Download - High Speed Parameter Estimation for a Homogenized Energy Model- Doctoral Defense Presentation
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High Speed Parameter Estimation for a Homogenized Energy Model
Final Oral Exam
Jon M. ErnstbergerAdvisor: Ralph C. Smith
June 23, 2008
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Presentation Outline Applications Motivation Employed Models Past density formulations Initial estimate techniques Galerkin expansion HEM formulation Incorporated Temperature dependence Results using gradient-based and stochastic searches to PZT data Future Work
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Applications• Jet Engine Chevrons
• 4 dB engine noise reduction• 3 dB reduction occurs if you
turn off a jet engine• Bio-medical applications (SMAs)
• Heart stents• Reconstructive surgery
• Energy harvesting• DARPA Initiative• Recharge devices
• THUNDER• Pumps• Valves
Courtesy of boeing.com
boeing.com
From crucibleresearch.com
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Motivation-Active Machining System• ETREMA Products, Inc.• Active Mat. Terfenol-D• High-Speed Milling (4,000
RPM)
Courtesy of http://www.etrema-usa.com/
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Motivation-PZT Actuated Devices
PZT Nanopositioning Atomic Force Microscope
THUNDER Actuator
AFM image from sciencegl.com
AFM schematic
THUNDER Actuator from faceinternational.com
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Energies-FerromagneticGibbs EnergyHelmholtz Energy
w. neg. thermal relaxation
Local Hysteron from
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Thermal Relaxation
Moment Fraction Evolution:
Local Avg. Magnetization:
Expected Magnetization:
Switching Likelihood:
Boltzmann Relation:
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Homogenized Energy Model
Subject to:
Where:
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Helmholtz Energy
Gibbs Energy
Local Polarization
Energies-Ferroelectric
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180° Switching-Thermal Relaxation
Boltzmann Relation
Switching Likelihood
Dipole Fraction Evolution
Expected Polarization
Local Average Polarization
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90° Switching-Energies/Local RelationsHelmholtz Energies
Gibbs Energy
Local Polarization 90°-Switch due to compressive stress
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90° Switching-Thermal Relaxation
Boltzmann Relation
Switching Likelihood
Dipole Fraction Evolution
Expected Polarization
Local Average Polarization
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Homogenized Energy Model-Ferroelectrics
Four Kernels 180°-Switching
Negligible relaxation Thermal relaxation
90°-Switching Negligible relaxation Thermal relaxation
Density Behaviors Exponential decay Interaction field symmetry Positive coercive field
domain Quadrature Decomposition
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Temperature DependenceUsing a Helmholtz Energy which incorporates Temperature
from which are yielded
through the relation
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Lumped Rod Model
Balance rod forces σA with restoring mechanism
or
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Density Choice-Normal/Lognormal
Runtime 52.90 seconds
100 Hz 200 Hz
300 Hz 500 Hz
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Parameter ID-Initial Estimate (E-P)
Remanence
Susceptibility
Density ParametersStandard deviationsCoercive field mean
(a)
(b)
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Parameter ID-Initial Estimate (E-P)
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Parameter ID-Initial Estimate/StrainRecall
Ignore Kelvin-Voigt damping, magnetostriction, and derivative termsPresume no applied stress, knowledge of remanence and Young's modulus, and simple magnetization
Determine suspectibility and piezomagnetization coefficientsDetermine coercive field mean and standard deviationDetermine interaction field standard deviation
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Parameter ID-Initial Estimate/Strain
Simplified model embedded into a “point-click” GUI (a) 100 Hz, (b) 200 Hz, (c ) 300 Hz, and (d) 500 Hz
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Constraints
Density points to estimate
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Densities-Constrained General Densities
Best fit10 quadrature intervals per density68 parameters to estimateRuntime 969.43 seconds
100 Hz 200 Hz
300 Hz 500 Hz
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Densities-Galerkin ExpansionsUse Galerkin expansion approximate to general densities
Advantages: 1. Smaller parameter space (8+3(N+1)/2 vs. 8+6N) 2. Decrease in runtime in comparison to general density
Disdvantages: 1. Fit will not be as good as general density fit 2. Still requires density constraints for physical behavior
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Densities-SQP/SQP Linear expansion100 Hz 200 Hz
300 Hz 500 Hz
•N=8 Intervals•4 Pt. Gauss. Quad.•Linear Expansion•2000 SQP Fcn Evals•Runtime: 164.7s
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Galerkin Normal/Lognormal Basis– Normally distributed basis elements for interaction field density– Lognormally distributed basis elements for the coercive field density– Removes decay constraints
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Densities: Galerkin normal/lognormal w. single mean
Runtime 250 seconds10 quadrature intervals5 interaction field bases7 coercive field bases
100 Hz 200 Hz
300 Hz 500 Hz
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Densities: Galerkin normal/lognormal w. multiple means
Runtime 244.7 seconds10 quadrature intervals5 interaction field bases9 coercive field bases (3 std. devs, 3 means)Lower residual than single mean
100 Hz 200 Hz100 Hz
300 Hz 500 Hz
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Temperature Dependence
Top: Terfenol-D Data M vs. H data taken at 292 and 363 K.Bottom: Fits to data using estimated parameters.
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Initiate via. GUI Employ Galerkin
normal/lognormal and normal/normal basis
Various data sets (inc. applied comp. stress)
Parameter Estimation
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180°-Negligible Relaxation (Gradient)
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180°-Thermal Relaxation (Gradient)
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90°-Negligible Relaxation (Gradient)
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90°-Negligible Relaxation (Single-Mean)16 Mpa 8 MPa
1 MPa
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90°-Negligible Relaxation (Multi-Mean)16 MPa 8 MPa
1 MPa
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180°-Thermal Relaxation (SA)
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90°-Negligible Relaxation (SA) Galerkin normal/normal basis
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90°-Negligible Relaxation Applied Compressive Stress (SA)
16 MPa 8 MPa
1 MPa
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Conclusions Augmented previous density formulations to
generate more physical approximates Eased estimation computation load w. linear and
cubic Galerkin expansion formulation of the HEM Successfully implemented Galerkin
normal/lognormal (normal/normal) basis Tools to determine initial parameter estimates for
field-polarization and field-strain data. Reduced parameter estimation runtime for the AMS
to about 4 minutes Performed parameter estimation to Terfenol-D/AMS
data with gradient-based and stochastic searches
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Conclusions (2) Achieved accurate PZT parameter estimates
employing for kernels with 90° and 180°-switching (including data w. applied compressive stress)
Validated toolset for initial estimates Estimated parameters with gradient-based routines
and simulated annealing Showed dissipativity of the HEM Generated a full GUI for the parameter estimation
process
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Future Work Ferroelastic model with thermal relaxation Combination Galerkin normal/lognormal basis
w. Temperature Dep. Terfenol-D/PZT data to fit. Hybrid gradient/stochastic searches Estimation for SMA model
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References
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If a system is dissipative, it loses energy. “The energy at final time is less than or equal to initial
energy plus input energy.” Showed dissipativity of
HEM with negligible thermal relaxation for supply ratesand
HEM with thermal relaxation for same supply rates Statement of stability and helps design controllers
Dissipativity of HEM
HM MH
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Parameter ID GUI
Easy front-end for deployment
Allows automated initiation or full manual control
Requires little expertise
MATLAB-based Appendix B