A Method of Weighted Residuals Approachto Machine Learning with Aerospace

Applications

New Professor Lecture Series

April 14, 2005

Andrew Meade

William Marsh Rice University, USA

Presentation Outline

•Background

• Approach

• Performance

•Applications

•Conclusions/Future Work

Many aerospace problems can be described with the following example from aerodynamics:

Dimensional parameter of interest =

= F (size, shape, fluid velocity, fluid properties, dimensional constants)

1. Configuration geometry (shape) ⇒ F

2. Angle of attack (aoa) , i.e., vehicles attitude in the pitch plane

relative to the flight direction. ⇒ α3. Vehicle size or model scale. ⇒ S

4. Free-stream velocity. ⇒ V∞

5. Density of the undisturbed air. ⇒ ρ∞

6. Reynolds number. ⇒ Rec

7. Mach number. ⇒ Ma∞

Dimensional parameters of interest (lift, drag, and moment) ⇒ L, D, and M

Background

We define the dynamic pressure as q∞ = 1

2ρ∞V∞

2 ,

the Reynolds number based on the chord as Rec =

ρ∞V∞c

µ∞

,

the freestream Mach number as Ma∞ =V∞

a∞

,

and the reference area (planform area) as S = bc .

Background

We can write the relationship

coefficient of lift: CL =

L

q∞S = F

1α , Re

c, Ma∞( )

coefficient of drag: CD

= D

q∞S = F

2α , Re

c, Ma∞( )

coefficient of moment: CM

= M

q∞ S( )c = F3α , Re

c, Ma∞( )

The functions F1, F

2, and F

3, which also depend on the shape of the aircraft configuration,

is the objective behind aerospace engineering.

Background

Our tools in approximating the multidimensional functions F1, F

2, and F

3, which we will

define as surrogates, are:

• Pure Theory

• Physical Experiments

• Computational Mechanics

Background

Background

• Surrogates can be a table lookup, a system of partial differential equations (PDEs) or

non-smooth simulation computer codes.

• Depending on their fidelity, surrogates may be very expensive to solve and may be

nondifferentiable and discontinuous.

Can we use all available information (theory, CFD, and experiments) to build surrogates and

hence physical knowledge that are valid over a wide range of conditions?

[Anderson (1995)]

Motivation

Relying exclusively on one approach over another won't cut it anymore. The rivalry between

experimental and computational approaches must end if we are to make any progress.

Hans Mark, et. al. 1975 Clifford Truesdell, 1984

Figure: NASP hypersonic aircraft design

Motivation

The design of future aircraft will require even greater coupling between

physical disciplines and better fidelity of their respective surrogates

(e.g., hypersonic aircraft)

Inlet CompressionLift

Pitching Moment Nozzle ThrustLift

Pitching Moment

AirframeFuel

Payload

Motivation

• Strong interactions between vehicle components

• Aerodynamics, propulsion, control, structure, tank, thermal protection, etc.

• Highly integrated engine and airframe

• Much of vehicle is engine inlet / nozzle

• Large propulsive lift and pitching moments – strong contributor to trim, stability & control

• Large Mach and dynamic pressure variations in flight

• Severe aerodynamic heating

• Thermal protection must be integrated with structure

• High fuel mass fraction required - majority of volume accommodates fuel

Figure: CFD in design of High Speed Civil Transport(HSCT) (NASA Langley)

A high fidelity CFD code by itself is used only in a very small region of

the flight envelop because of time and expense.

Motivation

• In addition, we must admit that there is no “universal” turbulence model

in fluid dynamics.

• There are just too many fine details in the flow to simulate it with any

computational efficiency (Direct Numerical Simulation).

• The nearest we have is Large Eddy Simulation (LES) which is computationally costly.

• No single turbulence model predicts all textures of a moderately complex flow.

Motivation

Figure: LES for flow around a cylinder: Re = 3900[www.inf.bauwesen.tu-muenchen.de/forschung/konwihr/konwi..]

I believe we can take advantage of theory, physical experiments and computational

methods, using tools from scattered data approximation machine learning tools( )

and Tikhnov Regularization (TR)

Specifically, the problem of mathematical analysis of experimental data is treated as an

ill-posed problem. Its regularization involves the introduction of additional information

regarding the physical system.

We will utilize a-priori mathematical models of physical systems at appropriate orders of

fidelity for regularization.

Approach

I take my inspiration from the nervous system of biological systems. Other researchersare investigating whether first order approximation of neural networks can be used asflight controllers and also used to detect and recover from fatigue, probability ofsubsystem failure, or damage to future aircraft.

Motivation

Figure: Israeli F-15. Half of right wing lost. Figure: DHL Bagdad Airport. Complete loss of hydraulics.

Example: The Dragonfly

• 300 million year old design• 97% success rate in capturing prey.• Flight relies entirely on 3-D unsteady aerodynamics.• Unstable in all axes.• Can fly from a forward speed of +100 body lengths/sec (60 mph) to - 3 bls/sec within 5 bls• Can alter heading by 90 degs in less than 0.1 secs• Flying insects have the fastest visual processing

system in the animal kingdom

These feats are accomplished through the neuro-circuitry of a brain smaller than asesame seed.

Motivation

The dragonfly’s flight control neuro-circuitry provides anonlinear mapping between input and output.

The neurons are modeled mathematically tofirst order approximation and assembled into an ANN.

Biological neuron. Neuron’s mathematical approximation. Artificial neural network.

Dragonfly’s flight control.

Plant

Motivation

Motivation

Research is presently underway at NASA and the DoD on this type of controller.

[Gundy-Burlet et al. (2004)]

The neural controller can then be trained with back-propagation using empirical datafrom the plant to approximate the nonlinear mapping and eventually replace the plant.

Direct control scheme (blank slate learning). Indirect control scheme (learning on top of instincts).

The neural controller for a physical system should be able to:

• Map multiple inputs to multiple outputs dynamically.

• Perform supervised classification.• Use as few neurons as possible and

with computational efficiency.

Scattered data approximation

Sparse approximation

}}

Motivation

Approach

The amount of experimental data is always limited by the cost of material resources,

instrumentation, and labor. Therefore, engineering analysis of experimental data

involves an approximation of the system response surface from a limited amount of

measurements. This leads to the classical problem of interpolating scattered data.

Figure: Hemsch wing-body-nacelle wind tunnel model for the AIAA Drag Prediction

Workshop (NASA Langley)

Approach

Conventional Regularization

Define:

d ≡ denotes the dimensionality of the approximation problem

xi≡ x

1,i,…, x

d ,i( ) = d dimensional input

hk

x( ) ≡ kth basis function

wk≡ coefficients corresponding to the basis functions

u(x) ≡ exact response of a mechanical system

uEXP (x) ≡ experimental data points

ua (x) ≡ approximation of u(x)

µi≡ random noise of measurements at coordinate x

i

Assume

ua (x) = hk

x( )wk

k∑

Approach

The function ua (x) is designed to approximately pass through the experimental data points

uEXP (x) at coordinates xi= x

1,i,…, x

d ,i( ). Also assume that uEXP (x) = u(x)+ µi

Define:

L i⎡⎣ ⎤⎦ ≡ differential operator

g x( ) ≡ source function

B i⎡⎣ ⎤⎦ ≡ boundary operator

Ω ≡ problem domain

∂Ω ≡ boundary of problem domain

A large class of mechanics problems can be described by the equations:

L u0

x( )⎡⎣ ⎤⎦ = g x( ) , x ∈Ω

B u0

x( )⎡⎣ ⎤⎦ = 0, x ∈∂Ω

Approach

Assuming we can write the differential equation in a variational form for an arbitrary function f ,

1

2 f , f − g, f + b f ,q where q is a Lagrange multiplier function

Then we can combine information from experimental data and a-priori mathematical models

through Tikhonov regularization,

ε ua⎡⎣ ⎤⎦ =uEXP (x

i)− ua (x

i)

λi

⎡

⎣⎢⎢

⎤

⎦⎥⎥i=1

s

∑2

+ 1

2 ua ,ua − g,ua + b ua ,q

The parameter λi is positive and represents the degree of reliance on the a-priori mathematical

model of the physical system relative to the reliability of the experimental data.

Approach

The minimum of ε ua( )satisfies

L ua x( )⎡⎣ ⎤⎦ = g x( ) + 2uEXP (x

i)− ua (x

i)

λi

⎡

⎣⎢⎢

⎤

⎦⎥⎥i=1

s

∑ δ x − xi( ) , x ∈Ω

B ua x( )⎡⎣ ⎤⎦ = 0, x ∈∂Ω

With u0

x( ) the solution to the differential equation, our solution to this

modified mathematical model is

ua x( ) = u0

x( ) + G x, xi( )w

ii=1

s

∑ with wi= G + 1

2Λ

⎛⎝⎜

⎞⎠⎟

−1

uEXP − u0( )

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪i

where G x, xi( ) is the Green's function to L i⎡⎣ ⎤⎦ , G x

j, x

i( ) = G{ }j ,i

, and Λ is a diagonal matrix.

Approach

In conventional Tikhnov Regularization Λ = λI and as

λ → 0, ua x( )→ uEXP x( ) and λ → ∞, ua x( )→ u0

x( )

The correct value for λ is such that for random noise that is statistically independent and

uniformly distributed in the interval −τ , τ⎡⎣ ⎤⎦ it should give,

uEXP (xi)− ua (x

i)⎡⎣ ⎤⎦

i=1

s

∑2

= s ⋅Var µ( ) = sτ 2

3

Note in this formulation that the value of λ that satisfies the above constraint is solved for

iteratively and requires the solution of ua (x) within each iteration.

The direct solution of λ is still an area of active research.

Approach

Though the conventional TR is a valid approach to seamlessly fusing experimental and

a-priori mathematical models, it is impractical in its present form.

• The Green's functions for practical models are ususally unavailable. Is it possible

to use existing numerical solutions in the literature for fusion?

• Is it possible to find a value for λ that satisfies sτ 2

3 without the trouble of solving for

ua (x) explicitly?

• Rather than use a single value for λ is it possible to use a distributed one?

• Is the full data set required to accomplish data-model fusion?

• Is it possible to do all of this with minimal user interaction?

Approach #1

In our first attempt to address the previous questions we returned to the variational form

ε ua⎡⎣ ⎤⎦ =uEXP (x

i)− ua (x

i)

λi

⎡

⎣⎢⎢

⎤

⎦⎥⎥i=1

s

∑2

+ 1

2 ua ,ua − g,ua + b ua ,q

Where ua ,ua is the quadratic symmetric energy form associated with L •⎡⎣ ⎤⎦ and b •,• is a

bilinear form of the boundary operator.

Using the constant λ and ua = hj(x)w

ij=1

N

∑ we can differentiate ε ua⎡⎣ ⎤⎦ with respect to the

coefficients wi and derive a Method of Weighted Residual MWR( ) form of the fusion problem

2 hj(x

i)h

k(x

i)+ λ h

j(x), h

k(x)

i=1

s

∑⎛⎝⎜

⎞⎠⎟

wj

j=1

N

∑ + λb hk(x), q =

2 uEXP (xi)h

k(x

i)

i=1

s

∑ + λ g, hk(x) for k = 1,…, N

Approach #1

The MWR can be described as a numerical approximation method that solves for the

coefficients wi by setting the inner product of the weighted equation residual to zero:

R,ψk

= 0, k = 1, ..., s

where L[ua ]− g = R and

ψk≡ weighting function

Figure: Distribution of the equation residual R.

Approach #1

With the MWR we have access to all of the popular numerical methods in a single framework.

Finite volume method: ψk=

1 in Ωk

0 outside Ωk

⎧⎨⎪

⎩⎪

Collocation method: ψk= δ x − x

k( ) where δ is the Dirac delta function

Least-squares method: ψk= ∂R

∂wk

Method of moments: ψk= xk

Generalized Galerkin method: ψk= F h

k( ) ⇒ finite elements

The form of the bases hi(x) and the integration quadrature can also give us access to the

finite difference and spectral methods.

Approach #1

Returning to our equation of interest,

2 hj(x

i)h

k(x

i)+ λ h

j(x), h

k(x)

i=1

s

∑⎛⎝⎜

⎞⎠⎟

wj

j=1

N

∑ + λb hk(x), q =

2 uEXP (xi)h

k(x

i)

i=1

s

∑ + λ g,hk(x) for k = 1,…, N

can be written as the well-conditioned system

C + λA( )w = f + λm

C ≡ matrix containing the independent variable information of the measurement points

A ≡ conventional discretization matrix from computational mechanics

f ≡ vector representing the observational data

m ≡ forcing term vector of the a-priori mathematical model with boundary conditions

We have had success with this formulation using recurrent artificial neural networktransfer functions as bases and using Duffing’s chaotic equation as the a-priorimodel.

[Meade (1996)]

Figure: Complete 2-layer 22 node RANN assembly

from network construction scheme λ → ∞( ).

Approach #1

Figure: Phase space trajectories and Poincare' maps for

left: chaotic higher frequency boundary ω = 0.865 and

right: chaotic lower frequency boundary ω = 0.78.

We have had success with this formulation using recurrent artificial neural networktransfer functions as bases and using Duffing’s chaotic equation as the a-priorimodel.

[Meade (2005)]

Figure: Chaotic phase space trajectory of the

RANN model for ω = 0.860.

Approach #1

Figure: Fused response using data from a

non-chaotic response λ = 0.8( ).

We have also had success with this formulation using a Thin-Layer Navier-Stokessolver in finite difference form fused with measured surface pressure and boundarylayer profile.

[Wang MS Thesis (1998)]

Approach #1

Approach

Recall our questions:

• The Green's functions for practical models are usually unavailable. Is it possible

to use existing numerical solutions in the literature for fusion?

• Is it possible to find a value for λ that satisfies sτ 2

3 without the trouble of solving for

ua (x) explicitly?

• Rather than use a single value for λ is it possible to use a distributed one?

• Is the full data set required to accomplish data-model fusion?

• Is it possible to do all of this with minimal user interaction?

Approach #2

We have developed an alternative method in hopes of addressing the remaining concerns.

Starting with a straightforward reformulation using the solution to a numerical model,

uCFD (x), from the literature

L ua x( )− uCFD (x)⎡⎣ ⎤⎦ = 0+ 2uEXP (x

i)− ua (x

i)

λi

⎡

⎣⎢⎢

⎤

⎦⎥⎥i=1

s

∑ δ x − xi( ) , x ∈Ω

B ua (x)− uCFD (x)⎡⎣ ⎤⎦ = 0, x ∈∂Ω

where L i⎡⎣ ⎤⎦ and B i⎡⎣ ⎤⎦ are now linear operators.

Approach #2

If we define e(x) ≡ uEXP (x)− uCFD (x) and use hi

x( ) = G x, xi( ) , our solution to this

modified form is

hi

x( )wi

i=1

s

∑ = ua (x)− uCFD (x) u(x)− uCFD (x) = e(x)− µ(x)

We will construct a scattered data approximation from the training data e(x1), e(x

2), …, e(x

s){ }

and train to a tolerance τ equal to the maximum estimated measurement error max |µi| ≤ τ( ).

These scattered data approximation techniques include artificial neural networks, specifically

radial basis function networks RBFN( ) and generalized regression neural networks GRNN( ) , support vector machines (SVM), and our own technique we call

Sequential Function Approximation SFA( ).

Approach #2

Our Sequential Function Approximation algorithm is a variation of Orr's Forward Selection

training method and Platt's Resource Allocating network that seeks to inprove the

computational efficiency through the MWR.

Since the radial basis function is also the Green's function for a Laplacian operator

in d dimensions we set

h(x,σn,c

n) = exp

− x − cn( )i x − c

n( )σ

n2

⎛

⎝⎜

⎞

⎠⎟ so then

Rn(x,σ

n,c

n) = u(x)− u

na (x)= u(x)− u

n−1a (x)− w

nh(x,σ

n,c

n)

= Rn−1

− wnh

n

Approach #2

The objective is to determine wn, σ

n, and c

n that minimize the residual R

n. Through the MWR

we can reformulate the residual equation as a minimization problem,

Rn, R

n= R

n−1, R

n−1− 2w

nR

n−1,h

n+ w

n2 h

n,h

n which for an optimum w

n; w

n=

Rn−1

,hn

hn,h

n

becomes ⇒ R

n, R

n

Rn−1

, Rn−1

= 1−R

n−1,h

n

2

hn,h

nR

n−1, R

n−1

or Rn 2

= Rn−1 2

sinθn

where θn is the angle between R

n−1 and h

n. So this formulation should give R

n 2< R

n−1 2

as long as θn≠ π

2.

Approach #2

This formulation is quite similar to Proper Orthogonal Decompositon POD( ) , empirical

basis function method, and the Krylov method.

We've used the SFA algorithm successfully with differential equations and conventional

finite element basis functions in the mesh-free solution of differential equations.

[Meade, Kokkolaras, and Zeldin 1997]

Approach #2

Discrete Ordinate Method (DOM) Thermal Radiation Problem

Non-dim. temp. approx. using 21 exp.bases by meshless method

Non-dim. temp. approx. using 40401finite volume bases

Approach #2

[Thomson, Meade, and Bayazitoglu (2001)]

Approach #2

The solution for discrete data sets is the nonlinear minimization of

1−(R

n−1⋅H

n)2

(Hn⋅H

n)(r

n−1⋅r

n−1)

with wn=

(Rn−1

⋅Hn)

(Hn⋅H

n)

, where σn, and c

n are unconstrained

The network parameters wn, σ

n, and c

n account for one network unit at the nth iteration.

With an initial value for σn set, the algorithm begins and with the determination of

wn, σ

n, and c

n a new basis function is allocated and R

n is updated.

The iterative process continues until either a pre-determined tolerance is reached

i.e., max |Rn| ≤ τ or R

niR

n≤ s

τ3

⎛⎝⎜

⎞⎠⎟

or n = s.

Convergence rate should be : RniR

n≤ Ca−n

This approach has been used successfully with RBF in the approximation and analysis of

high-dimensional scattered data problems.

Approach #2

Multidimensional Regression: Kinematics Problem

Table: LS-SVM Results with Optimization of Parameters

Robot arm problem. The 8 inputs correspond to arm joint orientations, while the single output

corresponds to the distance of the end effector from a fixed point in space.

Table: SFA Results with τ = 0

Approach #2

Binary Classification

Table: LS-SVM and SFA Classification Performance

The binary function z = sgn sin x( ) + y⎡⎣ ⎤⎦ , that takes the values of ±1, is approximated

using 100 data sets of two inputs.

In the SFA approximation τ is set equal to the maximum measurement error τ = 0.99( ).

LS-SVM SFA# of TrainingSets

# ofSupports

Accuracy # ofSupports

Accuracy

10 10 0.57 3 0.5230 30 0.98 5 0.9950 50 0.96 7 0.9775 75 0.97 11 0.97100 100 1.00 8 1.00

Approach #2

Binary Classification

Figure: LS-SVM Results with 30 Supports. Accuracy = 0.98 Figure: SFA Results with 5 Supports. Accuracy = 0.99

Self-Assembling ANNs Applied to A Numerical Data/Model Fusion Problem

Approach #2

Approach #2

Our previous work indicates:

• The SFA algorithm improves the efficiency of the standard RBF approximations

by determining one basis function for every training iteration.

• Optimal number of basis functions minimize the computational demand and storage.

• Requires little user interaction

• Can model multi-dimensional continuous and discontinuous functions.

Approach #2

Recall our questions:

• The Green's functions for practical models are ususally unavailable. Is it possible

to use existing numerical solutions in the literature for fusion?

• Is it possible to find a value for λ that satisfies sτ 2

3 without the trouble of solving for

ua (x) explicitly?

• Rather than use a single value for λ is it possible to use a distributed one?

• Is the full data set required to accomplish data-model fusion?

• Is it possible to do all of this with minimal user interaction?

Self-Assembling ANNs Applied to A Cavity Classification Problem

SFA (tol = 0.99)SVM

162,115,22,12,14,19

0.9813267

0.902642,32,18,10,12,12

0.9176200

0.831532,23,10,6,7,11

0.8352150

0.756614,11,5,4,10,12

0.7528100

0.53567,4,2,3,4,90.486950

AccuracyNumber ofsupports

AccuracyNumber oftrain

samples

Applications

Figures: HH-60H and U.S. Navy Amphibious Assault Ship

Applications

Physical experiments, especially flight tests, can be very expensive and tedious.

Design of launch/recovery envelope for a U.S. Navy helicopter requires:

4-5 days of ship-board flights

4 pilots, 2 aircrew, 4 test engineers, 5 maintenance personnel

Manuvering the ship to simulation various sea conditions

Hundreds of thousands of USD$

Identification / Classification of Naval Rotorcraft Recovery

The Effect of Parked Aircrafton LHD Class Deck Flow

Patterns

Parked helicopters appear to moveleading edge vortex closer toaircraft located at LHD spot 5

White regionrepresentsfootprint of

leading edge“wingtip”

vortex1/240 scale LHD 6

NASA Ames FML 48”x32” Tunnel Relative Winds: 30 deg starboard

H-46 at spots 5 and 6 (upwind helo omitted)

Kurtis R. LongNAWCAD Pax River MD

(At NASA Ames Research Center)(650)-604-0613

Interpreting Oil Flow Images:1) Pigment aligns with flow direction;2) Pigment piles up at regions of low

horizontal speed (or vertical flow)3) No pigment in high speed flow regions

Reverse and Separated Flow atBow

Turbulent Gusts Separatingfrom Unsteady Bow Flow

“Wingtip” Vortex Emanating from Deck Edge(Generally at deck edge, for bow winds)

Turbulent WakeFlow Behind Island

“Necklace” Vortex AroundBase of Ship

(Generally below deck edgeand very weak)

“Necklace” Vortex AroundBase of Island

Applications

Applications

Identification of Naval Rotorcraft Recovery

Figure: HH-60H control panel. Figure: Sea Knight mayhem

Identification / Classification of Naval Rotorcraft Recovery(Meade and Long, 2004)

Effort to replace the standard two-dimensional launch/recovery envelope.

The SFA approach is used to construct a manifold that approximates the quality ratings

of several HH-60H command pilots after recovery from Navy amphibious carriers.

369 data sets

13 dimensions

4 classes

Applications

CAUTION:UNRESTRAINED FLIGHT DECK SAFETYNETS MAY RISE UPRIGHT FOR WINDS035-325 EXCEEDING 30 KTS.

Entire Envelope: day.Shaded Area: night.

PITCH(+/-) 5ROLL(+/-) 8

5

1 0

1 5

2 0

2 5

3 0

3 5

40 KT

STERNAPPROACH

345

330

280

245

125

070

030

020

270 090

PRS #Pilot Effort Rating Description

1Slight

No problems; minimal pilot effortrequired to conduct consistently safeshipboard evolutions under theseconditions.

2Moderate

Consistently safe shipboardevolutions possible under theseconditions. These points define fleetlimits recommended by NAWCADPax River.

3 Maximum

Evolutions successfully conductedonly through maximum effort ofexperienced test pilots using proventest methods under controlled testconditions. Loss of aircraft or shipsystem likely to raise effort beyondcapabilities of average fleet pilot.

4Unsatisfactory

Pilot effort and/or controllabilityreach critical levels. Repeated safeevolutions by experienced test pilotsare not probable, even undercontrolled test conditions.

Table: Pilot Rating Scale (PRS) Figure: HH-60H Operational Recovery Envelope

Applications

InputIndex

Abbreviation Definition (Units)

1 Ship Type USN Ship Type: DD 967 (1), DDG 61 (2) DD 971 (3), DD976 (4)2 WOD Spd Wind Over Deck Speed. Relative wind speed (kts).3 WOD Dir Wind Over Deck Direction. Relative wind direction (degrees)4 Long CG Longitudinal CG station of helicopter. Length aft of datum (in).5 Wfuel Weight of fuel aboard helicopter (lb).6 GW Gross Weight of helicopter (lb).7 Qavg Average hover torque required during evolution (%).8 Qmax Maximum hover torque required during evolution (%).9 Pitch Pitch angle of ship during evolution (degrees)10 Roll Roll angle of ship during evolution (degrees)11 OAT Outside Air Temperature (degrees)12 Hp Pressure altitude (ft).13 Hd Density altitude (ft).

Identification of Naval Rotorcraft Recovery

Table: Classification Model Inputs

Applications

Identification of Naval Rotorcraft Recovery

Figure: SFA recovery model using τ=0.16 and 201 radial bases: (a) approximation of PRS for recovery,

(b) approximation error, (c) convergence rate of the residual, (d) input sensitivity.

Most sensitive to: GW, Wfuel, and Hd.

Classification of Naval Rotorcraft Recovery

Applications

SFA SFA LS-SVM LS-SVM # of Training Sets

# of Supports Accuracy # of Supports Accuracy

10 ( 4, 10, 5, 0) 0.17 Full 0.22 50 (16, 23, 8, 0) 0.82 Full 0.75 100 (17, 29, 12, 2) 0.84 Full 0.69 150 (25, 39, 13, 2) 0.88 Full 0.80 200 (31, 52, 19, 2) 0.92 Full 0.81 250 (36, 54, 16, 2) 0.93 Full 0.82 300 (46, 71, 23, 2) 0.96 Full 0.82 369 (49, 74, 23, 2) 1.00 Full 0.79

Table: PRS Classification Comparison using 1 Against All Scheme.

LS-SVM (Matlab v 1.5), SFA τ = 0.99( ).

Applications

The health monitoring system we have investigated belongs to the Full-Span Tilt-rotor

Aeroacoustic Model (TRAM) used in the NASA Ames 40 × 80 ft Windtunnel.

324 data sets

71 inputs

5 targets

τ = 0.001

Regression / Identification in a Health Monitoring System(Meade, 2003)

Figure: Full Span Tilt Rotor Aeroacoustic Model configuration and schematic

Acoustic Traverses

•0.25-scale V-22•R=4.75 ft•3 balances•2 electric motors, 300 Hp ea•Bayonet mount: -9 to 18 degAOA

Applications

Figure: FSTRAM in the tunnel.

Location of input sensors: (top) fuselage, (bottom) right hand nacelle.

Regression / Identification in a HMS

Applications

Regression / Identification in a HMS

Figure: Nacelle transmission #4 bearing with 265 supports and τ = 2

(a) time series model of the temperature, (b) approximation error, (c) convergence rate, and

(d) input sensitivity.

Most sensitive to motor RPM.

Applications

Regression Analysis in a HMS

Qconvection

Qconduction= 0.

D = 0.05 m,bearing diameter.

T∞environmenttemperature.

rotating shaft

x = 0.002 m, bearing bodythickness.

Qin

, is the frictional heat generated by the bearing.

T∞ = 25 °C, is the environmental temperature assumed to be constant.

h = 6.5 W/m2 °C, is the heat transfer coefficient (estimated for a 5-cm dia. horizontal cylinder in air).

A = !D2 /4= 0.00196 m2 , is the convective surface.

∀ = Ax = 3.927x10-6 m3, is the volume of the convective surface.

C = 465J/kg °C is the specific heat of the bearing material (carbon steel).

ρ = 7833 kg/m3, is the material density.

Applications

Regression Analysis in a HMS

Ti= T∞ +

Qin− Q

in− hA(T

i−1−T∞ )( )

hAexp −

hA(ti− t

i−1)

cρ∀⎡

⎣⎢

⎤

⎦⎥

The heat generated by the bearing, Qin

, is determined by the friction equation,

Qin= (µ ⋅ L ⋅r) ⋅(GR ⋅Ω

i⋅ 2π

60), Watts

where

µ = 0.002 : is the friction coefficient estimate (obtained from bearing manufacturer)

L : is the frictional load in Newtons (dependent on the bearing location)

r = 0.025 m : is the bearing radius

GR : is the transmission gear reduction constant and depends on the bearing location.

Ωi : is the operating RPM.

L is determined from our sensitivity analysis of the SFA model.

Applications

Regression Analysis in a HMS

Bearing Abbr. GR L (N)

− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −Gearbox BTCLG2_R 13.1 10

Static Mast Upper TEMPUB_R 1 100

Swashplate TEMPSPB_R 1 30

Static Mast Lower TEMPSMBL_R 1 100

Transmission BTNT4_R 4 30

Gearbox

0.00

50.00

100.00

150.00

200.00

250.00

0.00 20.00 40.00 60.00 80.00 100.00 120.00 140.00

time(min)

temp(F)

BTCLG2_R

equation

Conclusions & Future Work

The SFA used within a Tikhnov regularization framework shows some promise as a way to merge theory,

experimental observations, and computational fluid dynamics.

We have shown it is possible to form ua (x) with little user interaction.

Further investigation of the method and the applications are required:

• Perform meshless solution of ua (x) with the solution of uCFD (x) combined with uEXP (xi). This would

produce meshless and data-driven computational mechanics solvers.

• Investigate method in designing better experiments.

• Investigate the method with proxy data.

• Investigate other types of bases.

This work was partially supported under NASA Cooperative Agreement No. NCC-1-02038.

References

A. J. Meade, “Regularization of a Programmed Recurrent Artificial Neural Network.”

Submitted to IEEE Transactions on Artificial Neural Networks. 2005.

Wei Wang, “Exploration of Tikhonov Regularization for the Fusion of

Experimental Data and Computational Fluid Dynamics,” M.S., August 1998.

A. J. Meade, M. Kokkolaras, and B. Zeldin, “Sequential Function Approximation for

the Solution of Differential Equations,”

Communications in Numerical Methods in Engineering, 13 (1997): 977-986.

D. L. Thomson, A. J. Meade, and Y. Bayazitoglu, “Solution of the Radiative Transfer Equation

in Discrete Ordinate Form by Sequential Function Approximation,”

International Journal of Thermal Sciences, 40 (6) (2001): 517-527.

Jose Navarrete, “Construction of Airfoil Performance Tables by the Fusion of

Experimental Data and Numerical Data,” M.S., August 2004.