Conjugate Heat Transfer and POD for Inverse Problems

Conjugate Heat Transfer and

POD for Inverse Problems

Alain Kassab

Mechanical and Aerospace Engineering

University of Central Florida, Orlando, Florida (UCF)

Outline

• Conjugate heat transfer (CHT)

• Applications of hybrid and monolithic methods

• POD methods in Heat Transfer and Inverse

Problems

• Conclusions

CML lab cluster

3

• room 183 of the Engineering I building at UCF – 2 x air

conditioning units.

• 250 cores – not all the same age

• Dell PowerEdge 64 bit servers (Intel Xeon processors).

• 8Gb memory per processor.

• 14 TB of HD storage with tape back-up.

• Latest addition is a 64 core 1U blade with 2 x AMD EPYC 7501 (2.0GHz, 32-Core)

with 512Gb RAM (8GB/core).

• Supported by the College of Engineering and Computer Science Technical Support

Office who install, maintain, and monitor research clusters.

• Commercial codes available: Starccm+, Abaqus, Pointwise, Mimics, ...

• Cluster runs under CentOS 7.4 and uses Ganglia open source monitoring system.

• Conjugate Heat Transfer (CHT) arises naturally in most instances where

external and internal temperature fields are coupled:

• In addition, the solid may be subject to thermal stresses due to

temperature gradients and external loadings (thermoelasticity)

• Eliminates need for imposing heat transfer coefficients.

Conjugate Heat Transfer – with applications …

Tfluid=Tsolid

qfluid=qsolid

fluid/solid

Interface B.C.

Wq

T

uFlow

Wq

T

uFlow

Wq

T

uFlow

T

Solid

Fluidf

s

s

qs

• Conjugate Heat Transfer is relevant, for instance in the design

and analysis of:

• Cooled turbine blade/vanes.

• Fuel ejectors, nozzle or combustor walls.

• Heat exchangers.

• Automotive engine blocks.

• Pin fin cooling, rib turbulators....

• Thermal protection system for re-entry vehicles.

• Electronic chip cooling

• Microchannel heat transfer

• Shock tube for “long” ignition study test times.

• Others ….


• Conjugate Heat:

•Analytical methods for limited to simple geometries with

earliest attempts in the 1906’s, flat plate and channels.

• Numerical solutions in the 1970’s with the evolution of computational methods.

• Many commercial codes (Starccm, Fluent, ANSYS CFX…)

now are offering multi-physics capabilities including conjugate

heat transfer

• Specialized CFD codes (FVM or FEM) may not have the capability to solve conjugate problems: add on capability.


• Generally, the approaches to resolve conjugate heat transfer can be characterized as:

1. Hybrid coupling procedure: couple CFD solver to a conventional FVM, FEM or BEM solver to resolve the heat transfer within the solid walls with possible interpolation between two grids if they are disparate.

2. Homogeneous method: direct coupling of the fluid zone and the solid zone using the same discretization and numerical approach interpolation-free crossing of the heat fluxes between the neighboring cell faces.


Concept

Applications: Hybrid FVM/BEM

FVM/BEM Hybrid approach

Coupled FVM/BEM Conjugate Heat

Transfer and Thermoelasticity

• Cooled turbine blades analysis

• Cooling passage shape optimization

Conjugate Heat Transfer: coupling flow and solid

responses enforcing continuity of flux and

temperature at fluid/solid interface eliminates

heat transfer coefficients

Applications: Homogeneous FVM

Wq

T

uFlow

Wq

T

uFlow

Wq

T

uFlow

T

T , t

q

*

, u


• Film cooling effectiveness

• Shock tube heat transfer with reflected shock

DiaphragmDiaphragm

He/CO2 Driver

Coupled FVM/BEM for conjugate heat transfer: a Hybrid approach

• use most suitable method for each field:

- fluid : FVM

- solid : BEM

• BEM requires a surface discretization

• BEM nodal unknowns are: temperature and heat flux (heat transfer)

displacement and traction (thermoelasticity)

• BEM nodal unknowns are precisely the variables need to couple fluid and

thermal fields by continuity


• Achieve a cooled surface temperature at a uniform value minimizing coolant flow

rate and obeying design constraints

• Quasi-CHT: lumped Rayleigh-Fanno solver + BEM heat transfer solver (2-D cross-section

model for heat load)

• Non-linear Simplex of Nelder and Mead.

Applications: blade cooling channel shape optimization w/ Siemens channel

code + BEM +Non-Linear Simplex

Nordlund, R.S. and Kassab, A.J., "A Conjugate BEM Optimization Algorithm for

The Design of Turbine Vane Cooling Channels," Proc. BETECH96, R. C. Ertekin,

M. Tanaka,R.P.Shaw, and Brebbia, C.A. (eds.), 1996, Hawaii, pp. 237-246


Applications: internal blade cooling

• BEM/FVM coupling for cooled blades: 2-D analysis (FVM/BEM)

H. J. Li and A.J. Kassab, A Coupled FVM/BEM Solution to Conjugate Heat Transfer in Turbine Blades, AIAA 94-1981, AIAA/ASME 94-2933.

isotherms


Applications: Thrust vector control vanes:

• supersonic transient application

NAVY SBIR with Applied Technology

Associates

Rahaim, C.P., Kassab, A.J., and Cavalleri,

R., "A Coupled Dual Reciprocity Boundary

Element/Finite Volume Method for

Transient Conjugate Heat Transfer," AIAA

Journal of Thermophysics and Heat

Transfer, Vol. 14, No. 1, 2000, pp. 27-38.


• FVM/BEM coupling methodology:

transfer of nodal values from FVM and

BEM (and back) independent surface

meshes is performed with a localized

Radial-Basis Function (RBF)

interpolation.

Approach to Coupling FVM and BEM

T

q

*

Rmax

ri1y

x

TCFD,2

TCFD,3

TCFD,5

TCFD,4

TCFD,1

ri4

ri2

ri3

ri5

ri

Rmax

ri1y

x

TCFD,2

TCFD,3

TCFD,5

TCFD,4

TCFD,1

ri4

ri2

ri3

ri5

ri

• Temperature Forward/Flux Back (TFFB) coupling methodology: FVM provides

Dirichlet conditions to BEM and BEM provides Neuman conditions to FVM

• At the fluid solid interface require temperature and heat flux continuity


T

To

o

dTTkTk

TU ')'()(

1)(

Boundary Conditions:

on n

Tk(T)-

on

s

s

q

TT

Kirchhoff Transform:

02 U 0)(. TTkGoverning Equation:

on n

k-

on )(

o s

s

qU

TUU


BEM for non-linear problems

Note: a convective BC transforms to a non-linear boundary condition and that

requires iteration. In CHT there are no heat transfer coefficients

• Domain decomposition

1. direct solver - sparse block matrix

2. iterative solver – initial guess and interface

continuity of temperature and

heat flux (displacement and traction)

• Domain decomposition + iterative solution is ideally suited to parallelization


2

1

22

11

2

1

22

110

0

0

0q

q

q

GG

GG

T

T

T

HH

HH i

'i

i'

i

'i

i'

2W 1W 2W

i1

qGTH

• New algebraic system size based on K sub-domains: n 2N/(K+1) Boundary

Elements (K: # of subdomains)

• Sub-domain iteration to satisfy flux

continuity:

• Sub-domain iteration to satisfy

continuity of temperature:

qW1I = qW1

I -(qW1I +qW2

I )/2

qW2I = qW2

I -(qW1I +qW2

I )/2

TW1I = (TW1

I +TW2I )/2 + R” qW1

I /2

TW2I = (TW1

I +TW2I )/2 + R” qW2

I /2

22

42

W2

II32 I12

21

41

W1

I31 +


Divo, E.A., Kassab, A.J. and Rodriguez, F., "Parallel Domain

Decomposition Approach for large-scale 3D Boundary

Element Models in Linear and Non-Linear Heat

Conduction," Numerical Heat Transfer, Part B,

Fundamentals, Vol. 44, No.5, 2003, pp. 417- 437.


Homogeneous CHT modeling - localized collocation meshless method (MQ-RBF)

u = 0.25

v = 0

p = 0

T = 0

Air

1.225

= 1.79e-05

k = 2.42e-02

c = 1006.43

u = 0

v = 0

p = 0

q = 0

u = 0

v = 0

p = 0

q = 0

u = 0

v = 0

p = 0

q = 0

T = 100(0,0)

(0.11,0.01)

(0.05,0.005) (0.06,0.005)

Titanium

= 4850

k = 7.44

c = 544.25

x [m]

du

/dy

[1/s

]

0.06 0.07 0.08 0.09 0.1 0.11-300

-200

-100

0

100

200

Meshless 5x5

Meshless 10x10

Meshless 20x20

Meshless 40x40

FVM First Order

FVM Power Law

FVM QUICK

FVM Second Order

u [m/s] (Lines: Fluent - Symbols: Meshless)

y[m

]

0

0.002

0.004

0.006

0.008

0.01

LCMM CHT

Block temp

Fluent CHT

Block temp

Divo E and Kassab AJ., An Efficient Localized RBF Meshless Method for Fluid Flow and Conjugate Heat Transfer, ASME Journal of Heat Transfer,

2007, Vol. 129, pp. 124-136.


Temperature contours

Velocity contoursData center distribution for cooling plenum,

cooling hole ( width), and main cooling channel

Results from CHT modeling: film cooling


Gritsch, M., Schulz, A., and Wittig, S., 1998, "Adiabatic Wall Effectiveness

Measurements of Film Cooling Holes With Expanded Exits," ASME J.

Turbomachinery, Vol. 120, pp.549-556.

Plenum

Mainstream

Fan-Shaped Hole

(a)Plenum

Mainstream

Fan-Shaped Hole

(a)

Silieti, M., Kassab, A.J. and Divo, E.A., "Film Cooling Effectiveness: comparison of adiabatic and conjugate heat transfer

CFD models," International Journal of Thermal Sciences, 2009, Vol. 48, pp. 2237–2248

Mainstream

Plenum

Endwall

(b)

Mainstream

Plenum

Endwall

(b)

Film effectiveness

- local temperature

- stagnation temperature of coolant

at the injection point

- main flow recovery temperature


streamwise distance, x/D

ce

nte

rlin

ee

ffe

ctive

ne

ss

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gritsch et. al. 1998

RKE

SST kw

V2F

Comparison of computed centerline adiabatic

effectiveness with data of Gritsch et al.

Local adiabatic and conjugate effectiveness predicted by

three turbulence models (conjugate effectiveness with RKE)

streamwise distance, x/D

Ce

nte

rlin

eE

ffe

ctive

ne

ss

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gritsch et. al. 1998

Adiabatic Case

Conjugate Case

Comparison of computed centerline

effectiveness (RKE) model with data of

Gritsch et al.

(a) Adiabatic (b) Conjugate

Temperature magnitude contours in Kelvin along centerline

plane in the film cooling hole region predicted using the

RKE turbulence modell. Conjugate with high temperature

Polymer properties.

Motivation: Shock Tube Facility

The shock tube is used to investigate chemical kinetic behavior

A shock tube has a high-pressure driver and low-pressure driven section

Measurements conducted at the endwall behind the reflected shock

Properties of fuels such as ignition delay time are needed for operation of

gas turbines (blowout, flashback,…)


Lamnaouer, M. , Divo, E., Kassab, A. J. and Petersen, E., "A Conjugate Axi-symmetric Model of a High-Pressure Shock-Tube Facility," International Journal of Numerical Methods for Heat and Fluid Flow, 2014, Vol. 24 No. 4, pp. 873-890.

Frazier, C., Lamnaouer, M,. Divo, E., Kassab, A., and Petersen, E., "Effect of Wall Heat Transfer on Shock-Tube Test Temperature atLong Times", Shock Waves - An International Journal on Shock Waves, Detonations and Explosions ‘ 2011, Vo. 21, No. 1, pp.1-17

Mechanisms for non-ideal behavior in shock tubes:• Non-ideal rupture of the diaphragm

• Reflected shock/boundary layer interactions

• Driver gas contamination

• Contact surface instabilities

• Heat losses to the shock tube side walls CHT analysis

The shock tube is a transient test facility with unsteady and highly nonlinear physical processes that may be modeled with the means of Multi-Dimensional, CFD simulations

Background

Shock Tube CHT CFD Model

Boundary Conditions

CHT BC’s at

fluid/solid interface

Shock Tube CHT CFD Model

• Entire Shock Tube Geometry

• Structured Mesh

• Axi-symmetric Model

• Dynamic Grid adaption allows for a

better resolution of the shock and

contact discontinuities.

• Grid- independent solution

• Challenge of large mesh sizes,

small time steps overcame with

Parallel Computing

Mesh

2 4 6 8

0

5

10

15

20

25

30

35

Time (ms)

Pre

ssu

re (

atm

)

100000 nodes

150000 nodes

200000 nodes

250000 nodes

Driver gas : He

Driven gas : Ar

P4 = 25 atm

P1 = 0.5 atm

Incident shock

WaveContact Surface

Bifurcation Modeling in Air, T~950 K, P~1 atm

Pressure Contours Behind Reflected Shock Wave Depict the Bifurcation

Structure which Renders the Flow in the End-Wall Region Non-Uniform

computed 55 bifurcated foot angle is in agreement w/ observations in literature

Temperature

26

Conjugate Heat Transfer


Temperature Contours Behind Reflected Shock Wave show the Hot

and Cold Jets in the Endwall Region

Hot jets ~100 K

higher than

average flow

temperature

Cold jets carrying

the colder fluid

from the boundary

layer fluid impinge

on the slip line and

shock-tube walls

Weaker interaction

due to the thinner

boundary layer as a

result of energy loss

to the side and end-

walls captured by the

conjugate heat

transfer model


Vorticity Contours Behind the Reflected Shock Wave show the Co-rotating

organized Vortical Structures in the Shear Layer.

The embedded vortices

between the side wall and

the shear layer grow in size

and in number as the

reflected shock moves

away from the end wall

Vortices continue to

interact with the

boundary layer

Shock Tube Model Validation with Experiment

Experiments were Performed in Texas A&M Shock-Tube Lab

Viscous solution is Validated with Experimental Data in

N2 Test gas. Model displays a slightly more oscillatory

profile than the experiment.

T~900K, P~2.5 atm T~1600 K, P~17 atm

5 6 7 8 9

0.0

0.5

1.0

1.5

Driver: 100% He

Driven: 100% N2

model (950 K, 2.4 atm)

data (911 K, 2.48 atm)

Norm

aliz

ed P

Rela

tive t

o P

1

time, ms

3 4 5 6 7 8 9

0.0

0.5

1.0

1.5

model (1600 K, 17 atm)

data (1585 K, 16.7 atm)

Driver: 100% He

Driven: 100% N2

Norm

aliz

ed P

Rela

tive t

o P

1

time, ms

Bifurcation Modeling – CHT model findings

• Lingering pockets of elevated temperature near the endwall predicted to be about 100K higher than the average temperature behind the reflected shock wave could lead to a local ignition event.

• Chaos and Dryer have reviewed the occurrence of such non-homogeneous events in shock-tube studies in the literature.

• The conjugate heat transfer model results in a slightly weaker reflected

shock wave due to the energy loss to the side and end walls.

COMMENTS

• CHT: coupling of heat conduction in the solid with convection heat transfer in contacting fluid. Eliminates heat transfer coefficients.

• Hybrid coupling procedure: couple CFD solver to a conventional FVM, FEM or BEM solver to resolve the heat transfer within the solid walls with possible interpolation between two grids if they are disparate.

• Homogeneous method: direct coupling of the fluid zone and the solid zone using the same discretization and numerical approach interpolation-free crossing of the heat fluxes between the neighboring cell faces.

• In many cases heat conduction from nearby hot sources/sinks interacting with fluid should not be neglected.


Why use POD?

History

Methodology

Application to heat transfer and fluid mechanics

A derivation of POD

RBF interpolation network

Trained POD-RBF Network Inverse Methods

32

….and now for something completely different

POD- model reduction

Many ingredients to solve inverse problems effectively:

(a) reduces the degrees of freedom in the system (model reduction)

(b) Seems to add inherent regularization by filters excess error through truncation

of the eigenvalues?

Other approximation methods may not likely adequately represent the data as

they use an arbitrary set of (non-optimal) basis functions to model the data.

POD optimally chooses the best way to model the data

33

Why POD?

• FORWARD PROBLEM

GIVEN:

1. governing equation for field variable

2. physical properties

3. boundary conditions

4. initial condition(s)

5. Geometry

FIND: field variables, temperature, displacement, potential, ….

• INVERSE PROBLEM

GIVEN:

1. part of conditions 1-5 in a forward problem

2. Over-specified condition at the boundary or interior

FIND: unknown in 1-5

Forward and Inverse Problems

one may state the inverse problem as: from the observed effect then determine its cause 34

• APPLICATIONS IN HEAT TRANSFER

1. property evaluation: k,h, etc...

2. governing equations: u'''G

3. boundary condition: T,q, etc...

4. initial condition: T(r,0)

5. determine thermal contact resistance

6. Determine unknown (hidden) geometry

applied to nondestructive evaluation (NDE):

detection of subsurface flaws and cavities

Related problem of inverse design and optimization, e.g. shape optimization

q = 0

R (x)

k

k

y

Xx= 0

y = 0

y = l

Tu

= T hot

=500 K

ku

kb

y

xx= 0 x = L

y = L

y = l

Temperature measuring points

Tb

= T cold

=0 K

q = 0q = 0

R (x)

k

k

y

Xx= 0

y = 0

y = l

Tu

= T hot

=500 K

ku

kb

y

xx= 0 x = L

y = L

y = l

Temperature measuring points

Tb

= T cold

=0 K

q = 0

35

• inverse problems are intimately connected to measurement

• measured data provides over-specified condition

• ingredients of the inverse problem solution:

1. Objective function - measures the difference between

measured data Tm,i and computed values at under current

estimates of the sought-after unknown(s) at a number NMP of

measuring points

2. Forward problem solver (FDM, FVM, BEM, meshless…) - solves

the forward problem under current estimates of the sought-after

unknown(s)

3. Optimizer used to minimize the objective function – gradient based

CG, non-gradient based GA, evolutionary algorithms….

NMP

c ,i m,i~ ~

i

J(a) [T (a ) T ]

2

1

c ,i

~T (a )

36

•inverse problem is ill-posed

• noise in input data is amplified by the inverse solution process

Inverse solution

Output DataInput Data

37

EXAMPLE: heat transfer coefficient retrieval (not really an inverse

problem but cast as one to filer noise)

• regularization to stabilize result

1. regularize the functional, e.g. Thikhonov regularization, damped

least-squares,…

2. regularize via the solver, Conjugate Gradient with stopping criteria,

truncated SVD, discrepancy principle, genetic algorithms, POD...

203040Node

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

h-2D62s

h-2D124s

h-2D186s

h-2D248s

h-2D310s

38

α

one means of finding regularization parameter α is the L-curve method of

Hanson

0 0.05 0.1 0.15 0.2 0.2564.8

64.9

65

65.1

65.2

Alpha

Least Square Ap prox.

Alpha = 0.001

Alpha

Least Square Ap prox.

Alpha = 0.001

Alpha Optimization Curve

39

Development

POD was developed over a century ago by Pearson as a statistical tool to

correlate data: find a plane that is closet to points. Over the past 30 years POD

has been applied to many engineering applications.

Used in applications from signal and control theory, data processing, image

reconstruction, parameter estimation, heat transfer, fluid flow and many others.

Objective

Optimally choose a set of basis vectors that correlate known data.

History

Names

POD has been redeveloped over many years and is related to Principal

Component Analysis (PCA), Karhunen-Loéve Decomposition (KLD) …

Applications

40

• Consider the need to approximate a continuous function

• How to choose the basis vectors φ to capture the field of interest?

• PODs use a data set from the field and utilize it to generate the basis functions

• Approximates data set utilizing a matrix of the data itself

• and find φ that is optimal in sense of best approximation

41

First POD generates a snapshot vector by altering the desired

parameter(s) p and storing the output data inside a vector u.

Uu

u1

ui

uN

Sampling points

u refers to the recorded temperatures, velocities, reaction forces

etc…captured numerically or empirically.

The snapshot vectors are then stored in a matrix called the snapshot matrix

denoted by U

42

• Basis vectors in matrix Φ can then be defined as a linear combination of the

snapshots u and in matrix form

Φ U

V M

MMM

N N=

• Choose basis (expansion vectors) using the data itself since it is expected to

contain information about the field response to the variation of the parameter p

• Require basis to be orthonormal (useful property)

ΦTΦ=I(M)

VUΦUVΦ

M

j

i

ij

j

1

43

Wish to express all snapshots in a truncated basis

(approximate operation)

1min

Ki j

jijA K

U Φ U Φ A

U F

N N

MM K

K

how to construct the basis, to ensure minimum

error for a predefined K ?

A

44

Approximation problem

for a given find such that || || minK Φ U Φ A

POD Algorithm:

• find the optimal basis then

• evaluate the coefficients of the expansion

Standard Algorithm:

• for a given basis (guess optimal basis)

• evaluate find the coefficients of the expansion

POD compares with

Fourier analysis:

• first seek the eigenfunctions of the problem

• then the coefficients of expansion45

How to calculate the basis?

UUC T

equivalent to an

eigenvalue problem

j

j

jvvC

C is symmetric and positive definite

optimization problem for given || || minK U Φ A

subjected to constraints IΦΦ TVUΦ

U

M

N

UTM

N

= CM

M

• N>>M, dimension of

matrix C is small

v j - eigenvector, j-th column of V(M)

λj - eigenvalue real and positive

46

• eigenvalues in diagonal matrix Λ often drop off rapidly in value

• only most important can be used (check residual)

• truncated POD expansion can be used to represent the field

Exact solution:

Eigenvalues solve:L= 2 m

k = 3 W/mK

D = 0.05 m2/s

h = 10 W/m2K

to = 1 K

Analytical solution was

sampled at N=101 points

M= 100 snapshots every 10s

Building the POD snapshots

i λi

1 151.56

2 0.057

3 0.0602x10-6

First three eigenmodes and POD bases

47

express all snapshots in a truncated basis

with K<<M (approximate operation)

1

Ki j

jijA

U Φ U Φ A

U F

N N

MM K

K 𝐀

TA Φ U

and from orthogonality of POD basis formally

48

836

2679

3117

1400 K, 600 W/(mK)2

1400 K, 500 W/(m K)2

750 K, 650 W/(mK)2

0

1600

tem

pera

ture

,K

extrapolationsnapshots

Example: heating up a turbine bladeheat conductivity =20 W/mK,

specific heat of a unit volume cp =5 x 106 J/m3 K.

initial condition T0 = 300K. 200 snapshots every 0.1 s,

Crank-Nicholson time stepping

3,151 degrees of freedom reduced to 17 max error 13K (1.5%)

Comparing execution times: 200s to steady state with time step 0.1s

time to solution POD/FEM = 1/8

1200

800

400

0 100 200 300 400

Node 836

POD FEM

Node 2679

POD FEM

Node 3117

POD FEM

worst case

comparison of FEM POD results

49R. A. Bialecki, A. J. Kassab, and A. Fic, Proper orthogonal decomposition and modal analysis for Acceleration of

transient FEM thermal analysis, Int. J. Numer. Meth. Engng 2005; 62:774–797.

FEM 3,151 DOFS POD 17 DOFS

time 40 s

POD 17 DOFS FEM 3,151 DOFS

time 100 s

50

max

maxmax

max

)/( 2error local

/ 1error local

/ error average

TT

TT

TT

condition initialover excess max.

condition initialover excess

error absolutemean

error absolute max.

nodegiven at error absolute

max

max

T

T

T

T

T

error vs time

0 50 100 150 200 250time, s

0.0

0.4

0.8

1.2

1.6

2.0

rela

tive

err

or,

%

average

local1

local2

POD solution

51

• How to use this POD expansion in inverse problems or for other applications

to interpolate to non-sampled parameter values?

1. truncated POD

2. need to interpolate solution for new values of parameter(s) p

• process can begin with the use of Radial Basis Function Interpolation (RBF’s) ,

some of which are listed below

POD-RBF Interpolating Network

52

• The Hardy inverse multi-quadric Radial Basis Function (RBF) is defined as

• Each pi references the ith parameter values p used to create the snapshot of

U (temperatures , pressures, displacement…) uj for i = 1, 2 … M

• A POD-RBF network is created to estimate the dependence on the parameter vector P and to generate the dependent variable (temperatures, deformations …) of the system.

where

where M represents the total number of snapshots

POD-RBF Interpolating Network

r i = | p – p i |i

p1 p2 … pM

c is the shape factori

53

is used to define an interpolation matrix of constant coefficients B by solving the

collocation problem,

• An explicit approximation of the dependence of a column of the snapshot matrix

U on parameters p can be regenerated as

( ) ( )a ap pu ΦBf

1

1

2

2

(| |)

(| |)( )

(| |)

a

M

M

f p p

f p pp

f p p

f

1 2 1 2[ ( ) ( ) ( )] [ ( ) ( ) ( )]a a a M a a a Mp p p p p pu u u ΦB f f f

U ΦBF

requiring the left-hand to collocate the data matrix U, we have in matrix form,

54

• Where the RBF interpolating matrix F is

• Solving for the coefficient matrix B

• An explicit approximation of the dependence of a column of the snapshot matrix

U on parameters p is then

11 1 1

1

1

1 1 1( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

j M

ji i iM

jM M M M

i i i

M M M

f p p f p p f p p

f p p f p p f p p

f p p f p p f p p

F

TA Φ U

T TΦ U Φ ΦBF

1B A FTΦ Φ = I T T

Φ U Φ ΦBF

55

• We now have an approximation of the field variable snapshot vector for any

value of p he POD snapshot ~ numerical variation of parameters solution to

the field problem

• Can now use this reduced model to evaluate least-squares objective function,

for example the regularized objective function over i=1,2..N measuring points.

2 2

1 1

( )N N

i i i

i i

J u y u y

p p p

Which is minimized to

update the parameters p at

each iteration using an

optimization algorithm, GA,

Levenberg-Marquardt ….

u(p)=ΦB f(r(p))

56J(p) min

measuremen

tsmeasurements RBF-trained POD

Applications of RBF-trained POD

57

Estimating the heat transfer

coefficient distribution and thermal

conductivity of blade and coating

materials and here p = (kb,ks,h1,

h2,h3,h4) where the h’s are the values

at the Lagrange interpolating knots

along the blade perimeter

Ostrowski, Z., Bialecki, R.A. and Kassab, A.J., "Estimation of constant thermal conductivity by use of Proper Orthogonal Decomposition," Computational

Mechanics, 2005, 37:1, 52-59.

Ostrowski, Z., Bialecki, R.A.and Kassab, A.J., Solving inverse heat conduction problems using trained POD-RBF network inverse method, Inverse

Problems in Science and Engineering, 2008, 16:1, 39-54.

Performing Proper Orthogonal Decomposition produced the following

eigenvalues λ truncated after the 5th value.

A domain is sampled using 16 nodes to generate the POD snapshots using the exact solution to retrieve the temperature distribution.

A total of 100 snapshots were taken over 10 increments of a and b individually.

The conductivity constants to be determined are a and b using the linear

equation

Identifying conductivity in a square domain

9.065 106

1.715 104

1.348 103

23.903

0.026

T(0,y) T(L,y)

13 14 15 16

9 10 11 12

5 6 7 8

1 2 3 4

T(0,y) T(L,y)T(0,y) T(L,y)

13 14 15 16

9 10 11 12

5 6 7 8

1 2 3 4

13 14 15 16

9 10 11 12

5 6 7 8

1 2 3 4

Tempi j

T xi

yj

Temp

such that p = {a, b}58

0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6

8

Measured K(x)

POD Estimate K(x)

Measured K(x)

POD Estimate K(x)

Comparison of POD Estimate versus Measured K(x)

1 2 3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

Eigenfunction

POD Basis

Eigenfunction

POD Basis

Node

n = 2; m = 0

1 2 3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

Eigenfunction

POD Basis

Eigenfunction

POD Basis

Node

n = 1; m = 1

1 2 3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

Eigenfunction

POD Basis

Eigenfunction

POD Basis

Node

n = 7; m = 0

1 2 3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

Eigenfunction

POD Basis

Eigenfunction

POD Basis

Node

n = 4; m = 0

• Comparison of POD basis versus analytic eigenfunction

sin cosx y

n x m yeigenfunc c

L L

1, 2,3...

0,1, 2...

n

m

where

59

L-Shaped domain

.

T(0,y)

X=0 X=L

T(x,y)

T(0,y)

X=0 X=L

T(x,y)

11 12

9 10

5 6 7 8

1 2 3 4

11 12

9 10

5 6 7 8

1 2 3 4

λ

6.560 x 106

9.623 x 104

1.032 x 103

18.241

0.015

12nodes,

100 snapshots

Truncate after 5

1 2 3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

Eigenfunction

POD Basis

Eigenfunction

POD Basis

Node

n = 2; m = 0

1 2 3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

Eigenfunction

POD Basis

Eigenfunction

POD Basis

Node

n = 1; m = 1

1 2 3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

Eigenfunction

POD Basis

Eigenfunction

POD Basis

Node

n = 7; m = 0

1 2 3 4 5 6 7 8 9 10 11 121

0.5

0

0.5

1

Eigenfunction

POD Basis

Eigenfunction

POD Basis

Node

n = 4; m = 0

60

Objective: create an accurate real-time wind-load calculator and design tool for PV

systems.

◦ Rapid and accurate assessments of the uplift and down-force loads on a PV

mounting system.

◦ Optimize PV array configuration and position on the roof.

◦ Identify viable solutions from available mounting systems.

RBF-trained POD for wind load predictions of roof mounted PV solar panels

Problem: standard method (ASCE/SE 7) for calculating wind-loads on roof-

mounted PV systems is inaccurate based on simplified models that do not

account for full 3D effects, end effects, ….this method is performed manually and

therefore is slow and prone to inconsistencies.

61

Huayamave, V., Ceballos, A.,

Barriento, C., Seigneur, H.,

Barkaszi, S., Divo, A., and Kassab,

A., "RBF-Trained POD-Accelerated

CFD Analysis of Wind Loads on PV

systems," International Journal of

Numerical Methods for Heat and

Fluid Flow, 2017, Vol 27, No. 3, pp.

660-673.

RBF trained POD for wind load predictions of roof mounted PV solar panels Figures below illustrate the CFD rendered flow field using velocity

streamlines and pressure contours for sample cases in the

directions: 0˚/180˚, 30˚/330˚--150˚/210˚ and 60˚/300˚--120˚/240˚, at

160 mph.

62

CFD analysis of the Parameterized PV system Configurations

• A total of eighty-four (84) configurations for CFD analysis were defined by altering two design parameters, (1) wind speed (V=80mph to 200mph, in 20mph increments) and (2) wind angle (θ=360 around, in 30 increments), so that p={V,θ}

• The number of CFD runs was reduced to only twenty-eight (28) by taking advantage of symmetry and by placing PV panels on either side of the gable roof in order to render two or four solutions in one CFD run.

63

Pressure (psf)

Speed\Angle

0˚ 30˚

(330˚)

60˚

(300˚)

90˚

(270˚)

120˚

(240˚)

150˚

(210˚)

180˚

80 mph -10.39 -8.89 -5.44 -7.57 -8.36 -8.35 -6.44

100 mph -16.25 -13.89 -8.43 -11.83 -13.07 -13.05 -10.05

120 mph -23.42 -20.01 -12.15 -17.03 -18.90 -18.79 -14.47

140 mph -31.90 -27.24 -16.74 -23.17 -25.91 -25.58 -19.69

160 mph -41.69 -35.59 -21.83 -30.26 -33.87 -33.42 -25.72

180 mph -52.79 -45.05 -27.60 -38.29 -43.46 -42.31 -32.55

200 mph -65.20 -55.63 -34.50 -47.26 -53.76 -52.25 -40.19

60”

39”

1.5”

1.5”

3”

3”

The panel configuration consists of 9x3 PV modules arrangement: 7020

sample points for pressure and shear stress on panel

After the CFD computations were performed for all cases and the

7,020x91 POD snapshot matrix U was formed, the decomposition was

performed and tested

64

• The 91x91 covariance matrix C was formed as C = UTU followed by a standard eigenvalue decomposition which produced the results shown:

• To illustrate POD truncation, a test is performed comparing the CFD-generated pressure distribution and the POD-generated pressure distribution truncated after 12 eigenvalues for a wind speed of 140 mph and a wind angle of 30.

65

P Pw

• The POD-RBF interpolation compared with two CFD solutions that were not originally used as part of the POD snapshots: 90mph and 0:

• And at: 90mph and an angle of 180:

• We have confidence in the POD-RBF interpolation network’s ability to predict wind load distributions over PV panels at arbitrary wind velocities and angles dictated by installation requirements and codes.

66

PCFD PPOD

PCFD PPOD

Acknowledgment• This work is supported by the US Department of Energy under grant number DE-SC0010161.

67

• The trained POD-RBF interpolation network performs interpolation to

obtain the pressure distribution on the PV system surface and these

compare well to actual grid-converged fully-turbulent 3D CFD solutions at

the specified values of the design variables (wind speed and angle).

• Extend parameterization to include all possible design variables such as

roof slope, array configuration, location, supports, elevation, topography,

etc.

68

Conclusions

• RBF trained POD performed well as interpolators in several applications.

• Potential for parallelization to speed up the solution?

• Potential for new applications?

Documents

Conjugate Heat Transfer and POD for Inverse Problems