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Noname manuscript No.
(will be inserted by the editor)
Eect of Weighting Full-eld Residuals when Fitting Material
Model Parameters
Charles F. Jekel ·Martin P. Venter ·Gerhard
Venter ·Raphael T. Haftka
Received: date / Accepted: date
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 PVC-coated 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 replicated the full displace-
ment eld of the experimental test. Two dierent objective functions were consid-
ered to quantify the discrepancy between the FE model and test data. One function
considered equal weight among displacement components, while the other function
weighted the discrepancies to balance dierent displacement components. The result-
ing parameters, for isotropic and orthotropic models, were heavily dependent upon
which objective function was chosen. Additionally, cross validation was performed
to select between the two material models. Similar to the material parameters, the
cross validation results preferred dierent material models depending upon which
objective function was being used.
Keywords Inverse analysis ·Objective functions ·material parameter identication
1 Introduction
The nite element (FE) method has become an important design tool for membrane
structures, and selecting material parameters to represent complex material behavior
is a dicult task. This is especially true for the most complex and non-linear ma-
terials including coated woven fabrics. Inverse analyses, or iterative schemes for FE
model updating (FEMU) have been used to nd material parameters from complex
load cases. In this paper, two dierent objective functions were investigated when
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2 Charles F. Jekel et al.
estimating material parameters from bulge ination tests on PVC-coated 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).
PVC-coated polyester is a coated textile. It’s most commonly modeled as a con-
tinuous, homogeneous, orthotropic material (Shaw et al., 2010), which is largely
dependent on the warp-to-ll stiness ratio (Dinh et al., 2017). The typical weave
has the warp yarns pulled taut, while the ll yarns are woven in-between the warp
yarns. The ll yarns run orthogonal to the warp yarns. Various non-linear models
have been used in an attempt to better describe the behavior of the material. Galliot
and Luchsinger (2009) proposed a non-linear material model based on the load ratio
between the material warp and ll directions. Ambroziak and Kłosowski (2014) used
a piecewise linear orthotropic model, and Jekel et al. (2017) used polynomials to
describe the non-linear 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 (Rachik et al.,
2001; Charalambides et al., 2002; Machado et al., 2012; Tonge et al., 2013) have used
bulge (or bubble 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. The equal biaxial load occurs at the apex of the deected mem-
brane for an isotropic material. The measured pressure and displacements are then
used to infer material parameters, by comparing the results to a FE simulation.
A variety of applications (Becker et al., 2012; Murienne and Nguyen, 2016; Mejía
and Lantsoght, 2016; Chisari et al., 2017) have used Digital image correlation (DIC)
as non-invasive deformation measuring technique. The technique uses the correlation
between consecutive images, from multiple cameras, to calculate a full 3D displace-
ment eld. For bulge tests, DIC is an ideal tool to obtain a full 3D displacement eld
while not interfering with the ination or deection of the membrane material.
Material parameters in various models have been identied using FEMU (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 that 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
lead to responses that resemble the experimental response (Lee et al., 2019). This
process is often referred to as model calibration.
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 probalistically. Unfortunately, DIC displacement eld data is
typically treated as a black box where the uncertainties are not well understood. The
bulge ination tests is particularly challenging having out-of-plane 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
Eect of Weighting Full-eld Residuals when Fitting Material Model Parameters 3
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
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
multi-objective optimization. This process to compute the objective functions may
be computationally expensive with lots of full-eld data.
The previous studies (Rachik et al., 2001; Charalambides et al., 2002) used only
the height to determine material parameters from bulge ination tests. Machado
et al. (2012) used curvatures to determine a stress tensor from bulge ination 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 PVC-coated polyester. Jekel et al. (2016) showed that
it was possible to use the displacement eld of a bulge test to select parameters for
a non-linear 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. This work will nd material parameters by
tting the full experimental displacement eld directly. A new approach is describe
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. While (Jekel
et al., 2016) considered the discrepancy in the x,yand zdisplacements elds to be
equal weight, the objective function was dominated by the z displacements which
were an order of magnitudes larger than the xand ycomponents. Therefore, this
work explores the eect of weighting down the zdiscrepancy.
This work also investigates using cross validation to select the best material
model when performing FEMU. The idea is that cross validation could be used
to determine whether the isotropic or orthotropic material model generalizes the
PVC-coated polyester better in the complex load case. This is especially important
for PVC-coated polyester, which is not a homogeneous (or continuous) material,
but modeled as one in the FE method. Cross validation was performed using the
two dierent objective functions described to select material parameters from the
experimental data.
2 Methods
Bulge ination tests were performed using DIC on PVC-coated polyester.1A FE
model was created to replicate the boundary conditions of the bulge ination tests.
Two dierent objective functions are described. Each objective function represents
a dierent method to quantify the dierence between the experimental data and
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.
4 Charles F. Jekel et al.
DIC
200 mm
Pressure Sensor Line Pressure
Test
Fixture
Inflated
Material
Fig. 1 Bulge ination test overview.
FE model. The process to perform the inverse analysis is briey described. Cross
validation is also discussed as a method for selecting material models in this scheme.
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 PVC-coated 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 SPTE-P10R-S4-V-2.5K pressure transmitter.
Full eld bulge ination tests generated a large amount of data as shown in Ta-
ble 1. The xy-plane 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.
The number of data points ranged from test to test. The variation was largely de-
pendent on how nely the DIC data was processed. 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
∆
zvalues.
The DIC techniques were not perfect, as there are missing data points in some of
the test. This happens when correlations is lost between images, and becomes more
Eect of Weighting Full-eld Residuals when Fitting Material Model Parameters 5
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.
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
evident with larger deections. Additionally, the severity of missing data varies from
test to test. Figure 3 shows each x,ydata point at the last ination pressure of each
test, where each colored pixel is a DIC data point. The spacing of the x,ydata points
vary from test to test as a result of dierent DIC processing 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.
2.2 Finite Element model
An implicit non-linear FE model was constructed in ABAQUS which resembled the
physical boundary conditions of the bulge ination test. Sheplak and Dugundji (1998)
described dierential equations to solve for the displacement eld of bulge ination
tests for isotropic and orthotropic material models, but (Sheplak and Dugundji, 1998)
was not used as the geometry is simple enough to quickly construct in any non-linear
FE code. 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 load-case 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 linear elements prevent nonphysical out-of-plane
force imbalances. The displacement eld at the nodes of the FE solver are exported
to be used in further calculations when the model is run with a given material model.
The displacement eld of the FE model needs to be computed several times
at various pressures which correspond with the pressures of the experimental 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 to match the exact
6 Charles F. Jekel et al.
(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.
experimental pressures. Interpolation was chosen since it didn’t involve manually
editing the input deck for each test, and resulted in a xed number of exported
displacement elds. Specically linear interpolation was used to evaluate the FE
model’s displacement at the nodes for the exact pressures of the experimental test.
Linear interpolation is use to solve the displacement eld
∆
(x,y,p)by interpolating
the displacement eld at the two nearest 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 models’ load steps.
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 in-between 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 single precision was the same level of pre-
cision used by the FE model.
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
Eect of Weighting Full-eld Residuals when Fitting Material Model Parameters 7
considered. The orthotropic model was simplied as a three parameter mode, 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). Using a constant Poisson’s ratio simplies issues
with gradient magnitudes, as it is anticipated that gradients of Poisson’s ratio could
be orders of magnitude dierent than stiness moduli.
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 interpolate
the displacements from the initial (x,y) node locations at each outputted pressure.
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).
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.
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.
The radial basis functions are expressed as
y=Aλ (2)
8 Charles F. Jekel et al.
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.
where
A=
||x1−x1||2||x2−x1||2··· ||xn−x1||2
||x1−x2||2||x2−x2||2··· ||xn−x2||2
.
.
..
.
.....
.
.
||x1−xn||2||x2−xn||2··· ||xn−xn||2
(3)
which follows a simple linear kernel (Broomhead and Lowe, 1988). The RBF param-
eters λare solved for a general input of x, output of y, with ndata points. Then ˆn
predictions are generated for new ˆxlocations as
ˆy=
ˆ
Aλ (4)
where
ˆ
A=
||x1−ˆx1||2||x2−ˆx1||2· · · ||xn−ˆx1||2
||x1−ˆx2||2||x2−ˆx2||2· · · ||xn−ˆx2||2
.
.
..
.
.....
.
.
||x1−ˆxˆn||2||x2−ˆxˆn||2··· ||xn−ˆxˆn||2
.(5)
For this problem, xrefers to the in-plane (x,y) location of the FE model nodes, ˆx
refers to the in-plane (x,y) location of the DIC data, ydenotes the displacement from
the FE model, and ˆyrepresents the experimental displacements. This type of RBF
is ideal for interpolating surfaces of FE models since it is exact at node locations,
and results in a smooth interpolation in-between the FE node locations.
2.3 Objective functions
The objective function which quanties the discrepancy between the physical bulge
ination tests and the FE model dierence is fundamental for the optimization in the
inverse analysis. The average absolute deviation between the xdisplacement of the
FE model and ination test is denoted r
∆
x(j,β)for the jtest and βset of material
Eect of Weighting Full-eld Residuals when Fitting Material Model Parameters 9
parameters. These average absolute deviations are expressed as
r
∆
x(j,β) = 1
nj
nj
∑
i=1
|
∆
x(xi,yi,pi)t−
∆
x(xi,yi,pi,β)f|(6)
r
∆
y(j,β) = 1
nj
nj
∑
i=1
|
∆
y(xi,yi,pi)t−
∆
y(xi,yi,pi,β)f|(7)
r
∆
z(j,β) = 1
nj
nj
∑
i=1
|
∆
z(xi,yi,pi)t−
∆
z(xi,yi,pi,β)f|(8)
where njis the total number of data points in the jtest. A simple discrepancy is
then formulated as the average L1distance in mm as
e(β) = 1
nt
nt
∑
j
r
∆
x(j,β) + r
∆
y(j,β) + r
∆
z(j,β)(9)
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 zdirections to
have an equal weight. This formulation could be problematic if the discrepancy in one
displacement components dominates the others. In the bulge ination test data, the
∆
zcomponent was roughly ten times larger than the in-plane (xor y) components.
This creates the potential for the r
∆
zdiscrepancies to be larger than 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 in-plane and out-of-plane 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)(10)
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)(11)
which represents the ratio of the average xand ydisplacement to the maximum z
displacement. This resulted in w=0.1for the bulge ination tests in consideration.
Two dierent objective functions are presented to quantify the dierence between
the full displacement eld of the FE model and DIC tests. One objective function is
an average L1norm between the x,y,zdisplacements, while the other function is a
relative L1which considers the weighted dierence between x,yand zdisplacements.
A zero for both functions would indicate that the FE model’s displacement eld
exactly matches the experimental data.
The optimization requires the objective function eor ewto be computed multiple
times. There are many steps required to automate this using software, and the process
2The software available online allows a weight to be specied for each directional component
of the displacement eld.
10 Charles F. Jekel et al.
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 eor ewfrom inputted 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 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)(12)
subject to:
β
l≤
β
k≤
β
u,k=1,2,···,np.(13)
where βis the vector of material parameters which are restricted to some reasonable
lower and upper bounds. The isotropic parameters are expressed as β= (E)or β=
(E,G), and the simplied orthotropic parameters are expressed as β= (E1,E2,G12).
Note that ewis substituted for ewhen minimizing the weighted objective function.
The rst optimization strategy used a global optimizer with an allocated number
of function evaluations. When the global optimizer exhausted the specied budget,
a local optimizer was used for nal convergence to a local optimum. The global
optimization strategy used was Ecient Global Optimization (EGO), which utilized
the expected improvement from a Gaussian process to minimize the function (Jones
et al., 1998; The GPyOpt authors, 2016). 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). Initially 50 EGO
function evaluations (calculations of e) were performed before switching to the BFGS
implementation. A budget of 200 function evaluations for the BFGS appeared to be
sucient at nding a local optimum.
After several runs of the EGO to BFGS strategy, it became evident that the
optimizer was failing at nding a global optimum. A multi-start optimization strat-
egy was adopted to better deal with the presence of multiple local minima. The
multi-start process ran ve BFGS optimizations from dierent 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. Converge
considered either relative changes in the objective function, absolute changes in the
Eect of Weighting Full-eld Residuals when Fitting Material Model Parameters 11
objective function, or gradient magnitude. This multi-start optimization was able to
consistently nd better optimums than the EGO to BFGS strategy, but at the cost
of additional objective functions.
There is the possibility that some combination of material parameters may cause
the FE analysis to not converge. This is problematic when the optimization algo-
rithm requires a discrepancy for a particular set of parameters that lead to failed
convergence. To deal with this problem, the maximum objective value from the run-
time history was passed to the optimization algorithm when the FE analysis failed to
converge. Additionally, a discrepancy value of 30 mm was passed if the rst function
evaluation in a given run failed to converge. This strategy works well with EGO,
however it creates a non-dierential objective function which can be problematic for
gradient based optimization algorithms. There are also problems with the use of a
L1based objective function, where gradients can vanish when the residual of a single
points goes to zero.
While there were a number of potential issues using gradient based optimization
with this application, in practice the gradient optimization was able to successfully
minimize the objective function. Looking at the optimization history, the FE analysis
would only fail to converge during the line search stage of the gradient based opti-
mization algorithm, and not the nite dierences which approximate the gradients.
This is less problematic because the gradients were accurate. Lastly, there were no
observed issues with the optimization caused by the L1objective function.
2.5 Cross validation
Cross validation is a model selection or validation tool used in various regression
problems to assess the quality of models (Queipo et al., 2005). Cross validation
provides for a nearly unbiased estimate of the modeling error, and can be used to
diagnose overtting or bias errors. Cross validation can be used to compare the
performance of one material model to another in the context of tting material
models with an inverse analysis (FEMU). This may be important in practice when
the ideal material model is unknown. In this case, cross validation will be used to
quantitatively compare how the linear isotropic and orthotropic material models
represent the behavior of PVC-coated polyester from these bulge ination tests.
The processes proposed is similar to leave-one-out cross validation, and is de-
scribed in Table 3. This cross validation score was computed for both the linear
isotropic and orthotropic material models. The model with the lower cross validation
score is assumed to be a better generalized representation of the material behavior.
Table 3 Process to compute leave-one-test-out cross validation error.
Step Description
1 Perform optimization without test j
2 Calculate the discrepancy eor ewon the left-out test j
3 Repeat 1 & 2 for all tests
4 Cross validation score is the average discrepancy eor ewfrom the left-out tests
12 Charles F. Jekel et al.
3 Results
Inverse analyses were performed on the bulge ination tests to nd material pa-
rameters for the linear isotropic and linear orthotropic models. The results show
parameters when each test were t separately (such that nt=1). Lastly, the cross
validation errors were computed by tting parameters to all combinations of the
three tests (nt=3). The focus of the results is to demonstrate the eect of the two
objective functions.
3.1 Linear isotropic material model
Resulting parameters for the single parameter (E) linear isotropic material model are
found in Table 4, and results for the two parameter model (Eand G) are shown in
Table 5. The one parameter model used a Poisson’s ratio of 0.24, while optimization of
the two parameter resulted in a Poisson’s ratio near 0.5. Eectively the shear modulus
in the two parameter model approached the lower limit3, and the FE model is unable
to run when the Poisson ratio exceed 0.5. Poisson’s ratio appears to signicantly
eect the stiness modulus E, in which the one parameter model resulted in a larger
stiness modulus. For both models there was little dierence between parameters
from minimizing eor ewwith the rst test. The second and third test produced
larger stiness moduli when minimizing ew, and the degree of the increase was larger
with the one parameter model. In general, it appeared that minimizing ewresulted in
stier material parameters. Parameters of the two parameter isotropic model appears
less eected by the weighted objective function than the single objective function.
Table 4 One parameter isotropic material results from each inverse analysis. Note
ν
was xed
to 0.24.
Minimizing eMinimizing ew
E(GPa) E(GPa)
Test 1 0.279 0.283
Test 2 0.222 0.292
Test 3 0.242 0.282
Test 4 0.218 0.253
Table 5 Resulting isotropic material parameters from each inverse analysis. Note
ν
is calculated
from Eand G.
Minimizing eMinimizing ew
E(GPa) G(GPa)
ν
E(GPa) G(GPa)
ν
Test 1 0.279 0.113 0.24 0.283 0.114 0.24
Test 2 0.160 0.054 0.48 0.170 0.057 0.48
Test 3 0.162 0.054 0.50 0.170 0.057 0.49
Test 4 0.154 0.052 0.48 0.155 0.052 0.49
The objective values from the various ts are shown in Table 6. For both objective
functions, the fourth test resulted in the smallest objective values (or the best t).
3The lower limit of Gis E/3, and Poisson’s ratio is expressed as
ν
=E/(2G)−1.
Eect of Weighting Full-eld Residuals when Fitting Material Model Parameters 13
The largest objective values (or the worst t test) depends upon which objective
function was minimized, but appears to be either the second or third test. In most
cases, the two parameter model resulted in lower objective function values than the
one parameter model.
Table 6 Resulting objective values when tting the linear isotropic models to each bulge in-
ation test.
One parameter (E) Two parameter (E,G)
e(mm) ewe(mm) ew
Test 1 1.554 0.497 1.554 0.497
Test 2 1.881 0.795 1.699 0.725
Test 3 1.870 0.588 1.716 0.549
Test 4 1.195 0.457 1.054 0.403
3.2 Linear orthotropic material model
The simplied linear orthotropic material parameters resulting from inverse analysis
on the individual tests are shown in Table 5. The parameters of the orthotropic model
are very dierent depending upon which objective function was minimized. There are
sizable changes to both E1and E2depending upon whether eor ewwas minimized.
The most interesting changes occur in the second and third test, where the choice of
objective function reversed the stiness directions. Minimizing eresulted in E2>E1,
but minimizing ewresulted in E1>E2. Overall, minimizing ewresulted in parameters
that were more consistent from test to test which is expected since the test material
was nominally identical. While the stiness moduli were very dierent depending on
eor ew, the shear modulus was nearly the same in all conditions.
Table 7 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)
E1E2G12 E1E2G12
Test 1 0.343 0.248 0.005 0.303 0.229 0.005
Test 2 0.212 0.241 0.004 0.306 0.230 0.005
Test 3 0.217 0.257 0.004 0.306 0.229 0.005
Test 4 0.239 0.215 0.004 0.280 0.215 0.005
The objective values resulting from each parameter set with the linear orthotropic
model are shown in Table 8. The table shows the resulting values of ewwhen e
was minimized, and vice versa. The denition of the worse t depends upon which
objective function was minimized. In all cases, the value of ewwas worse when e
was minimized than when ewwas minimized. A similar statement can be made
for minimizing ew. If we consider test two, the objective values when minimizing
echanged up to 15% when minimizing ewwhile E1changed over 40% when the
objective function was changed.
Comparisons of the dierences between the two parameter isotropic material
model and the linear orthotropic material model are shown in Figures 7 through
14 Charles F. Jekel et al.
Table 8 Objective values when minimizing eor ewfor the linear orthotropic model to each
bulge ination test.
Minimizing e(GPa) Minimizing ew(GPa)
e(mm) ewe(mm) ew
Test 1 1.550 0.512 1.773 0.490
Test 2 1.660 0.765 1.917 0.710
Test 3 1.702 0.570 1.759 0.527
Test 4 1.033 0.406 1.101 0.380
10. The chosen displacement locations occur at the approximate maximums for the
linear orthotropic FE model, as previously shown in Figures 4 through 6. The
∆
x
displacements occur at [x=56,y=0], the
∆
ydisplacements occur at [x=0,y=63],
and the
∆
zdisplacements occur at [x=0,y=0]on the surface of the material. The
displacements were plotted with the ination pressure. The results of the one param-
eter isotropic material model have been omitted, because they were nearly identical
to the two parameter isotropic results with the same objective function.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Pressure, bar
0
1
2
3
4
x
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(a)
∆
xdisplacement
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Pressure, bar
0
1
2
3
4
5
6
y
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(b)
∆
ydisplacement
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Pressure, bar
5
10
15
20
25
30
35
40
z
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(c)
∆
zdisplacement
Fig. 7 Resulting displacements from the two parameter isotropic and the linear orthotropic
material models compared with test 1. The
∆
xresults shown in A) occur at [x=56,y=0]. The
∆
yresults shown in B) occur at [x=0,y=63]. The
∆
zresults shown in C) occur at [x=0,y=0].
The linear orthotropic FE model appears to match the selected test data better
than the isotropic model in all of the cases presented in Figures 7 through 10. Al-
though it is unclear whether the choice of eor ewas the objective function resulted
in a better ts. It appears that the linear orthotropic model with ewas the objective
function matched the
∆
xand
∆
ydisplacements better, while the eobjective function
matched the
∆
zdisplacements better. There are exceptions to both of these cases,
where the reverse is seen, depending on which test is considered.
The dierences between eand ewfor the two parameter isotropic model were
very subtle. However, there is a more noticeable dierence between eand ewfor
the linear orthotropic material model. This is most evident in the maximum
∆
z
displacement pressure curves for all tests. The non-linearity of the
∆
zdisplacements
is more prevalent than the
∆
xor
∆
ydisplacements.
3.3 Cross validation material model comparison
Inverse analyses were performed to t isotropic and orthotropic material models
to the bulge ination tests. Additional inverse analyses were performed such that
Eect of Weighting Full-eld Residuals when Fitting Material Model Parameters 15
0.0 0.5 1.0 1.5 2.0
Pressure, bar
0.5
1.0
1.5
2.0
2.5
3.0
3.5
x
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(a)
∆
xdisplacement
0.0 0.5 1.0 1.5 2.0
Pressure, bar
0.5
1.0
1.5
2.0
2.5
3.0
3.5
y
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(b)
∆
ydisplacement
0.0 0.5 1.0 1.5 2.0
Pressure, bar
10
15
20
25
30
35
40
z
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(c)
∆
zdisplacement
Fig. 8 Resulting displacements from the two parameter isotropic and the linear orthotropic
material models compared with test 2. The
∆
xresults shown in A) occur at [x=56,y=0]. The
∆
yresults shown in B) occur at [x=0,y=63]. The
∆
zresults shown in C) occur at [x=0,y=0].
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Pressure, bar
0
1
2
3
4
x
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(a)
∆
xdisplacement
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Pressure, bar
0
1
2
3
4
5
y
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(b)
∆
ydisplacement
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Pressure, bar
5
10
15
20
25
30
35
40
45
z
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(c)
∆
zdisplacement
Fig. 9 Resulting displacements from the two parameter isotropic and the linear orthotropic
material models compared with test 3. The
∆
xresults shown in A) occur at [x=56,y=0]. The
∆
yresults shown in B) occur at [x=0,y=63]. The
∆
zresults shown in C) occur at [x=0,y=0].
0.0 0.5 1.0 1.5 2.0 2.5
Pressure, bar
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
x
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(a)
∆
xdisplacement
0.0 0.5 1.0 1.5 2.0 2.5
Pressure, bar
0
1
2
3
4
5
y
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(b)
∆
ydisplacement
0.0 0.5 1.0 1.5 2.0 2.5
Pressure, bar
5
10
15
20
25
30
35
40
z
Displacement, mm
Two Iso. Obj:
e
Two Iso. Obj:
ew
Lin. Ortho. Obj:
e
Lin. Ortho. Obj:
ew
Test data
(c)
∆
zdisplacement
Fig. 10 Resulting displacements from the two parameter isotropic and the linear orthotropic
material models compared with test 4. The
∆
xresults shown in A) occur at [x=56,y=0]. The
∆
yresults shown in B) occur at [x=0,y=63]. The
∆
zresults shown in C) occur at [x=0,y=0].
a leave-one-test-out cross validation score was computed for each material model.
The resulting discrepancy values are presented in Table 9. The orthotropic material
model had the lowest cross validation error when ewwas minimized, but the two
parameter isotropic material model had the lowest cross validation error when e
was minimized. The dierent objective functions appear to prefer dierent material
models. This implies that the material model which generalizes the bulge ination
test better depends on the denition of the discrepancy between the test data and
FE model.
The two parameter isotropic material model parameters from the full inverse
analysis and the cross validation runs are presented in Table 10. When ewas min-
16 Charles F. Jekel et al.
Table 9 Resulting discrepancy from the inverse analysis and leave-one-test-out cross validation.
Cross validation error Left out test values
Model e(mm) ewe[1, 2, 3, 4] ew[1, 2, 3, 4]
isotropic (E) 1.995 0.590 [2.47, 2.10, 1.87, 1.54] [0.62, 0.52, 0.59, 0.63]
isotropic (Eand G) 1.833 0.554 [2.37, 1.85, 1.72, 1.40] [0.57, 0.50, 0.56, 0.60]
orthotropic 1.856 0.533 [2.47, 1.88, 1.73, 1.35] [0.49, 0.72, 0.53, 0.40]
imized, the stiness modulus varied from 0.155 to 0.170 GPa. However, when ew
was minimized the stiness modulus varied from 0.170 to 0.193 GPa. and the shear
modulus varied from 0.057 to 0.070 GPa. Overall, ewresulted in more consistent and
stier moduli.
Table 10 Resulting two parameter isotropic material parameters from each inverse analysis.
Minimizing e(GPa) Minimizing ew(GPa)
E G E G
Leaving test 1 out 0.155 0.052 0.170 0.057
Leaving test 2 out 0.167 0.056 0.170 0.057
Leaving test 3 out 0.164 0.055 0.170 0.056
Leaving test 4 out 0.170 0.057 0.193 0.066
The orthotropic material parameters from the cross validation study are shown
in Table 11. There is a signicant dierence between the parameter variance depend-
ing upon which objective function was used. For instance, E1varied from 0.224 to
0.280 GPa when ewas minimized. However, when ewwas minimized E1varied from
0.303 to 0.305 GPa. A similar trend occurs for E2and G12, where minimizing ew
resulted in more consistent parameters.
Table 11 Resulting orthotropic material parameters from each inverse analysis.
Minimizing e(GPa) Minimizing ew(GPa)
E1E2G12 E1E2G12
Leaving test 1 out 0.224 0.235 0.003 0.303 0.229 0.005
Leaving test 2 out 0.297 0.231 0.004 0.304 0.229 0.005
Leaving test 3 out 0.284 0.224 0.004 0.304 0.229 0.005
Leaving test 4 out 0.280 0.244 0.003 0.305 0.229 0.005
4 Discussion
The most striking result was the eect of the two objective functions on selecting
parameters for the linear orthotropic material model. When ewas 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 mini-
mized, in which the parameters from test to test were fairly consistent with E1>E2.
The dierence between the two objective functions was that econsidered the dis-
crepancies in the x,yand zdirections to be of equal weight, while ewconsidered the
Eect of Weighting Full-eld Residuals when Fitting Material Model Parameters 17
zdiscrepancies to be one tenth the weight. This weighting factor corresponds to an
imbalance between the test maximum zdisplacement being about ten times larger
than the maximum xor ydisplacements. It’s important to clarify that both eand
ewwould approach zero if it was possible for a perfect t, however resulting material
parameters occur with the inability to perfectly t the data. In these circumstances
it appears that one may need to careful consider how to eventuate the discrepancy
of the displacement eld, especially under similar imbalanced displacement data.
Tests two and three 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 dierent objective functions had an interesting eect on using leave-one-
test-out cross validation for material model selection. When ewas minimized, the
cross validation error favored the two parameter isotropic material model. Though
when ewwas minimized, the cross validation error favored the orthotropic material
model by a much larger margin. This reemphasizes the importance of selecting an
appropriate objective function, as the choice of objective function not only eects
the resulting parameters, but also eects the perceived generalization error of the
model.
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 material
parameters in a FE model that best matched the experimental displacement eld
from tests on PVC-coated polyester. Material parameters were determined for a lin-
ear isotropic and simplied linear orthotropic material models. Two dierent objec-
tive functions were considered to describe the discrepancy between the experimental
data and numerical model. The rst objective function considered equal weight be-
tween the displacement components, while the other function gave more weight to
the xand ydisplacements. The weighting scheme was chosen to compensate for the
fact that the majority of the deections within the bulge ination test occur out-of-
plane. 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 full eld data.
Cross validation was performed to determine whether the isotropic or orthotropic
material model was a better representation of the material behavior. There was little
dierence in the cross validation error according to the equally weighted objective
function, however the weighted objective function heavily favored the orthotropic
18 Charles F. Jekel et al.
material model. This indicates that the choice of objective function was not only
important in material parameter selection, but can also impact material model se-
lection.
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 script to process the full displacement eld data.
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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