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 Transfer – with applications …
• 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.
Conjugate Heat Transfer – with applications …
• 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.
Conjugate Heat Transfer – with applications …
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
Conjugate Heat Transfer – with applications …
• 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
Conjugate Heat Transfer – with applications …
• 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
Conjugate Heat Transfer – with applications …
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
Conjugate Heat Transfer – with applications …
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.
Conjugate Heat Transfer – with applications …
• 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
Conjugate Heat Transfer – with applications …
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
Conjugate Heat Transfer – with applications …
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
Conjugate Heat Transfer – with applications …
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 +
Conjugate Heat Transfer – with applications …
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.
Conjugate Heat Transfer – with applications …
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.
Conjugate Heat Transfer – with applications …
Temperature contours
Velocity contoursData center distribution for cooling plenum,
cooling hole ( width), and main cooling channel
Results from CHT modeling: film cooling
Conjugate Heat Transfer – with applications …
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
Conjugate Heat Transfer – with applications …
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,…)
Conjugate Heat Transfer – with applications …
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
Bifurcation Modeling in Air, T~950 K, P~1 atm
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
Bifurcation Modeling in Air, T~950 K, P~1 atm
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.
Conjugate Heat Transfer – with applications …
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?