High Speed Parameter Estimation for a Homogenized Energy Model

Final Oral Exam

Jon M. ErnstbergerAdvisor: Ralph C. Smith

June 23, 2008

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

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

Motivation-Active Machining System• ETREMA Products, Inc.• Active Mat. Terfenol-D• High-Speed Milling (4,000

RPM)

Courtesy of http://www.etrema-usa.com/

Motivation-PZT Actuated Devices

PZT Nanopositioning Atomic Force Microscope

THUNDER Actuator

AFM image from sciencegl.com

AFM schematic

THUNDER Actuator from faceinternational.com

Energies-FerromagneticGibbs EnergyHelmholtz Energy

w. neg. thermal relaxation

Local Hysteron from

Thermal Relaxation

Moment Fraction Evolution:

Local Avg. Magnetization:

Expected Magnetization:

Switching Likelihood:

Boltzmann Relation:

Homogenized Energy Model

Subject to:

Where:

Helmholtz Energy

Gibbs Energy

Local Polarization

Energies-Ferroelectric

180° Switching-Thermal Relaxation

Boltzmann Relation

Switching Likelihood

Dipole Fraction Evolution

Expected Polarization

Local Average Polarization

90° Switching-Energies/Local RelationsHelmholtz Energies

Gibbs Energy

Local Polarization 90°-Switch due to compressive stress

90° Switching-Thermal Relaxation

Boltzmann Relation

Switching Likelihood

Dipole Fraction Evolution

Expected Polarization

Local Average Polarization

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

Temperature DependenceUsing a Helmholtz Energy which incorporates Temperature

from which are yielded

through the relation

Lumped Rod Model

Balance rod forces σA with restoring mechanism

or

Density Choice-Normal/Lognormal

Runtime 52.90 seconds

100 Hz 200 Hz

300 Hz 500 Hz

Parameter ID-Initial Estimate (E-P)

Remanence

Susceptibility

Density ParametersStandard deviationsCoercive field mean

(a)

(b)

Parameter ID-Initial Estimate (E-P)

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

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

Constraints

Density points to estimate

Densities-Constrained General Densities

Best fit10 quadrature intervals per density68 parameters to estimateRuntime 969.43 seconds

100 Hz 200 Hz

300 Hz 500 Hz

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

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

Galerkin Normal/Lognormal Basis– Normally distributed basis elements for interaction field density– Lognormally distributed basis elements for the coercive field density– Removes decay constraints

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

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

Temperature Dependence

Top: Terfenol-D Data M vs. H data taken at 292 and 363 K.Bottom: Fits to data using estimated parameters.

Initiate via. GUI Employ Galerkin

normal/lognormal and normal/normal basis

Various data sets (inc. applied comp. stress)

Parameter Estimation

180°-Negligible Relaxation (Gradient)

180°-Thermal Relaxation (Gradient)

90°-Negligible Relaxation (Gradient)

90°-Negligible Relaxation (Single-Mean)16 Mpa 8 MPa

1 MPa

90°-Negligible Relaxation (Multi-Mean)16 MPa 8 MPa

1 MPa

180°-Thermal Relaxation (SA)

90°-Negligible Relaxation (SA) Galerkin normal/normal basis

90°-Negligible Relaxation Applied Compressive Stress (SA)

16 MPa 8 MPa

1 MPa

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

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

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

References

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

Parameter ID GUI

Easy front-end for deployment

Allows automated initiation or full manual control

Requires little expertise

MATLAB-based Appendix B