of patch
Outputs:SPL at passengers’ heads
FE Model:multi-field, ~100.000 DOFs
Inputs:piezo voltage & volume velocity of loudspeaker
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Slide 7
Step 1: create reduced parametric model using Fraunhofer modelreduction toolbox
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Slide 8
1. Write usual APDL scriptwithout solution partAPDL script describing
parametric fully coupled
multi-field ANSYS®
model
ANSYS®
full file
with original
system matrices
and row-dof
assignment
Reduced
parametric
state space
model and
info on I/O
channels
MATLAB®
toolbox
MRT (Fraunhofer
Model Reduction
Toolbox)
ANSYS®
multiphysics
Read and modify
parameters
Call in batch mode
write
Set reduction
parameters,
import and reduce
ANSYS®
model
How to generate a parametric reduced model with MRT
future inputs
future parameters
potential input/output nodes
no export commands required!
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Slide 9
How to generate a parametric reduced model with MRT
FolientextAPDL script describing
parametric fully coupled
multi-field ANSYS®
model
ANSYS®
full file
with original
system matrices
and row-dof
assignment
Reduced
parametric
state space
model and
info on I/O
channels
MATLAB®
toolbox
MRT (Fraunhofer
Model Reduction
Toolbox)
ANSYS®
multiphysics
Read and modify
parameters
Call in batch mode
write
Set reduction
parameters,
import and reduce
ANSYS®
model
2. Change prametersin Matlab GUI
Select load inputsSelect parameters
Set parameter rangesSet order of interpolatingpolynomial
Select nodal inputs
Select nodal outputsChoose reduction parameters
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Slide 10
How to generate a parametric reduced model with MRT
APDL script describing
parametric fully coupled
multi-field ANSYS®
model
ANSYS®
full file
with original
system matrices
and row-dof
assignment
Reduced
parametric
state space
model and
info on I/O
channels
MATLAB®
toolbox
MRT (Fraunhofer
Model Reduction
Toolbox)
ANSYS®
multiphysics
Read and modify
parameters
Call in batch mode
write
Set reduction
parameters,
import and reduce
ANSYS®
model
3. Start reduction
APDL script is automatically modified
ANSYS is automatically called in batchmode and system written to full-file
ANSYS model is automatically importedand reduced to state space model
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Slide 11
APDL script describing
parametric fully coupled
multi-field ANSYS®
model
ANSYS®
full file
with original
system matrices
and row-dof
assignment
Reduced
parametric
state space
model and
info on I/O
channels
MATLAB®
toolbox
MRT (Fraunhofer
Model Reduction
Toolbox)
ANSYS®
multiphysics
Read and modify
parameters
Call in batch mode
write
Set reduction
parameters,
import and reduce
ANSYS®
model
How to generate a parametric reduced model with MRT
This is repeated for different automatically chosen parameters covering design space
1 1
1 1
2 2
2 2
3 3
3 3
4 4
4 4
5 5
5 5
6 6
6 6
9 9
9 9
7 7
7 7
8 8
8 8
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Slide 12
by direct reduction ?
How to get reduced models for new parameters ?
• Needs another ANSYS license
• Needs calculations taking minutes to hours
• Does not use reductions created before
�NO !
New reduced model is interpolated fromexisting ones
• without ANSYS
• within fractions of a second
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Slide 13
Step 2: Form optimal H2 controller
• Work done by colleague A. Wirsen
• Weighting functions put emphasis on range from 20 to 200 Hz
• To be published at ECCM 2010 (also Paris)
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Slide 14
LBF acoustic aquarium
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Slide 15
LBF acoustic aquarium
• Matthias Kurch (Fraunhofer LBF-Darmstadt) provides Ansys FE-model and measurements of “acoustic aquarium” via Tiziana and me.
• Plans: Common paper on fitting damping and stiffness of screw-rubber layer to measurements using parametric reduced model. (LBF: M. Kurch, ITWM: J. Mohring, S. Lefteriu, A. Wirsen)
cavity: filled with air,rigid walls
alu-plate
steel-frame
piezo-actuatorsrubber-layer
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Slide 16
LBF acoustic aquarium : aluminum-plate
• Dimensions: 300x280x2 mm3
• 3 x 2 actuators (A1, A2, A3)
• PIC 155 50x25x0.5
• 3 sensors (S1, S2, S3)
• PIC 155 10x10x0.1
With permission of M. Kurch (LBF)
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Slide 17
LBF acoustic aquarium: laser-scanning-vibrometer
• 99 measurement positions
• 3 measurements per excitation: microphone, vibrometer, piezo-sensor S2
• 298 frequency responses
• Hammer excitation
With permission of M. Kurch (LBF)
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Slide 18
• Aluminum-plate, steel-frame, rubber-layer: SOLID186
• Actuators and sensors: SOLID226
• Air volume: FLUID30
• Measurement positions coincide with nodes of FE-mesh
• 35537 nodes
LBF acoustic aquarium: finite element model
With permission of M. Kurch (LBF)
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Slide 19
LBF acoustic aquarium: need for fitting parameters of screw-rubber-layer
621,30 Hz 655,51 Hz
With permission of M. Kurch (LBF)
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Slide 20
Numerical improvements in Fraunhofer
model reduction toolbox
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Slide 21
Numerical improvements
• Depending on number of parameters and order of polynomial we have to build and reduce several 10 or 100 FE models
• This can be done in parallel without any communication,if enough cores and Ansys licenses are available
• Full parametric LMS car model takes about 45 min(12 evaluations, 32 cores, 5 Ansys licenses)
Parallel version
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Slide 22
Numerical improvements
• Continuation of common work with Julian
• Goal: reduce
• Instead of standard moment matching, i.e. computing
form incomplete SVD of operator T without forming matrix explicitly
U essential states to project on.
Efficient reduction of thermal problems with many inputs (area sources) ,Ex Ax Bu y Cx= + =&
( )21 1 1 1 1orth , , ,V A B A EA B A E A B− − − − − = L
( )1 1 1 1 1, , ,k
T A B A EA B A E A Bξ ξ− − − − − = L
*T USV≈
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Slide 23
New theoretical results on
parametric reduction via matrix
matching
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Slide 24
Idea of matrix matching
Interpolate matrices
…Bring realizations close to firstone by solving gen. Sylvester eqns. for similarity transform
…Reduced models
Independent reduction
…FE-models for selected
parameters
( )1 1 1 1, , ,E A B C% %% %
( )1 1 1 1( ), ( ), ( ), ( )E p A p B p C p ( )( ), ( ), ( ), ( )n n n nE p A p B p C p
( ), , ,k k k kE A B C% %% %
( )1 1 1 1, , ,E A B C( (( ( 1 1 1, , ,
k
k k k k k k k k k k
A
T E T T A T T B C T− − −
(
% %% %
14243
( ) ( )i ii
A p p Aω= Σ( (
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Slide 25
Success of matrix matching
• Poles move correctly with the parameters, even in case of eigenvalue-crossing
• FE-mesh may change topology
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Slide 26
Success of matrix matching
• Poles move correctly with the parameters, even in case of eigenvalue-crossing
• FE-mesh may change topology
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Slide 27
Assume . Expect .
Keep , but replace realization of by equivalent one:
with , i.e. .
Linear interpolation of matrices yields nonsense:
Choosing realizations optimal for interpolation
Consider linear interpolation of one-parametric class of standard state space systems
Models are already reduced, i.e. typical state space dimension < 100.
( ) [ ]( ), ( ), ( ) : ( ) ( ) , ( ) , 0,1p A p B p C p x A p x B p u y C p x pΣ = = + = ∈&
For illustration
1 0 ( , , )A B CΣ = Σ =Quality of interpolation depends strongly on chosen realization
Example 1
0.5 0 1Σ = Σ = Σ
( )1 11 , ,X AX X B CX− −′Σ = X I= − ( )1 , ,A B C′Σ = − −
( ) ( )0.5 0.5 , , 0. ( ,05 ,0), .A B C A B AC′Σ = + − − =
0Σ 1Σ
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Slide 28
Assume . Expect
Keep , but replace realization of by equivalent one:
Linear interp. is not minimal
Choosing realizations optimal for interpolation
Consider linear interpolation of one-parametric class of standard state space systems
Models are already reduced, i.e. typical state space dimension < 100.
( ) [ ]( ), ( ), ( ) : ( ) ( ) , ( ) , 0,1p A p B p C p x A p x B p u y C p x pΣ = = + = ∈&
For illustration
1 0
0 1 0 1 0, ,
0 0 1 0 1
αβ
Σ = Σ =
Quality of interpolation depends strongly on chosen realization
Example 2
0.5 0 1Σ = Σ = Σ
1
0 0 1 0 1, ,
0 1 0 1 0
βα
′Σ =
0.5
( ) / 2 0 1/ 2 1/ 2 1/ 2 1/ 2, ,
0 ( ) / 2 1/ 2 1/ 2 1/ 2 1/ 2
α βα β
+ ′Σ = +
0Σ 1Σ
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Slide 29
Choosing realizations optimal for interpolation
1( ) , ( ) , 1,2i i i i i iH s C M B M sI A i−= = − =
( )( )( )0 0 0( )pH s C p C M p M B p B= + ∆ + ∆ + ∆
Known : Interpolating transfer functions works fine for saway from poles
Idea: compare this interpolation with transfer-function of interpolated matrices at “save” s
Difference can be estimated
( )( )( )0 1 1 0
0 0 0 0 0 0
( ) (1 ) ( ) ( ), i.e. with
(1 )
pH s p H s pH s B B B
p C M B p C C M M B B
= − + ∆ = −
= − + + ∆ + ∆ + ∆
%
( ) ( )30 0 0
( ) ( )
(1 )
p pH s H s
p p C M B CM B C M B O∆
−
= − ∆ ∆ + ∆ ∆ + ∆ ∆ + ∆ ⋅
%
144444424444443
2 2 2
1 0 1 0 1 0
0 0 0 0
0 0
,
with , etc.
M M B B C C
B C M C
M B
α β γ
α β
∆ ≤ − + − + −
= =
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Slide 30
Choosing realizations optimal for interpolation
Idea: Minimize bound for difference over all equi-valentrealizations of
For (almost) unitary X this is (almost) equivalent to minimizing
Claiming necessary cond. with Frobenius-norm leads to generalized Sylvester equation for X
1Σ
2 2 21 11 0 1 0 1 0( )J X X M X M X B B C X Cα β γ− −= − + − + −%
2 2 2
1 0 1 0 1 0( )J X M X XM B XB C X Cα β γ= − + − + −
( )[ ] 0J X H H′ = ∀
( )* *, ,F
A A tr A A AB BA A A= = = =
( ) ( )( )
* * * *1 1 1 1 0 0 0
* *1 0 1 0
0
* *1 0 1 0M X M M
M M C C X X M
X
M B B
B B C CM
α γ α β
βα γ− = ++
+ + +
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Slide 31
Open questions
Stability
1pΣ2pΣ
3pΣ4pΣ
• Assume are stable . Are so the interpolated systems?
• Answered by Sanda
ipΣ
?pΣ
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Slide 32
Open questions
Unitarity • True error bound 2 2 21 1
1 0 1 0 1 0( )J X X M X M X B B C X Cα β γ− −= − + − + −%
2 2 2
1 0 1 0 1 0( )J X M X XM B XB C X Cα β γ= − + − + −
and the one used for finding optimal Transformation X
coincide only, if X is unitary .
• If some modes enter/leave with changing parameters, then X might become singular
• Robust construction of X by claiming unitarity explicitly
• Answered by Sanda
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Slide 33
Open questions
Influence of interpolationscheme ?
• global polynomial
• piecewise polynomial
• spline
• Answered by Sanda
How to scale different i/o’s (e.g. pressures, voltages)
• under investigation
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Slide 34
To be continued by Sanda
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Slide 35
Introduction
It is demonstrated
• How to optimize a smart structures application modeled by ANSYS®
(e.g. optimal position of piezo patch on firewall)
• which requires many expensive evaluations (e.g. frequency responses)
• by parametric reduced models
• using our Model Reduction Toolbox for ANSYS® / Matlab®
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Slide 36
Sample problem: optimal patch position for concrete car
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Slide 37
Sample problem: optimal patch position for concrete car
setup
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Slide 38
Sample problem: optimal patch position for concrete car
parametric
optimization• (x,y) of patch
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Slide 39
Sample problem: optimal patch position for concrete car
From multi-field Ansys model
to
parametric reduced Matlab
state space model
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Slide 40
Influence of patch position on efficiency
Pressure field by loudspeaker to be compensated at driver‘s ear (50 Hz)
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Slide 41
Influence of patch position on efficiency
Associated normal displacementof plate (firewall)
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Slide 42
Influence of patch position on efficiency
Admissible patch positions
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Slide 43
Influence of patch position on efficiency
Efficient position:pattern resembles the oneinduced by loudspeaker
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Slide 44
Influence of patch position on efficiency
Inefficient position:pattern “orthogonal” to the
one induced by loudspeaker
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Slide 45
Need for parametric reduced model
• In general, patch position must be uniformly effective on a whole frequency range
• Evaluating position requires computing frequency responseof transfer function loudspeaker ���� driver’s ear
• ANSYS® requires 160s for single frequency (Xeon 2.5 GHz)
� We need reduced model
• Many different patch positions must be tried:several local minima , Newton iteration about each minimum
� We need parametric model which can be evaluated at new positions without creating and reducing new ANSYS ®
models
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Slide 46
Parametric reduced model
Admissible region
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Slide 47
Parametric reduced model
Multi-field Ansys model for parameters p1 = (x1,y1)
create
Admissible region
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Slide 48
Parametric reduced model
state space model for parameters p1 = (x1,y1)
reduce
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Slide 49
Parametric reduced model
Multi-field Ansys model for parameters p2 = (x2,y2)
create
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Slide 50
Parametric reduced model
state space model for parameters p2 = (x2,y2)
reduce
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Slide 51
Parametric reduced model
state space model for parameters p9 = (x9,y9)
reduce
Multi-field Ansys model for parameters p9 = (x9,y9)
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Slide 52
Parametric reduced model
Reduced state space model for other parameters
by reduction
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Slide 53
Parametric reduced model
Reduced state space model for other parameters
xx
x
by reduction
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Slide 54
Parametric reduced model
Reduced state space model for other parameters
by interpolation
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Slide 55
Challenge 1: reduction of multi-field FE models
• Consider multi-field Ansys model of concrete car for fixed parameter set
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Slide 56
Challenge 1: reduction of multi-field FE models
• Consider multi-field Ansys model of concrete car for fixed parameter set
• Built-in reduction schemes of Ansys and Matlab fail due to non-symmetric/singular matrices or non-Rayleigh damping and size (~100.000 DOFs), respectively.
singular non_Rayleigh damping non-symmetric
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Slide 57
Challenge 1: reduction of multi-field FE models
Solution
• Turn 2nd order system into 1st order system
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Slide 58
Challenge 1: reduction of multi-field FE models
Solution
• Turn 2nd order system into 1st order system
• Turn to shift and invert form (largest eigenvalues correspond to those of original system close to shift s0). With T = A –s0E:
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Slide 59
Challenge 1: reduction of multi-field FE models
Solution
• Turn 2nd order system into 1st order system
• Turn to shift and invert form (largest eigenvalues correspond to those of original system close to shift s0). With T = A –s0E:
• Project on nxk matrices V,W with W*V=I, k<<n from modal analysis or Arnoldi process
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Slide 60
Challenge 1: reduction of multi-field FE models
Contribution by fellows
• Sanda: structure preserving reduction: Reduced model is again of 2nd order.
• Julian: hybrid reduction scheme for systems with many inputs and outputs combining modal truncation and moment matching.
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Slide 61
Challenge 1: reduction of multi-field FE models
Contribution by fellows
• Sanda: structure preserving reduction: Reduced model is again of 2nd order.
• Julian: hybrid reduction scheme for systems with many inputs and outputs combining modal truncation and moment matching.
J. Stoev and J. Mohring. Hybrid reduction of non-symmetric mimo systems. Submitted to Systems & Control Letters (Status: second revision)
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Slide 62
Challenge 2: interpolation of reduced state space m odels
Polynomial interpolation
Polynomial interpolation of scalar function (example: 1 parameter x, 2nd order)
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Slide 63
Challenge 2: interpolation of reduced state space m odels
Polynomial interpolation
Polynomial interpolation of scalar function (example: 1 parameter x, 2nd order)
System matrices may be interpolated in exactly the same way:
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Slide 64
Challenge 2: interpolation of reduced state space m odels
How to choose realizations?
• Realizations and
have same transfer function
i.e. they describe same system.
• Interpolating badly chosen realizations may completely fail:
Consider one-parametric family of single state SISO systems
Realizations of same system (T=-1), but linear interpolation gives .
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Slide 65
Challenge 2: interpolation of reduced state space m odels
Looking for way out
• Interpolate (unique) transfer function
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Slide 66
Challenge 2: interpolation of reduced state space m odels
Looking for way out
• Interpolate (unique) transfer function
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Slide 67
Challenge 2: interpolation of reduced state space m odels
Looking for way out
• Interpolate (unique) transfer function
• Consider normal forms based onsorted eigenvalues:
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Slide 68
Challenge 2: interpolation of reduced state space m odels
Looking for way out
• Interpolate (unique) transfer function
• Consider normal forms based onsorted eigenvalues:
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Slide 69
Challenge 2: interpolation of reduced state space m odels
Looking for way out
• Interpolate (unique) transfer function
• Consider normal forms based onsorted eigenvalues:
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Slide 70
New approach
• Select arbitrary realization for first parameter as reference
• Choose realizations for other parameters as close as possible to reference
Challenge 2: interpolation of reduced state space m odels
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Slide 71
New approach
• Select arbitrary realization for first parameter as reference
• Choose realizations for other parameters as close as possible to reference
• Formally: find transform T minimizing
Challenge 2: interpolation of reduced state space m odels
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Slide 72
New approach
• Select arbitrary realization for first parameter as reference
• Choose realizations for other parameters as close as possible to reference
• Formally: find transform T minimizing
leading to matrix equation for T
Challenge 2: interpolation of reduced state space m odels
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Slide 73
• ANSYS® model is created and reduced at 9 positions
• Reduced models are interpolated bilinearly
• Efficiency function is evalutaed at
50x50=2500 positions, which takes 100s altogether
Optimal patch positions for different frequency ran ges
min max
1/ 22
ker( ) / ( )k
piezo k spea kh i h iω ω ω
ε ω ω≤ ≤
=
∑
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Slide 74
Optimal patch positions for different frequency ran ges
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Slide 75
Optimal patch positions for different frequency ran ges
Response to loudspeaker:- independent of patch position- counterchecked with ANSYS®
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Slide 76
Optimal patch positions for different frequency ran ges
Response to patch- in 9 positions where ANSYS® modelswere created and reduced
- difference > 20dB
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Slide 77
Optimal patch positions for different frequency ran ges
Response for patch position optimally effective at 50Hz computed- for interpolated reduced model (small effort)
- for directly created and reduced model (big effort)
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Slide 78
instead of Conclusion:
Request for collaboration
Are there fellows interested in optimizing their smart structures design with MRT, possibly at Kaiserslautern?