Introduction to Design of Experiments & Other Stuff . . .

Nathan RolanderMETTL Lab Meeting PresentationToday

Systems Realization Laboratory

Microelectronics & Emerging Technologies Thermal Laboratory

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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

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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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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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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

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L1

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Feasable experiemnt pointsOptimal experiment points

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D-Optimal Design Points

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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