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Noname manuscript No.
(will be inserted by the editor)
Importance of Weighting FullField Displacement Components
when Fitting Material Parameters
Charles F. Jekel? ?? ·Martin P. Venter ·Gerhard
Venter ·Raphael T. Haftka
Received: date / Accepted: date
?This document was prepared as an account of work sponsored by an agency of the United
States government. Neither the United States government nor Lawrence Livermore National
Security, LLC, nor any of their employees makes any warranty, expressed or implied, or
assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of
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government or Lawrence Livermore National Security, LLC. The views and opinions of authors
expressed herein do not necessarily state or reect those of the United States government or
Lawrence Livermore National Security, LLC, and shall not be used for advertising or product
endorsement purposes. LLNLJRNL835048DRAFT
?? Lawrence Livermore National Laboratory is operated by Lawrence Livermore National
Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration
under Contract DEAC5207NA27344.
C. F. Jekel *corresponding author*
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL
32611, USA
Email: jekel1@llnl.gov
Present address: of C. F. Jekel
Lawrence Livermore National Laboratory, L227, PO Box 808, Livermore, CA, 94551, USA
M. P. Venter
Email: mpventer@sun.ac.za
Department of Mechanical and Mechatronic Engineering, Stellenbosch University, Private
Bag X1, Matieland, 7602, South Africa
G. Venter
Email: gventer@sun.ac.za
Department of Mechanical and Mechatronic Engineering, Stellenbosch University, Private
Bag X1, Matieland, 7602, South Africa
R. T. Haftka
Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL
32611, USA
Supplementary material available online that includes data and videos covered in this
paper: https://github.com/cjekel/inv_bubble_opt
2
Abstract A methodology is presented to t material parameters in nite element
(FE) models using full displacement eld data. Four bulge ination tests were per
formed on a PVCcoated polyester material. Digital image correlation (DIC) was
used to capture the full displacement eld of the material. An inverse analysis was
set up to nd material parameters in a FE model which best matched the full dis
placement eld of the experimental test. Three dierent objective functions were
considered to quantify the discrepancy between the FE model and experiments. The
rst function only matched the maximum outofplane deection, representative of
an experiment with a single point deection measure. The other functions consider
matching the numerical model to the full three dimensional displacement eld, where
one function considers a weighted scheme to balance the order of magnitude dier
ence between inplane and outofplane displacements. The resulting material param
eters were heavily dependent upon which objective function was chosen. The best
parameters are thought to come from the weighted objective function considering
the fulleld data, which matched the inplane displacements better than the other
objective functions.
Keywords Inverse analysis ·Objective functions ·material parameter identication ·
bulge ination test
1 Introduction
The nite element (FE) method is an important tool used to design membrane
structures. These FE models rely on the ecacy of material models for design insight,
and unfortunately selecting appropriate material parameters to represent complex
behavior can be dicult. This is especially true for materials like coated fabrics,
which typically consist of a vinyl polymer coating over woven yarns bundled from
several bers. Inverse analyses, or iterative schemes for FE model updating (FEMU)
have been used to nd material parameters from complex load cases. In this paper,
three dierent objective functions were investigated with FEMU to estimate material
parameters from bulge ination tests on PVCcoated polyester. Objective function
refers to how the discrepancy between the FE model and test data is quantied, and
material parameters are typically selected to minimize this function. The objective
function has an important eect on the material parameters selected using this type
of inverse analysis (Zhan et al., 2011; Jekel et al., 2019; Tam, 2020).
PVCcoated polyester is a coated textile. It’s most commonly modeled as a con
tinuous, homogeneous, orthotropic material (Shaw et al., 2010), which can have large
material direction dependent strength due to the warptoll stiness ratio (Dinh
et al., 2017). The typical weave has the warp yarns pulled taut, while the ll yarns
are woven inbetween the warp yarns. The ll yarns run orthogonal to the warp
yarns. Various nonlinear models have been used in an attempt to better describe
the behavior of the material. Galliot and Luchsinger (2009) proposed a nonlinear
material model based on the load ratio between the material warp and ll direc
tions. Ambroziak and Kłosowski (2014) used a piecewise linear orthotropic model,
and Jekel et al. (2017) used polynomials to describe the nonlinear elastic behavior
of the material.
Biaxial tests are commonly used to characterize material parameters for struc
tural membrane materials (Stranghöner et al., 2016). Several studies have used bulge
3
(or bubble, or membrane) ination tests to induce an equal biaxial load on the
material. These bulge tests typically involve inducing a pressure on one side of a
circularly clamped membrane material. Reuge et al. (2001) performed these bulge
ination tests on a rubber material and outlined the assumptions that lead to an
equal biaxial load occurring at the apex of the deected membrane material. Sasso
et al. (2008) provides a good review of how bulge ination tests compare to other
forms of tension tests. Much of these early bulge ination tests only recorded the
apex height (Rachik et al., 2001; Charalambides et al., 2002). More recent bulge
ination tests have adopted the use of 3D digital image correlation to measure the
entire surface deformation of the membrane material (Machado et al., 2012). Tonge
et al. (2013a) stated that the full displacement eld allows for the determination
of inplane material directions and was later used to calibrate a material model for
human skin.
It is often the case that FEMU is used to calibrate material parameters by formu
lating and inverse problem to match a numerical model to some physical experiment
(Lovato et al., 1993; Cailletaud and Pilvin, 1994; Drass and Schneider, 2016). The
process can be generalized by using optimization to nd parameters in a FE model
which minimize the discrepancy between the model and experimental data (Asaadi
et al., 2017). In a forward problem, parameters can be directly inferred from an
experimental test. While an inverse problem iteratively updates the FE model to
nd parameters that best match an experimental response (Lee et al., 2019). This
process is often referred to as model calibration. The FE model must be accurate
enough to get reasonable parameters when using FEMU. Oversimplied assumptions
or incorrect boundary conditions may result in nonsensible parameters. As such, a
validation step is necessary to ensure the parameters calibrated with FEMU are
sensible before being used in other practical tasks. We refer the reader to strategies
by Oh et al. (2016) and Son et al. (2020) to statistically improve the accuracy of
FE models to both blind and known uncertainties. Additionally, there may be some
cases where unknown boundary conditions may be included into the optimization
formulation of the material calibration problem as performed by van Rensburg et al.
(2018).
Model calibration can be performed using deterministic or probabilistic meth
ods (Lee et al., 2019). A deterministic model calibration does not include the mea
surement uncertainty, while probabilistic method can account for both model and
experimental uncertainty. Jung et al. (2016) stated that models calibrated determin
isticly do not account for material uncertainty, and thus may result in unreliable
solutions. The uncertainty of the experiments needs to be quantied in order to cal
ibrate material models probabilistically. Unfortunately, DIC displacement eld data
is typically treated as a black box where uncertainties are not well understood. The
bulge ination test is particularly challenging having outofplane displacements on
the curved material surface. While it may be more reliable to calibrate the mate
rial model parameters using a probabilistic method, this work uses a deterministic
method because the uncertainty due to DIC errors of the displacement measurements
is currently unknown. However, the experimental measurements are made available,
so that a probabilistic calibration can be performed in the future.
There are many dierent choices of possible objective functions to calibrate ma
terial models. Zhan et al. (2011) proposed the EARTH method to calibrate model
parameters into phase, magnitude, and slope errors from time history data. A sin
gle DIC bulge ination test may generate over a hundred thousand unique time
4
histories. In order to apply EARTH, each of these time histories would need to be
broken into their respective errors and treated as separate objective functions in a
multiobjective optimization. This process to compute the objective functions may
be computationally expensive with fulleld data.
The previous studies (Rachik et al., 2001; Charalambides et al., 2002) used
only the height to determine material parameters from bulge ination tests. Tonge
et al. (2013b) only used a single stress resultant as the cost function for calibration.
Machado et al. (2012) used curvatures to determine a stress tensor from bulge ina
tion tests to infer material parameters for an elastic material. While these previous
methods work well for a planar isotropic material, they are dicult to extend to a
directional dependent woven textile like PVCcoated polyester. Additionally, it is un
clear how important normalization will be to the objective function when calibrating
to multiple displacement components. This work will calibrate a linear orthotropic
material model from bulge ination tests. Calibrating material parameters using the
bugle height (displacement at maximum deection) will be compared to not only the
full displacement eld, but also with weighted (or normalized) displacements.
Jekel et al. (2016) showed that it was possible to use the displacement eld of
a bulge test to select parameters for a nonlinear orthotropic material model using
simulated experimental data. Jekel (2016) then used this process on polynomial
displacement elds tted to experimental data at selected ination pressures. These
polynomial surfaces introduce an extra layer of error in the material parameters,
as the error in the polynomial t was not properly accounted for. While previous
work used a surrogate model to represent the experimental data, this work will
nd material parameters by tting the full experimental displacement eld directly.
A new approach is described which allows for any data point in the experimental
displacement eld be compared to a FE model, for any initial surface location and
ination pressure.
Jekel et al. (2016) considered the discrepancy in the x,yand zdisplacements
elds. When only the zdisplacements were utilized, there appeared to be multi
ple equivalent displacement solutions for a given material parameter set, possibility
hinting that the FEMU problem was undetermined. However, with the inclusion of
xand ydisplacements, there appeared to be a unique set of material parameters
to represent the entire displacement eld. While the previous work only considered
numerical results, this work now considers experimental data. The experiments show
that the outofplane deformations were an order of magnitudes larger than the in
plane displacements, which we believe this weighting may play an important role
when calibrating parameters to real experimental data.
This paper is organized into ve sections: 1) Introduction, 2) Methods, 3) Re
sults, 4) Discussion, and 5) Conclusion. The Methods section details the process
used to obtain material parameters from the bulge ination tests. Subsections are
used within the method section to draw attention to experimental test setup, FE
model, mathematical discrepancy between the experiment and FE model (objective
function), and optimization procedure. The results section is split into three parts
to separate the isotropic material parameters, orthotropic material parameters, and
the cross validation study. The discussion goes on to focus on the orthotropic results,
and conclusions are presented.
5
2 Methods
This section describes the FEMU method utilized in this research to understand
how the objective function aects resulting material parameters. The bulge ina
tion tests were performed using DIC on PVCcoated polyester.1This experiment
outputs fulleld displacement data (
∆
x,
∆
y,and
∆
z) from a at circular piece of
membrane material. A FE model was created replicating the boundary conditions
of the bulge ination tests. Three dierent objective functions are described. Each
objective function represents a dierent method to quantify the dierence between
the experimental data and FE model. One quantity considers only the height of
the bulge experiment, the other considers the fulldisplacement eld, and the last
considers a weighted variant of the fulldisplacement eld. Then the optimization
process to perform the inverse analysis is briey described, which results in linear
orthotropic material parameters.
2.1 Experimental tests
The bulge ination tests involve clamping a sample of membrane material into a
circular clamp. Pressure is then induced on one side of the material. The material
deections were recorded using DIC. The DIC system used was the StrainMaster
with DaVis (LaVision GmbH, 2014), which was capable of syncing the ination
pressure with the recorded images. A visual representation of a bulge ination test
is provided in Figure 1.
Four bulge ination tests were performed at Stellenbosch University in South
Africa. Details of the process and test xture are found in section 5.1 of (Jekel, 2016).
The diameter of the circular bulge test was 200 mm. The PVCcoated polyester tested
was Mehler Texnologies VALMEX®7318 (the same material tested in(Jekel et al.,
2017)). Four samples of material were cut from the same roll, with each sample being
a 250 mm square. Spray paint was added to the surface of each specimen, with a
random pattern, to increase the surface contrast of the material for DIC processing.
Each test was inated from zero to three bar, by manually opening a compressed air
valve to the bulge ination test xture. This resulted in each test being inated at
a unique load rate as seen in Figure 2. The internal pressure was recorded with a
Festo SPTEP10RS4V2.5K pressure transmitter.
Fulleld bulge ination tests generated a large amount of data as shown in
Table 1. The xyplane was oriented with the surface of the material prior to ination,
with the xdirection occurring in the warp material direction, while the ydirection
occurred in the ll material direction. The material was inated in the zdirection.
Here the largest test generated nearly two million data points, while the smallest
test generated only two hundred thousand data points. Each data point represents
a unique combination of ination pressure pand initial x,ylocation. There are three
deformations recorded for each data point, represented as separate
∆
x,
∆
y,and
∆
z
values.
The postprocessing parameters relating to the DIC technique were changed from
test to test which resulted in a dierent number of data points being captured on
1An online repository is available at https://github.com/cjekel/inv_bubble_opt which
includes the source code to perform the inverse analysis, test data, and procedures to reproduce
this work.
6
DIC
200 mm
Pressure Sensor Line Pressure
Test
Fixture
Inflated
Material
Fig. 1 Bulge ination test overview.
Table 1 Number of unique (x,y,p)data points from each test and ination pressures.
Test # of data points
1 1,836,961
2 729,718
3 1,201,509
4 289,312
the material specimen from test to test. The postprocessing parameters include a
window size, which is used to match the correlation between subsequent images.
The size of the window directly inuences the number of data points that occur on
the material specimen. This window size needed to be changed from test to test
because of variation in the random speckle pattern on the material. In an industrial
application this variation can be controlled through the use of tools (e.g. multiple
paint stencils) to produce a consistent pattern from test to test. A more consistent
speckle pattern would allow for the same postprocessing parameters, and a more
consistent number of data points on the surface of the material specimen.
The DIC techniques were not perfect, as there are missing data points in some
of the tests which is a common phenomenon with these techniques. This happens
when correlations are lost between images, and becomes more evident with larger
deections. The tested PVCcoated polyester had a glossy nish, which would cause
reections to move on the surface of the material as the material deformed outof
plane. These moving reections would cause the DIC technique to lose correlation of
patterns with subsequent images, resulting in ”holes” on the postprocessed images.
Additionally, the severity of missing data varies from test to test. Figure 3 shows
7
02468
Time, seconds
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Pressure, bar
Test 1
Test 2
Test 3
Test 4
Fig. 2 Pressure time curves from each bulge ination test.
each x,ydata point at the last ination pressure of each test, where each colored
pixel is a DIC data point. While Tests 1 and 2 have the highest density of data
points, Test 1 doesn’t have any missing data points while Test 2 has several small
holes on the surface of the material. The worst of the missing data occurs with Test
3, which has large holes on one quarter of the test. Test 4 only has two small holes
on the surface of the data, and also has the least number of x,ydata points on the
surface of the material.
For a recorded pressure, the experimental test data includes a matrix of values,
where each column represents either the initial position (x,y,z), or the deformation
in each direction (
∆
x,
∆
y,
∆
z). A single row of this matrix represents a single point
in three dimensional space on the surface of the material. The kth point was initially
at xk,yk, and zkprior to ination. Then for a given pressure, the point is physically
located at xk+
∆
xk,yk+
∆
yk, and zk+
∆
zkin the threedimensional space. Videos for
each test are included as supplementary material to help visualize the experimental
data. There is a video for each test showing the
∆
x,
∆
y,or
∆
zdeections of the
material. There is also a video showing the threedimensional shape of the material
([xk+
∆
xk,yk+
∆
yk,zk+
∆
zk]). Take note that the videos are not in a real physical
scale. The videos show the raw data points captured from the experimental test,
where each data point is used in the calibration process of the material.
8
(a) Test 1 (b) Test 2
(c) Test 3 (d) Test 4
Fig. 3 Plots of the x,ydata points of each bulge ination test. The darker color is related to
having a higher density of data points.
2.2 Finite Element model
An implicit nonlinear FE model was constructed in ABAQUS replicating the bound
ary conditions of the bulge ination test. The material is modeled as a membrane
material. Clamped boundary conditions are applied to the edge of the material. A
surface pressure is applied as a function of time to inate the membrane material.
A nonlinear model is needed to describe the geometric nonlinearity as Sheplak and
Dugundji (1998) demonstrated, with large deections the center load displacement
curve quickly transitions from linear to nonlinear as the midplane force quickly
grows with respect to geometry.
The FE model uses an implicit solver, with 201 load steps between zero and three
bar. Each load step uses adaptive time stepping. The adaptive time stepping allows
the model to solve a single loadcase in one step, or cut back to smaller increments
if needed. Over 900 linear Q4 membrane elements are used to represent the surface
of the material. The displacement eld at the nodes are exported to be compared to
the experimental tests for a candidate set of material parameters.
The displacement eld of the FE model needs to be available at various pressures
of the current experimental data and potential future data. Either load steps can
be added to the FE model at the exact experimental pressures, or the displacement
eld of the FE model can be interpolated at the observed experimental pressures.
9
Interpolation was chosen since it didn’t involve manually editing the input deck
for each experiment. Specically linear interpolation was used to evaluate the FE
model’s displacement at the nodes for the exact pressures of the experimental test.
The displacement of the FE model can be expressed as
∆
(x,y,p). The experimen
tal data records the displacements at the initial position x,yand pressure p. Linear
interpolation is used to solve the displacement eld
∆
(x,y,p)at the exact experimen
tal pressure pby interpolating the nodal FE solution at the two nearest computed
pressures as
∆
(x,y,p)−
∆
(x,y,p1)
p−p1
=
∆
(x,y,p2)−
∆
(x,y,p1)
p2−p1
(1)
where p2and p1represent the nearest pressures of the FE model’s load steps. The
advantages of this interpolation scheme is that both the FE deck and postprocessing
routines do not need to be altered in order to obtain displacements of the model for
any obscure pressure from future experiments.
Overall, the linear interpolation scheme proved very accurate when interpolating
between the FE model load steps. The linear interpolation accuracy was compared
to FE models with load steps halfway inbetween the previous 201 load steps (rep
resenting the worse possible interpolation case). The linear interpolation error was
negligible, with the interpolation error following on the order of single precision
(10−8mm) numerical noise. This level of noise was the same level of precision used
by the FE model. The maximum possible error from this interpolation scheme has
a hard bound as dened by Conte and Boor (1980). For any given p2and p1, the
maximum error from interpolation in deection is bounded by
Emax =(p2−p1)2
8max
∂
2
∆
∂
p2
.(2)
Linear isotropic and orthotropic material models were investigated. An isotropic
model with one unknown parameter (stiness modulus E), and an isotropic model
with two unknown parameters (stiness modulus Eand the shear modulus G) were
considered. The orthotropic model was simplied as a three parameter model, with
parameters for the stiness moduli (E1&E2), and the shear modulus (G12 ). The one
parameter isotropic model and the orthotropic model use a Poisson’s ratio of 0.24 as
measured in (Jekel et al., 2017). We believe that our experimental setup would require
through thickness measurements in order to accurately characterize the Poisson’s
ratio. There are additional compelexities While previous work of Jekel et al. (2017)
used nonlinear material models to characterize the PCVcoated polyester, this work
used linear material models which seemed sucient to match the loaddisplacement
response observed in the bulge experiments.
The displacement eld of the FE model for an orthotropic material model at
2.0 bar is shown in Figures 4 through 6. The maximum
∆
zvalue is about ten times
larger than the
∆
xor
∆
yvalues. Radial Basis Functions (RBF) are used to interpo
late the displacements from the initial (x,y) node locations of the FE model to be
evaluated at the experimental locations (Broomhead and Lowe, 1988). The RBFs are
exact at the node locations, and result in a smooth displacement eld from the linear
four node FE elements. The InterpolateSimpleRBF object construct these RBFs to
the full displacement eld of the FE analysis. The RBFs were inspired by the SciPy
rbf function (Virtanen et al., 2020).
10
100 75 50 25 0 25 50 75 100
x
(mm)
100
75
50
25
0
25
50
75
100
y
(mm)
0
4
8
12
16
20
24
28
32
z
(mm)
Fig. 4 Displacement
∆
zof FE model at 2.0 bar with orthotropic properties E1=0.8GPa,
E2=0.15 GPa, G12 =0.025 GPa, and
ν
12 =0.24.
2.3 Objective functions
The objective function is the mathematical quantity of interest which quanties the
discrepancy between the physical bulge ination tests and the FE model. An opti
mization procedure minimizes the objective function to nd the best set of material
parameters. A simple objective function can be constructed which only minimizes
the dierence in
∆
zdisplacement at the (x=0,y=0)center of the membrane as
ez(β) =
np
∑
i=1

∆
z(pi)t−
∆
z(pi,β)f(3)
where npis the number of ination pressures for from a given test. This is a L1norm
between the experimental heights and observed heights.
Quantifying the dierence between the fulleld experimental data is more com
plicated than only considering a single point. Let the average absolute deviation be
tween the xdisplacement of the FE model and ination test be denoted by r
∆
x(j,β)
for the jtest and βset of material parameters. The L1deviations for the displace
11
100 75 50 25 0 25 50 75 100
x
(mm)
100
75
50
25
0
25
50
75
100
y
(mm)
2.4
1.8
1.2
0.6
0.0
0.6
1.2
1.8
2.4
x
(mm)
Fig. 5 Displacement
∆
xof FE model at 2.0 bar with orthotropic properties E1=0.8GPa,
E2=0.15 GPa, G12 =0.025 GPa, and
ν
12 =0.24. Note the symmetry about x=0.
ment components are expressed as
r
∆
x(j,β) = 1
nj
nj
∑
i=1

∆
x(xi,yi,pi)t−
∆
x(xi,yi,pi,β)f(4)
r
∆
y(j,β) = 1
nj
nj
∑
i=1

∆
y(xi,yi,pi)t−
∆
y(xi,yi,pi,β)f(5)
r
∆
z(j,β) = 1
nj
nj
∑
i=1

∆
z(xi,yi,pi)t−
∆
z(xi,yi,pi,β)f(6)
where njis the total number of data points in the jtest. A simple discrepancy is
then formulated as the cumulation of the L1discrepancies in mm as
e(β) = 1
nt
nt
∑
j
r
∆
x(j,β) + r
∆
y(j,β) + r
∆
z(j,β)(7)
where ntis the total number of tests. The subscripts tis for the physical ination
test data, while the subscript fis from the FE model.
The formulation of econsiders the discrepancy in the
∆
x,
∆
y, and
∆
zdisplace
ments to be of equal weight. This formulation could be problematic if the discrep
ancy in one displacement components dominates the others. In the bulge ination
test data, the
∆
zcomponent was roughly ten times larger than the inplane (xor
y) components. This creates the potential for the r
∆
zdiscrepancies to be larger than
12
100 75 50 25 0 25 50 75 100
x
(mm)
100
75
50
25
0
25
50
75
100
y
(mm)
3.2
2.4
1.6
0.8
0.0
0.8
1.6
2.4
3.2
y
(mm)
Fig. 6 Displacement
∆
yof FE model at 2.0 bar with orthotropic properties E1=0.8GPa,
E2=0.15 GPa, G12 =0.025 GPa, and
ν
12 =0.24. Note the symmetry about y=0.
the other two directions, because the
∆
zvalues have the potential to be at least ten
times larger than
∆
xor
∆
y.
A second objective function is proposed as ewto deal with the imbalance between
the maximum inplane and outofplane displacements. The function is just a slight
modication of e, and is expressed as
ew(β) = 1
nt
nt
∑
j
r
∆
x(j,β) + r
∆
y(j,β) + wr
∆
z(j,β)(8)
where wis a weighting component2. While there can be many ways to select w, a
simple scheme was chosen as
wz=1
nt
nt
∑
j
max
∆
x(j)+max
∆
y(j)
2 max
∆
z(j)(9)
which represents the ratio of the average xand ydisplacement to the maximum z
displacement. This resulted in w=0.1for the bulge ination tests under consider
ation. This equation to weight displacement components dierently is similar to a
relative error formulation used by Yun et al. (2018), and also similar to normalization
schemes used in multiobjective optimizations as described by Tam (2020).
The optimization requires the objective function to be computed multiple times
in the FEMU procedure. There are many steps required to automate this using
2The software available online allows a weight to be specied for each directional component
of the displacement eld.
13
software which is described in Table 2. Several Python functions were created to
interface with the the ABAQUS solver and post processor for this application. The
function essentially returns the objective from an inputted candidate set of material
parameters.
Table 2 Process to compute the objective function for given material parameters.
Step Description
1 Write the material model parameters to the ABAQUS input le
2 Run ABAQUS solver on the input le
3 Run ABAQUS post processor to export displacement eld of FE model
4 Load the FE displacement eld into memory
5 Compute the discrepancy between FE model and DIC data:
i) Linearly interpolate the FE model to match the pressures of the bulge test data
ii) Construct and evaluate RBFs to the FE model displacement eld
iii) Compute either ezor r
∆
x,r
∆
y, and r
∆
zfor each set of test data
6 Compute the nal objective function of eor ew
2.4 Optimization
The inverse analysis is the process of nding the material parameters of the FE model
to match the bulge ination test data. The optimization problem can be stated as
minimize: e(β)(10)
subject to:
β
l≤
β
k≤
β
u,k=1,2,···,np.(11)
where βis the vector of material parameters which are restricted to some reasonable
lower and upper bounds, and e(β)represents some objective function describing the
discrepancy between experiment and numerical model. Bounds will depend upon the
material model, and in this case were selected to occur just beyond the observed range
in Jekel et al. (2017). In cases where there is no previous literature on the material
or material model, bounds can be selected based on the feasible domain of the FE
model. For this FE model, there is some lower bound to the stiness parameter which
will cause the FE model not to converge, however such upper bound does not exists.
The isotropic parameters are expressed as β= (E)or β= (E,G), and the simplied
orthotropic parameters are expressed as β= (E1,E2,G12).
A multistart optimization strategy was utilized in order to avoid the presence
of multiple local minima. A variant of the BFGS (Broyden, 1970; Fletcher, 1970;
Goldfarb, 1970; Shanno, 1970) gradient based optimization was used as the local
optimizer (Virtanen et al., 2020; Byrd et al., 1995). The multistart process ran
ve BFGS optimizations from dierent random starting points in the design space
(Schutte et al., 2006). Each of the ve BFGS runs were limited to either 200 function
evaluations, or satisfying the convergence criteria. The convergence considered either
relative changes in the objective function, absolute changes in the objective function,
or gradient magnitude.
There is the possibility that some combination of material parameters may cause
the FE analysis to not converge. As an exaggerated example, consider the case when
14
stiness moduli approach zero. In such a case the deformation becomes too large that
nite elements become poorly shaped and unable to converge. There were observed
instances in our optimizations when the optimizer would try a material parameter
combination that the FEM would fail to reach nal convergence. The boundary of
the feasible set of parameters that does not run into this type of FE issues is not
known a priori. To deal with this problem, the maximum objective value from the
runtime history was passed to the optimization algorithm when the FE analysis
failed to converge.
While there were a number of potential issues using gradient based optimization
with this application, in practice the gradient optimization was able to produce
objective functions lower than an initial random guess. Looking at the optimization
history, the FE analysis would only fail to converge during the line search stage
of the gradient based optimization algorithm, and not the nite dierences which
was used to approximate gradients. Lastly, there were no observed issues with the
optimization caused by the L1objective function, noting that the gradients won’t
exist if a zero objective value occurs.
3 Results
Inverse analyses were performed on the bulge ination tests to nd material param
eters for the linear orthotropic model. The focus of the results is to demonstrate
the eect of minimizing three dierent objective functions. These objective func
tions are summarized as the sum of fulleld displacement residuals (e), the sum
of weighted displacement fulleld residuals (ew), and the bulge height residual (ez)
which considered only a single point.
The simplied linear orthotropic material parameters resulting from inverse anal
ysis on the individual tests are shown in Table 3. There are dierent material param
eters depending upon which objective function was minimized. The most noticeable
parameter dierences can be seen in the primary E1and secondary E2stiness mod
uli. The most interesting changes occur in the second and third test, where the
choice of objective function reversed the stiness directions. Minimizing eor ezre
sulted in E2>E1, but minimizing ewresulted in E1>E2. Overall, minimizing ew
resulted in parameters that were more consistent from test to test as evident by
having the lowest range for the set of tests. As a reminder each test was conducted
on a nominally identical material specimen, and it was expected that there would be
minimum parameter dierences from test to test. While E1and E2stiness moduli
were very dierent depending on eor ew, the shear modulus was nearly the same in
all conditions.
The objective values resulting from each parameter set with the linear orthotropic
model are shown in Table 4. The table shows the resulting values of ewand ez
when ewas minimized, and vice versa. The denition of the worst t depends upon
which objective function was minimized. It is important to point out that for each
test, the minimum value of each objective only occurred when that objective was
minimized. If this was not the case, it would indicate that an optimization had failed
because a better optimum existed. The other important aspect of this table is that
it demonstrates that objective functions are fundamentally dierent. In other words,
nding the best ezdoes not yield the best eor ew, and vice versa.
15
Table 3 Resulting orthotropic material parameters from minimizing tests independently with
each inverse analysis. Note that
ν
12 was xed to 0.24.
Minimizing e(GPa) Minimizing ew(GPa) Minimizing ez(GPa)
E1E2G12 E1E2G12 E1E2G12
Test 1 0.343 0.248 0.005 0.303 0.229 0.005 0.343 0.246 0.005
Test 2 0.212 0.241 0.004 0.306 0.230 0.005 0.213 0.219 0.003
Test 3 0.217 0.257 0.004 0.306 0.229 0.005 0.206 0.209 0.003
Test 4 0.239 0.215 0.004 0.280 0.215 0.005 0.210 0.214 0.003
Mean 0.252 0.240 0.004 0.299 0.226 0.005 0.243 0.222 0.003
Range 0.131 0.042 0.001 0.026 0.015 0.000 0.137 0.037 0.002
Table 4 Objective values when minimizing e,ew, or ezfor each bulge ination test.
Minimizing e(GPa) Minimizing ew(GPa) Minimizing ez(GPa)
e eweze eweze ewez
Test 1 1.55 0.51 1.75 1.77 0.49 2.81 1.55 0.51 1.75
Test 2 1.66 0.77 1.03 1.92 0.71 2.25 1.78 0.83 0.86
Test 3 1.70 0.57 2.04 1.76 0.53 2.71 2.19 0.74 0.71
Test 4 1.03 0.41 0.82 1.10 0.38 1.27 1.11 0.45 0.76
The objective values were compared with a zero displacing FE model to draw
comparison at a poor design point. This can be thought of as an much stier ma
terial model (i.e. two orders of magnitude larger), and represents the worst possible
calibration for each test. Additionally, this could be used as the starting point for
an optimization algorithm, because it should always be possible to select a material
model that does not displace. The objective values are shown in Table 5, and can be
seen to be approximately four times worse than the best optimized objective values.
The dierent objective functions do not agree to which test has the best t to the
zero displacement model. The third test has the lowest eand ezvalues, while the
second test has the lowest ewvalue. Again, this demonstrates that objective functions
are fundamentally dierent.
Table 5 Objective values for a zero displacement numerical model (innitely sti material).
e ewez
Test 1 5.62 2.03 3.99
Test 2 5.59 1.53 4.51
Test 3 4.50 1.67 3.14
Test 4 5.75 1.74 4.76
Comparisons between the FE models and the test data are shown in Figure 7.
The gures show slices along the lines y=0and x=0at a selected ination pressure.
The mean squared error (mse) is shown in each gure, which can be used to quantify
which result best matches the experimental data. The results from other ination
pressures are qualitatively similar to the dierences in objective functions shown
here. Additionally, gures from all ination pressures as well as ination videos are
included in the supplementary material. In general the FE models had a tendency
to overestimate the deection at low pressures, but underestimate the deection at
high pressures.
It appears that the objective function ezmatches the apex of the bulge test better
than eor ew. The apex is where the largest
∆
zdeection occurs, and it appears that
16
eis much better at matching the apex than ew. While ewdoes a poor job matching
the deections at the apex, it potentially does a better job of matching the prole
(or shape) of the bulge ination test along the outer edges of the test (x ≥ 50 and
y ≥ 50). It can be noted that the results from eand ezare more similar, while the
results from ewappears to be the most dierent.
100 75 50 25 0 25 50 75 100
x
0
5
10
15
20
25
30
35
40
45
z
Test 1, 2.964 bar
test
e
, MSE: 0.48
ew
, MSE: 2.87
ez
, MSE: 0.49
100 75 50 25 0 25 50 75 100
y
0
5
10
15
20
25
30
35
40
45
z
test
e
, MSE: 3.78
ew
, MSE: 1.10
ez
, MSE: 3.71
100 75 50 25 0 25 50 75 100
x
0
5
10
15
20
25
30
35
40
45
z
Test 2, 2.347 bar
test
e
, MSE: 0.84
ew
, MSE: 5.65
ez
, MSE: 0.35
100 75 50 25 0 25 50 75 100
y
0
5
10
15
20
25
30
35
40
45
z
test
e
, MSE: 0.67
ew
, MSE: 2.60
ez
, MSE: 1.62
100 75 50 25 0 25 50 75 100
x
0
5
10
15
20
25
30
35
40
45
z
Test 3, 2.66 bar
test
e
, MSE: 1.52
ew
, MSE: 3.79
ez
, MSE: 3.59
100 75 50 25 0 25 50 75 100
y
0
5
10
15
20
25
30
35
40
45
z
test
e
, MSE: 3.13
ew
, MSE: 5.13
ez
, MSE: 6.68
100 75 50 25 0 25 50 75 100
x
0
5
10
15
20
25
30
35
40
45
z
Test 4, 2.506 bar
test
e
, MSE: 0.72
ew
, MSE: 1.45
ez
, MSE: 1.70
100 75 50 25 0 25 50 75 100
y
0
5
10
15
20
25
30
35
40
45
z
test
e
, MSE: 1.08
ew
, MSE: 3.35
ez
, MSE: 0.30
Fig. 7 Proles through the lines y=0and x=0comparing the results using each objective
function for the four tests.
The previous gures highlighted the
∆
zdisplacements. This is the most intuitive
displacement component, as
∆
zresembles the shape one sees while performing this
test. However,
∆
zis only onethird of the displacement components. Proles of
∆
x
and
∆
yare shown in Figure 8. The
∆
xdisplacement was plotted along the line
y=0, and the
∆
ydisplacement along the line x=0. These proles cut through
the interesting behavior of the maximum inplane deformations, which somewhat
resemble a sinusoidal wave. Again a lower mse means the result is a better match
to the experimental data. The same ination pressures form the previous gures are
highlighted.
It should not be a surprise to report that the ewobjective function matches the
∆
xand
∆
ydisplacements better than the other objective functions. Essentially ew
puts more importance on matching the inplane displacement components correctly,
and as a result the peaks of the
∆
xand
∆
ydisplacement are better represented with
ew. While the results for ezand eare similar, a slight edge can be given to eas it
appears to matches the
∆
xand
∆
yslightly better than ezthrough the shown ination
pressures.
17
100 75 50 25 0 25 50 75 100
x
5.00
3.89
2.78
1.67
0.56
0.56
1.67
2.78
3.89
5.00
x
Test 1, 2.964 bar
test
e
, MSE: 0.06
ew
, MSE: 0.06
ez
, MSE: 0.06
100 75 50 25 0 25 50 75 100
y
5.00
3.89
2.78
1.67
0.56
0.56
1.67
2.78
3.89
5.00
y
test
e
, MSE: 0.69
ew
, MSE: 0.39
ez
, MSE: 0.68
100 75 50 25 0 25 50 75 100
x
5.00
3.89
2.78
1.67
0.56
0.56
1.67
2.78
3.89
5.00
x
Test 2, 2.347 bar
test
e
, MSE: 0.57
ew
, MSE: 0.25
ez
, MSE: 0.75
100 75 50 25 0 25 50 75 100
y
5.00
3.89
2.78
1.67
0.56
0.56
1.67
2.78
3.89
5.00
y
test
e
, MSE: 0.37
ew
, MSE: 0.36
ez
, MSE: 0.43
100 75 50 25 0 25 50 75 100
x
5.00
3.89
2.78
1.67
0.56
0.56
1.67
2.78
3.89
5.00
x
Test 3, 2.66 bar
test
e
, MSE: 0.27
ew
, MSE: 0.07
ez
, MSE: 0.83
100 75 50 25 0 25 50 75 100
y
5.00
3.89
2.78
1.67
0.56
0.56
1.67
2.78
3.89
5.00
y
test
e
, MSE: 0.25
ew
, MSE: 0.17
ez
, MSE: 0.07
100 75 50 25 0 25 50 75 100
x
5.00
3.89
2.78
1.67
0.56
0.56
1.67
2.78
3.89
5.00
x
Test 4, 2.506 bar
test
e
, MSE: 0.21
ew
, MSE: 0.08
ez
, MSE: 0.39
100 75 50 25 0 25 50 75 100
y
5.00
3.89
2.78
1.67
0.56
0.56
1.67
2.78
3.89
5.00
y
test
e
, MSE: 0.14
ew
, MSE: 0.20
ez
, MSE: 0.11
Fig. 8 Proles of
∆
xand
∆
ythrough the lines y=0and x=0comparing the results using each
objective function.
4 Discussion
The results characterized the E1,E2, and G12 stiness moduli from a bulge ination
test. Fundamentally these material components all lie in the material plane, however
the bulge test is likely to produce larger outofplane than inplane displacements.
The dierent objective functions compare three cases: i) only matching the maximum
outofplane displacement (ez), ii) all displacement components equally (e), and iii)
a weighted balance between inplane and out of plane components (ew).
The most striking result was the eect of the objective functions on selecting
parameters for the linear orthotropic material model. When eor ezwas minimized
the linear orthotropic parameters varied signicantly from test to test, with some
tests resulting in E2>E1and others E1>E2. This was not the case when ewwas
minimized, in which the parameters from test to test were consistent with E1>E2.
We know that E1>E2given our understanding of the material’s construction and
manufacturer’s statements. Additionally, the results from ewbetter match quasi
static uniaxial tests for E2performed by Jekel et al. (2017). These tests show that
E1≈0.3and E2≈0.2at 0.1 strain, which agree well with the ewresults presented
here.
It’s important to clarify that ez,e, and ewwould all approach zero if it was pos
sible for a perfect t to the experimental data. Given that the model was incapable
of a perfect t, we obtained dierent material model parameters depending upon
objective function which each quantify the best t dierently. This demonstrates the
importance of fulleld data over single point data for a complicated loadcase, as
characterizing only the maximum deection point was unable to accurately charac
terize the E1and E2moduli. This work also demonstrates that work with full 3D
eld data may need to take special consideration of objective functions. Despite e
18
considering the fulleld data, it resulted in nearly the same parameters as ezwhich
only consider a single point, because of the dominance of the
∆
zdisplacements.
Test 2 and Test 3 are of particular interest with the linear orthotropic model,
because these tests resulted in E2>E1when ewas minimized. It could be reasonable
to assume that the test directions were incorrect by a factor of 90◦, and that this
mistake in the xand ydirections resulted in the set of parameters. Additional inverse
analyses were performed on these tests, where the test data was rotated by factors
of 45◦and 90◦. What is interesting is that when ewas minimized the resulting
parameters resulted in E2>E1regardless of the rotation, and there was little change
to the parameters. However, when ewwas minimized both E2and E1would change
signicantly based on the rotation. This hints that giving more weight to the xand
ydisplacements, like in ew, would be better if an inverse analyses was required to
identify orthotropic parameters without knowing the primary and secondary material
directions.
The linear orthotropic material in the FE model appears to be provide a reason
able representation of the test data. One of the benets of the formulation of the
eobjective function is that it’s value is an easily interpretable physical length (as
opposed to an L2norm). Here the largest value of eseen in table 4 was about 2 mm,
so the average deviation between of all models and experiments was a Manhattan
distance less than 2 mm.
5 Conclusion
An inverse analysis was described to nd material parameters by matching the full
displacement eld from bulge ination tests. Optimization was used to nd ma
terial parameters in a FE model that best matched the experimental displacement
eld from tests on PVCcoated polyester. Material parameters were determined for a
simplied linear orthotropic material model. Three dierent objective functions were
considered to describe the discrepancy between the experimental data and numerical
model. One objective function considered only a single point, another considered the
full 3D displacement eld, and the last objective function considered a weighted resid
ual of the full displacement eld. The weighting scheme was chosen to compensate
for the fact that the majority of the deections within the bulge ination test occur
outofplane, and thus weighted the outofplane deections equal to the inplane
deections. Resulting material parameters for the linear orthotropic material model
were very dierent depending on which objective function was minimized. Thus the
choice of objective function being considered is very important when performing such
an optimization on the fulleld data. The denitions of the simplied orthotropic
material parameters occur in the material plane. Matching only the maximum single
outofplane bulge deection, or matching the full 3D displacement components re
sulted in similar material parameters that were thought to be invalid. The resulting
inplane stiness moduli calibrated were potentially nonphysical. However, better
inplane stiness moduli were identied when the inplane deections were weighted
equal to the outofplane deections using the fulleld measurements. This hope
fully serves as an important lesson when working with fulleld data that has one or
two dominating components, as potentially the smaller value measurements may be
signicantly more important.
19
Acknowledgements Thanks to Sudharshan Udhayakumar for help constructing the FE model
and comparing the FE model to analytical solutions. Thanks to Andrés Bernardo for helping
set up the script to process the full displacement eld data. We are grateful to the anonymous
reviewers who have provided a number of comments that greatly improved the quality of this
manuscript.
Replication of results
The FE models require a commercial code to run. Experimental data, Python scripts
to perform the optimization, and FE model input decks are available online with
instructions at https://github.com/cjekel/inv_bubble_opt
Declarations
Funding
Charles F. Jekel has received the following funding for his PhD research which has
supported this work: University of Florida Graduate Preeminence Award, U.S. De
partment of Veterans Aairs Educational Assistance, and Stellenbosch University
Merrit Bursary.
Conicts of interest
The authors declare that they have no conict of interest.
Availability of data and material
The experimental data is available online with instructions on how to download at
https://github.com/cjekel/inv_bubble_opt
Code availability
The scientic Python ecosystem was used to produce these results. Specic scripts
used to perform the optimizations are available online at https://github.com/
cjekel/inv_bubble_opt
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