On-line uncertainty estimation in composites manufacturing
K. I. Tifkitsis1, A. A. Skordos1
1
School of Aerospace, Transport and Manufacturing, Cranfield University, Bedford,
MK43 0AL, UK
Corresponding author’s email: [email protected]
Abstract. This paper addresses the integration of composites process monitoring signals and
stochastic simulation into an inverse heat transfer scheme for the estimation of thermal properties
and boundary conditions and their uncertainty. The Kriging method was implemented to
construct a computationally efficient surrogate model of the cure process based on Finite
Element (FE) modelling results. Process monitoring experiments were carried out in order to
measure the temperature evolution during the process of composites cure of a glass fibre
reinforced epoxy composite part. The scheme developed is based on Markov Chain Monte Carlo
(MCMC) and uses the surface heat transfer coefficient and thermal conductivity as the unknown
stochastic variables. The resulting distribution of surface heat transfer coefficient of the inverse
procedure is compared with statistical properties of experimental measurements in order to
examine the contribution of process monitoring in narrowing down the variability of boundary
condition estimation. The thermal conductivity estimation is used to develop a constitutive
model which is compared to existing models. The utilisation of an inverse scheme with real time
monitoring signals driving the estimation of process parameters during composites
manufacturing can result in on-line estimation of the probability of the formation processinduced defects.
1. Introduction
Composites manufacturing involves several stages such as lay-up, draping, resin impregnation and
curing. The cure process constitutes a non-linear heat transfer problem where the thermosetting resin
reacts and is transformed from an oligomeric liquid to a glassy solid. The quality of the final part is
mainly dependent on process parameters and boundary conditions. Several sources of uncertainty affect
process time and product quality [1]. Boundary conditions such as tool temperature and heat transfer
coefficient present significant variations during cure resulting in uncertainty of part quality and the
possibility of defects formation during the process [2]. Also, thermal properties such as resin thermal
conductivity are difficult to entirely define due to experimental scatter [3].
Inverse procedures have been developed to address heat transfer problems based on Bayesian
inference for the estimation of thermal properties and boundary conditions [4–6]. Markov Chain Monte
Carlo (MCMC) method is based on Bayes’ theorem calculating the posterior probability distribution
which is proportional to product likelihood distribution and prior probability distribution [7]. MCMC
can compute the uncertainty associated with the parameter estimation by incorporating process
monitoring measurements [8–10] and modelling into the inverse scheme.
In the present paper an inverse heat transfer scheme is developed incorporating process monitoring
signals and modelling for the estimation of resin thermal conductivity and heat transfer coefficient
variability during the cure of a flat composite part. A surrogate model is used based on Kriging
substituting the FE model and minimising the computational time of the inverse solution. The inversion
procedure is applied in the case of Resin Transfer Moulding (RTM) of a glass fibre reinforced epoxy
composites.
2. Methodology
2.1. Processing
This study focuses on the curing of a glass fibre reinforced composite in a RTM process. In this process,
resin impregnates a dry preform placed in a sealed rigid mould under pressure and the curing occurs
with further heating of the mould. An RTM facility, which is shown in Figure 1, has been utilised for
the experimental measurements. The dimensions of the mould cavity are 800x340x3 mm. The sides of
the cavity were sealed using silicone rubber while the tool was closed using a glass plate and a set of
stiffeners. Heating is achieved by an array of heating elements placed under the mould cavity. The
specific experimental configuration was selected in order to reduce the heat transfer problem to onedimension. An E-TX1769 (BTI Europe) E-glass fabric with a surface density of 1769 g/m2 and sequence
of +45/-45/0 has been used. The resin matrix was Hexcel RTM6 epoxy resin. Two layers of this material
(total sequence +45/-45/0/0/-45/+45) were utilised in order to achieve a fibre weight fraction of 65%
and thickness of 3mm. Resin filling was carried out at 120 °C. After filling completion, heating up at
1.5 °C/min was applied up to 160 °C, and the temperature was kept constant. Three K-type
thermocouples were placed at the bottom, at the mid-thickness and on the top layer of the curing
component.
2.2. Cure model
The only heat transfer mechanism in cure process is heat conduction, since forced convection plays a
negligible role. The governing energy balance is:
𝜌𝑐𝑝
𝜕𝑇
𝑑𝑎
= 𝛻 ∙ 𝑲𝛻𝑇 + (1 − 𝑣𝑓 )𝜌𝑟 𝐻𝑡𝑜𝑡
𝜕𝑡
𝑑𝑡
(1)
where 𝜌 is the density of the composite, 𝑐𝑝 is the specific heat capacity, 𝑇 the temperature, 𝑲 the thermal
conductivity tensor, 𝑣𝑓 the fibre volume fraction, 𝜌𝑟 the resin density, 𝐻𝑡𝑜𝑡 the total heat of the curing
reaction, and 𝑎 the degree of cure reaction. The second term of the right side of the Eq. (1) expresses the
heat generated due to the exothermic resin reaction. More details of the heat transfer problem can be
found in [10].
A thermal cure simulation model was implement in the finite element solver MSC.Marc, to simulate
the curing stage of RTM process. Figure 2 depicts a schematic representation of the model geometry.
The model comprises two parts; a composite flat laminate and the glass top plate. The composite part
comprises 6 3D iso- parametric eight-node composite brick elements 175 MSC.Marc element type for
thermal analysis [11].
Figure 1 The resin transfer moulding tool.
Air convection
boundary condition
Glass plate
Part
Insulator
Insulator
Prescribed temperature
boundary condition
Tool
Figure 2 Schematic representation of the model.
Each element represents one ply of E-glass with 0.5 mm nominal thickness. Consequently, the overall
thickness of the flat laminate is 3 mm. The boundary conditions illustrated in Figure 2 are applied using
user subroutines FORCDT and UFILM for time dependent prescribed temperature and natural air
convection respectively [12]. The predefined profile consisted of a ramp up at 1.5 °C/min from 130 °C
to 160 °C and then dwell at 160 °C for 90 min. Due to the symmetry across the width of the laminate,
the heat transfer model was solved as a 1D problem. The initial condition was considered to be 2%
degree of cure and uniform temperature after the end of filling. The specific heat capacity of the top
glass plate is 0.84 J/g/°C, the thermal conductivity 0.78 W/m/°C, the density 2.7 g/cm3 and the nominal
heat transfer coefficient 8.5 W/m2/°C [10]. The boundary conditions such as heat transfer coefficient
can show considerable variability which can induce significant variation in process outcome [2].
Therefore surface heat transfer coefficient of top glass plate ℎ has been considered as unknown
parameter in the inverse problem.
User subroutines UCURE, USPCHT, and ANKOND were used for the integration of material submodels, cure reaction kinetics, specific heat capacity and thermal conductivity respectively [12]. The
cure kinetics model is a combination of an nth order model an autocatalytic model [13]. The cure reaction
rate in the cure kinetic models is calculated as follows:
𝑑𝑎
(2)
= 𝑘1 (1 − 𝑎)𝑛1 + 𝑘2 (1 − 𝑎)𝑛2 𝑎𝑚
𝑑𝑡
where 𝑎 is the current degree of cure, 𝑚, 𝑛1 , 𝑛2 the reaction orders, 𝑘1 and 𝑘2 the reaction rate constants
following an Arrhenius law:
(3)
𝑘1 = 𝐴1 exp(−𝐸1 /𝑅𝑇)
(4)
𝑘2 = 𝐴2 exp(−𝐸2 /𝑅𝑇)
where 𝐴1 , 𝐴2 denote pre-exponential factors, 𝐸 the activation energy for chemical reaction and 𝑅 the
universal gas constant. Model constants values are reported in [14]. The specific heat capacity of the
resin and the fibre is calculated in a similar way using experimental data obtained by modulated
differential scanning calorimetry [10]. The specific heat capacity of the composite is computed making
use of rule of mixtures as follows:
(5)
𝑐𝑝 = 𝑤𝑓 𝑐𝑝𝑓 + (1 − 𝑤𝑓 )𝑐𝑝𝑟
where 𝑤𝑓 is the fibre weight fraction, 𝑐𝑝𝑓 the fibre specific heat capacity and 𝑐𝑝𝑟 the specific heat
capacity of the resin. The thermal conductivity of the anisotropic composite material in the longitudinal
direction is computed using an appropriate geometry-based model [15] and can be expressed as follows:
(6)
𝐾11 = 𝑣𝑓 𝐾𝑙𝑓 + (1 − 𝑣𝑓 )𝐾𝑟
where 𝑣𝑓 is the volume fraction of the fibre, 𝐾𝑙𝑓 and 𝐾𝑟 are the thermal conductivity of the fibre in the
longitudinal direction and of the resin, respectively. In the transverse direction the thermal conductivity
is calculated as follows
2
𝐾𝑡𝑓
( 𝐾 + 1)
𝐾𝑡𝑓
𝐾𝑡𝑓
1 𝐾𝑡𝑓
𝑟
𝐾22 = 𝐾33 = 𝑣𝑓 𝐾𝑟 (
− 1) + 𝐾𝑟 ( −
) + 𝐾𝑟 (
1) √𝑣𝑓 2 − 𝑣𝑓 +
2
𝐾𝑟
2 2𝐾𝑟
𝐾𝑟
2𝐾𝑡𝑓
( 𝐾 − 2)
𝑟
(7)
where 𝐾𝑡𝑓 is the thermal conductivity of the fibre in the transverse direction. The relationship between
thermal conductivity of the resin, temperature, and degree of cure can be expressed as follows:
(8)
𝐾𝑟 = 𝑘5 𝑇𝑎2 − 𝑘4 𝑇𝑎 − 𝑘3 𝑇 − 𝑘2 𝑎2 + 𝑘1 𝑎 + 𝑘
The thermal conductivity values for the resin were obtained experimentally making use of a transient
technique that measures the thermal conductivity of the resin during cure [3]. However, the intercept
𝑘 from Eq. (8) is difficult to be estimated due to the noise of the experimental data. Details corresponding
in Eq. (2)-(8) such as sub-models constants are presented in [16].
2.3. Surrogate model
Cure process simulation using non-linear FE analysis requires high computational time. Therefore in
inversion procedures such as the MCMC algorithm, the use of FE models is computational cumbersome.
A surrogate model was used to address this by replacing the FE model. Figure 3 presents the procedure
that has been followed. The construction of the surrogate model requires as input a set of design points
and their responses generated using FE analysis. Latin Hypercube Sampling (LHS) [17] , a random
sample generation method, was selected for generating a sample of 2000 input values and their
responses. The inputs of the surrogate model and their ranges are; the thermal conductivity level (0.050.2 W/m/°C) of the resin, the surface heat transfer coefficient of the glass top plate (2-14 W/m2/°C), and
the cure time (0-110min). The outputs of the surrogate model are the temperature at mid-thickness
(𝑇𝑚𝑖𝑑 ) and on the top (𝑇𝑡𝑜𝑝 ) of the composite component.
A Kriging, was implemented to build the surrogate model. Kriging enables a prediction of untried
sample values to be made without bias with minimum variance and more accurately than low-order
polynomial regression models [18].
Given a set of 𝑚 design sites
(9)
𝑆 = [𝑠1 𝑠2 ⋯ 𝑠𝑚 ]Τ with 𝑠𝑖 ∈ ℝ𝑛
and responses
(10)
𝑌 = [𝑦1 𝑦2 ⋯ 𝑦𝑚 ]Τ with 𝑦𝑖 ∈ ℝ𝑞
the data is assumed to satisfy the normalisation conditions
(11)
𝜇[𝑆:,𝑗 ] = 0,
𝑉[𝑆:,𝑗 , 𝑆:,𝑗 ] = 1, 𝑗 = 1, . . . , 𝑛
(12)
𝜇[𝑌:,𝑗 ] = 0,
𝑉[𝑌:,𝑗 , 𝑌:,𝑗 ] = 1, 𝑗 = 1, . . . , 𝑞
where 𝜇[∙] and 𝑉[∙,∙] denote the mean and the covariance respectively.
The Kriging model treats the deterministic response vector 𝑦(𝑥) ∈ ℝ𝑞 , for a 𝑛 dimensional input
𝑥 ∈ 𝒟 ⊆ ℝ𝑛 as a realisation of a regression model ℱ and a random field,
(13)
𝑦̂𝑙 (𝑥) = ℱ(𝛽:,𝑙 , 𝑥) + 𝑧𝑙 (𝑥), 𝑙 = 1, . . . , 𝑞
The regression model ℱ is a linear combination of 𝑝 chosen functions 𝑓𝑗 (𝑥): ℝ𝑛 ⟼ ℝ,
ℱ(𝛽:,𝑙 , 𝑥) = 𝛽1,𝑙 𝑓1 (𝑥) + ⋯ 𝛽𝑝,𝑙 𝑓𝑝 (𝑥)
= [𝑓1 (𝑥) ⋯ 𝑓𝑝 (𝑥)]𝛽:,𝑙
≡ 𝑓(𝑥)Τ 𝛽:,𝑙
where the coefficients {𝛽𝑝,𝑙 } are regression parameters.
(14)
FEA model
MSC.Marc
Latin Hypercube
Sampling
(LHS)
Validation
Kriging
Surrogate model
=
Figure 3 Surrogate model construction methodology.
The random field 𝑧 is assumed to have mean zero and covariance
(15)
𝐸[𝑧𝑙 (𝑤)𝑧𝑙 (𝑥)] = 𝜎𝑙2 𝑅(𝜃, 𝑤, 𝑥),
𝑙 = 1, . . . , 𝑞
2
th
where 𝜎𝑙 is the field variance for the 𝑙 component of the response and 𝑅(𝜃, 𝑤, 𝑥) is the correlation
model with parameter vector 𝜃.
For the set 𝑆 of design sites, a 𝑚 × 𝑝 design matrix 𝐹 can be constructed with 𝐹𝑖𝑗 = 𝑓𝑗 (𝑠𝑖 ),
(16)
𝐹 = [𝑓(𝑠1 ) ⋯ 𝑓(𝑠𝑚 )]Τ
The 𝑚 × 𝑝 correlation matrix 𝑅 can be constructed as
(17)
𝑅𝑖𝑗 = ℛ(𝜃, 𝑠𝑖 , 𝑠𝑗 ),
𝑖, 𝑗 = 1, . . . , 𝑚
Considering the 𝐹 and 𝑅 matrices, the fitted regression parameter 𝛽 ∗ , a 𝑝 × 𝑞 matrix, can be
calculated using least squares as
(18)
𝛽 ∗ = (𝐹 T 𝑅−1 𝐹)−1 𝐹 T 𝑅 −1 𝑌
For any untried design point 𝑥, the vector 𝑟(𝑥) of correlations between different 𝑧 at design sites
and 𝑥, can be defined as
(19)
𝑟(𝑥) = [ ℛ(𝜃, 𝑠1 , 𝑥) ⋯ ℛ(𝜃, 𝑠𝑚 , 𝑥)]T
Therefore, the Kriging predictor is
(20)
𝑦̂(𝑥) = 𝑓(𝑥)T 𝛽 ∗ + 𝑟(𝑥)T 𝛾 ∗
∗
where the 𝑚 × 𝑞 matrix 𝛾 can be calculated through the residuals,
(21)
𝑅𝛾 ∗ = 𝑌 − 𝐹𝛽 ∗
Matrices 𝛽 ∗ and 𝛾 ∗ are fixed for a fixed set of design data. For every new 𝑥 only the vectors 𝑓(𝑥) ∈
ℝ and 𝑟(𝑥) ∈ ℝ𝑚 have to be computed.
A Gaussian function was selected for the construction of the surrogate model and a 2nd order
polynomial was selected for the regression model. The Matlab toolbox for Kriging modelling [19] was
utilised to construct the model calculating the coefficients of Eq. (21) 𝛽 ∗ and 𝛾 ∗ of the 2nd order
regression and of the Gaussian correlation function respectively.
𝑝
2.4. Inversion algorithm
Inverse heat transfer problems are often ill-posed in nature[20]. Bayesian inference [6] addresses illposed problems, by incorporating experimental data and prior beliefs about the parameters values to
estimate unknown variables. Unlike deterministic methods, such as regularisation methods [20],
Bayesian inference operates as a sampler, estimating the average and the standard deviation of the
unknown parameters. The Markov Chain Monte Carlo (MCMC) method is based on Bayes’ theorem
and is used in many inverse heat transfer problems due to its simplicity [13–16]. Bayes’ theorem
connects the experimental and model values 𝑌 and 𝑥 respectively as follows:
𝑃(𝑌|𝑥)𝑃(𝑥)
(22)
𝑃(𝑌)
where 𝑃(𝑥|𝑌) is the posterior probability density function, 𝑃(𝑌|𝑥) is the likelihood density function,
𝑃(𝑥) is the prior density function and 𝑃(𝑌) is the normalizing constant. Bayes’ theorem can be written
in a proportional form as follows, where the posterior probability depends on the likelihood and prior
distribution.
(23)
𝑃(𝑥|𝑌) ∝ 𝑃(𝑌|𝑥)𝑃(𝑥)
Eq. (23) can be used to describe in which way the model needs to be modified taking into account
experimental data. The Metropolis Hasting (MH) algorithm was utilised to generate samples 𝑋̅ from a
proposal distribution 𝑞(∙). An acceptance criterion is applied in each proposed sample and by accepting
or rejecting it, the posterior distribution converges to the target distribution 𝑃(𝑓(𝑋̅ )|𝑌̅). Here 𝑋̅ is a
vector representing the unknown parameters 𝑘 and ℎ used to compute the model response 𝑓(𝑋̅), and 𝑌̅
represents a matrix of the experimental data. The acceptance criterion 𝑎 can be described as follows:
̅̅̅̅̅̅
𝑃(𝑓(𝑋̅𝑖 )|𝑌̅) ∙ 𝑞(𝑋̅𝑖 |𝑋
𝑖−1 )
(24)
𝑎 = 𝑚𝑖𝑛 {1,
}
̅̅̅̅̅̅
̅
̅̅̅̅̅̅
𝑃(𝑓(𝑋𝑖−1 )|𝑌) ∙ 𝑞(𝑋𝑖−1 |𝑋̅𝑖 )
where 𝑋̅𝑖 and ̅̅̅̅̅̅
𝑋𝑖−1 is the sample of MCMC iteration 𝑖 and 𝑖 − 1 respectively.
The random walk Metropolis-Hastings algorithm which is a modification of the conventional MH
algorithm was implemented in this study. In this method the proposal distribution 𝑞(∙) is symmetric.
Due to the symmetry the new sample can be calculated from a noise level 𝜀 in the form of a Gaussian
distribution with mean value 0 and standard deviation 𝜎𝜀 , which is applied to the parameter value ̅̅̅̅̅̅
𝑋𝑖−1
from the previous step. The algorithm operates in the following steps:
1. Initialise ̅̅̅
𝑋0
2. For 𝑖 = 1 to 𝑛 do
i.
Draw a sample 𝑢~𝑈(0,1) from a uniform distribution between 0, 1.
ii.
Draw sample 𝜀~𝛮(0, 𝜎𝜀 ) ⟶ 𝑋̅𝑖 = ̅̅̅̅̅̅
𝑋𝑖−1 + 𝜀
iii.
Calculate acceptance probability 𝑎
iv.
If 𝑢 ≤ 𝑎 then accept 𝑋̅𝑖
v.
Else go to step 2 with 𝑋̅𝑖 = ̅̅̅̅̅̅
𝑋𝑖−1
In this algorithm 𝑛 is the number of MCMC iterations, and 𝑎 is defined as:
𝑃(𝑓(𝑋̅𝑖 )|𝑌̅)
(25)
𝑎 = 𝑚𝑖𝑛 {1,
}
̅̅̅̅̅̅
̅
𝑃(𝑓(𝑋
𝑖−1 )|𝑌 )
The posterior probability in Eq. (25) can be calculated using the Bayes’ theorem. The acceptance
probability 𝑎 can be calculated on the basis of the likelihood and prior using the logarithmic values of
𝑋̅𝑖 and ̅̅̅̅̅̅
𝑋𝑖−1 for each iteration 𝑖:
𝑃(𝑌̅|𝑓(exp{𝑋̅𝑖 }))𝑃(𝑋̅𝑖 )
(26)
𝑎 = 𝑚𝑖𝑛 {1,
}
̅̅̅̅̅̅
̅̅̅̅̅̅
𝑃(𝑌̅|𝑓(exp{𝑋
𝑖−1 }))𝑃(𝑋𝑖−1 )
In this context 𝑌̅ represents the temperature experimental data of the two thermocouples placed at
the mid-thickness and on top of the composite part, whilst 𝑓(exp{𝑋̅𝑖 }) is the model response
(𝑇𝑚𝑖𝑑 , 𝑇𝑡𝑜𝑝 ) using the transformed parameters 𝑘 and ℎ. The likelihood is calculated as follows:
𝑃(𝑥|𝑌) =
𝑙
𝑃(𝑌̅|𝑓(exp{𝑋̅𝑖 })) = ∑ ln{𝑁(𝑌̅𝑘 ; 𝑓𝑘 (exp{𝑋̅𝑖 }) , 𝜎)}
(27)
𝑘=1
where 𝑙 denotes the total number of experimental data. The likelihood incorporates all the distributions
which are computed with experimental data 𝑌̅𝑘 using a normal distribution with the model values
𝑓𝑘 (exp{𝑋̅𝑖 }) as a mean and a standard deviation 𝜎. The prior distribution is computed in a similar way:
𝑛
𝑃(𝑋̅𝑖 ) = ∑ ln{𝑁(𝑋𝑗 ; 0, 𝜎𝑋 𝑗 )}
(28)
𝑗=1
where 𝑛 denotes the number of unknown parameters and 𝜎𝑋 𝑗 the standard deviation of the prior
distribution.
In the MCMC algorithm, standard deviations (𝜎, 𝜎𝑋 𝑗 , 𝜎𝜀 ) operate as tuning parameters and need to
be adjusted before the initiation of the inversion procedure. The standard deviation 𝜎 used in the
likelihood term, is assigned with a relative small value as this parameters corresponds to the quality of
the theoretical response [23]. The standard deviation 𝜎𝑋 𝑗 , included in prior distribution is set to a large
value allowing the algorithm to explore a larger parameter region [24]. The standard deviation 𝜎𝜀 defines
the noise level 𝜀 and determines the sampling behaviour of the chain [25]. The right choice of these
standard deviations depends on acceptance probability rate which must be between 30% and 50% for
low-dimensional models [25].
Simulations of a single chain may be trapped in a local mode and fail to explore the remaining modes
of notable probability. Parallel tempering method was applied to address this problem [20,21]. In this
method a temperature parameter 𝑇 with the property 1 ≤ 𝑇 ≤ ∞ is introduced where 𝑇 = 1 denotes the
desired target distribution and is referred as cold sample. Values with 𝑇 ≫ 1, which are referred to as
hot samples, flatten the target distribution and allow the acceptance of wider range of proposed
parameters. Hence, these distributions explore a larger parameter region. In parallel tempering a
tempering parameter defined as 𝛽 = 1/𝑇. The tempering parameter is assigned as follows
(29)
𝜋(𝑓(𝑋̅𝑖 )|𝑌̅, 𝛽) = exp{𝛽𝑙𝑛𝑃(𝑌̅|𝑓(𝑋̅𝑖 ))} 𝑃(𝑋̅𝑖 ) for 0 < 𝛽 < 1
This tempering posterior distribution is calculated using Bayes’ theorem. In each chain a different
discrete value of 𝛽 is assigned resulting in a ladder with different temperatures. After a certain number
of iterations (𝑛𝑠 ) a parameter swap algorithm is initiated which exchanges parameters between two
1
chains, if 𝑈1 ~𝑈[0,1] ≤ 𝑛 where 𝑈1 is a random number form a uniform distribution. If the swap occurs,
𝑠
a chain 𝑚 is randomly selected to swap the parameter set with the chain 𝑚 + 1. A swap is accepted if
𝑠 ≥ 𝑈2 where 𝑈2 ~𝑈[0,1] and 𝑠 is the acceptance probability defined as follows:
̅̅̅̅̅̅̅
̅
𝜋(𝑓(𝑋
𝑚+1 )|𝑌, 𝛽𝑚 )
(30)
𝑠 = 𝑚𝑖𝑛 {1,
}
̅̅̅̅
̅
𝜋(𝑓(𝑋
𝑚 )|𝑌, 𝛽𝑚+1 )
Chains with higher temperatures can explore different modes, whilst chains within the ladder allow
the possibility to refine these sets. Only the results of the cold chain are considered for the final sample
whilst the results from the remaining chains are usually disregarded.
3. Results and discussion
3.1. Validation of surrogate model
Validation tests were carried out in order to investigate the surrogate model accuracy. Responses
surfaces, showing the relationship between model outputs (𝑇𝑚𝑖𝑑 , 𝑇𝑡𝑜𝑝 ) and inputs (𝑘, ℎ, 𝑡), were
constructed in order to compare the surrogate model with the FE results. Figures 4a and 4b illustrate the
dependence of 𝑘 and ℎ on 𝑇𝑚𝑖𝑑 and 𝑇𝑡𝑜𝑝 respectively at 𝑡 = 60 𝑚𝑖𝑛. The surrogate model is in
agreement with the FE results with a mean squared error (MSE) less than 0.03. It can be observed that
the heat transfer coefficient causes greater changes in 𝑇𝑚𝑖𝑑 and 𝑇𝑡𝑜𝑝 than the thermal conductivity level.
Also, the temperature at the top of the part is more sensitive than the temperature at the mid-thickness
to parameter changes. Temperatures both at mid-thickness and the top of the part decrease when the
heat transfer coefficient and the thermal conductivity level increase.
b)
[ C]
[ C]
a)
Figure 4 a) Responses surfaces for 𝑡 = 60[𝑚𝑖𝑛]: a) Temperature at the mid thickness as a function of the heat
transfer coefficient and the thermal conductivity level; b) Temperature at the top as a function of the heat transfer
coefficient and the thermal conductivity level.
3.2. Uncertainty parameters estimation
Four parallel chains were set up for 50.000 iterations starting from the initial values of k =
0.12 W/m/°C and h = 8.5 𝑊/𝑚2 /°𝐶. The standard deviation 𝜎 for the likelihood distribution defining
the Gaussian error between experimental and model value was set to 0.8. In order to induce an informal
prior distribution a standard deviations 𝜎𝑋 𝑗 of 0.6 and 30 for 𝑘 and ℎ were introduced respectively.
A number of short sequences were simulated to tune the standard deviation 𝜎𝜀 for the applied noise
level. An acceptance probability in the region of 30% -40% was achieved for the four parallel chains.
Figure 5 illustrates the evolution of posterior distributions of parallel chains. It can be observed that the
cold chain (chain 1) converges in a mode after 4000 MCMC iterations. The first 4000 samples
highlighted in grey are the burn in zone and are disregarded from the final sample.
The values within the stationary sequence are highly correlated as depicted in Figures 6a and 6b due
to the nature of the MH algorithm. Consequently, a step size of 10 and 200 for 𝑘 and ℎ respectively was
used for the thinning of the sampling
Figure 5 Convergent assessment via posterior distribution plot.
a)
b)
Figure 6 Sample autocorrelation a) heat transfer coefficient b) thermal conductivity level.
Figures 7 and 8 show the uncorrelated sampling without the burn-in zone and the cumulative density
function of ℎ and 𝑘 respectively. The mean value of thermal conductivity level is 0.136 W/m/°C which
does not differ significant from the original value, whilst the standard deviation is very low and equals
to 0.0006 W/m/°C. The heat transfer coefficient average is 4.2 W/m2/°C with a standard deviation of
0.06 W/m2/°C, where the nominal value is 8.5 W/m2/°C. This indicates the uncertainty of this variable
which may vary from experiment to experiment and strongly depends on manufacturing environment
conditions. In terms of variability, the inversion procedure narrowed down the uncertainty of ℎ reducing
coefficient of variation from 12% [2] to 1.4%. Figure 9a depicts the experimental measurements results
at the bottom, the mid-thickness and the top of the curing part alongside the model responses calculated
by using the nominal values of thermal conductivity level and surface heat transfer coefficient. It can be
observed that there are significant discrepancies between model and experimental data for both 𝑇𝑚𝑖𝑑
and 𝑇𝑡𝑜𝑝 .
b)
a)
Figure 7 a) Uncorrelated sample of the heat transfer coefficient b) Cumulative density function of the heat
transfer coefficient.
a)
b)
Figure 8 a) Uncorrelated sample of the thermal conductivity level b) Cumulative density function of the thermal
conductivity level.
The results indicate that the nominal values of resin thermal conductivity level and heat transfer
coefficient need to be estimated using the online experimental data. The 𝑇𝑚𝑖𝑑 , and 𝑇𝑡𝑜𝑝 calculated with
the estimated mean values of 𝑘 and ℎ are in good agreement with the experimental data as illustrated in
Figure 9b with an average error of 0.44 °C. It can be observed that the model ignores experimental data
fluctuations in the region between 20 and 30 min and errors after 70 min at the mid-thickness of the
composite part.
4. Conclusions
An inversion procedure based on MCMC method was developed in this study to estimate the uncertainty
of thermal properties in curing stage of manufacturing process. The utilisation of a surrogate model
reduces significantly the computational time whilst representing accurately the heat transfer problem.
The findings highlight the efficiency of the MCMC method in terms of estimating the statistical
properties of thermal conductivity and heat transfer coefficient.
b)
a)
Figure 9 Experimental data and model responses comparison a) prior knowledge b) estimated values.
A validation test was carried out comparing the model with the new values with the existing
experimental data. The comparison has shown that the model is in good agreement with the experimental
data. Therefore, this model can be utilised for the process outcomes and process induced defects
estimation once the uncertainty of input parameters was narrowed down. This work sets the basis for
real time prediction using MCMC method in both the curing and flow stage of composite manufacturing
process.
5. Acknowledgments
This work was supported by the European Commission through the Clean Sky 2 project ‘Simulation
tool development for a composite manufacturing process default prediction integrated into a quality
control system’ (199349) and the Engineering and Physical Sciences Research Council, through the
grant ‘Robustness performance optimisation for automated composites manufacture’ (EP/K031430/1).
References
[1]
T. S. Mesogitis, A. A. Skordos, and A. C. Long, “Uncertainty in the manufacturing of fibrous
thermosetting composites: A review,” Compos. Part A Appl. Sci. Manuf., vol. 57, pp. 67–75,
2014.
[2]
T. S. Mesogitis, A. A. Skordos, and A. C. Long, “Stochastic heat transfer simulation of the cure
of advanced composites,” J. Compos. Mater., vol. 50, pp. 2971–2986, 2016.
[3]
Alexandros A. Skordos, “Modelling and monitoring of resin transer moulding,” PhD thesis,
Cranfield university, 2000.
[4]
J. Wang and N. Zabaras, “Using Bayesian statistics in the estimation of heat source in radiation,”
Int. J. Heat Mass Transf., vol. 48, pp. 15–29, 2005.
[5]
J. Wang and N. Zabaras, “A Bayesian inference approach to the inverse heat conduction
problem,” Int. J. Heat Mass Transf., vol. 47, pp. 3927–3941, 2004.
[6]
J. P. Kaipio and C. Fox, “The Bayesian Framework For Inverse Problems In Heat Transfer,”
Heat Transf. Eng., vol. 32, pp. 83, 2011.
[7]
N. Gnanasekaran and C. Balaji, “Markov chain Monte Carlo (MCMC) approach for the
determination of thermal diffusivity using transient fin heat transfer experiments,” Int. J. Therm.
Sci., vol. 63, pp. 46–54, 2013.
[8]
G. M. Maistros and I. K. Partridge, “Monitoring autoclave cure in commercial carbon
fibre/epoxy composites,” Compos. Part B Eng., vol. 29, no. 3, pp. 245–250, 1998.
[9]
G. Lebrun, R. Gauvin, and K. N. Kendall, “Experimental investigation of resin temperature and
pressure during filling and curring in a flat steel RTM mould,” Compos. Part A Appl. Sci. Manuf.,
vol. 27, pp. 347–356, 1996.
[10] A. A. Skordos and I. K. Partridge, “Inverse heat transfer for optimization and on-line thermal
properties estimation in composites curing.,” Inverse Probl. Sci. Eng., vol. 12, pp. 157–172,
2004.
[11] “Marc® volume B: Element library. www.mscsoftware.com, 2011.”
[12] “Marc® volume D: User subroutines and Special Routines. www.mscsoftware.com, 2011.”
[13] P. I. Karkanas and I. K. Partridge, “Cure modeling and monitoring of epoxy/amine resin systems.
I. Cure kinetics modeling,” J. Appl. Polym. Sci., vol. 77, pp. 1419–1431, 2000.
[14] T. S. Mesogitis, A. A. Skordos, and A. C. Long, “Stochastic simulation of the influence of cure
kinetics uncertainty on composites cure,” Compos. Sci. Technol., vol. 110, pp. 145–151, 2015.
[15] J. D. Farmer and E. E. Covert, “Thermal conductivity of a thermosetting advanced composite
during its cure,” J. Thermophys. heat Transf., vol. 10, pp. 467–475, 1996.
[16] G. Struzziero and A. A. Skordos, “Multi-objective optimisation of the cure of thick components,”
Compos. Part A Appl. Sci. Manuf., vol. 93, pp. 126–136, 2017.
[17] W. J. C. M. D. McKay R. J. Beckman, “A Comparison of Three Methods for Selecting Values
of Input Variables in the Analysis of Output from a Computer Code,” Technometrics, vol. 21,
pp. 239–245, 1979.
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
M. A. Oliver and R. Webster, “A tutorial guide to geostatistics : Computing and modelling
variograms and kriging,” Catena, vol. 113, pp. 56–69, 2014.
S. N. Lophaven, H. B. Nielsen, and J. Søndergaard, “DACE-A Matlab Kriging toolbox, version
2.0,” 2002.
N. S. Mera, L. Elliott, D. B. Ingham, and D. Lesnic, “A comparison of different regularization
methods for a Cauchy problem in anisotropic heat conduction,” Int. J. Numer. Methods Heat
Fluid Flow, vol. 13, pp. 528–546, 2003.
T. D. Fadale, A. V. Nenarokomov, and A. F. Emery, “Uncertainties in parameter estimation: the
inverse problem,” Int. J. Heat Mass Transf., vol. 38, pp. 511–518, 1995.
S. Somasundharam and K. S. Reddy, “Inverse estimation of thermal properties using Bayesian
inference and three different sampling techniques,” Inverse Probl. Sci. Eng., vol. 5977, pp. 1–
16, 2016.
B. B. and C. T. RC Aster, Parameter estimation and inverse problems. 2005.
P. Gustafson, D. Aeschliman, and A. R. Levy, “A Simple Approach to Fitting Bayesian Survival
Models,” Lifetime Data Anal., vol. 9, pp. 5–19, 2003.
G. O. Roberts, A. Gelman, and W. R. Gilks, “Weak convergence and optimal scaling of random
walk Metropolis algorithms,” Ann. Appl. Probab., vol. 7, pp. 110–120, 1997.
D. J. Earl and M. W. Deem, “Parallel Tempering: Theory, Applications, and New Perspectives,”
Phys. Chem. Chem. Phys., vol. 7, pp. 3910–3916, 2005.
R. M. Neal, “Sampling from multimodal distributions using tempered transitions,” Stat. Comput.,
vol. 6, pp. 353–366, 1996.