optimal_design_of_experiment.lyx~

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\begin_body

\begin_layout Title
Advanced instrumentation strategy and test matrix selection for turbines
 case
\end_layout

\begin_layout Standard
Ability to design instrumentation of any type of experiment is crucial to
 cost, effectiveness and robustness of the experimental setup.
 Growing computational power allows to predict phenomenon that is being
 measured.
 Obtained information can be used to perform optimization and design an
 experiment that will reduce measurement errors.
 The main subject of this work was to find formal procedure, that will 
\shape italic
a priori
\shape default
 provide instrumentation guidelines, like where to measure pressure on an
 airfoil.
 The proposed optimization algorithm is based on statistical criterion called
 
\shape italic
A-criterion
\shape default
, which is constructed from surrogate linear model.
 The model is obtained from Proper Orthogonal Decomposition (POD) applied
 to numerical results set.
 Additionally, thanks to proposed approach, collected data from an experiment
 together with numerical results allows to reconstruct all flow fields in
 computational domain.
 Numerical tests have shown that reconstruction gives best approximation
 of pressure profile in comparison to other approaches.
 
\end_layout

\begin_layout Standard
The proposed method can be described in the following steps: 
\end_layout

\begin_layout Enumerate
Perform set of numerical simulations a number of flow parameters and/or
 geometry configurations (e.g.
 different pressure drop, different angles of attack).
 This base solutions should define subspace in physical parameters where
 all experiments should lie.
\end_layout

\begin_layout Enumerate
Construct linear model using POD applied for all numerical data
\begin_inset Formula 
\[
V^{i}=W\cdot\beta^{i}
\]

\end_inset


\begin_inset Newline newline
\end_inset

where 
\begin_inset Formula $V^{i}$
\end_inset

 is the vector of nodal results for certain flow configuration (
\begin_inset Formula $i$
\end_inset

 index), 
\begin_inset Formula $W$
\end_inset

 is the matrix with POD modes as columns (orthogonal vectors), 
\begin_inset Formula $\beta^{i}$
\end_inset

 is coefficients vector.
 
\end_layout

\begin_layout Enumerate
Perform model reduction by leaving only 
\begin_inset Formula $m$
\end_inset

 first columns of 
\begin_inset Formula $W$
\end_inset

 matrix and 
\begin_inset Formula $m$
\end_inset

 coefficients of 
\begin_inset Formula $\beta^{i}$
\end_inset

 vector
\begin_inset Formula 
\[
\tilde{V}^{i}=W_{:,m}\cdot\beta_{m}^{i}
\]

\end_inset


\begin_inset Newline newline
\end_inset

where 
\begin_inset Formula $W_{:,m}$
\end_inset

 is reduced 
\begin_inset Formula $W$
\end_inset

 matrix and 
\begin_inset Formula $\beta_{m}^{i}$
\end_inset

 is reduced coefficients vector.
 Note that 
\begin_inset Formula $\tilde{V}^{i}$
\end_inset

 is still of the same size as 
\begin_inset Formula $V^{i}$
\end_inset

, but requires less data to be computed.
 
\end_layout

\begin_layout Enumerate
The goal is to calculate estimator of reAfteral parameters 
\begin_inset Formula $\hat{\beta_{m}}$
\end_inset

 from an experimental data.
 The smaller error will appear in 
\begin_inset Formula $\hat{\beta_{m}}$
\end_inset

 the more information will be gained.
 Dimension of 
\begin_inset Formula $\beta_{m}$
\end_inset

 vector is relatively small since POD reduction usually requires to chose
 from 10 up to 20 modes.
 Therefore the number of measurement points required to calculate 
\begin_inset Formula $\hat{\beta_{m}}$
\end_inset

 will need to be at least equal to 
\begin_inset Formula $m$
\end_inset

, what is rather ease to achieve.
 Each row of 
\begin_inset Formula $W_{:,m}$
\end_inset

 matrix refers to numerical mesh node and measurement should be performed
 in the same locations.
 The number of rows 
\begin_inset Formula $W_{:,m}$
\end_inset

 is equal to 
\begin_inset Formula $a\cdot N$
\end_inset

 where 
\begin_inset Formula $a$
\end_inset

 is number of flow variables (pressure, velocity components, etc.) and 
\begin_inset Formula $N$
\end_inset

 is number of mesh nodes.
 Thus for finding least square estimator 
\series bold

\begin_inset Formula $\hat{\beta_{m}}$
\end_inset

 
\series default
only some small subset of 
\begin_inset Formula $W_{:,m}$
\end_inset

 rows is required.
 However the way rows would be chosen it will affect the covariance of 
\begin_inset Formula $\hat{\beta_{m}}$
\end_inset

 and thus affect measurement error.
 A-criterion in this case allows to test which rows (and measurement points)
 from 
\begin_inset Formula $W_{:,m}$
\end_inset

 results in the smallest possible covariance of 
\begin_inset Formula $\hat{\beta_{m}}$
\end_inset

.
 The A-criterion in this case is formulated as choosing 
\begin_inset Formula $W_{:,m}$
\end_inset

 from 
\begin_inset Formula $W$
\end_inset

 so trace of covariance matrix 
\begin_inset Formula $(W_{:,m}W_{:,m}^{T})^{-1}$
\end_inset

 is minimal.
 Optimization algorithm can be constructed as follows
\end_layout

\begin_deeper
\begin_layout Enumerate
Equally distribute 
\begin_inset Formula $p$
\end_inset

 points along profile for instance.
\end_layout

\begin_layout Enumerate
Take into consideration one point from 
\begin_inset Formula $p$
\end_inset

 and calculate A-criterion for each possible position (consider all other
 points as fixed).
\end_layout

\begin_layout Enumerate
Move the point to the position which was referring to smallest value of
 A-criterion.
\end_layout

\begin_layout Enumerate
Repeat steps from (b) and (c) for all initial points.
 
\end_layout

\end_deeper
\begin_layout Enumerate
After conducting experiment solution reconstruction can be performed.
 New estimator 
\begin_inset Formula $\hat{\beta_{m}}$
\end_inset

 can be calculated using least square method and experimental data.
 Values of 
\begin_inset Formula $\hat{\beta_{m}}$
\end_inset

 can be used to calculate solution in each mesh node by using formula 
\begin_inset Formula $V=W_{:,m}\cdot\hat{\beta_{m}}$
\end_inset

.
\end_layout

\begin_layout Standard
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Presented method was validated using virtual experiment.
 The experiment consisted of results from numerical simulation based on
 high and low density meshes.
 High density mesh results with addition of uncorrelated error with normal
 distribution was used as 
\begin_inset Quotes eld
\end_inset

real
\begin_inset Quotes erd
\end_inset

 experimental data, and coarse mesh was used for measurement locations optimizat
ion.
 The goal was to reconstruct pressure profile in the best possible way from
 
\begin_inset Quotes eld
\end_inset

real
\begin_inset Quotes erd
\end_inset

 data applied to the sensors positions.
 The results obtained from optimal positions were then compared with a profile
 reconstructed from equally distributed points along the profile.
 
\end_layout

\begin_layout Standard
\begin_inset Float figure
wide false
sideways false
status collapsed

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\begin_inset Graphics
	filename optimized_cascasde_1/profil.png
	lyxscale 25
	width 75text%

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
Resulting instrumentation points distribution.
 Red crosses denotes uniform points distribution (uniform in spline parameter
 space) and green dots denotes optima ones.
 In the design process of real experiment the size of experimental part
 will be known, and minimum distance between points should be take into
 account, as long as possible correlation between measurements.
\begin_inset CommandInset label
LatexCommand label
name "fig:Instrumentation-positions-optimi-1-1"

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\begin_inset Float figure
wide false
sideways true
status collapsed

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	filename optimized_cascasde_1/Aoa45P0.359_pressur.png
	lyxscale 25
	height 80text%

\end_inset


\begin_inset Caption Standard

\begin_layout Plain Layout
The result of virtual experiment.
 Blue line denotes pressure profile obtained for angle of attack equal to
 0 and pressure on outlet equal to 0.7.
 It is considered as 
\begin_inset Quotes eld
\end_inset

real
\begin_inset Quotes erd
\end_inset

.
 Thin gray lines denotes optimal and uniform instrumentation points positions.
 Color field around reconstructed lines denotes standard deviation of each
 reconstructed solution.
 It should be noted that for most of the figure black dots representing
 reconstructed solution overlaps with blue line representing pressure profile
 to be reconstructed.
 
\begin_inset CommandInset label
LatexCommand label
name "fig:pMa0.7-1-1"

\end_inset


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wide false
sideways false
status collapsed

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	filename optimized_cascasde_1/small_velocity2_rec.png
	lyxscale 25
	width 80text%

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	filename optimized_cascasde_1/small_velocity2.png
	lyxscale 25
	width 80text%

\end_inset


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\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Velocity magnitude around cascade element.
 The velocity field reconstructed form pressure measurements on the airfoil
 surface is presented in upper picture.
 The virtual experiment result, computed on much finer mesh, is presented
 in lower figure
\begin_inset CommandInset label
LatexCommand label
name "fig:Velocity-magnitude-around"

\end_inset


\end_layout

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\end_body
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