Available via license: CC BY 4.0

Content may be subject to copyright.

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:

dφx=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

2π(φdef (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:

dαx=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

pw∗dS =Dxx

Sκxxκ∗

xx +κyyκ∗

yy +2κxyκ∗

xydS

+Dxy

Sκxxκ∗

yy +κyyκ∗

xx −2κxyκ∗

xydS

+ρtS

S

aw∗dS.(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, w∗the virtual deflection and

κ∗the 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 w∗and κ∗one 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

xxκ∗i

xx +κi

yyκ∗i

yy +2κi

xyκ∗i

xy

+Dxy

N

i=1

κi

xxκ∗i

yy +κi

yyκ∗i

xx −2κi

xyκ∗i

xy

+ρtS

N

i=1

aiw∗iN

i=1

w∗i−1

.(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 [18–20,22]. 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 m−3

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, 6σα−1 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 s−2. The observed

standard deviations varied with the position along the plate

surface and reached up to 1.4 m s−2. 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 0.7rD−1. 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

pG−cos 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 =210−3km−1.

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.

References

1. Usherwood JR (2009) The aerodynamic forces and pressure

distribution of a revolving pigeon wing. Exp Fluids 46(5):991–

1003

2. Livingood NBJ, Hrycak, P (1973) Impingement heat transfer

from turbulent air jets to flat plates: A literature survey. Tech.

Rep. NASA-TM-x-2778, E-7298, NASA Lewis Research Center;

Cleveland, OH

3. Corcos GM (1963) Resolution of pressure in turbulence. J Acoust

Soc Amer 35(2):192–199

4. Corcos GM (1964) The structure of the turbulent pressure field in

boundary-layer flows. J Fluid Mech 18(3):353–378

5. Tropea C, Yarin A, Foss J (2007) Springer handbook of

experimental fluid mechanics. Springer, Berlin

6. Yang L, Zare-Behtash H, Erdem E, Kontis K (2012) Application

of AA-PSP to hypersonic flows: The double ramp model. Sens

Actuators B: Chem 161(1):100–107

7. van Oudheusden BW (2013) PIV-based pressure measurement.

Measur Sci Technol 032(3):001

8. Ragni D, Ashok A, van Oudheusden BW, Scarano F (2009)

Surface pressure and aerodynamic loads determination of a

transonic airfoil based on particle image velocimetry. Measur Sci

Technol 20(7):074,005

9. Brown K, Brown J, Patil M, Devenport W (2018) Inverse

measurement of wall pressure field in flexible-wall wind tunnels

using global wall deformation data. Exp Fluids 59(2):25

10. Timoshenko S, Woinowsky-Krieger S (1959) Theory of plates

and shells. Engineering societies monographs. McGraw-Hill,

New York

11. Pezerat C, Guyader JL (2000) Force Analysis Technique:

Reconstruction of force distribution on plates. Acta Acust United

Acust 86(2):322–332

12. Lecl´

ere Q, P´

ezerat C (2012) Vibration source identification using

corrected finite difference schemes. J Sound Vibr 331(6):1366–

1377

13. Lecoq D, P´

ezerat C, Thomas JH, Bi W (2014) Extraction of

the acoustic component of a turbulent flow exciting a plate by

inverting the vibration problem. J Sound Vibr 333(12):2505–

2519

14. Pierron F, Gr´

ediac M (2012) The virtual fields method. Extracting

constitutive mechanical parameters from full-field deformation

measurements. Springer, New York

15. Martins J, Andrade-Campos A, Thuillier S (2018) Comparison of

inverse identification strategies for constitutive mechanical mod-

els using full-field measurements. Int J Mech Sci 145:330–345

16. Moulart R, Pierron F, Hallett SR, Wisnom MR (2011) Full-

field strain measurement and identification of composites moduli

at high strain rate with the virtual fields method. Exp Mech

51(4):509–536

17. Pierron F, Sutton MA, Tiwari V (2011) ultra high speed DIC and

virtual fields method analysis of a three point bending impact test

on an aluminium bar. Exp Mech 51(4):537–563

18. Robin O, Berry A (2018) Estimating the sound transmission loss

of a single partition using vibration measurements. Appl Acoust

141:301–306

19. Berry A, Robin O, Pierron F (2014) Identification of dynamic

loading on a bending plate using the virtual fields method. J Sound

Vibr 333(26):7151–7164

20. Toussaint E, Gr´

ediac M, Pierron F (2006) The virtual fields

method with piecewise virtual fields. Int J Mech Sci 48(3):256–

264

21. Berry A, Robin O (2016) Identification of spatially correlated

excitations on a bending plate using the virtual fields method. J

Sound Vibr 375:76–91

22. O’Donoughue P, Robin O, Berry A (2017) Time-resolved

identification of mechanical loadings on plates using the

virtual fields method and deflectometry measurements. Strain

54(3):e12,258

23. Surrel Y, Fournier N, Gr´

ediac M, Paris PA (1999) Phase-stepped

deflectometry applied to shape measurement of bent plates. Exp

Mech 39(1):66–70

24. Devivier C, Pierron F, Wisnom M (2012) Damage detection

in composite materials using deflectometry, a full-field slope

measurement technique. Compos Part A: Appl Sci Manuf

43(10):1650–1666

25. Giraudeau A, Pierron F, Guo B (2010) An alternative to modal

analysis for material stiffness and damping identification from

vibrating plates. J Sound Vib 329(10):1653–1672

26. Devivier C, Pierron F, Glynne-Jones P, Hill M (2016) Time-

resolved full-field imaging of ultrasonic Lamb waves using

deflectometry. Exp Mech 56:1–13

27. O’Donoughue P, Robin O, Berry A (2019) Inference of random

excitations from contactless vibration measurements on a panel

or membrane using the virtual fields method. In: Ciappi E, De

Rosa S, Franco F, Guyader JL, Hambric SA, Leung RCK, Hanford

AD (eds) Flinovia—flow induced noise and vibration issues

and aspects-II. Springer International Publishing, Cham, pp 357–

372

28. Kalifa RB, Habli S, Sa¨

ıd NM, Bournot H, Palec GL (2016) The

effect of coflows on a turbulent jet impacting on a plate. Appl

Math Modell 40(11):5942–5963

Exp Mech

29. Zuckerman N, Lior N (2006) Jet impingement heat transfer:

physics, correlations, and numerical modeling. Adv Heat Transfer

39(C):565–631

30. Beltaos S (1976) Oblique impingement of circular turbulent jets. J

Hydraul Res 14(1):17–36

31. Gr´

ediac M, Sur F, Blaysat B (2016) The grid method for in-

plane displacement and strain measurement: a review and analysis.

Strain 52(3):205–243

32. Ritter R (1982) Reflection moire methods for plate bending

studies. Opt Eng 21:21–29

33. Balzer J, Werling S (2010) Principles of shape from specular

reflection. Measurement 43(10):1305–1317

34. Surrel Y, Pierron F (2019) Deflectometry on curved surfaces. In:

Proceedings of the 2018 Annual Conference on Experimental and

Applied Mechanics, pp 217–221

35. Dai X, Xie H, Wang Q (2014) Geometric phase analysis based

on the windowed Fourier transform for the deformation field

measurement. Opt Laser Technol 58:119–127

36. Surrel Y (2000) Photomechanics, chap. Fringe Analysis. Springer,

Berlin, pp 55–102

37. Poon CY, Kujawinska M, Ruiz C (1993) Spatial-carrier phase

shifting method of fringe analysis for moir´

e interferometry. J

Strain Anal Eng Des 28(2):79–88

38. Surrel Y (1996) Design of algorithms for phase measurements by

the use of phase stepping. Appl Opt 35(1):51–60

39. Hibino K, Larkin K, Oreb B, Farrant D (1995) Phase shifting for

nonsinusoidal waveforms with phase-shift errors. J Opt Soc Am A

12(4):761–768

40. Badulescu C, Gr´

ediac M, Mathias JD (2009) Investigation of the

grid method for accurate in-plane strain measurement. Measur.

Sci. Technol 20(9):095,102

41. Dym C, Shames I (1973) Solid mechanics: a variational approach.

Advanced engineering series. McGraw-Hill, New York

42. Zienkiewicz O (1977) The finite element method. McGraw-Hill,

New York

43. Devivier C (2012) Damage identification in layered composite

plates using kinematic full-field measurements. Ph.D. thesis

Universit´

e de Technologie de Troyes

44. Barbero E (2013) Finite element analysis of composite materials

using ANSYS®, 2nd edn. Composite Materials. CRC Press, Boca

Raton

45. Rossi M, Pierron F (2012) On the use of simulated experiments

in designing tests for material characterization from full-field

measurements. Int J Solids Struct 49(3):420–435

46. Ashby M (2011) Materials selection in mechanical design, 4th

edn. Butterworth-Heinemann, Oxford

47. Toniuc H, Pierron F (2018) Infrared deflectometry for sur-

face slope deformation measurements Experimental Mechanics.

(submitted)

Publisher’s Note Springer Nature remains neutral with regard to

jurisdictional claims in published maps and institutional affiliations.