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Abstract

This work presents a methodology for reconstructing full-field surface pressure information from deflectometry measurements on a thin plate using the Virtual Fields Method (VFM). Low-amplitude mean pressure distributions of the order of few O(100) Pa from an impinging air jet are investigated. These are commonly measured point-wise using arrays of pressure transducers, which require drilling holes into the specimen. In contrast,the approach presented here allows obtaining a large number of data points on the investigated specimen without impact on surface properties and flow. Deflectometry provides full-field deformation data on the specimen surface with remarkably high sensitivity. The VFM allows extracting information from the full-field data using the principle of virtual work. A finite element model is employed in combination with artificial grid deformation to assess the uncertainty of the pressure reconstructions.Both experimental and model data are presented and compared to show capabilities and restrictions of this method.
Experimental Mechanics
https://doi.org/10.1007/s11340-019-00530-2
Full-Field Surface Pressure Reconstruction Using the Virtual Fields
Method
R. Kaufmann1·B. Ganapathisubramani1·F. Pierron1
Received: 14 December 2018 / Accepted: 14 June 2019
©The Author(s) 2019
Abstract
This work presents a methodology for reconstructing full-field surface pressure information from deflectometry
measurements on a thin plate using the Virtual Fields Method (VFM). Low-amplitude mean pressure distributions of the
order of few O(100)Pa from an impinging air jet are investigated. These are commonly measured point-wise using arrays
of pressure transducers, which require drilling holes into the specimen. In contrast, the approach presented here allows
obtaining a large number of data points on the investigated specimen without impact on surface properties and flow.
Deflectometry provides full-field deformation data on the specimen surface with remarkably high sensitivity. The VFM
allows extracting information from the full-field data using the principle of virtual work. A finite element model is employed
in combination with artificial grid deformation to assess the uncertainty of the pressure reconstructions. Both experimental
and model data are presented and compared to show capabilities and restrictions of this method.
Keywords Deflectometry ·Virtual Fields Method ·Surface pressure reconstruction ·Full-field measurement ·
Fluid-structure interaction
Introduction
Full-field surface pressure measurements are highly relevant
for engineering applications like material testing, compo-
nent design in aerodynamics and the use of impinging jets
for cooling, de-icing and drying. Surface pressure informa-
tion can be used to determine aerodynamic loads [1]and
to evaluate the performance of impinging jets used for heat
and mass transfer [2]. They are however difficult to achieve,
as available methods are not universally applicable. Most
commonly, large numbers of pressure transducers are fitted
into the investigated surface. This is an invasive technique
as it requires one to drill holes into the sample. Further,
it yields limited spatial resolution [3,4]. Pressure sensitive
paints allow obtaining full-field data, but are not suited for
low-range differential pressure measurements [5, chapter
4.4; 6]. They further require extensive calibration efforts, as
well as a controlled experimental environment. Calculating
R. Kaufmann
rene.kaufmann@southampton.ac.uk
1University of Southampton, Highfield, Southampton
SO17 1BJ, UK
pressure from Particle Image Velocimetry (PIV) is a non-
invasive method that yields full-field data in the flow field
[7,8]. This allows estimations of pressure along lines on
which the surface coincides with the field of view.
Another approach is the reconstruction of pressure infor-
mation from full-field surface deformation measurements
by solving the local equilibrium equations. Recently, wall
pressure was calculated from 3D-Digital Image Correla-
tion (DIC) measurements on a flexible Kevlar wind-tunnel
wall in an anechoic chamber [9]. This was achieved by
projecting the measured deflections onto polynomial basis
functions and inserting their derivatives into the correspond-
ing equilibrium equations. The obtained pressure coeffi-
cients compared well to transducer data for the relatively
large spatial scales that were investigated. Many problems
in the field of fluid-structure interactions can be simplified
to low amplitude loads acting on thin plates. This allows
employing the Love-Kirchhoff thin plate theory [10]to
write the local equilibrium of the plate. The required full-
field deformation information on the test surface can be
obtained using a number of measurement techniques, e.g.
DIC, Laser Doppler Vibrometers (LDV) or interferometry
techniques. However, the fourth order deflection derivatives
required to solve the Love-Kirchhoff equilibrium equation
make an application in the presence of experimental noise
Exp Mech
challenging, particularly for low signal-to-noise ratios. To a
degree, this issue can be addressed by applying regularisa-
tion techniques. In studies based on solving the equilibrium
equation locally by employing a finite difference scheme,
regularisation was achieved by applying wave number fil-
ters [11] or by adapting the number of data points used
for the finite differences [12]. This allowed an identifica-
tion and localisation of external vibration sources acting on
the investigated specimen. Similarly, the acoustic compo-
nent of a flow was identified using wave number filters in
an investigation of a turbulent boundary layer [13]. Gener-
ally, the accuracy of this approach in terms of localisation
and amplitude identification depends strongly on the chosen
regularisation.
An alternative for solving the thin plate problem using
full-field data is the Virtual Fields Method (VFM), which
is based on the principle of virtual work and only requires
second order deflection derivatives. The VFM is an inverse
method that uses full-field kinematic measurements to
identify mechanical material properties from known loading
or vice versa. A detailed overview of the method and
the range of applications is given in [14]. It notably does
not require detailed knowledge of the boundary conditions
and does not rely on computationally expensive iterative
procedures. A study comparing Finite Element Model
Updating, the Constitutive Equation Gap Method and the
VFM for constitutive mechanical models using full-field
measurements found that the VFM consistently performed
best in terms of computational cost with reasonable results
[15]. The VFM has been adapted for load reconstruction in a
number of studies, including dynamic load identification in
a Hopkinson bar [16,17]. The data were found to compare
reasonably well to standard measurement techniques. The
VFM was also used to reconstruct spatially-averaged sound
pressure levels from an acoustic field using a scanning
Laser Doppler Vibrometer (LDV) [18]. Dynamic transverse
loads, as well as vibrations caused by acoustic pressure
were identified using the same technique in [19]. The
results were found to be accurate for distributed loads. The
latter used a VFM approach based on piecewise virtual
fields, which allows more accurate descriptions of boundary
conditions for complex shapes and heterogeneous materials
[20]. This approach was extended to random spatial wall
pressure excitations in [21], reconstructing power spectral
density functions from measured data and using the VFM
to describe the plate response. The authors found that this
method requires piecewise virtual fields to be defined over
small regions. Recently, the VFM approach was combined
with deflectometry for the identification of mechanical
point loads of several O(1)N[22]. Deflectometry is a highly
sensitive technique for slope measurement [23]. It was
successfully used in a range of applications like damage
detection of composites [24], the analysis of stiffness
and damping parameters of vibrating plates [25] and for
imaging of ultrasonic lamb waves [26]. Since deflectometry
measurements yield surface slopes, the combination with
the VFM reduces the required order of derivatives of
experimental data for pressure reconstruction to one.
Known loads were reconstructed in [22] with good accuracy
for certain reconstruction window sizes which were found
empirically. Deflectometry and the VFM were also used to
identify pressure auto-spectra of spatially averaged random
excitations in [27]. The results agreed well with microphone
array measurements, except at the structural resonance
frequencies and for poor signal-to-noise-ratios. In the same
study, the VFM approach was extended to membranes
and the applicability was investigated using a simulated
experiment. A shortcoming of these previous studies was
that the accuracy was not assessed for unknown input loads.
This is an important step because neither the resolution
in space nor the uncertainty in pressure amplitude can be
predicted directly as they depend on the signal amplitude
and distribution, the noise level and the reconstruction
parameters.
The main focus of the work presented here is the deter-
mination of static low-amplitude pressure distributions
with peak values of few O(100)Pa from time-averaged
full-field slope measurements, as well as an assessment
of the uncertainties of the method. In the following
sections “Theory”and“Experimental Methods”, a brief
overview of the theoretical background and experimental
setupisgiven.In“Experimental Results”, experimental
results are presented for two different specimen and for
several reconstruction parameters. The pressure reconstruc-
tions are compared to pressure transducer measurements.
Section “Simulated Experiments” introduces a numerical
model for simulated experiments. This allows an assess-
ment of the uncertainty of the method in terms of both
systematic errors and the influence of random
noise. In “Simulated Experiments”, a finite element updating
procedure is proposed to compensate for systematic errors.
Theory
Impinging Jets
A fan-driven, round air jet was used to apply a load on the
specimen. The flow generated by this impinging jet can be
divided into the free jet, stagnation and wall region [28].
These regions, shown in Fig. 1, consist of subregions with
distinct flow features which are governed by the ratio
between downstream distance and nozzle diameter H/D
and Reynolds number Re. Directly downstream from the
nozzle exit, the free jet develops for sufficiently large
H/D 2[29]. The velocity profile spreads as it moves
Exp Mech
Fig. 1 Impinging jet regions
downstream due to entrainment and viscous diffusion caus-
ing a transfer of momentum to surrounding fluid particles.
Upon approaching the impingement plate a stagnation
region forms, characterized by an increase in static pres-
sure up to the stagnation point on the plate surface. The
rising static pressure results in pressure gradients diverting
the flow radially away from the jet centerline. The laterally
diverted flow forms the wall region. The pressure distribu-
tion on the impingement surface is approximately Gaussian
[30]. This study focuses on the measurement of the mean
load distribution on the impingement plate.
Deflectometry
Deflectometry is an optical full-field measurement tech-
nique for surface slopes [23]. Figure 2shows a schematic of
the setup. A camera measures the reflected image of a peri-
odic spatial signal, here a cross-hatched grid, on the surface
of a specular reflective sample. The distance between the
grid and sample is denoted by hGand the grid pitch by pG.
The angle θhas to be sufficiently small to minimize grid dis-
tortion in the recorded image. A pixel directed at point M on
the specimen surface will image the reflected grid at point P
in an unloaded configuration. If a load is applied to the sur-
face, it deforms locally and the same pixel will now image
the reflected grid at point P. It is assumed here that rigid
body movements and out of plane deflections are negligible
(for details see “Error Sources”below).
The displacement ubetween P and Prelates to the
phase difference dφin the grid signal in x- and y-direction
respectively as follows:
x=2π
pG
ux,dφ
y=2π
pG
uy(1)
A spatial shift by one grid pitch pGcorresponds to a phase
shift of 2π. However, a direct displacement estimation from
the phase difference between a reference and a deformed
Fig. 2 Deflectometry setup, top view
Exp Mech
image does not take into account that the physical point on
the plate surface is subject to a displacement. An iterative
procedure to improve the displacement results given in
[31, section 4.2] is employed here:
un+1(x)=−pG
2πdef (x+un(x)) φref (x)) (2)
A relationship between slopes and displacement is derived e.g.
in [32]. It is based on geometric considerations and assumes
that θis sufficiently small, so the camera records images in
normal incidence and hGis large against the shift u:
x=ux
2hG
,dα
y=uy
2hG
(3)
Otherwise, a more complex calibration is required [33,34].
Equation (3) will be used here.
The spatial resolution of the method is driven by pG.The
phase resolution is noise dependent and can be defined as
the standard deviation of a phase map detected between two
stationary images. Consequently, slope resolution depends
on pG,hGand the phase resolution.
Phase detection
The literature describes a number of methods for retrieving
phase information from grid images, e.g., [31,35,36].
Here, a spatial phase-stepping algorithm is employed which
allows investigating dynamic events [37,38]. One phase
map is calculated per image. The chosen algorithm needs
to be capable of coping with miscalibration, i.e. a slightly
non-integer number of pixels per grid period. This can
occur due to imperfections in the printed grid, misalignment
between camera, sample and grid, lens distortion, as well
as fill factor issues. In addition, the investigated signal
is not generally sinusoidal. This requires an algorithm
suppressing harmonics and sets a lower limit to the required
number of samples, i.e. pixels recorded per grid pitch [39].
A windowed discrete Fourier transform algorithm using
triangular weighting and a detection kernel size of two grid
periods as used in e.g., [36]and[40] will be used in this
study.
Pressure Reconstruction
The problem investigated here is a thin plate in pure bending,
which allows the Love-Kirchhoff theory to be employed
[41]. Assuming that the plate material is linear elastic,
isotropic and homogeneous, the principle of virtual work is
expressed by:
S
pwdS =Dxx
Sκxxκ
xx +κyyκ
yy +2κxyκ
xydS
+Dxy
Sκxxκ
yy +κyyκ
xx 2κxyκ
xydS
+ρtS
S
awdS.(4)
S is the surface area, p the investigated pressure, Dxx
and Dxy the plate bending stiffness matrix components, κ
the curvatures, ρthe plate material density, tSthe plate
thickness, athe acceleration, wthe virtual deflection and
κthe virtual curvatures. Here, the parameters Dxx,Dxy ,
ρand tSare known from the plate manufacturer. κand a
are obtained from deflectometry measurements, see “Data
Acquisition and Processing” below. For the selection of the
virtual fields wand κone needs to take into account
theoretical as well as practical restrictions of the problem
like continuity, boundary conditions and sensitivity to noise.
The problem can be simplified by assuming the pressure
p to be constant over the investigated area and by approxi-
mating the integrals with discrete sums.
p=Dxx
N
i=1
κi
xxκi
xx +κi
yyκi
yy +2κi
xyκi
xy
+Dxy
N
i=1
κi
xxκi
yy +κi
yyκi
xx 2κi
xyκi
xy
+ρtS
N
i=1
aiwiN
i=1
wi1
.(5)
Here, N is the number of discretised surface elements dSi.
Virtual Fields
For the present problem of identifying an unknown load
distribution, it is beneficial to choose piecewise virtual
fields due to their flexibility [1820,22]. In this study, the
virtual fields are defined over a window of chosen size
which is then shifted over the surface S until the entire
area is covered. One pressure value is calculated for each
window. In the following, this window will be referred to
as pressure reconstruction window PRW. This procedure
also allows for oversampling in the spatial reconstruction by
shifting the window by less than a full window size.
Here, the only theoretical requirements for the virtual
fields are continuity and differentiability. Since curvatures
relate to deflections through their second spatial derivatives
for a thin plate in pure bending, the virtual deflections are
required to be C1continuous. It is further necessary to
eliminate the unknown contributions of virtual work along
the plate boundaries. This is achieved by choosing virtual
displacements and slopes that are zero around the window
borders. 4-node Hermite 16 element shape functions as used
in FEM [42] fulfill these requirements. The full equations
defining these functions can be found in [14, chapter 14].
Figure 3shows example virtual fields. 9 nodes are defined
for a PRW. All degrees of freedom are set to zero except for
the virtual deflection of the center node, which is set to 1.
Exp Mech
Fig. 3 Example Hermite 16 virtual fields with superimposed virtual elements and nodes (black). ξ1,ξ2are parametric coordinates. The example
window size is 32 points in each direction. Full equations can be found in [14, chapter 14]
The size of the PRW is an important parameter for the
pressure reconstruction. Generally, the presence of random
noise requires a larger PRW in order to average out the
effect of noise on the pressure value within the window. A
smaller PRW however can perform better at capturing small
scale spatial structures, as large windows may average out
amplitude peaks. One challenge in varying the window size
is that the systematic error varies with it, as well as the effect
of random noise on pressure reconstruction. This problem is
investigated numerically in “Simulated Experiments”.
Experimental Methods
Setup
Figure 4shows a schematic of the experimental setup.
A round, fan-driven impinging air jet was used to apply
pressure on the specimen. The jet was fully turbulent at a
downstream distance of 0.5 cm from the nozzle exit. The
specimen was glued on a square acrylic frame. The grid was
printed on transparency and fixed between two glass plates
in the setup. A white light source was placed behind it. The
camera was placed next to the grid at the same distance from
the sample such that the reflected grid image is recorded at
normal incidence. The distance between the sample and grid
was chosen to be as large as possible in order to minimise
the angle θ(see Fig. 2). Two different glass sample plates
were investigated, one with thickness of 1mm and the other
3 mm. All relevant experimental parameters are listed in
Table 1.
Grid
A cross-hatched grid printed on a transparency was used
as the spatial carrier. Sine grids printed in x- and y-
direction would be preferable for phase detection as they
do not induce high frequency harmonics in the phase
detection. Printing these in sufficient quality is however
difficult to achieve with standard printers. Using a hatched
grid and slightly defocusing the image achieves a similar
result because the discrete black and white areas become
blurred, effectively yielding a grey scale transition between
minimum and maximum intensity. This does however result
in a slightly lower signal to noise ratio. It should be noted
that when printing the grid, an integer number of printed
Fig. 4 Experimental setup
Exp Mech
Tab le 1 Setup parameters
Optics
Camera Photron Fastcam
SA1.1
Technology CMOS
Camera pixel size 20 μm
Surface fill factor 52%
Dynamic range 12 bit
Settings
Resolution 1024 ×1024 pixels
Frame rate f 50 fps
Exposure 1/100 s
Region of interest 64 ×64 mm2
Magnification M 0.32
f-number NLens 32
Focal length fLens 300 mm
Light source Halogen, 500 W
Sample
Type First-surface mirror
Material Glass
Young’s modulus E74 GPa
Poisson’s ratio ν0.23
Density ρ2.5 103kg m3
Thickness tS1 mm, 3 mm
Side length lsca. 90 mm, 190 mm
Grid
Printed grid pitch pG1.02 mm
Grid-sample distance hG1.03 m
Pixels per pitch ppp 8
Jet
Nozzle shape Round
Nozzle diameter D 20 mm
Area contraction ratio 0.13
Nozzle exit dynamic pressure pexit 630 Pa
Reynold’s number Re 4·104
Sample-nozzle distance hN40 mm
dots per half pitch is required to avoid aliasing (e.g., [43]).
For the current setup, grids with 1 mm pitch were printed on
transparencies using a Konica Minolta bizhub C652 printers
at 600 dpi.
Sample
The choice of the sample plate material and finish proved
crucial for the investigation of small pressure amplitudes
and spatial scales. The surface slopes under loading need to
be large enough for detection, while at the same time the
sample surface has to be plane enough for the grid image
to be sufficiently in focus over the entire field of view.
Perspex mirrors, polished aluminium and glass plates with
reflective foils proved either too diffusive due to the
Rayleigh criterion or insufficiently plane, resulting in a lack
of depth of field when trying to image the reflected grid.
Optical glass mirrors were chosen instead, as they provide
adequate stiffness parameters and remain sufficiently plane
when mounted. As it was possible to estimate the slope
resolution from the noise level observed when recording two
undeformed images on any sample thickness, deformation
estimations based on the expected experimental load were
used as input for finite element simulations to select suitable
plate parameters. It was found that plates with thickness of
3 mm or lower were required. Good results were achieved
using a 1 mm thick first-surface glass mirror as specimen.
Still, fitting the 1 mm glass mirror on the frame caused it
to bend slightly, resulting in small deviations from a perfect
plane and subsequent local lack of depth of field. This was
addressed by closing the aperture. A second, 3 mm thick
mirror was used for comparison as it did not bend notably
when mounted, though signal amplitudes for this case
proved to be very low. The sample plates were glued onto a
perspex frame along all edges.
Transducer Measurements
Pressure transducer measurements allowed a validation of
the pressure reconstructions from deflectometry and the
VFM. Endevco 8507C-2 type transducers were fitted in
an aluminium plate along a line from the stagnation point
outwards. The transducers have a diameter of 2.5 mm and
were fitted with a spacing of 5 mm. They were fitted to be
flush with the surface to within approximately 0.5 mm. Data
was acquired at 10 kHz over 20 s using a NI PXIe-4330
module.
Data Acquisition and Processing
One reference image was taken in an unloaded configuration
before activating the jet. The jet required approximately
20 s to settle, after which a series of images was recorded.
One data point was calculated per grid pitch during
phase detection. Slopes were calculated relative to the
reference image. Time averaged mean slope maps were
calculated over N =5400 measurements at 50 Hz, limited
by camera storage. From the slope maps the curvatures
were obtained through spatial differentiation using centered
finite differences. This requires knowledge of the physical
distance between two data points on the specimen. It
corresponds to the portion of the mirror required to observe
the reflection of one grid pitch, which can be determined
geometrically assuming θis sufficiently small (see Fig. 2).
In the present setup, camera sensor and grid were at the
Exp Mech
same distance from the mirror hG, such that the distance
was half a printed grid pitch. Since differentiation tends
to amplify the effect of noise, it can be beneficial to filter
slope data before calculating curvatures. Here, the mean
slopes were filtered using a 2D Gaussian filter, performing
a convolution in the spatial domain. The filter kernel is
characterized by its side length which is determined by the
standard deviation, here denoted σα, and truncated at 3σαin
both directions. Because of its size, the filter kernel cannot
be applied to the data points at the border of the field of view
without padding. As padding should be avoided to prevent
bias, 6σα1 data points were cropped along the edges of
the field of view. While acting as a low-pass filter which
reduces the effect of random noise, this technique also tends
to reduce signal amplitude.
For the investigated problem of a mean flow profile,
the accelerations average out to zero. This was confirmed
with vibrometer measurements on several points along the
test surface using a Polytec PDV 100 Portable Digital
Vibrometer. Data was acquired at 4 kHz over 20 s. The noise
level in LDV measurements was 0.3 m s2. The observed
standard deviations varied with the position along the plate
surface and reached up to 1.4 m s2. Therefore, the term
involving accelerations in Eq. 4is zero as well and will
therefore be neglected in the following.
Pressure reconstructions were conducted for several
PRW sizes. The results were oversampled by shifting the
PRW over the investigated field of view by one data point
per iteration. Note that due to the finite size of these
windows, half a PRW of data points is lost around the edges
of the field of view.
Experimental Results
Slope maps obtained from deflectometry measurements
were processed and temporally averaged as described in
Data Acquisition and Processing”. Results for both speci-
mens are presented in the following, one plate with 1 mm
thickness and 90 mm side length, and one with 3 mm thick-
ness and 190 mm side length. The region of interest is
64 mm in both directions for each test cases. Figure 5(a)–(d)
show the measured mean slope maps for both test plates.
Distances are given in terms of radial distance from the
impinging jet’s stagnation point r, normalized by the noz-
zle diameter D, in x- and y-direction respectively. Note that
the region of interest showing the jet center does not coin-
cide with the plate center, so the slope amplitudes are not
necessarily symmetric. The signal amplitudes for the 3 mm
test case are significantly lower than for the 1 mm case.
Slope shapes are different for both cases because the plates
have different side length while the field of view remains the
same size. Further, the stagnation point is off-center in the
3mmtest.
Figure 5(e)–(p) shows mean curvature maps with and
without Gaussian filter. Stripes are visible in all curvature
maps for the unfiltered 1mm test data. This indicates the
presence of a systematic error source in the experimen-
tal setup. Without slope filter, curvatures obtained from
the 3mm plate test are governed by noise. The curvature
map for ¯κxx (Fig. 5(g)) additionally shows fringes. These
disappear after slope filtering, though filtered data still
appear asymmetric, again indicating a systematic error. To
assure that this issue occurring in for both plates does not
originate from a lack of convergence, mean and instanta-
neous curvature maps were calculated and compared. All
maps show the same bias, with small variations in amplitude.
This may be caused by misalignment between grid and
image sensor due to imperfections in the printed grid,
combined with the CMOS chip’s fill factor. This results
in a slightly varying number of pixels per grid pitch over
the field of view, which leads to errors in phase detection
and fringes. While this issue could be mitigated by careful
realignment of camera and grid as well as slightly defo-
cusing the image to address the low camera fill factor, it
could not be fully eliminated. Another possible error source
is the deviation of the plate surface from a perfect plane,
e.g. due to deformations of the sample during mounting.
Since differentiation amplifies the impact of noise, filtering
the slope maps yields much smoother curvature maps. The
downside is a possible loss of signal amplitude and of data
points along the edges (see “Data Acquisition and Processing”).
Figure 6(a)–(d) show pressure reconstructions using
different PRW sizes. Pressure is given in terms of
difference to atmospheric pressure, p . Here, one data
point corresponds to a physical distance of 0.5 mm, such
that a PRW of 28 points corresponds to a window side
length of 14 mm or 0.7rD1. The large number of data
points is a result of oversampling by shifting the PRW
over the investigated area by one point per iteration. The
expected Gaussian shape of the distribution is found to be
well reconstructed for filtered data and sufficiently large
PRW, here above ca. 22 data points, for the 1mm plate.
Reconstructions from 3mm plate tests are less symmetric.
The position of the stagnation point is visible for all shown
parameter combinations, but the shape of the distribution
shows a recurring pattern which stems from the systematic
error already observed in curvature maps. For both tests,
some reconstructions show areas of negative differential
pressure, which is unexpected for the mean distributions in
this flow. This is likely to be a consequence of random noise,
as similar patterns were observed in simulated experiments
for noisy model data (see “Grid Deformation Study
below). For comparisons with the transducer measurements,
Exp Mech
Fig. 5 Measured mean slope and curvature maps
pressure reconstructions were averaged circumferentially
for each corresponding radial distance from the stagnation
point. Figure 6(e) and (f) show the results. The vertical
error bars on transducer data represent both the systematic
errors of the equipment as well as the random error of the
mean pressure value. The horizontal error bars indicate the
uncertainty in placing the transducers relative to the jet.
Results from the 1mm plate measurements appear to show
Exp Mech
Fig. 6 Comparison of VFM pressure reconstruction with pressure transducer data
a systematic underestimation of the pressure amplitude at
all points. Possible sources for this error are discussed in
detail in “Error Sources” below. However, the shape of the
distribution is captured reasonably well. The 13mm plate
results show a good reconstruction of the peak amplitude,
but the shape of the pressure distribution deviates due to the
influence of random noise patterns. The results clearly show
that the effects of the size of the PRW and the Gaussian
smoothing kernel σαon the reconstruction outcome are
significant. Therefore, the influence of the reconstruction
parameters is investigated numerically in the following
section.
Simulated Experiments
Comparisons of the VFM pressure reconstruction with the
pressure transducer data shows that there are discrepancies
between the results. Furthermore, it is unclear what parts
of the reconstructed pressure amplitude stems from signal,
random noise or systematic error. Processing experimental
data with noise can produce pressure distributions that
are indistinguishable from the signal of interest. It is also
important to note that the complex measurement chain from
images to pressure does not allow for analytical expressions
to be obtained and only numerical simulations can shed light
on the problem.
Numerical studies allow addressing this problem and
estimating the effects of random and systematic error [31].
As a first step, a finite element model of the investigated
thin plate problem is created. By applying a model load, the
local displacements and slopes that result from the bend-
ing experiment can be simulated. For the next step, the grid
image recorded with the camera is modelled numerically.
The simulated displacements are used to calculate the defor-
mations of the model grid image. Experimentally observed
grey level noise is added to these grids. The simulated
grids serve as input for a study of the influence of process-
ing parameters on the pressure reconstruction. Comparisons
with the model load allow an assessment of the uncertain-
ties of the processing technique in the presence of random
noise. In the last subsection, a finite element correction pro-
cedure is introduced to compensate for the reconstruction
error.
Finite Element Model
Numerical data of slope maps from a thin plate bending
under a given load distribution was calculated using a finite
element simulation. This was conducted using the software
Exp Mech
Fig. 7 ANSYS model in- and output for 1mm plate model
ANSYS APDLv181. SHELL181 elements were chosen as
they are well suited for modelling the investigated thin plate
problem [44]. Both experimental test plates were simulated
as homogeneous with the parameters detailed in Table 1.
All degrees of freedom were fixed along the edges. For both
plates a square mesh was used with 1440 elements for the
1 mm thick plate and 2280 elements for the 3 mm thick
plate. This allowed obtaining 1024 points in a window cor-
responding to 64 mm, which corresponds to the experimen-
tal number of camera pixels and field of view. Figure 7(a)
shows the Gaussian pressure distribution used as input,
with an amplitude of 630 Pa and σload =9 mm. Figure 7(b)
shows the resulting deflections, Fig. 7(c) and (d) the model
slopes for the 1mm plate case.
Systematic Error
The simulated slopes can be used as input for the VFM
pressure reconstruction the same way as those obtained
experimentally. This allows an assessment of the system-
atic error of the processing technique independent from
experimental errors. A metric for estimating the error of a
reconstruction was defined taking into account the differ-
ence between reconstructed and input pressure amplitude in
terms of the local input amplitude at each point:
=1
N
N
i=1(prec,i pin,i)2/pin,i
(6)
prec, i is the reconstructed and pin, i the input pressure at
each point i with a total number of points N. Pressure
values below 1Pa were omitted for this metric. Figure 8(a)
shows the results for the accuracy estimate for pressure
reconstructions from noise free slope data for different
PRW. The results are oversampled as in the experimental
case by shifting the PRW by one point per iteration. A
minimum exists at PRW =22 with =0.12, which
indicates an average accuracy of ca. 88% of the local
amplitude. The corresponding pressure reconstruction map
is shown in Fig. 8(b). It should be noted that the local
pressure amplitudes are underestimated for all investigated
cases. For increasing PRW sizes, the peak amplitude is
Fig. 8 Systematic error estimate
for VFM
Exp Mech
underestimated because the virtual fields act as a weighted
average over the entire window. Small PRWs were expected
to yield best results in a noise free environment since they
average over fewer data points. This is not confirmed here.
Different finite element mesh sizes were tested to rule out
model convergence issues. The low accuracy obtained for
small windows is probably due to a lack of heterogeneity
of (real) curvature in small windows. If curvatures are
constant, they can be taken out of the integral in equation
(4). Because the virtual curvatures average out to zero
over one window, the integral then yields zero. For small
windows, this situation is approached, likely leading to
wrong pressure values. Choosing heterogeneous virtual
curvature fields could be used to address this issue in future
studies. One approach could be to defined more nodes on
each virtual field and a non-zero virtual deflections on a
node other than the center one to increase heterogeneity.
Another way could be to employ higher order approaches
for pressure calculation within one window, which is
expected to yield higher accuracy for large PRW.
Grid Deformation Study
Artificial grid deformation allows for a more comprehensive
assessment of error propagation by including the effects
of camera resolution and noise. Following the approach
described in [45], a periodic function with a wavelength
corresponding to the experimental grid pitch was used in x-
and y-direction to generate the artificial grid.
I(x,y) =Imin +Imin Imax
2+Imax
4
·cos 2πx
pG+cos 2πy
pG
cos 2πx
pGcos 2πy
pG(7)
Here, Imin and Imax are the minimum and maximum
intensity values of the experimental grid images. The signal
amplitude values were discretised to match the camera’s
dynamic range. All simulated image parameters were set to
replicate the experimental conditions as described in
Table 1. This spatial grid signal was oversampled by a
factor of 10 and spatially integrated to simulate the signal
recording process of the camera, as detailed in [45]. To
further assess the actual experiment, random noise was
added to the artificial grid images based on the grey
level noise measured during experiments, here 0.95% and
0.61% of the used dynamic range in case of the 1 mm
and 3 mm plate tests respectively. It varies because the
illumination varied between both experiments, such that the
used dynamic range was different. The amount of random
noise is reduced with the number of measurements over
which the mean value is calculated. However, the reduction
of noise is not described by 1/Nas would be expected.
The same observation was made in [43]. It was investigated
by taking a series of images without applying a load to
the specimen. It was found that the amount of noise in
phase maps increases with the time that has passed between
two images being taken. It is likely that this is a result of
small movements or deformations of the sample, printed
grid and camera due to vibrations and temperature changes
during the measurement. This does not fully account for the
observed effect however. As a consequence, the amount of
random noise for averages over multiple measurements has
to be determined experimentally. For 5400 measurements
on the undeformed sample, it was found that the random
noise in phase was reduced by a factor of ca. 2.5 compared
to two measurements. The values are statistically well
converged after 30 realisations of simulated noise.
The simulation neglects the effects of grid defects, lens
imperfections, inhomogeneous illumination and imperfec-
tions of the specimen. However, it does account for any
systematic errors associated with the number of pixels on
the camera sensor and the random errors coming from grey
level noise in the images. Figure 9shows a close-up view of
simulated and experimental grid images. Simulated slopes
yield corresponding deformations of the artificial grid at
every point using equations (1)and(3). The obtained artifi-
cial grids for deformed and undeformed configurations can
now be used as input for the phase detection algorithm.
Areas with negative pressure amplitude were observed in
reconstructions from noisy model data, very similar to
those observed experimentally. A lower limit for pressure
resolution was determined by adding noise to two unde-
formed artificial grids and processing them. The standard
deviation of pressure values obtained from this reconstruc-
tion can be interpreted as a metric for the lower detection
limit of the pressure reconstruction for the corresponding
parameter combination. Values below the obtained thresh-
old are neglected in all reconstructions in the following.
Phases obtained from artificial, deformed grids were
processed and the reconstructed and input pressure were
compared using the metric introduced in equation (6). This
allows quantifying the systematic error of phase detection
and VFM for all combinations of the relevant processing
parameters. Oversampling in the phase detection algorithm,
i.e. calculating more than one phase value per grid pitch,
was found to improve the results, though at high compu-
tational cost. Particularly in combination with larger PRW
and slope filter kernels, phase oversampled slope maps
yield diminishing improvements in accuracy in terms of the
overall cost. In the VFM pressure reconstruction, oversam-
pling provided a significant improvement at acceptable cost.
The slope filter kernel size σαalso increases computational
Exp Mech
Fig. 9 Example grid sections
cost, but mitigates the effects of random noise efficiently.
The influence of both the size of σαand PRW are inves-
tigated in the following as they yield the most significant
improvements.
Figure 10(a) and (b) show the findings for varying param-
eters σαand PRW for each plate. These allow selecting
parameter combinations with highest precision in terms of
amplitude over the entire field of view (Fig. 11). Figures 12
and 13 show example comparisons of pressure reconstruc-
tions for different . Figure 12 shows experimental data
with two different parameter combinations for both plates
and Fig. 13 below shows the corresponding results obtained
using model data. For reference, Fig. 11 shows on top the
model input distribution sections in the respective field of
view. As expected, reconstructions using larger smoothing
kernels tend to yield lower peak amplitudes. However, the
amplitudes in other areas are be captured better, as noise
induced peaks are filtered more efficiently. The fact that
some numerical reconstructions do not represent Gaussian
distributions well shows that noise effects are not averaged
out entirely. For the low signal to noise ratio encountered in
the 3 mm plate case, some reconstructions overestimate the
peak pressure amplitude. This is a consequence of the dif-
ferentiation of slope noise, which leads to large curvature
and thus pressure values. Since this also leads to areas in
which the pressure amplitude is underestimated, the effect
averages out for sufficiently large slope smoothing kernel
and PRW.
Fig. 10 Pressure reconstruction accuracy analysis
Exp Mech
Fig. 11 Model input pressure distribution sections for comparison with reconstruction results
Fig. 12 Comparison of pressure reconstructions from experimental data for different parameter combinations
Fig. 13 Comparison of pressure reconstructions from noisy model data for different parameter combinations
Exp Mech
Fig. 14 FE corrected results for noise free model data
Finite Element Correction
The systematic error caused by the reconstruction technique
which was identified above shows an underestimation of the
input pressure for noise free data. In the presence of noise,
a similar observation is made for large enough signal to
noise ratio as in the 1 mm plate case. This error source can
be mitigated with a finite element correction procedure. For
this approach, an initial reconstructed pressure distribution
is used as input for the numerical model described above.
In practice, this is the experimentally identified distribu-
tion from the VFM. Processing the resulting slope maps
obtained using the finite element model (see “Finite Element
Model”) yields the first iterated pressure distribution. The
difference between this iteration and the original pressure
reconstruction corresponds to the systematic error at every
point of the pressure map. This difference is generally lower
in amplitude than that between the original reconstruction
and the real pressure distribution caused by systematic
error, but it serves as a first estimation of that difference.
Adding this difference to the original reconstruction yields
an updated approximation of the real pressure distribution:
dpupdate,n =prec +(prec pit,n)(8)
This procedure can be repeated until (prec pit,n)falls
below a chosen threshold. Figure 14 shows how the input
load is well recovered after only few iterations for modelled,
noise free data. For the shown case, the second iteration
result is already well converged and much closer to the input
distribution, with an improvement from ca. 15% average
error to below 6%. Similar results were found for the other
investigated PRW sizes.
An application to experimental data is more challenging.
Each iteration tends to amplify noise patterns in pressure
maps from both random and systematic error sources.
Reconstructions from smoothed slope maps mitigate this
issue, but suffer from a reduced number of available data
points. Note that for each iteration, the size of one smooth-
ing window, i.e. 6σα, plus half a PRW of data points is lost
around the edges (see also “Data Acquisition and Processing”).
Here, this can be mitigated by using reconstructions with
small slope smoothing kernels and by calculating circum-
ferential averages from the stagnation point outwards, thus
averaging out some of the random noise. These are then
extrapolated to 2D distributions to obtain a suitable input for
the finite element updating procedure. The entire process is
applied to both numerical and experimental data, allowing
Fig. 15 Error estimates for
circumferentially averaged
pressure reconstructions for
varying slope filter kernel and
PRW size for 1mm plate test and
with grey level noise 0.95% of
the dynamic range
Exp Mech
for a comparison of the results and thus further assessment
of the influence of systematic experimental errors.
To select the correct reconstruction parameters for this
approach, the accuracy assessment was repeated using cir-
cumferential averages instead of the entire field of view. The
results vary, because low amplitude pressures are now aver-
aged over a larger number of data points. Further, part of the
field of view with low pressure amplitude is not taken into
account as it is rectangular. The result is shown in Fig. 15.
Figure 16 shows the results for iterations of experimental
data and noisy model data. A 10% error bar corresponding
to the estimated uncertainty resulting from the material’s
Young’s modulus is shown for the iterations on experi-
mental data at the positions of transducers for comparison.
Figure 16(a) shows that for σα=3andPRW=28 the peak
amplitude from transducer measurements is approximated
to about 10% after 2 iterations of the experimental data.
Since slope smoothing leads to a significant loss in data
points, no further iterations are possible for this case. The
corresponding numerical case, see Fig. 16(b), shows a close
approximation of the input load.
For experimental data and σα=0andPRW=34, see
Fig. 16(c), the influence of noise patterns becomes visible.
These patterns are amplified by the correction procedure.
Numerical data show a very good approximation of the
input load, whereas experimental VFM data deviate from
transducer data by ca. 10% after correction.
For σα=0andPRW=22, see Fig. 16(e), noise effects
in experimental data are significant. Therefore, regularisa-
tion is necessary before iterating the results. Here, a fourth
order polynomial was fitted to the averaged results. The iter-
ated corrections once again approximate the transducer data
Fig. 16 Finite element updating
results. Error bars on VFM
represent the estimated
uncertainty resulting from the
material’s Young’s modulus.
Error bars on transducer data
represent both the systematic
errors of the equipment as well
as the random error of the mean
pressure value
Exp Mech
to within ca. 10% of the peak amplitude. Figure 16(f) shows
that for noisy model data an acceptable original estimation
of the input amplitude is obtained. The corresponding cor-
rected pressure distribution overestimates the peak and low
range pressure amplitudes of the input distribution by ca.
5% of the peak amplitude. The in comparison to numer-
ical data more pronounced noise patterns in experimental
data (see also Figs. 11(b) and 12) were found to stem not
only from random but also from systematic error sources
(see “Experimental Results”). They may also be the reason
for the large difference between experimental and numer-
ical data in the initial reconstruction amplitude, here for
PRW =22 ca. 15%.
All iterations appear reasonably well converged after the
second iteration. Notably, the difference in peak amplitude
is reduced to around 10% or better for all investigated cases.
The outcome depends on the prevalence of noise patterns,
which is more pronounced for small PRWs and small or no
slope filters. However, larger reconstruction windows and
filter kernels do not allow for many iterations since the loss
of data points around the edges increases with PRW size.
Error Sources
The presented comparisons between real and simulated
experiments have shown the influence of random noise
and processing parameters on the pressure reconstruc-
tion. Experimental random noise patterns were qualitatively
reproduced with the modelled data for all investigated cases.
The presence of random noise was found to have a significant
impact on the reconstruction results. A systematic error in
the processing method was found to result in an underes-
timation of pressure amplitudes for noise-free model data.
This error varies with the processing parameters. Further,
a systematic experimental error appears between recon-
structed and transducer-measured pressures. It was found
that reconstructions from model data were consistently
closer to the input data than the experimental reconstruc-
tions were to pressure transducer data, which are an estab-
lished measurement technique. Based on the comparisons of
numerical and experimental data shown in “Finite Element
Correction”, this error resulted in an additional underesti-
mation of approximately 10% of the peak amplitude.
There are several possible sources for this experimen-
tal error. Miscalibration, i.e. non-integer numbers of pixels
per pitch in the recorded grid, can lead to errors in the
detected phases. It can be caused by misalignments between
camera sensor and printed grid. Even with careful arrange-
ment, small deformations of the specimen surface can cause
misalignment issues. Note that these can also occur due
to the deformations of the specimen under the investigated
(dynamic) load. Misalignment can particularly result in
fringes which can lead to the unexpected patterns observed
in curvature maps in “Experimental Results”. Irregularities
and damages in the printed grid can also result in errors dur-
ing phase detection. The influence of these error sources
on pressure amplitude is however difficult to quantify.
Another possible error source is wrong material param-
eter values, particularly the Young’s modulus. The data
information provided by the manufacturer gives a value of
E=74 GPa, but values between 47 and 83 GPa are found
for glass in the literature (e.g. [46, table 15.3]). 3- and
4-point bending tests on the specimen yielded values
between 69 and 83 GPa before the sample broke. Note that
the relationship between Young’s modulus and plate stiff-
ness matrix components, and thus pressure amplitudes (see
equation (4)), is linear, i.e. a 10% higher value of E would
increase all pressure amplitudes by 10%, compensating for
the discrepancy observed here. Deviations of the Poisson’s
ratio from the manufacturer information would have a sim-
ilar impact. Since the plate stiffness matrix components are
proportional to the third power of the plate thickness, errors
in its determination have a higher impact than is the case
for the other material parameters. Several measurements
did however confirm the thickness values provided by the
manufacturer. Assuming an error of 0.1% in the plate thick-
ness as worst case estimate, one obtains a 3% error in the
pressure amplitude.
Also, the assumptions of negligibility of rigid body
movement and out of plane displacement need to be consid-
ered. LDV measurements on the frame holding the speci-
men showed no results above noise level, which corresponds
to 0.1 μm here. Rigid body movement can therefore be
ruled out as a relevant error source. The effect of out of
plane displacements can be estimated based on the expected
deflections, w, and the distance between grid and speci-
men. A detailed derivation of this relationship is given in
[43, chapter 2.1.2]. The resulting error on curvature maps is
κoop =w
hS. The finite element simulations from “Simulated
Experiments” showed that the deflections for the 1mm plate
test can be expected to be smaller than 2μm, which would
correspond to an error in curvature of κoop =2103km1.
This worst-case estimate corresponds to an error of only
0.05% of the peak curvature signal amplitude. Finally, the
thin plate assumptions were tested using the finite ele-
ment simulation introduced in “Finite Element Model”.
The chosen SHELL181 elements are suited for linear as
well as for large rotation and large strain nonlinear applica-
tions. This means that simulated slopes and curvatures could
deviate from those calculated from the deflections using
thin plate assumptions (see e.g., [10]), if the latter were in
fact not applicable. The simulated and the calculated slopes
and curvatures were compared to verify the validity of the
assumptions. For the 1mm thick plate it was found that the
difference was five orders of magnitude below the signal
Exp Mech
amplitude in case of slopes and thee orders of magnitude in
case of curvatures.
Limitations and Future Work
This study shows that it is possible to obtain full-field
pressure measurements of the order of few O(100)Pa ampli-
tude with the described setup and processing technique. A
number of experimental limitations were encountered from
applying this method to low amplitude loads. Small grid
pitches are required to provide the required slope resolution.
These require a very smooth and plane specular reflective
specimen surface. Further decreasing the grid pitch would
require more camera pixels to investigate the same region
of interest, as the phase detection algorithm requires a mini-
mum amount of pixels per pitch. Alternatively, the distance
between grid and sample could be increased, which would
require a different lens to achieve the same magnification.
Furthermore, the specimen has to be stiff enough to pro-
vide a plane surface when mounted to avoid bias errors, but
is required to deform sufficiently to provide enough signal
for the measurement technique. The issue of misalignment
could be addressed by using high precision components like
micro stages with stepper motors to arrange camera, sample
and grid.
Another approach is the use of infrared instead of vis-
ible light for deflectometry, with heated grids as spatial
carrier [47]. Since infrared light has a longer wavelength
than visible light, it allows achieving specular reflection
on specimens that do not have mirror-like but reasonably
smooth surfaces with up to about 1.5 μm of RMS rough-
ness, like perspex and metal plates. However, available
cameras are limited in terms of spatial and temporal reso-
lutions. Further issues are the lack of an aperture ring and
that the lenses required to achieve comparable magnifica-
tion are more expensive. An extension of the application of
deflectometry to moderately curved surfaces was presented
recently [34]. This approach requires a calibration for defor-
mation measurement. Furthermore, the required depth of
field is a restricting factor for the use of small grid pitches.
A successful combination of deflectometry measurements
on curved surfaces with VFM pressure reconstruction would
be of great value, as it would allow direct measurements
on practically relevant surfaces like e.g. aerofoils, fuselages
and ship hulls.
In future studies, the turbulent fluctuations that occur in
many practical flows like the impinging jet used here will
be investigated. Typically they have pressure amplitudes
of the order of few O(10)Pa and below. These could not
be resolved in this study. Preliminary analyses of time
resolved data taken at 4 kHz show that this is in parts due
to a systematic experimental error, which results in spatial
distributions fluctuating at low frequency and relatively
high amplitude. The application of Fourier analyses and
Dynamic Mode Decomposition (DMD) are currently being
investigated with promising first results. Dynamic full-
field pressure reconstruction of turbulent fluctuations are a
continuous challenge for current experimental measurement
techniques due to their low amplitudes and small spatial
scales, rendering the further development of the technique
presented here highly relevant.
Another currently investigated improvement involves
employing the aforementioned higher resolution cameras
and smaller grid pitches to increase slope sensitivity and
spatial resolution. This approach does not allow for time
resolved measurements due to frame rate limitations of high
resolution cameras, but first tests using phase averaging
for periodic flows generated by synthetic jets are very
promising.
Finally, the selection of virtual fields is an important fac-
tor in improving the quality of reconstructions. Particularly
higher order approaches in pressure identification are likely
to reduce the systematic error.
Conclusion
This work presents a method for surface pressure recon-
structions from slope measurements using a deflectometry
setup combined with the VFM. Experimental and numerical
methods have been introduced to assess the pressure recon-
structions.
Low amplitude pressure distributions were recon-
structed from full-field slope measurements using the
material constitutive mechanical parameters.
Experimental results are presented and compared for
several reconstruction parameters and for two different
specimen.
VFM pressure reconstructions were compared to
pressure transducer measurements.
Simulated experiments employing a finite element
model and artificial grid deformation were used to
assess the uncertainty of the method.
The numerical results were used to select optimal
reconstruction parameters, taking into account experi-
mentally observed noise.
A finite element correction procedure was proposed
to mitigate the systematic error of VFM pressure
reconstructions.
Error sources were discussed based on the findings of
both the experimental and the simulated results.
A systematic processing error leading to an underestimation
of the pressure amplitude was identified. Since the shape
of the distribution is still reconstructed well, it is possible
Exp Mech
to compensate for this error using the proposed numerical
approaches as long as noise patterns are not too pronounced.
A systematic experimental error was found to result in an
additional underestimation of the pressure amplitude by ca.
10% more than simulated reconstructions. Yet, the results
stand out in terms of the low pressure amplitudes and the
large number of data points obtained.
Data Provision
All relevant data produced in this study is available under
the DOI https://doi.org/10.5258/SOTON/D0973.
Acknowledgements This work was funded by the Engineering and
Physical Sciences Research Council (EPSRC). F. Pierron acknowl-
edges support from the Wolfson Foundation through a Royal Society
Wolfson Research Merit Award (2012-2017). Advice and assistance
given by C´
edric Devivier, Yves Surrel, Manuel Aguiar Ferreira and
Lloyd Fletcher has been a great help in conducting simulations and
planning of experiments. The comments provided by Manuel Aguiar
Ferreira and Lloyd Fletcher have greatly improved this paper.
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unrestricted
use, distribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a
link to the Creative Commons license, and indicate if changes were
made.
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... These techniques were previously combined to reconstruct dynamic mechanical point loads [10]. Pressure reconstructions of air jets impinging on a thin glass mirror were investigated in a series of studies [11][12][13]. In these previous studies it was found that shapes, locations and time histories of the relatively small external loads of ca. ...
... 10-1000 Pa could be identified qualitatively well, but that the accuracy of the approach is highly dependent on signal-to-noise ratio, experimental bias and processing parameters. Simulated experiments were conducted in [11] to assess the accuracy of static reconstruction results. The present study applies this methodology to shock waves impacting steel plates to investigate its capabilities in this extreme experimental environment in terms of potential experimental bias from vibrations, temporal and spatial resolution. ...
... Since the material parameters are known a priori, p can then be identified once suitable virtual fields are chosen. 4-node hermite 16 element shape functions as used in the finite element method [11] are well suited here because they provide C 1 continuity of w à and therefore C 0 continuity of the virtual slopes [9, chap. 15]. ...
Chapter
This study investigates full-field, dynamic pressure reconstruction during shock-structure interactions using optical measurements and the virtual fields method (VFM). Shock wave impacts pose severe challenges to experimental measurement techniques due to the substantial, almost instantaneous pressure rises they induce. Their effects are typically measured pointwise using pressure transducers or as total force using load cells. Here, surface deformations were measured on the blind side of a flat steel plate in pure bending using a deflectometry setup. Pressure was reconstructed from the deformations induced by an impacting shock wave using a piecewise VFM approach. Different shock wave symmetries were used in order to investigate the capabilities of identifying spatial distributions reliably under the experimental conditions in the shock tube. Pointwise pressure transducer measurements allowed a validation of the results. It was found that different shapes of load distributions on the sample surface can be identified qualitatively, but that the comparability of both measurement techniques is limited due to filter and sampling capabilities.
... Spatially averaged random excitations were identified with this method in O' Donoughue et al. (2019). In Kaufmann et al. (2019) it was used to measure mean pressure distributions of an impinging air jet with differential pressure amplitudes of several O(100) Pa on thin glass plates of 1 mm thickness. The study also proposes a methodology to assess the accuracy of pressure reconstructions and to select optimal reconstruction parameters. ...
... However, in several aerodynamic and hydrodynamic applications, it is important to obtain surfacepressure fluctuations (both broadband as well as at certain frequencies). In the present study, the work of Kaufmann et al. (2019) is extended to measure the spatio-temporal evolution of low differential pressure events which are generated by the flow on a surface. The method is demonstrated in a canonical flow problem: a jet impinging on a flat surface. ...
... The mean pressure distributions obtained from timeresolved measurements in the present study are compared to the mean distributions obtained from uncorrelated snapshots in Kaufmann et al. (2019). Figure 6 shows the azimuthally averaged mean pressure distribution for both methods and pressure transducer data for comparison. ...
Article
Full-text available
This study presents an approach for obtaining full-field dynamic surface-pressure reconstructions with low differential amplitudes. The method is demonstrated in a setup where an air jet is impinging on a flat plate. Deformations of the flat plate under dynamic loading of the impinging jet were obtained using a deflectometry setup that allows measurement of surface slopes with high accuracy and sensitivity. The measured slope information was then used as input for the virtual fields method to reconstruct pressure. Pressure fluctuations with amplitudes of down to O(1)Pa\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(1)~\text {Pa}$$\end{document} were extracted from time-resolved deflectometry data using temporal band-pass filters. Pressure transducer measurements allowed comparisons of the results with an established measurement technique. Even though the identified uncertainties in fluctuations were found to be as large as 50%, the spatial distributions of dynamic pressure events were captured well. Dynamic mode decomposition was used to identify relevant spatial information that correspond to specific frequencies. These dynamically important spatio-temporal events could be observed despite their low differential amplitudes. Finally, the limitations of the proposed pressure determination method and strategies for future improvements are discussed. Graphic abstract
... Dynamic mechanical point loads were measured this way in [18]. In [15], pressure reconstructions of an impinging air jet on a flat plate were obtained for mean distributions. The study also addressed the accuracy and systematic processing error of the approach, as well as a procedure to mitigate the latter. ...
... In order to determine what size to choose and to estimate what systematic error this results in, the experimental parameters can be used to simulate the experiment. A methodology using finite element simulations and artificial grid deformation for this purpose is introduced in [15]. Figure 3 shows a schematic of the experimental setup. ...
... Large PRW sizes act as more efficient low-pass filter, but can lead to an underestimation of pressure amplitude and average out small-scale spatial distributions. [15] introduced a methodology for selecting optimal processing parameters in terms of local pressure amplitudes. It addresses the complex interactions between pressure signal, random noise and systematic processing errors. ...
Article
Full-text available
In this study, pressure distributions were reconstructed from phase-locked surface deformation measurements on a thin plate. Slope changes on the plate surface were induced by an external flow interacting with the specimen and measured with a highly sensitive deflectometry setup. The Virtual Fields Method (VFM) was used to obtain pressure reconstructions from the processed surface slopes and the plate material constitutive mechanical parameters. The applicability of the approach in combination with phase-locked measurements is demonstrated using a synthetic jet setup generating a periodic flow in air. Phase-averaging slope data allows mitigating random noise effects and resolving low-range differential pressure amplitudes despite the turbulent flow. The size of the spatial structures of the investigated low amplitude flow events identified in full-field with the present method are \(\mathcal {O}(1)~\text {mm}\), which is beyond the capabilities of other available surface pressure measurement techniques. Challenges and limitations in achieving the metrological performance for resolving the observed surface slopes of \(\mathcal {O}(0.1)~\text {mm km}^{-1}\) are described and improvements for future applications are discussed.
... A fan-driven, round air jet was used to generate the investigated flow, which can be divided into a free jet, stagnation and wall region, e.g. Kalifa et al., 2016; Figure 2: Experimental setup (redrawn from (Kaufmann et al., 2019a)). Gaussian Beltaos, 1976. ...
... This is comparable to or better than other optical based pressure determination techniques like PSP. Further details of the mean flow comparison and the selection of optimal pressure reconstruction parameters required to achieve this can be found in Kaufmann et al., 2019a. Given the agreement in the mean flow, the time-resolved data can now be further examined to obtain surface pressure fluctuations. ...
... Using the LDV to measure vibrations of the camera, it was also found that the camera cooling fans caused vibrations at several frequencies below 100 Hz, which were also identified in the measured slope spectrum. Based on comparisons with numerical data presented in Kaufmann et al., 2019a, the effect of these experimental errors could amount to up to 10% of the peak pressure value for the mean flow. ...
Thesis
This thesis presents a technique for the reconstruction of full-field surface pressure distributions with low differential amplitudes. The method is demonstrated in two different setups with air jets impinging on flat plates. Surface deformations were obtained using deflectometry, a highly sensitive technique for slope measurement. The surface slopes in combination with the plate material constitutive mechanical parameters were used as input for pressure reconstructions with the Virtual Fields Method (VFM), which is an application of the principle of virtual work. Both static and dynamic pressure distributions were reconstructed in full-field with this technique. Results were compared with pressure transducer measurements at discrete points. The observed pressure amplitudes were between O(1) Pa – O(100) Pa. The spatial extent of the investigated flow structures was O(1) mm – O(10) mm. Dynamic pressure information was extracted from time-resolved deflectometry data using temporal band-pass filters or phaselocked measurements. Dynamic Mode Decomposition (DMD) was used on time-resolved data to identify relevant spatial information that correspond to specific frequencies. Despite the low differential amplitudes, dynamically important spatio-temporal events could be observed. Full-field measurements of such small-scale, low-amplitude pressure events are not possible with established pressure measurement techniques. A finite element model was employed in combination with artificial grid deformation to assess the uncertainty of the pressure reconstructions. Finally, challenges and limitations in achieving the metrological performance for resolving the observed surface slopes of O(0.1) mm km−1 – O(10) mm km−1 as well as in the pressure reconstruction approach are described and strategies for future applications are discussed.
... A fan-driven, round air jet was used to generate the investigated flow, which can be divided into a free jet, stagnation and wall region, e.g. Kalifa et al., 2016; Figure 2: Experimental setup (redrawn from (Kaufmann et al., 2019a)). Gaussian Beltaos, 1976. ...
... This is comparable to or better than other optical based pressure determination techniques like PSP. Further details of the mean flow comparison and the selection of optimal pressure reconstruction parameters required to achieve this can be found in Kaufmann et al., 2019a. Given the agreement in the mean flow, the time-resolved data can now be further examined to obtain surface pressure fluctuations. ...
... Using the LDV to measure vibrations of the camera, it was also found that the camera cooling fans caused vibrations at several frequencies below 100 Hz, which were also identified in the measured slope spectrum. Based on comparisons with numerical data presented in Kaufmann et al., 2019a, the effect of these experimental errors could amount to up to 10% of the peak pressure value for the mean flow. ...
Thesis
This thesis presents a technique for the reconstruction of full-field surface pressure distributions with low differential amplitudes. The method is demonstrated in two different setups with air jets impinging on flat plates. Surface deformations were obtained using deflectometry, a highly sensitive technique for slope measurement. The surface slopes in combination with the plate material constitutive mechanical parameters were used as input for pressure reconstructions with the Virtual Fields Method (VFM), which is an application of the principle of virtual work. Both static and dynamic pressure distributions were reconstructed in full-field with this technique. Results were compared with pressure transducer measurements at discrete points. The observed pressure amplitudes were between O(1) Pa – O(100) Pa. The spatial extent of the investigated flow structures was O(1) mm – O(10) mm. Dynamic pressure information was extracted from time-resolved deflectometry data using temporal band-pass filters or phaselocked measurements. Dynamic Mode Decomposition (DMD) was used on time-resolved data to identify relevant spatial information that correspond to specific frequencies. Despite the low differential amplitudes, dynamically important spatio-temporal events could be observed. Full-field measurements of such small-scale, low-amplitude pressure events are not possible with established pressure measurement techniques. A finite element model was employed in combination with artificial grid deformation to assess the uncertainty of the pressure reconstructions. Finally, challenges and limitations in achieving the metrological performance for resolving the observed surface slopes of O(0.1) mm km−1 – O(10) mm km−1 as well as in the pressure reconstruction approach are described and strategies for future applications are discussed.
... When pressure loads are imposed on a deformable structure, fluid-structure interaction (FSI) effects are known to cause non-trivial loading scenarios which are difficult to quantify (see e.g., (Aune et al., 2021)). This project aims at reconstructing the full-field surface pressure loads acting on a deforming structure employing the virtual fields method (VFM) on full-field kinematic measurements (Kaufmann et al., 2019). Even though the current framework is limited to reconstructions of full-field pressure information from deformation data of thin plates in pure bending, it also allows for future extensions to other loading and deformation scenarios. ...
... RECOLO contains a collection of tools enabling the user to perform virtual experiments on synthetically generated data as well as performing pressure reconstruction on experimental datasets. The pressure reconstruction algorithm is based on the work by (Kaufmann et al., 2019). The implementation is based on numerical operations provided by NumPy (Oliphant, 2015) and SciPy (Jones et al., n.d.) as well as visualization by Matplotlib (Hunter, 2007). ...
Article
Full-text available
In experimental mechanics, conducting non-intrusive measurements of surface pressure distributions acting on blast-loaded structures remains a challenge even in controlled, laboratory environments (see e.g., (Pannel et al., 2021)). Still, for the design of tomorrow’s sustainable and material-efficient structures, detailed knowledge of how pressure loads from extreme loading events interact with deformable structures is essential. When pressure loads are imposed on a deformable structure, fluid-structure interaction (FSI) effects are known to cause non-trivial loading scenarios which are difficult to quantify (see e.g., (Aune et al., 2021)). This project aims at reconstructing the full-field surface pressure loads acting on a deforming structure employing the virtual fields method (VFM) on full-field kinematic measurements (Kaufmann et al., 2019). Even though the current framework is limited to reconstructions of full-field pressure information from deformation data of thin plates in pure bending, it also allows for future extensions to other loading and deformation scenarios. Provided that the properties of the structure are known, the transient pressure distribution on the plate can be reconstructed. To understand the capabilities and accuracy associated with the pressure reconstruction methodology, the package provides tools for performing virtual experiments based on analytical data or data from finite element simulations. The current implementation is based on the deflectometry technique, using the grid method to obtain the deformation measurements and corresponding kinematics of the structure. This Python package is made for RECOnstructing surface pressure LOads, RECOLO, acting on plated structures based on deformation measurements using the VFM (Pierron & Grédiac, 2012). The current implementation determines the surface pressure acting on a thin plate undergoing small deformations, assuming linear, elastic material behaviour. However, the framework will be extended to large plastic deformations, allowing the two-way interaction between the pressure loading and the deformation of the plate to be studied. Other VFM toolkits such as PeriPyVFM are readily available but typically aimed at determining material properties from deformation and load measurements. Hence, as opposed to other VFM toolkits, RECOLO assumes that the material properties are known and use the full-field deformation measurements to reconstruct the pressure loading. RECOLO contains a collection of tools enabling the user to perform virtual experiments on synthetically generated data as well as performing pressure reconstruction on experimental datasets. The pressure reconstruction algorithm is based on the work by (Kaufmann et al., 2019). The implementation is based on numerical operations provided by NumPy (Oliphant, 2015) and SciPy (Jones et al.) as well as visualization by Matplotlib (Hunter, 2007).
... Dans plusieurs études, les méthodes ont été utilisées dans le domaine fréquentiel pour des problèmes stationnaires [107,108]. Plus récemment, ces méthodes ont été appliquées au domaine temporel [141,142,143,144,145], afin d'obtenir directement la dépendance temporelle de la distribution des forces. Les données d'entrée (champ de vitesse ou de déplacement) doivent alors être acquises dans le domaine temporel de manière synchrones. ...
Thesis
L’avènement de nouvelles technologies amène à élargir le champ d’étude possible dans le domaine vibroacoustique amenant à étudier des excitations instationnaires et non-répétables nécessitant d’être caractérisées. Cependant les phénomènes résultant de ces excitations ne peuvent pas être mesurés de manière synchrone et non-intrusive avec des moyens expérimentaux classiques. Ce travail de thèse propose de coupler une technique de mesure vibratoire plein-champ, à haute résolution spatiale, sans contact et non-intrusive, l’holographie numérique ultra-rapide, à l’identification d’efforts pour différents régimes d'excitations de structure mécaniques. L’identification d’efforts est basée sur la méthode de résolution inverse (RI) où le champ des déplacements issu de la vibrométrie holographique numérique ultra-rapide est utilisé comme entrée dans l'équation de mouvement de la structure. Grâce aux mesures plein-champ de l'holographie numérique, l'analyse de vibrations stationnaires et transitoires est effectuée à la fois dans le domaine spatial et dans le domaine temporel. Le maillage dense des points de données est utilisé pour résoudre le problème inverse de l'identification des forces, qui est régularisé selon une approche Bayésienne permettant une régularisation optimale et automatique de la méthode. Des résultats expérimentaux pour des tests en régimes stationnaires et transitoires sont présentés. Dans le cas d’une excitation en régime stationnaire, les résultats sont comparés à ceux obtenus avec un vibromètre à balayage et présentent une très bonne concordance entre les profils de force observés ainsi qu’entre les amplitudes, validant ainsi l'approche proposée. En régime transitoire, les profils de force mesurés sont comparés aux mesures effectuées à l'aide d'un capteur de force montrant un bon accord entre les résultats obtenus.
Article
Full-text available
The use of high-speed cameras permits to visualize, analyze or study physical phenomena at both their time and spatial scales. Mixing high-speed imaging with coherent imaging allows recording and retrieving the optical path difference and this opens the way for investigating a broad variety of scientific challenges in biology, medicine, material science, physics and mechanics. At high frame rate, simultaneously obtaining suitable performance and level of accuracy is not straightforward. In the field of mechanics, this prevents high-speed imaging to be applied to full-field vibrometry. In this paper, we demonstrate a coherent imaging approach that can yield full-field structural vibration measurements with state-of-the-art performances. The method is based on high-speed on-line digital holography and recording a short time sequence. Validation of the proposed approach is carried out by comparison with a scanning laser Doppler vibrometer and by realistic simulations. Several error criteria demonstrate measurement capability of yielding amplitude and phase of structural deformations.
Article
Full-text available
This paper presents an implementation of deflectometry in the infrared spectrum. Deflectometry consists in recording the specular image of a reference grid pattern onto the mirror-like surface of a test specimen. This technique has two main advantages, high sensitivity and direct measurement of surface slopes, which in the case of thin plate bending is only one spatial differentiation away from surface strains. The objective of imaging in the infrared spectrum is to mitigate the main limitation of deflectometry in the visible spectrum, which is to require an extremely smooth surface to provide dominant specular reflection. This paper explores IR deflectometry for the first time for deformation measurements. Two different infrared cameras were assessed for use in IR deflectometry, a short wave quantum detector one, and a long wave microbolometer (MB) array one. Different materials of varying surface roughness were imaged and it was verified that the Rayleigh criterion was appropriate to determine whether IR deflectometry was feasible on a given surface. With the MB camera, most off-the-shelf material surfaces proved reflective enough to perform IR deflectometry. Finally, several bending tests were performed on aluminium plates and the deformation fields were shown to compare remarkably well with finite element simulations. The experimental data were then used in the Virtual Fields Method (VFM) and the elastic stiffness components of aluminium were retrieved with excellent accuracy, further validating IR deflectometry.
Article
Full-text available
The Kevlar-wall anechoic wind tunnel offers great value to the aeroacoustics research community, affording the capability to make simultaneous aeroacoustic and aerodynamic measurements. While the aeroacoustic potential of the Kevlar-wall test section is already being leveraged, the aerodynamic capability of these test sections is still to be fully realized. The flexibility of the Kevlar walls suggests the possibility that the internal test section flow may be characterized by precisely measuring small deflections of the flexible walls. Treating the Kevlar fabric walls as tensioned membranes with known pre-tension and material properties, an inverse stress problem arises where the pressure distribution over the wall is sought as a function of the measured wall deflection. Experimental wall deformations produced by the wind loading of an airfoil model are measured using digital image correlation and subsequently projected onto polynomial basis functions which have been formulated to mitigate the impact of measurement noise based on a finite-element study. Inserting analytic derivatives of the basis functions into the equilibrium relations for a membrane, full-field pressure distributions across the Kevlar walls are computed. These inversely calculated pressures, after being validated against an independent measurement technique, can then be integrated along the length of the test section to give the sectional lift of the airfoil. Notably, these first-time results are achieved with a non-contact technique and in an anechoic environment.
Conference Paper
p>This paper presents an application of deflectometry to measure the deformation of a thin cylindrical shell in bending. The principle of the calibration method is briefly outlined. Then, the experimental set-up is presented, followed by comparison of slopes, deflections and curvatures with results from a finite element model of the test. The results are satisfactory though limited by the quality of the reflective surface of the test samples. Future work will focus on more complex shapes to extend the technique.</p
Article
The sound transmission loss of a homogeneous, isotropic, thin panel under a diffuse acoustic field excitation is derived from a measurement of its airborne induced vibration field. Using this single dataset, the virtual fields method allows identifying the wall pressure field exciting the panel and estimating the corresponding incident acoustic power, provided that the differential operator governing the plate dynamics and its material properties are known a priori. Using the same dataset, the radiated acoustic power is calculated using the radiation resistance matrix method. For an aluminium plate, a comparison of transmission loss values obtained using this approach and a standardized measurement shows good agreement off resonance but large discrepancies on resonance and close to resonance, due to ill conditioning of the virtual fields method. A simple correction is proposed on resonance.
Article
The calibration of phenomenological constitutive material models has been a constant need, because the parameters differ for each material and the ability of a model to mimic the real behaviour of a material is highly dependent on the quality of these parameters. Classically, the parameters of constitutive models are determined by standard tests under the assumption of homogeneous strain and stress fields in the zone of interest. However, in the last decade, Digital Image Correlation techniques and full-field measurements have enabled the development of new parameter identification strategies, such as the Finite Element Model Updating, the Constitutive Equation Gap Method, the Equilibrium Gap Method and the Virtual Fields Method. Although these new strategies have proven to be effective for linear and non-linear models, the implementation procedure for some of them is still a laborious task. The aim of this work is to give a detailed insight into the implementation aspects and validation of these methods. Detailed flowcharts of each strategy, focusing on the implementation aspects, are presented and their advantages and disadvantages are discussed. Moreover, these modern strategies are compared for the cases of homogeneous isotropic linear elasticity and isotropic plasticity with isotropic hardening. A simple numerical example is used to validate and compare the different strategies.
Chapter
This paper aims at identifying random excitations acting on thin, plane structures from their measured vibration response. For random pressure fields such as the diffuse acoustic field (DAF) and turbulent boundary layer (TBL), two quantities of interest are to be determined, namely the wall pressure auto-spectral and cross-spectral density functions. These quantities are reconstructed using the virtual fields method, an identification technique based on the principle of virtual work. Numerical identification results for the auto-spectral and cross-spectral density functions are presented for both a plate and a membrane submitted to DAF and TBL excitations. Experimental identification results are then presented for the pressure auto-spectrum applied to an aluminum panel under DAF excitation from vibration response measurements that were obtained using deflectometry, a full-field optical measurement technique.
Article
This paper describes an approach for identifying the magnitude and location of both stationary and transient mechanical loadings applied to a thin rectangular simply supported plate. Full-field deflectometry measurements and the virtual fields method are used with the local equilibrium equation of the plate in the time domain to solve the force reconstruction problem, whereas previous work by the authors used this last equation in the frequency domain. As a result, it is possible to reconstruct load time history in addition to magnitude and location. Experimental results of this complete identification are presented for two different instrumented mechanical exciters: electrodynamic shaker and impact hammer for stationary and transient excitations, respectively. The approach is then applied to determine the location and time of multiple unknown transient excitations produced by a set of impacting metal marbles.
Article
The grid method is a technique suitable for the measurement of in-plane displacement and strain components on specimens undergoing a small deformation. It relies on a regular marking of the surfaces under investigation. Various techniques are proposed in the literature to retrieve these sought quantities from images of regular markings, but recent advances show that techniques developed initially to process fringe patterns lead to the best results. The grid method features a good compromise between measurement resolution and spatial resolution, thus making it an efficient tool to characterise strain gradients. Another advantage of this technique is the ability to establish closed-form expressions between its main metrological characteristics, thus enabling to predict them within certain limits. In this context, the objective of this paper is to give the state of the art in the grid method, the information being currently spread out in the literature. We propose first to recall various techniques that were used in the past to process grid images, to focus progressively on the one that is the most used in recent examples: the windowed Fourier transform. From a practical point of view, surfaces under investigation must be marked with grids, so the techniques available to mark specimens with grids are presented. Then we gather the information available in the recent literature to synthesise the connection between three important characteristics of full-field measurement techniques: the spatial resolution, the measurement resolution and the measurement bias. Some practical information is then offered to help the readers who discover this technique to start using it. In particular, programmes used here to process the grid images are offered to the readers on a dedicated website. We finally present some recent examples available in the literature to highlight the effectiveness of the grid method for in-plane displacement and strain measurement in real situations.
Article
This paper aims at identifying the autospectral density and spatial correlation functions of random excitations acting on the surface of a thin plate, from its measured vibration response. The general framework is the Virtual Fields Method (VFM), which was previously applied by the authors to the identification of deterministic excitations on plates. In the present paper, the VFM framework is extended to the case of spatially correlated excitations. It is shown that extraction of the loading power spectral density requires measuring power spectral density functions of transverse displacements and bending curvatures, which can be typically derived from contactless Laser Doppler Vibrometry measurements. The paper details the implementation of the VFM for random excitations, presents numerical simulations and experimental results for diffuse acoustic field excitation of a plate.