pod/mac-based modal basis selection for a reduced · pdf filea reduced order nonlinear...

POD/MAC-Based Modal Basis Selection fora Reduced Order Nonlinear Response Analysis
Stephen A RizziStephen A. RizziNASA Langley Research Center, Hampton, VA
and
Adam PrzekopNational Institute of Aerospace, Hampton, VA
Presented byKim S BeyKim S. Bey
NASA Langley Research Center, Hampton, VA
Fundamental Aeronautics Annual MeetingFundamental Aeronautics Annual MeetingNew Orleans, LA, October 30 November 1, 2007
Probabilistic Analysis and Design ToolsProbabilistic Analysis and Design Toolsfor High-Cycle Fatigue Resistant Hypersonic Vehicle Structures
NNX07AC26A

Motivation: High Cycle Fatigue Analysis
Fatigue life estimates require long-time stress/strain histories Analysis methods must be capable of handling complex
structures under combined high-intensity thermal and acoustic environments Large deflection nonlinear response Composite material properties
S ti ll d t ll li t d l di Spatially and temporally complicated loadings Computationally efficient methods are required Reduced order modeling is one approachg pp
Representative problem of interestRepresentative problem of interest58 x 25 in. multi-bay panel with stiffenersMixed element modelA i t l 16 000 d (96 000 D F )
2
Approximately 16,000 nodes (96,000 DoFs)

Objective
Prior work has shown that careful selection of normal modes as the basis functions for reduced order models results in accurate approximations
Development of a robust modal basis selection criterion for reduced order nonlinear simulation of random response is in progress
Proper Orthogonal Decomposition (POD) accuracy
Modal Assurance Criteria (MAC)
applicability to a wide range ofloading conditions
3

Outline
Nonlinear Reduced Order Formulation
Modal Basis Selection Procedure
Numerical Results
Concluding Remarks
4

Semi-discrete equations of motion in physical DoF (nDoF)(t)(t)(t))((t)(t) FXXKXCXM =++
(t))( XFNL nonlinear restoring force
Modal transformation
(t))( XFNL nonlinear restoring force
= matrix of basis functions
Equation of motion in reduced-order DoF (L

FFqCqM NL~~~~ =++
For each component of nonlinear restoring modal force vector
qqqbqqaqd)q,,q,(qF kjL L L
rjkkj
L Lrjkj
LrjL21
rNL ++=~ qqqqqq)qq(q kj
1j 1k 1jkkj
1j 1kjkj
1jjL21NL
= = == ==
linear quadratic cubic
Construct a sufficient number of static displacement fields qXq =
Obtain restoring forces via static nonlinear FE solution
NLT
NLNL FFFFXXK ==~)(
Solve algebraic system of equations for coefficients L,1,k,j, ,bL,1,kj, ,aL,1,j ,d rjk
rjk
rj ===
6

Work to Obtain Reduced Nonlinear System
qqqbqqaqd)q,,q,(qF kjL
1j
L
1k
L
1
rjkkj
L
1j
L
1k
rjkj
L
1j
rjL21
rNL ++= ~
L,1,r1j 1k 11j 1k1j
== = == ==
es
30003000
f S Clin
earC
ase
20002000
3L
2L
1L
+
+
33
Number of Static Nonlinear Cases
Sta
ticN
onl
L!L
321
where
Num
bero
fS 10001000
n)!-(Ln!L!
nL
=
Total Number of Modes
N
0 4 8 12 16 20 240
T t l N b f M d8 16 24
7
Total Number of ModesTotal Number of Modes

Modal Basis Selection Procedure
(t)(t))q,(t),q(t),(q(t)(t) L21 FFqCqM NL~~~~ =++
Perform system identification to characterize
(t)(t) qX = How is selected?Perform system identification to characterize
nonlinear dynamic response for a particular load case Proper Orthogonal Decomposition (POD)
Proper Orthogonal Modes (POM)
Determine most significant proper orthogonal modes (POM)modes (POM)
Cumulative POM Participation
Select normal modes that best represent properSelect normal modes that best represent proper orthogonal modes normal modes are independent of loads
M d l A C it i (MAC)
8
Modal Assurance Criteria (MAC)

System Identification to Characterize Nonlinear Response
Proper Orthogonal DecompositionObtain sample of full-order nonlinear response in physical DoF for some tn

Selection of Most Significant Proper Orthogonal Modes
Determine proper orthogonal mode participation Determine proper orthogonal mode participation normalized POVs
nDoF1ii == nDoF,1,inDoF
1jj
i ===
Specify desired level of cumulative POM participationM
1j=
1
0 1 #M
ii
M of selected POVs =
= < =
Th b f l t d POM M iwhere
The number of selected POMs, M, is determined by user specification of
10

Selection of Modal Basis Functions for Reduced Order Analysis
Find normal modes that best represent proper orthogonal modes
Perform normal modes analysis of undamped linear t
orthogonal modes
system [ ]{ } { }[ ] { } { } { }[ ]
i2
i
nDoF,1,i0MK
==][ 2
Form modal assurance criterion (MAC) matrix for basis l ti 2
[ ] { } { } { }[ ]nDoF21nDoFnDOF =][
selection
{ } { }{ } { }
{ } { }( ){ } { }( ) nDoF1M,1,kp
),pMAC(TT
Tk
k
==
=
2
Choose normal modes with MAC > 0.5
{ } { }( ){ } { }( ) nDoF,1,pp TkTk =
11

Numerical ExampleClamped Aluminum Beam
Transverse Displacement
p(t)y
152 dBm)
pPost-buckled Random Response
T = 35F
T
x
152 dB
cem
ent (
m 0.010.00
-0.011 x 0.9 cross-section
T18
158 dB
se D
ispl
ac
0.010.00
1 x 0.9 cross section144 B21 elements
Tcr = 6.6F
170 dBTr
ansv
ers
-0.01
0.01
Excitation
Uniformly distributed, band-limited
Mid
-Spa
n
0.00
-0.01
0 2 4 6 8
white noise pressure (0 1500 Hz)
Uniform temperature distribution
M 0 2 4 6 8Time (s)
12

Numerical ExampleClamped Aluminum Beam
Transverse Displacement
p(t)y
152 dBm)
pPost-buckled Random Response
T = 35F
T
x
152 dB
cem
ent (
m 0.010.00
T18
158 dB
se D
ispl
ac
0.010.00
1 x 0.9 cross section144 B21 elements
Tcr = 6.6F
170 dBTr
ansv
ers
-0.01
0.01
Excitation
Uniformly distributed, band-limited POD Load Case
Mid
-Spa
n
0.00
-0.01
0 2 4 6 8
M 0 2 4 6 8Time (s)
13

POM Selection using Cumulative Participation
POD Results for 35F and 170 dB
=99.90 %, 14 modes (5T + 9I)
=99.99 %, 17 modes (7T + 10I)
=99.00 %, 9 modes (3T + 6I)
=90.00 %, 5 modes (2T + 3I)
14

Response at Quarter-span Location for POD Load Case (35F and 170 dB)
In-Plane ResponseTransverse Response
2 /Hz
10-10 Physical DoF24 modes17 modes (99.99%)
m2 /H
z
10-7
10-6 Physical DoF24 modes17 modes (99.99%)
p
men
tPS
D,m
2
10-11
men
tPS
D,m
10-8
neD
ispl
acem
10-13
10-12
rse
Dis
plac
e
10-10
10-9
In-p
lan
10-14
10 13
Tran
sve
10-12
10-11
Frequency, Hz0 500 1000 1500
10
Frequency, Hz0 500 1000 1500
10
Reduced order analysis accurately captures nonlinear response
15
Reduced order analysis accurately captures nonlinear response

Computational Efficiency of POD/MAC Modal Basis Selection Procedure
Reduced order simulation cost- System reduction
Metric: Number of static nonlinear cases24 d 2 92424 modes 2,92417 modes 1,139 61% reduction 61% reduction
- Reduced system integrationMetric: Runge-Kutta 4th order scheme FLOPS/time step
24 modes ~126k17 modes ~45k 64 % reduction 64 % reduction
Milestone metric of 50% computational cost reduction achieved
16
Milestone metric of 50% computational cost reduction achieved.

Robustness Study
p(t)y Apply reduced order analysis to alternate acoustic load level
152 dBT
x
152 dB0.010.00
T18
Transverse DisplacementPost-buckled Random Response
T = 35F
158 dB0.010.00
1 x 0.9 cross section144 B21 elements (> 400 DoFs)
Tcr = 6.6F
170 dB-0.01
0.010 2 4 6 8
Time (s)
Excitation
Uniformly distributed, band-limited 0 00.00
-0.01
17

Robustness Study:Nonlinear Response at a Reduced Thermal-Acoustic Loading Level (35F and 158 dB)g ( )
Reduced order analysis derived from 35F and 170 dB loading level
Quarter-span location, transverse response
Hz
7
10-5 Physical DoF5 modes (90.00%)9 modes (99.00%)14 modes (99.90%)
5 5 modes (90.00%)9 modes (99.00%)14 modes (99.90%)17 modes (99.99%)
cem
ent,
m2 /H
10-9
10-7( )
17 modes (99.99%)24

pod/mac-based modal basis selection for a reduced · pdf filea reduced order nonlinear...

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