FEM
FOR THE TEST ENGINEER
Christopher C. Flanigan
Quartus Engineering Incorporated
San Diego, California USA
18th International Modal Analysis Conference (IMAC-XVIII)
San Antonio, Texas
February 7-10, 2000
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DOWNLOAD FROM THE
QUARTUS ENGINEERING WEB SITE
http://www.quartus.com
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FEM PEOPLE ARE REALLY SMART
• Or so they would have you believe!
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FEM for the Test Engineer
TOPICS
•
•
•
•
There’s reality, and then there’s FEM
FEM in a nutshell
FEM strengths and challenges
Pretest analysis
– Model reduction
– Sensor placement
• Posttest analysis
– Correlation
– Model updating
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There’s Reality, and Then There’s FEM
REALITY IS VERY COMPLICATED!
•
•
•
•
•
•
•
•
Many complex subsystems
Unique connections
Advanced materials
Broadband excitation
Nonlinearities
Flight-to-flight variability
Chaos
Extremely high order behavior
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There’s Reality, and Then There’s FEM
FEM ATTEMPTS TO
SIMULATE REALITY
• Fortunately, reality is
surprisingly linear
–
–
–
–
Material properties ( vs. )
Tension vs. compression
Small deflections (sin  = )
Load versus deflection
• Allows reasonable
opportunity simulate reality
using FEM
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
-1
-0.5
0
0.5
1
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There’s Reality, and Then There’s FEM
REMEMBER THAT FEM
ONLY APPROXIMATES REALITY
• Reality has lots of hard challenges
– Nonlinearity, chaos, etc.
• FEM limited by many factors
– Engineering knowledge and capabilities
– Basic understanding of mechanics
FEM
Ahead!
– Computer and software power
• But it’s the best approach we have
– Experience shows that FEM works well when used properly
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FEM Strengths and Challenges
TEST IS NOT REALITY EITHER!
• Test article instead of flight article
– Mass simulators, missing items, boundary conditions
• Excitation limitations
– Load level, spectrum (don’t break it!)
– Nonlinearities
• Testing limitations
– Sensor accuracy and calibration
– Data processing
• But it’s the best “reality check” available
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FEM
in a Nutshell
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FEM for the Test Engineer
FEM IN A NUTSHELL
•
•
•
•
•
•
Divide and conquer!
Shape functions
Elemental stiffness and mass matrices
Assembly of system matrices
Solving
Related topics
– Element library
– Superelements
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FEM in a Nutshell
CLOSED FORM SOLUTIONS, ANYONE?
• Consider a building
– Steel girders
– Concrete foundation
• Can you write an equation to
fully describe the building?
– I can’t!
• Even if possible, probably not
the best approach
– Very time consuming
– One-time solution
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FEM in a Nutshell
DIVIDE AND CONQUER!
• Behavior of complete
structure is complex
– Example: membrane
1.00
• Divide the membrane
into small pieces
0.80
0.60
0.40
0.20
– Buzzword: “element”
• Feasible to calculate
properties of each piece
• Collection of pieces
represents structure
0.00
-0.20
-0.40
-0.60
-0.80
-1.00
S1
S3
1
3
5
7
S5
9
S7
11
S9
S11
13
S13
15
S15
17
S17
19
S19
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FEM in a Nutshell
SHAPE FUNCTIONS ARE THE
FOUNDATION OF FINTE ELEMENTS
• Shape function
– Assumed shape of element when deflected
Spring
• Some element types are simple
– Springs, rods, bar
K
• Other elements are more difficult
– Plates, solids
• But that’s what Ph.D.’s are for!
– Extensive research
– Still evolving (MSC.NASTRAN V70.7)
F
X
F=KX
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FEM in a Nutshell
ELEMENT STIFFNESS MATRIX
FORMED USING SHAPE FUNCTIONS
• Element stiffness matrix
– Relates deflections of elemental DOF
to applied loads
Spring
• Forces at element DOF when unit
deflection imposed at DOFi and
other DOFj are fixed
• Example: linear spring (2 DOF)
K
K spring  
 K
 K
K 
K
F
X
F=KX
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FEM in a Nutshell
ELEMENT MASS MATRIX
HAS TWO OPTIONS
• Lumped mass
– Apply 1/N of the element mass to each node
1/4
1/4
1/4
1/4
• Consistent mass
– Called “coupled mass” in NASTRAN
– Use shape functions to generate mass matrix
• In practice, usually little difference
between the two methods
– Consistent mass more accurate
– Lumped mass faster
0 
0.5 M
Mspring  

0
0
.
5
M


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FEM in a Nutshell
SYSTEM MATRICES FORMED
FROM ELEMENT MATRICES
 2  2
K1  

 2 2 
 5  5
K2  

 5 5 
 1 1
K3  

1 1 
0
 2 2 0
 2 7  5 0 

K
 0  5 6  1


0 1 1 
 0
0
0
0.5 0
 0 1.5 0
0

M 
 0
0 2.5 0 


0
0 1.5
 0
M=1
K=2
M=2
K=5
M=3
K=1
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FEM in a Nutshell
CALCULATE SYSTEM STATIC
AND DYNAMIC RESPONSES
• Static analysis
P KX
• Normal modes analysis
K   i Mi  0
• Transient analysis
  T C  q  T K  q  T P
T M q
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FEM in a Nutshell
COMMERCIAL FEM ISSUES
• Element libraries
– Springs, rods, beams, shells, solids, rigids, special
– Linear and parabolic (shape functions, vertex nodes)
• Commercial codes
– NASTRAN popular for linear dynamics (aero, auto)
– ABAQUS and ANSYS popular for nonlinear
• Superelements (substructures)
– Simply a collection of finite elements
– Special capabilities to reduce to boundary nodes
– Assemble system by addition I/F nodes
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FEM in a Nutshell
HONORARY DEGREE IN FEM-OLOGY!
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FEM for the Test Engineer
FEM STRENGTHS AND CHALLENGES
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FEM Strengths and Challenges
FEM IS VERY POWERFUL FOR
WIDE ARRAY OF STRUCTURES
• Regular structures
– Fine mesh
• Sturdy connections
– Seam welds
• Well-defined mass
– Smooth distributed
– Small lumped masses
• Linear response
– Small displacements
General Dynamics
Control-Structure Interaction Testbed
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FEM Strengths and Challenges
FEM HAS MANY CHALLENGES
• Mesh refinement
– How many elements required?
– Stress/strain gradients, mode shapes
• Material properties
– A-basis, B-basis, etc.
– Composites
• Dimensions
– Tolerances, as-manufactured
• Joints
– Fasteners, bonds, spot welds
continued...
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FEM Strengths and Challenges
FEM HAS MANY CHALLENGES
• Mass modeling
– Accuracy of mass prop DB
– Difficulty in test/weighing
• Secondary structures
– Avionics boxes, batteries
– Wiring harnesses
• Shock mounts
• Nonlinearities
– (large deformation, slop, yield, etc.)
• Pilot error!
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FEM Strengths and Challenges
FEM ASSISTED BY ADVANCES
IN H/W AND S/W POWER
• Computers
– Moore’s law for CPU
– Disk space, memory
• Software
–
–
–
–
Sparse, iterative
Lanczos eigensolver
Domain decomposition
Pre- and post-processing
• Increasing resolution
– Closer to reality
Moravec, H., “When Will Computer Hardware Match the Human Brain?”
Robotics Institute Carnegie Mellon University
http://www.transhumanist.com/volume1/moravec.htm
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FEM Strengths and Challenges
FEM CONTINUES TO IMPROVE
ABILITY TO SIMULATE REALITY
• Model resolution
– Local details
• Some things still
very difficult
– Joints
• Expertise
– Mesh size, etc.
• FEM is not exact
– Big models do not guarantee accurate models
– That’s why testing is still required!
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FEM for the Test Engineer
PRETEST ANALYSIS
Develop
FEM
Pretest
Analysis
Test
Posttest
Correlation
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Pretest Analysis
MODAL SURVEY OFTEN PERFORMED
TO VERIFY FINITE ELEMENT MODEL
• Must be confident that structure will survive
operating environment
• Unrealistic to test flight structure to flight loads
• Alternate procedure
– Test structure under controlled conditions
– Correlate model to match test results
– Use test-correlated model to predict operating responses
• Modal survey performed to verify analysis model
– “Reality check”
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Pretest Analysis - TAM
TEST AND ANALYSIS DATA HAVE
DIFFERENT NUMBER OF DOF
• Model sizes
– FEM = 10,000-1,000,000 DOF
– Test = 50-500 accelerometers
• Compare test results to
analysis predictions
Ortho  T M 
• Need a common basis for
comparison
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Pretest Analysis - TAM
TEST-ANALYSIS MODEL (TAM)
PROVIDES BASIS FOR COMPARISON
• Test-analysis model (TAM)
– Mathematical reduction of finite element model
– Master DOF in TAM corresponds to accelerometer
• Transformation (condensation)
Kaa  Tga T Kgg Tga
Maa  Tga T Mgg Tga
• Many methods to perform reduction transformation
• Transformation method and sensor selection critical
for accurate TAM and test-analysis comparisons
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Pretest Analysis - TAM Transformation Methods
GUYAN REDUCTION IS THE
INDUSTRY STANDARD METHOD
• Robert Guyan, Rockwell, 1965
– Pronounced “Goo-yawn”, not “Gie-yan”
• Implemented in many commercial software codes
– NASTRAN, I-DEAS, ANSYS, etc.
• Start with static equations of motion
K oo K oa  Uo  Po 
  
K

 ao Kaa  Ua  Pa 
• Assume forces at omitted DOF are negligible
Po  0
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Pretest Analysis - TAM Transformation Methods
GUYAN REDUCTION IS A
SIMPLE METHOD TO IMPLEMENT
• Solve for motion at omitted DOF
Uo   Koo 1 Koa Ua
• Rewrite static equations of motion
Uo   K oo 1 K oa 
 
 Ua
U
Iaa
 a 

• Transformation matrix for Guyan reduction
 K oo 1 K oa 
TGuyan  

Iaa


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Pretest Analysis - TAM Transformation Methods
TRANSFORMATION VECTORS
ESTIMATE MOTION AT “OTHER” DOF
1.0
0.8
Displacement
0.6
0.4
0.2
0.0
-0.2
1
2
3
4
Node ID
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Pretest Analysis - TAM Transformation Methods
TRANSFORMATION VECTORS CAN
REDUCE OR EXPAND DATA
TAM
Display
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Pretest Analysis - TAM Transformation Methods
DISPLAY MODEL RECOVERED USING
TRANSFORMATION VECTORS
Standard Display
0.75
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1.00
Enhanced Display
0.75
1
2
3
4
1
2
3
4
0.50
0.25
0.00
-0.25
-0.50
-0.75
-1.00
Node ID
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Pretest Analysis - TAM Transformation Methods
IRS REDUCTION ADDS
FIRST ORDER MASS CORRECTION
• Guyan neglects mass effects at omitted DOF
• IRS adds first order approximation of mass effects
TGuyan
GGuyan  GIRS 


Iaa


GGuyan   Koo 1 Koa


GIRS  Koo 1 Moa  Moo GGuyan Maa 1 Kaa
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Pretest Analysis - TAM Transformation Methods
DYNAMIC REDUCTION ALSO
ADDS MASS CORRECTION
• Start with eigenvalue equation
i
K oo K oa  o 
i Moo


K
  
M
K
 ao
 ao
aa   a 
i
Moa  o 
 
Maa  a 
Replace eigenvalue with constant value L
 K oo  L Moo 1 K oa  L Moa 
TDyn Re d  

I


aa
• Equivalent to Guyan reduction if L = 0
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Pretest Analysis - TAM Transformation Methods
MODAL TAM BASED ON
FEM MODE SHAPES
• Partition FEM mode shapes
Uo  o 
Ua   a 
• Pseudo-inverse to form transformation matrix
Uo  Tmod al Ua

  T 
TModal   o a a
Iaa


1
T
a 


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Pretest Analysis - TAM Transformation Methods
EACH REDUCTION METHOD HAS
STRENGTHS AND WEAKNESSES
Guyan
ADVANTAGES
DISADVANTAGES
Easy to use, efficient
Limited accuracy
Works well if good A-set
Bad if poor A-set
Widely accepted
Unacceptable for high M/K
Better than Guyan
Requires DMAP alter
IRS
Errors if poor A-set
Better than Guyan
Dynamic
Requires DMAP alter
Choice of Lamda?
Limited experience
Modal
Exact within freq. range
Requires DMAP alter
Hybrid TAM option
Sensitivity
Limited experience
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Pretest Analysis - TAM Transformation Methods
STANDARD PRACTICE FAVORS
GUYAN REDUCTION
• Guyan reduction used most often
–
–
–
–
Easy to use and commercially available
Computationally efficient
Widely used and accepted
Good accuracy for many/most structures
• Use other methods when Guyan is inadequate
– Modal TAM very accurate but sensitive to FEM error
– IRS has 1st order mass correction but can be unstable
– Dynamic reduction seldom used (how to choose L)
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Pretest Analysis - Sensor Placement
SENSOR PLACEMENT IMPORTANT
FOR GOOD TAM AND TEST
• Optimize TAM
– Minimize reduction error
• Optimize test
– Get as much independent data as possible
• Focus on uncertainties
– High confidence areas need only modest instrumentation
– More instrumentation near critical uncertain areas (joints)
• Common sense and engineering judgement
– General visualization of mode shapes
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Pretest Analysis - Sensor Placement
MANY ALGORITHMS FOR
SENSOR PLACEMENT
• Kinetic energy
– Retain DOF with large kinetic energy
• Mass/stiffness ratio
– Retain DOF with high mass/stiffness ratio
• Iterated K.E. and M/K
– Remove one DOF per iteration
• Effective independence
– Retain DOF that maximize observability of mode shapes
• Genetic algorithm
– Survival of the fittest!
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Pretest Analysis - Sensor Placement
SENSOR PLACEMENT ALGORITHM
CLOSELY LINKED TO TAM METHOD
• Guyan or IRS reduction
– Must retain DOF with large mass
– Iterated K.E. or M/K
– Mass-weighted effective independence
• Modal or Hybrid reduction
– Effective independence
• Genetic algorithm offers best of all worlds
– Examine tons of TAMs!
– Seed generation from other methods
– Cost function based on TAM method
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Pretest Analysis - Sensor Placement
PRETEST ANALYSIS ASSISTS
PLANNING AND TEST
• Best estimate of modes
– Frequencies, shapes
• Accelerometer locations
– Optimized by sensor placement
studies
• TAM mass and stiffness
– Real-time ortho and x-ortho
• Frequency response functions
– Dry runs/shakedown prior to test
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FEM for the Test Engineer
TEST CONSIDERATIONS
Develop
FEM
Pretest
Analysis
Test
Posttest
Correlation
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Test Considerations
PRETEST DATA ALLOWS
REAL-TIME CHECKS OF RESULTS
• Traditional comparisons
ORTHO  test MTAM  test
T
XORTHO  
T
TAM
MTAM test
• What if test accuracy goals aren’t met?
– Keep testing (different excitement levels, locations, types)
– Stop testing (FEM may be incorrect!)
– Decide based on test quality checks
• Experienced test engineer extremely valuable!
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FEM for the Test Engineer
POSTTEST CORRELATION
Develop
FEM
Pretest
Analysis
Test
Posttest
Correlation
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Posttest Correlation
CORRELATION MUST BE FAST!
• FEM almost always has some differences vs. test
• Very limited opportunity to do correlation
– After structural testing and data processing complete
– Before operational use of model
• First flight of airplane
• Verification load cycle of spacecraft
• Need methods that are fast!
– Maximum insight
– Accurate
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Posttest Correlation
NO UNIQUE SOLUTION FOR
POSTTEST CORRELATION
• More “unknowns” than “knowns”
• Knowns
– Test data (FRF, frequencies, shapes at
test DOF, damping)
– Measured global/subsystem weights
• Unknowns
– FEM stiffness and mass (FEM DOF)
• No unique solution
• Seek “best” reasonable solution
“When you
have
eliminated
the
impossible,
whatever
remains,
however
improbable,
must be
the truth.”
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Posttest Correlation
MANY CORRELATION METHODS
• Trial-and-error
– Stop doing this! It's (almost)
the new millenium!
– Too slow for fast-paced projects
– Not sufficiently insightful for
complex systems
• FEM matrix updating
• FEM property updating
• Error localization
FEM
Updates
Test
OK?
Done
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Posttest Correlation
MATRIX UPDATE METHODS
ADJUST FEM K AND M ELEMENTS
• Objective
– Identify changes to FEM K and M so that analysis
matches test
•
•
•
•
•
•
Baruch and Bar-Itzhack (1978, 1982)
Berman (1971, 1984)
 2
 2
Kabe (1985)
K
 0

Kammer (1987)
 0
Smith and Beattie (1991)
0.5
 0
… and many others
M 
 0

 0
2
0
0
0
0
7 5 0 

 5 6  1

0 1 1 
0
1.5 0
0

0 2.5 0 

0
0 1.5
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Posttest Correlation
MATRIX UPDATE METHODS
HAVE LIMITATIONS
• Lack of physical insight
– What do changes in K, M coefficients mean?
• Lack of physical plausibility
– Baruch/Berman method doesn't enforce connectivity
• Limitations for large problems
– Great for small “demo” models, but ...
– “Smearing" caused by Guyan reduction/expansion
• What if test article different than flight vehicle?
• Requires very precise mode shapes (unrealistic)
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Posttest Correlation
PROPERTY UPDATE METHODS
ADJUST MATERIALS AND ELEMENTS
• Objective
– Identify changes to element and material
properties so that FEM matches test
•
•
•
•
•
•
Hasselman (1974)
Chen (1980)
Flanigan (1987, 1991)
Blelloch (1992)
Smith (1995)
… and many others
FEM
Updates*
Test
OK?
Done
* Calculate updates using
design sensitivity and optimization
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Posttest Correlation
COMMERCIAL SOFTWARE
FOR CORRELATION
• SDRC/MTS
– I-DEAS Correlation (MAC, ortho, x-ortho, mapping)
• LMS
– CADA LINK (parameter updating, Bayesian estimation)
• MSC
– SOL 200 design optimization (modes, FRF)
• Dynamic Design Solutions (DDS)
– FEMtools (follow-on to Systune)
• Others (SSID, ITAP, etc.)
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Posttest Correlation
MODE SHAPE EXPANSION
FOR CORRELATION IMPROVEMENT
TAM
Display
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Posttest Correlation
SHAPE EXPANSION IS AN
ALTERNATIVE TO MATRIX REDUCTION
• Expand test mode shapes to FEM DOF
Ug  Tga Ua
• Expansion and reduction give same results if same
matrices used
• Dynamic expansion based on eigenvalue equation



oi   Koo  i Moo Koa  i Moa ai
Computationally intensive
– But computers are getting faster all the time!
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FEM for the Test Engineer
SUMMARY
• FEM is a simple yet powerful method
– Complex structures from simple building blocks
• FEM must make many assumptions
– Joints, tolerances, linearity, mass, etc.
– Big models do not guarantee accuracy
• Testing provides a valuable “reality check”
– Within limits of test article, excitation levels, etc.
• FEM can work closely with test for mutual benefit
– Pretest analysis to optimize sensor locations
– TAM for providing test-analysis comparison basis
– Correlation and model updating for validated model
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FEM PEOPLE REALLY ARE SMART!
• And maybe test people are smart too!
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Copyright Quartus Engineering Incorporated, 2000.
FEM for the Test Engineer
RECOMMENDED READING
• Finite element method
– Concepts and Applications of Finite Element Analysis, 3rd ed.; Cook,
Robert D./Plesha, Michael E./Malkus, David S.; John Wiley & Sons; 1989
– Finite Element Procedures, Klaus-Jurgen Bathe; Prentice Hall; 1995
• Correlation and model updating
– Finite Element Model Updating in Structural Dynamics; M. I. Friswell,
J. E. Mottershead; Kluwer Academic Publishers; 1995.
• Optimization
– Numerical Optimization Techniques for Engineering Design, 3rd edition
(includes software); Garret N. Vanderplaats, Vanderplaats Research &
Development, Inc., 1999
Quartus Engineering
Copyright Quartus Engineering Incorporated, 2000.