introduction to design of experiments & other stuff .
DESCRIPTION
Introduction to Design of Experiments & Other Stuff . . . Nathan Rolander METTL Lab Meeting Presentation Today. S ystems R ealization L aboratory. M icroelectronics & E merging T echnologies T hermal L aboratory. METTL. Mock Blade Server Cabinet. Cabinet Diagram. Velocity Inlet - PowerPoint PPT PresentationTRANSCRIPT
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Introduction to Design of Experiments & Other Stuff . . .
Nathan RolanderMETTL Lab Meeting PresentationToday
Systems Realization Laboratory
Microelectronics & Emerging Technologies Thermal Laboratory
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04/22/23
Mock Blade Server Cabinet
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Cabinet Diagram
Velocity Inlet Outlet Fan Internal Fan Servers & FR4
Server rack containing 10
blade units
Blank rack to block airflow
Exhaust fan
Inlet vent
Server rack fans (x 4)
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Thermocouple Locations
50 Omega type T thermocouples used
Ice bath calibrated Cold junction
compensated Isothermal
junction Repeatability
tested
Server rack inlet air
temperature
Exhaust air temperature
Inlet air temperature
Server rack exhaust air temperature
Exhaust air TC measurement point
Inlet air TC measurement point
Server rack chip
temperatures
Foil Heater TC measurement point
How do these results compare to the
FLUENT model?
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Normalized Temperature Cabinet Response
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Graphical Explanation of the POD
The POD can be viewed as finding the principle axes of a cloud of multi-dimensional data
This is practiced in Principle Component Analysis for making sense of large quantities of data
Has been used in Turbulence to find coherent structures (Holmes)
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Example in 2D
Given 2D scatter of Data: Principle axes are found
through orthogonal regression
Usually x-y does not have physical meaning as not in the same units, therefore only fit in y
Orthogonal fit is independent of axis fit
0 5 10-2
0
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xy
raw dataleast squares fit to yleast squares fit to xorthogonal fit
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Orthogonal Residuals vs. y Residuals
Orthogonal fit residuals are always smaller than other linear regression fits
“The shortest distance between 2 points is a straight line”
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y residuals
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orthogonal residuals
raw dataleast squares fitorthogonal fit
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2D Principle Axis Computation
Mean Center Data Set:
Think of rotating the entire set of points around the origin about an angle θ:
1 1
1 1,n n
i ii i
X x and Y yn n
, , 1,...,i i i ix x X and y y Y for i n
' cos( ) sin( )x x y
' sin( ) cos( )y x y
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2D Principle Axis Computation
For the angle θ the sum of the squared of the vertical heights of the data is:
To find the best fit, this is minimized, therefore take the derivative with respect to θ and set to zero:
2
1
'n
i
S y
'2 ' dyyd
1
2 sin( ) cos( ) cos( ) sin( )n
i
dS x y x yd
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2D Principle Axis Computation
Set to 0 and manipulate algebraically:
This yields a quadratic in tan(θ):
Solution of tan(θ) is straightforward using the quadratic formula.
2 2 2
1 1 1
tan ( ) tan( ) 0n n n
i i i
xy x y xy
2tan ( ) tan( ) 1 0A 2 2
1
1
where,
n
in
i
x yA
xy
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2D Principle Axis Computation
The principle axis can also be found using the POD, recall that the POD can be computed as the SVD of U:
The rotational transformation matrix L is:
The computation of the angle of the principle axis angle,θ is identical with both approaches
{ , } n mU x y TU L V
cos( ) sin( )sin( ) cos( )
L
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Computing the Modes
The Principle Axes find the direction of maximum scatter in the data
This is the same as finding the minimum distance between the orthogonal regression line and the data points
Note that if the data is not mean centered, this will simply return a line from the origin to the centroid of the data set!
The direct analytical approach is only applicable in 2 dimensions, so SVD is better
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Computing the Modes
The POD modes are the rotation of the observed data set onto the found principle axes, and re-scaled such that the norm = 1
Therefore the direction of maximum variation is found 1st, followed by the next most direction of scatter, constrained to be orthogonal to the 1st, and so on for the number of dimensions = the number of observations
2s.t. 1L U
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Why does the PODc rock?
The complimentary POD augments the normal POD by influencing the direction of the first principle axis found
By forcing the first principle axis to find the maximum variation close to the solution to be reconstructed, the solution is much more locally accurate, but still retains the greater dynamics of the whole system
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A General Transformation Approach
Sometimes the flux function cannot be computed to find the POD mode’s contribution towards the desired goal
This flux computation can be circumvented by a general transformation from the Observation space to the POD space
ˆ( , )F u u nds
T U
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General Transformation Approach
The transformation is computed as:
This is the pseudo-inverse of the observation ensemble crossed with the ensemble of the POD modes (must be over-determined)
This transformation applied to any parameter in the observation space will transform it to the equivalent parameter value of that POD mode
T U
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General Transformation Application
For example, the range of inlet velocities used to generate the observations:
The inlet velocities of the POD modes, as would be computed by the flux function can be computed as:
This enables the computation of the POD mode heat fluxes for non-conjugate problems, or any other hard to compute phenomena
1 2, ,...,omV V V V
( )C F
oC T V
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Introduction to Design of Experiments
Design of Experiments (DOE) is an approach for obtaining the maximum value for the minimum number of experimental runs
Often paired with Response Surface Modeling (RSM) to build statistical models (multi dimensional curve fits)
Useful for initial screening of important control parameters, noise factors, and response – (partial factorial designs etc.)
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More Detailed DOE
DOE can also be used to build higher order response models, such as quadratic or higher order
These are more useful at a latter stage of work/design for the characterization of a system, after initial screening
Examples include Central Composite, Box-Benheken, Plackett-Burman
Today’s talk on Central Composite
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Central Composite Designs (CCD)
Central Composite designs are two-level full or partial factorial designs augmented to estimate 2nd order effects:
Quadratic response model:2 2
0 1 2 11 22 12y x z x z xz Linear Terms
Quadratic Terms
Interaction Terms
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Central Composite Designs
Central Composite designs are two-level full or partial factorial designs augmented to estimate 2nd order effects:
Quadratic response model:2 2
0 1 2 11 22 12y x z x z xz Linear Terms
Quadratic Terms
Interaction Terms
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-200
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400
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0
100
200
300
Central Composite Designs
Central Composite designs are two-level full or partial factorial designs augmented to estimate 2nd order effects:
Quadratic response model:2 2
0 1 2 11 22 12y x z x z xz Linear Terms
Quadratic Terms
Interaction Terms
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Central Composite Designs
Central Composite designs are two-level full or partial factorial designs augmented to estimate 2nd order effects:
Quadratic response model:2 2
0 1 2 11 22 12y x z x z xz Linear Terms
Quadratic Terms
Interaction Terms
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Central Composite Designs
(a) Initial 2 level full factorial design(b) Central composite design – added star (axial) and center points
to create 32 factorial design(c) Central composite design – α > 1 can test for cubic & quartic
effects (5 levels per variable)
α
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General CCD Formula
CCDs have 3 components (for k factors): 2k-f corner points – the base of any CCD is a 2 level
full or partial factorial design. These estimate the main and interaction linear effects.
2k star points – These estimate the quadratic main effects or higher if α > 1 .
n0 center points – If n0 > 1 a pure estimate of the variance, σ2 is possible.
Number total runs : nT = 2k-fnc + 2kns + n0
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Example
Three Factor CCD with n0 = 4 replications of center point:
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Commonly used Designs
(rotability discussed next)
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Rotable CCDs
The rotability criterion is concerned with the accuracy of the estimator ŷ
Rotable designs have the property that for any distance from the center point the variance σ2 will be the same
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Rotable CCDs
Rotability can be important because it is unknown what values of the system variables X will be used in the model evaluation
A design is rotable if:
Therefore, for a rotable 2 factor design:
142 ( )k f
c
s
nn
12 0 42 (1) 2
1
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Inscribed CCDs
What do I do if I don’t have a square region, or I can’t test values outside of a certain range?
Scale the design such that it does:
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Generating Optimal Designs
How can I find the optimal experimental points to fit if: I have a non-uniform design parameter space? I want to fit a different response model? I can’t run as many experiments as the normal
designs dictate? Use D-optimal designs to find the most efficient
points for your specific problem
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Example of non-uniform space
In this case, the two factors x1 and x2 cannot both be at the high level simultaneously:
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D-Optimal Design Approach
You need: The number of experiments you can perform nT
The response function of interest (usually quadratic) A candidate list of feasible points, C
The design criterion for D-optimal designs is to find the points that yield the smallest volume of the confidence interval of the fitted response function:
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D-Optimal Design Approach
This confidence region (as may be multi-dimensional) is given by the set of coefficients β that satisfy the inequality:
This is the same as the minimization of:
There are several algorithms to minimize D given C, nT, and the model to fit. MATLAB has the 2 most popular of these “rowexch” and “cordexch”.
( ) ' ' ( ) (1 ; , )b X X b F p n ppMSE
1( ' )D X X
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Data Center Tile Flow Measurements
Want to find how perf. tile positions affect flow
Constraints: W1,2 < 5 L1 > 2 L2 – L1 > 3 W1 = W2
Discrete tile locations: a total candidate set C of 600 points,
3 variables: L1, L2, W nT = 12
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Candidate & Optimal L1 & L2
Note odd triangular constrained design space
4 5 6 7 8 911
11.5
12
12.5
13
13.5
14
14.5
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15.5
16L1 & L2 independent
L1
L2
Feasable experiemnt pointsOptimal experiment points
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D-Optimal Design Points
4 11 5
4 16 0
9 16 0
9 16 5
4 16 5
7 14 0
4 11 0
4 14 0
6 13 3
6 16 5
7 16 2
4 14 3
1
2
3
4
5
6
7
8
9
10
11
12
L2 WL1Run
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DOE Summary & Questions?
For more detailed info on Design of Experiments and Response Surface Modeling there are many good Statistics Texts that cover the material (where I learned it from)
Design of Experiments really excels when there are larger numbers of design variables
The Data center fun could be performed with only 24 runs for full quadratic estimation of 5 variables! (best would be 30)
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And now . . .
CRACUnit
PerforatedTile
Server Cabinet
Cold Aisle
Hot Aisle
Under floor Plenum
CRACUnit
PerforatedTile
Server Cabinet
Cold Aisle
Hot Aisle
Under floor Plenum
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For something completely . . .
CRACUnit
PerforatedTile
CRACUnit
PerforatedTile
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Different (pretty pictures)
PerforatedTile
PerforatedCabinetFront Panel
PerforatedCabinet
Rear Panel PerforatedTile
PerforatedCabinetFront Panel
PerforatedCabinet
Rear PanelInlet Vent
Exit Vent
Inlet Vent
Exit Vent