FEM FOR THE TEST ENGINEER

Quartus EngineeringCopyright Quartus Engineering Incorporated, 2000.

FEMFOR THE TEST ENGINEER

Christopher C. FlaniganQuartus Engineering Incorporated

San Diego, California USA

18th International Modal Analysis Conference (IMAC-XVIII)San Antonio, TexasFebruary 7-10, 2000


DOWNLOAD FROM THEQUARTUS ENGINEERING WEB SITE

http://www.quartus.com


FEM PEOPLE ARE REALLY SMART

• Or so they would have you believe!


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



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



FEM ATTEMPTS TOSIMULATE 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

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REMEMBER THAT FEMONLY APPROXIMATES REALITY

• Reality has lots of hard challenges– Nonlinearity, chaos, etc.

• FEM limited by many factors– Engineering knowledge and capabilities

– Basic understanding of mechanics

– Computer and software power

• But it’s the best approach we have– Experience shows that FEM works well when used

properly

FEMAhead!


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


FEMin a Nutshell



FEM IN A NUTSHELL

• Divide and conquer!• Shape functions• Elemental stiffness and mass matrices• Assembly of system matrices• Solving• Related topics

– Element library– Superelements


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


FEM in a Nutshell

DIVIDE AND CONQUER!

• Behavior of complete structure is complex– Example: membrane

• Divide the membraneinto small pieces– Buzzword: “element”

• Feasible to calculate properties of each piece

• Collection of pieces represents structure

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FEM in a Nutshell

SHAPE FUNCTIONS ARE THE FOUNDATION OF FINTE ELEMENTS

• Shape function– Assumed shape of element when deflected

• Some element types are simple– Springs, rods, bar

• Other elements are more difficult– Plates, solids

• But that’s what Ph.D.’s are for!– Extensive research– Still evolving (MSC.NASTRAN V70.7)

Spring

F = K X

FX

K


FEM in a Nutshell

ELEMENT STIFFNESS MATRIXFORMED USING SHAPE FUNCTIONS

• Element stiffness matrix– Relates deflections of elemental DOF

to applied loads

• Forces at element DOF when unit deflection imposed at DOFi and other DOFj are fixed

• Example: linear spring (2 DOF)

Spring

F = K X

FX

K

KK

KKKspring


FEM in a Nutshell

ELEMENT MASS MATRIXHAS TWO OPTIONS

• Lumped mass– Apply 1/N of the element mass to each node

• 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

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0M5.0Mspring

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FEM in a Nutshell

SYSTEM MATRICES FORMEDFROM ELEMENT MATRICES

K = 2

K = 5

K = 1

M = 1

M = 2

M = 3

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FEM in a Nutshell

CALCULATE SYSTEM STATICAND DYNAMIC RESPONSES

• Static analysis

• Normal modes analysis

• Transient analysis

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0MK ii

XKP


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


FEM in a Nutshell

HONORARY DEGREE IN FEM-OLOGY!



FEM STRENGTHS AND CHALLENGES



FEM IS VERY POWERFUL FORWIDE 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



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...



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!



FEM ASSISTED BY ADVANCESIN 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



FEM CONTINUES TO IMPROVEABILITY TO SIMULATE REALITY

• Model resolution– Local details

• Some things stillvery difficult– Joints

• Expertise– Mesh size, etc.

• FEM is not exact– Big models do not guarantee accurate models– That’s why testing is still required!



PRETEST ANALYSIS

Develop

FEM

Pretest

AnalysisTest

Posttest

Correlation


Pretest Analysis

MODAL SURVEY OFTEN PERFORMEDTO 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”


Pretest Analysis - TAM

TEST AND ANALYSIS DATA HAVEDIFFERENT NUMBER OF DOF

• Model sizes– FEM = 10,000-1,000,000 DOF– Test = 50-500 accelerometers

• Compare test results to analysis predictions

• 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)

• 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 THEINDUSTRY 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

• Assume forces at omitted DOF are negligible

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GUYAN REDUCTION IS ASIMPLE METHOD TO IMPLEMENT

• Solve for motion at omitted DOF

• Rewrite static equations of motion

• Transformation matrix for Guyan reduction

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TRANSFORMATION VECTORSESTIMATE MOTION AT “OTHER” DOF

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TRANSFORMATION VECTORS CANREDUCE OR EXPAND DATA

TAM

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DISPLAY MODEL RECOVERED USING TRANSFORMATION VECTORS

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IRS REDUCTION ADDSFIRST ORDER MASS CORRECTION

• Guyan neglects mass effects at omitted DOF• IRS adds first order approximation of mass effects

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DYNAMIC REDUCTION ALSOADDS MASS CORRECTION

• Start with eigenvalue equation

Replace eigenvalue with constant value

• Equivalent to Guyan reduction if = 0

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MODAL TAM BASED ONFEM MODE SHAPES

• Partition FEM mode shapes

• Pseudo-inverse to form transformation matrix

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EACH REDUCTION METHOD HASSTRENGTHS AND WEAKNESSES

ADVANTAGES DISADVANTAGES

Easy to use, efficient Limited accuracy

Guyan 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 Requires DMAP alter

Dynamic Choice of Lamda?

Limited experience

Exact within freq. range Requires DMAP alter

Modal Hybrid TAM option Sensitivity

Limited experience



STANDARD PRACTICE FAVORSGUYAN 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 )


Pretest Analysis - Sensor Placement

SENSOR PLACEMENT IMPORTANTFOR 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



MANY ALGORITHMS FORSENSOR 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!



SENSOR PLACEMENT ALGORITHMCLOSELY 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



PRETEST ANALYSIS ASSISTSPLANNING 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



TEST CONSIDERATIONS

Develop

FEM

Pretest

AnalysisTest

Posttest

Correlation


Test Considerations

PRETEST DATA ALLOWSREAL-TIME CHECKS OF RESULTS

• Traditional comparisons

• 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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POSTTEST CORRELATION

Develop

FEM

Pretest

AnalysisTest

Posttest

Correlation


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



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.”



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

Test OK?

Done

Updates



MATRIX UPDATE METHODSADJUST 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)• Kabe (1985)• Kammer (1987)• Smith and Beattie (1991)• … and many others

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MATRIX UPDATE METHODSHAVE 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)



PROPERTY UPDATE METHODSADJUST 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 * Calculate updates using

design sensitivity and optimization

FEM

Test OK?

Done

Updates*



COMMERCIAL SOFTWAREFOR 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.)



MODE SHAPE EXPANSIONFOR CORRELATION IMPROVEMENT

TAM

Display



SHAPE EXPANSION IS ANALTERNATIVE TO MATRIX REDUCTION

• Expand test mode shapes to FEM DOF

• Expansion and reduction give same results if same matrices used

• Dynamic expansion based on eigenvalue equation

Computationally intensive– But computers are getting faster all the time!

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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


FEM PEOPLE REALLY ARE SMART!

• And maybe test people are smart too!



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

Documents

FEM FOR THE TEST ENGINEER