Introduction to Design of Experiments & Other Stuff . . .

Introduction to Design of Experiments & Other Stuff . . . Nathan Rolander METTL Lab Meeting Presentation Today Systems Realization Laboratory Microelectronics & Emerging Technologies Thermal Laboratory METTL

Mock Blade Server Cabinet

Cabinet Diagram • Velocity Inlet • Outlet Fan • Internal Fan • Servers & FR4

Thermocouple Locations • 50 Omega type T thermocouples used • Ice bath calibrated • Cold junction compensated • Isothermal junction • Repeatability tested How do these results compare to the FLUENT model?

Normalized Temperature Cabinet Response

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)

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

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”

2D Principle Axis Computation • Mean Center Data Set: • Think of rotating the entire set of points around the origin about an angle θ:

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:

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.

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

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

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

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

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

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

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

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.)

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

Central Composite Designs (CCD) • Central Composite designs are two-level full or partial factorial designs augmented to estimate 2nd order effects: • Quadratic response model: Linear Terms Quadratic Terms Interaction Terms

Central Composite Designs • Central Composite designs are two-level full or partial factorial designs augmented to estimate 2nd order effects: • Quadratic response model: Linear Terms Quadratic Terms Interaction Terms

Central Composite Designs • Initial 2 level full factorial design • Central composite design – added star (axial) and center points to create 32 factorial design • Central composite design – α > 1 can test for cubic & quartic effects (5 levels per variable) α

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

Example • Three Factor CCD with n0 = 4 replications of center point:

Commonly used Designs (rotability discussed next)

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

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:

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:

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

Example of non-uniform space • In this case, the two factors x1 and x2 cannot both be at the high level simultaneously:

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:

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”.

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

Candidate & Optimal L1 & L2 • Note odd triangular constrained design space

D-Optimal Design Points Run L1 L2 W 1 2 3 4 5 6 7 8 9 10 11 12 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

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)

And now . . .

For something completely . . .

Different (pretty pictures)

Introduction to Design of Experiments & Other Stuff . . .