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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa

Three-dimensional reconstruction of resin ﬂow using capacitance sensor data assimilation during a liquid composite molding process: A numerical study Masayuki Murata a, Ryosuke Matsuzaki b,⇑, Akira Todoroki a, Yoshihiro Mizutani a, Yoshiro Suzuki a a b

Department of Mechanical Sciences and Engineering, Tokyo Institute of Technology, 2-12-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan Department of Mechanical Engineering, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan

a r t i c l e

i n f o

Article history: Received 16 April 2014 Received in revised form 5 September 2014 Accepted 31 January 2015 Available online 17 February 2015 Keywords: B. Electrical properties C. Computational modeling D. Process monitoring E. Resin ﬂow

a b s t r a c t Liquid composite molding (LCM) is a method to manufacture ﬁber-reinforced composites, where dry fabric reinforcement is impregnated with a resin in a molding apparatus. However, the inherent process variability changes resin ﬂow patterns during mold ﬁlling, which in turn may cause void formation. We propose a method to reconstruct three-dimensional resin ﬂow in LCM, without embedding sensors into the composite structure. Capacitance measured from pairs of electrodes on molding tools and the stochastic simulation of resin ﬂow during an LCM process are integrated by a sequential data assimilation method based on the ensemble Kalman ﬁlter; then, three-dimensional resin ﬂow and permeability distribution are estimated simultaneously. The applicability of this method is investigated by numerical experiments, characterized by different spatial distributions of permeability. We conﬁrmed that changes in resin ﬂow caused by spatial permeability variations could be captured and the spatial distribution of permeability could be estimated by the proposed method. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Liquid composite molding (LCM), such as resin transfer molding (RTM) and vacuum assisted resin transfer molding (VaRTM), is a method for manufacturing ﬁber-reinforced plastics (FRPs) used for wind turbine blades, marine vessels, airplanes and other industrial products. In an LCM process, dry fabric reinforcement is impregnated with liquid resin in a molding apparatus. However, inherent process variabilities, such as spatial variations in permeability caused by variability in the fabric architecture, can change the resin ﬂow patterns during mold ﬁlling, which in turn may result in the formation of voids, including large dry spots or micro voids that signiﬁcantly degrade the mechanical properties of the composite structure. The application of LCM is currently limited because of its poor quality. Recent studies proposed optimization of resin injection gates and vents based on resin ﬂow simulation [1–4] to address quality issues in LCM. Statistical modeling of permeability variations [5] or stochastic simulation of resin ﬂow [6] has also been studied to further improve optimization of the processing conditions. However,

⇑ Corresponding author. Tel./fax: +81 4 7124 1501. E-mail address: [email protected] (R. Matsuzaki). http://dx.doi.org/10.1016/j.compositesa.2015.01.031 1359-835X/Ó 2015 Elsevier Ltd. All rights reserved.

it is almost impossible to control and optimize many factors involved in an LCM process prior to manufacturing. Therefore, monitoring of resin ﬂow patterns during an LCM process and implementing control systems are preferable to reliably predict and prevent formation of voids. This paper focuses on ﬂow monitoring methods in an LCM process. In previous studies, various methods for monitoring of resin ﬂow have been proposed. Optical ﬁber sensors [7–9], including ﬁber Bragg grating (FBG) sensors [10,11], are used for monitoring of resin arrival to the sensor location on the basis of changes in the refractive index or strength of the reﬂected light. Electric time-domain reﬂectometry (E-TDR) [12–14], is also used to monitor resin ﬂow by measuring reﬂected waves on a transmission line embedded in composites or molding tools. Although the optical ﬁber and E-TDR methods can monitor resin ﬂow with high accuracy, they only measure resin ﬂow along sensor lines. Further, the possible degradation of the mechanical properties of parts and interference with the smooth resin ﬂow due to embedded sensor lines may pose problems. Permittivity [15] or conductivity [16– 18] methods have been proposed for monitoring resin ﬂow by measuring the relative permittivity or conductivity of the polymer resin at the sensor location. However, these sensors are point or linear sensors on molding tools and only measure the resin ﬂow close to the sensors. Although grid sensor [19], area array sensor

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[20], or pressure sensor-based methods [21] enable monitoring of an arbitrary resin ﬂow process, the resin ﬂow was assumed to be two-dimensional and ﬂow in the thickness, or out-of-plane direction, has not been considered. This paper presents a method to reconstruct three-dimensional resin ﬂow in an LCM process without embedding sensors into composite parts. Electrical capacitance measured from pairs of electrodes on molding tools and stochastic simulation of resin ﬂow during an LCM process are integrated by a sequential data assimilation method based on the ensemble Kalman ﬁlter (EnKF), and then, three-dimensional resin ﬂow and permeability distribution are estimated simultaneously. This improves the accuracy of the ﬂow front prediction by simulation and can provide valuable information for formulating a strategy to control resin ﬂow and reduce the formation of voids during an LCM process.

where M is the number of measurements, and t is an index of the discrete time. The capacitance ci between a pair of electrodes is the ratio of the charge on electrode Qi to the potential difference between the electrode pair DV i , i.e., ci ¼ Q i =DV i . Fiber reinforcements placed in the mold cavity are impregnated with polymer resin during an LCM process. When the electric ﬁeld excited by the measurement electrodes is applied to a polar polymer such as polyester or epoxy resin, the dipoles in the resin are aligned to the direction of the electric ﬁeld, which causes dielectric polarization. Thus, polymer resin acts as a dielectric material. The electrical capacitance vector changes with resin impregnation during an LCM process because the dielectric constant in the space between a pair of electrodes increases as the resin impregnates into ﬁber reinforcements in the mold cavity. Therefore, the capacitance vector provides the information about the resin-ﬁlled region.

2. Flow reconstruction method Several types of sensors based on different physical phenomena have been used to capture resin ﬂow during LCM processes in previous studies, as summarized in the preceding section. In this study, an electrode array sensor was employed for ﬂow monitoring because of its ability to obtain information unobtrusively on resin ﬂow away from the sensor. In fact, it has been reported that an electrode array sensor can detect both in-plane and out-of-plane resin impregnation [22]. However, the monitoring of three-dimensional resin ﬂow, in which in-plane and out-of-plane ﬂow exist simultaneously during the molding of a thick part, has not been addressed. To address this problem, we propose in this paper a method to reconstruct three-dimensional resin ﬂow using values measured with an electrode array sensor, complemented by simulation modeling. The details of the method are presented in this section. 2.1. Electrical capacitance measurement

2.1.2. Governing equations of electric ﬁeld and sensitivity calculation To reconstruct resin ﬂow from measured capacitance, the relationship between resin distribution and capacitance must be established. Here, we consider this relationship in a manner similar to electrical capacitance tomography [24,25], which is the methodology used to reconstruct the permittivity distribution of an object from electric capacitance measurements. Because the wavelength of the electromagnetic ﬁeld substantially exceeds the dimensions of the sensors in typical measurement systems for ECT, the electric ﬁeld distribution obeys the electrostatic ﬁeld theory. The governing equation of the electric ﬁeld can be derived as the electrostatic approximation of the Maxwell equations in the absence of internal charges, i.e., r D ¼ 0, where D is the electric displacement. The electric displacement D is related to the electric ﬁeld intensity vector E via the permittivity of a material e, i.e., D ¼ eE. The permittivity e reﬂects the ability of a material to store electrical energy under the inﬂuence of an electric ﬁeld and is a second-order tensor when the material is anisotropic. We can then assume E ¼ rV for electric potential V and ﬁnally the governing equation of the electric ﬁeld can be written as

2.1.1. Electrical capacitance sensor and measurement Plate electrodes are arrayed on the molding tool and used as sensors to obtain information on resin ﬂow in the mold cavity, as shown in Fig. 1. Sensors of this type can be fabricated by embedding copper electrodes in a molding tool [19] or by making a pattern of electrodes on a ﬂexible substrate that can be attached to a molding tool [20,23]. Capacitances are measured by pairs of electrodes, and the measured values (c1 ; c2 ; ; ci ; ; cM ) are organized in an electrical capacitance vector deﬁned as

where X is the region inside the mold cavity. Once the electrical potential distribution is obtained by solving Eq. (2) with boundary conditions for the electrical potential V at the measurement electrodes, the capacitance between the electrodes can be calculated using the electric potential distribution. Because the induced charge on the electrode can be obtained from a volume integral,

ct ¼ ðc1 ; c2 ; ; ci ; ; cM ÞT ;

Qi ¼

ð1Þ

r erV ¼ 0 in X;

1 DV i

Z

ð2Þ

ðrVÞT eðrVÞdX;

ð3Þ

X

the capacitance can be obtained from its deﬁnition,

ci ¼

c1 Molding tool

ci c2

Qi 1 ¼ DV i DV 2i

Fig. 1. Schematic of electrical capacitance measurements. Capacitance is measured from a pair of electrodes on molding tools. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

ðrVÞT eðrVÞdX:

ð4Þ

X

Let Xj be a small region in X, and let fj be the ﬁll fraction f of that region, which ranges from 0 to 1 such that the limits f = 0 and f = 1 signify the region being completely unﬁlled and completely ﬁlled with resin, respectively. The sensitivity of fj to ci, i.e., the change in the capacitance ci induced by a change in ﬁll fraction fj in a small volume Xj, is obtained from the following equation:

@ci 1 sij ¼ ¼ @f j DV 2i

Electrode

Z

Z Xj

T

ðrVÞ

! @ ej ðrVÞdXj ; @f j

ð5Þ

where ej is the permittivity tensor in Xj. By exploiting the sensitivity, the relationship between small changes in the resin distribution and small changes in the capacitance vector can be described quantitatively. However, the information obtained by the

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M. Murata et al. / Composites: Part A 73 (2015) 1–10

capacitance vector is insufﬁcient to reconstruct three-dimensional resin distribution. Therefore, the governing equation of the resin ﬂow is employed to compensate for this insufﬁciency and provide reasonable estimation of the resin-ﬁlled region.

and the eigenvalues are sorted in descending order (k1 P k2 P P ki P ). The stochastic process, aðxÞ, can be expanded by the Karhunen–Loève (KL) expansion [29]

aðxÞ ¼ a þ 2.2. Stochastic simulation of resin ﬂow

u¼

l

rp;

ð6Þ

and the continuity equation for incompressible ﬂow,

r u ¼ 0;

ð7Þ

where u is the average ﬂuid velocity vector, K is the preform permeability tensor, l is the resin viscosity, and p is the pressure ﬁeld. The advancement of the resin ﬂow front can be described by the volume of ﬂuid (VOF) method [26], which solves the advection equation of the ﬁll fraction, f:

@f þ ur rf ¼ 0; @t

ð8Þ

where ur is the resin’s superﬁcial velocity, which is related to the average ﬂow velocity u via the porosity / by the equation u ¼ /ur . The control volume/ﬁnite element method was used to solve these governing equations [27]. The ﬁnite element method was used to solve Eqs. (5) and (7) for the pressure distribution. Once the ﬂow velocity distribution is determined, the resin ﬂow front is advanced by explicit integration of Eq. (8) in the time domain. This process is repeated until the entire fabric region is saturated. It should be noted that Eq. (7) is satisﬁed only for fully saturated ﬂow and the model is limited to isothermal conditions. In future work, not only the Darcy ﬂow, which is the main phenomenon in LCM, but also other phenomena, including ﬁber tow saturation [27] and non-isothermal ﬁlling, should be incorporated into the ﬂow model. 2.2.2. Stochastic representation of random permeability ﬁeld Accurate simulation of an LCM process requires accurate permeability distribution in a simulation model. However, permeability is not spatially constant in a ﬁber preform because of the inherent variability in the local fabric architecture [28]. In order to represent the spatial variation of permeability, permeability distribution needs to be modeled as a stochastic random ﬁeld. Since this study aims at estimating resin ﬂow by combining simulation and capacitance measurements, and does not focus on rigorous modeling of the stochastic properties of the permeability ﬁeld, the permeability ﬁeld is simply modeled as

KðxÞ ¼ KaðxÞ;

ð9Þ

where x is a position vector, K is the mean permeability, and aðxÞ and covariance denotes a stochastic process with mean E½aðxÞ ¼ a function covðaðxÞ; aðx0 ÞÞ. The covariance function has the spectrum decomposition 1 X covðaðxÞ; aðx0 ÞÞ ¼ ki /i ðxÞ/i ðx0 Þ;

ð10Þ

i¼1

where ki and /i ðxÞ are the eigenvalue and eigenfunction of the covariance function, respectively. They are the solution to the integral equation

Z D

covðaðxÞ; aðx0 ÞÞ/i ðx0 Þdxdx0 ¼ ki /i ðxÞ;

ð12Þ

i¼1

2.2.1. Governing equations of resin ﬂow during an LCM process The ﬂow velocity of the resin through ﬁber preforms during an LCM process can be assumed to follow Darcy’s law [1]:

K

1 pﬃﬃﬃﬃ X ki /i ðxÞbi ;

ð11Þ

where bi is a random coefﬁcient, which follows a mutually independent standard normal distribution. Truncating the series at the d th term gives the approximation of a stochastic permeability ﬁeld

þ Kðx; bÞ ¼ K a

! d pﬃﬃﬃﬃ X ki /i ðxÞbi ;

ð13Þ

i¼1

where b ¼ fb1 ; ; bd gT is referred to a coefﬁcient vector, which characterizes the approximated permeability ﬁeld. The spatially correlated permeability ﬁeld is modeled with the covariance function

" 2

0

covðaðxÞ; aðx ÞÞ ¼ ra exp

jx x0 j

gx

jy y0 j

gy

jz z0 j

gz

# ;

ð14Þ

where ra is the standard deviation of a; gx, gy, and gz are the correlation lengths in coordinate directions. Substituting Eq. (14) into Eq. (11) and solving the integral yields a stochastic permeability ﬁeld represented by Eq. (13). The governing equations of resin ﬂow in Section 2.2.1 are solved under the stochastic permeability ﬁeld to obtain pressure and resin ﬂow velocity distribution under variations of permeability. 2.3. Sequential data assimilation method In this study, a data assimilation method is used to estimate the unobservable time-varying state of the resin ﬂow. In order to apply a data assimilation technique to this problem, electric capacitance measurements and numerical simulations of resin ﬂow are represented as a nonlinear state space representation in Section 2.3.1. Based on that state space representation, capacitance is assimilated into a stochastic simulation of resin ﬂow during an LCM process by the EnKF-based algorithm, which employs the ensemble square root ﬁlter (EnSRF) and deterministic state sampling to reduce the sampling errors introduced by Monte Carlo sampling in the original EnKF. The EnKF algorithm is introduced in Section 2.3.2. The EnSRF and deterministic state sampling are brieﬂy introduced in Sections 2.3.3 and 2.3.4, respectively. 2.3.1. State space representation The spatial distribution of resin at timestep t is represented as a discrete form of ﬁll fraction T

f t ¼ ðf 1 ; ; f N Þ ;

ð15Þ

where N is the number of nodes in a numerical simulation. Capacitance measurements can be described as nonlinear mapping true

from a true resin distribution vector, f t , to a capacitance vector, ct. By linearizing the nonlinear mapping in reference to a point of interest, f t , the linearized observation equation can be obtained as

true

ct ¼ hðf t Þ þ St ðf t

f t Þ þ vt ;

ð16Þ

where St is a sensitivity matrix, and v t is an observation and linearization error vector. Elements of the sensitivity matrix are obtained by Eq. (5), and vt is assumed to follow the normal distribution with zero-mean and observation error covariance R. On the other hand, a numerical simulation of the resin ﬂow in an LCM process can be represented as a state equation

xtþ1 ¼ Fðxt ; wt Þ;

ð17Þ

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M. Murata et al. / Composites: Part A 73 (2015) 1–10

where F is a nonlinear operator that represents time evolution of a ﬂow simulation; wt is a system noise vector to represent the uncertainty in the time evolution; and xt is a state vector deﬁned as T

T

xt ¼ ðf t bTt Þ , which consists of a time-dependent resin distribution, ft, and a coefﬁcient vector, bt, determining the approximated permeability ﬁeld deﬁned in Eq. (13). The linearized observation of Eq. (16) is also modiﬁed with the state vector representation as

ct ¼ hðxt Þ þ Ht ðxtrue xt Þ þ v t ; t

ð18Þ

where Ht is an observation matrix, deﬁned as Ht ¼ ½ St O . Based on the state space representation of Eqs. (17) and (18), a data assimilation technique is applied to estimate xt, by integrating the state and the observation equations. 2.3.2. Ensemble Kalman ﬁlter The EnKF [30], a sequential Monte Carlo extension of the Kalman ﬁlter, is a sequential data assimilation technique known for its simple formulation and applicability to nonlinear systems. The algorithm of the EnKF is brieﬂy described below. First, an initial ensemble, a group of state vectors, is generated randomly. eð1Þ

eð2Þ

eðlÞ

eðLÞ

Xe0 ¼ ½x0 ; x0 ; ; x0 ; ; x0 ; where

eðlÞ xt

ð19Þ

is an estimate of the state vector, and L is the number of eðlÞ

the ensemble members. In the initial ensemble member x0 , a ﬁll eðlÞ

fraction vector f 0

is set based on resin inlets, that is, f = 1 only at eðlÞ b0

is set as a sample vector of zero-mean indethe resin inlet, and pendent standard normal random variables. A forecast step of the state vector is obtained by the state equation fðlÞ

xt

eðlÞ

eðlÞ

¼ F t ðxt1 ; wt1 Þ ðl ¼ 1; ; LÞ;

ð20Þ

fðlÞ xt

L 1X fðlÞ x : L l¼1 t

ð21Þ fðlÞ

Next, the forecast state vector, xt , is updated by the observaeðlÞ

tion vector, ct, and the estimated state vector, xt , is obtained as eðlÞ

xt

fðlÞ

ðlÞ

¼ xt þ Kt et ;

ð22Þ

where Kt is the Kalman gain obtained by 1

Kt ¼ Pft HTt ðR þ Ht Pft HTt Þ ; Pft

ð23Þ

L T 1 X fðlÞ ft ÞðxfðlÞ ft Þ ; ðx x x t L 1 l¼1 t

2.3.3. Ensemble square root ﬁlter In the update stage of the EnKF in Eq. (25), an ensemble of observations is generated and added to a capacitance vector. The perturbed observation approach introduces an additional sampling error because the number of ensemble members is ﬁnite. Therefore, a deterministic implementation of the update scheme, the ensemble square root ﬁlter (EnSRF), was proposed to reduce the sampling error [31]. The EnSRF updates the ensemble mean and the deviation from the mean separately. fðlÞ

The forecasted state vectors, xt , are separated into the ensemft , and the deviation from the mean, dxfðlÞ ble mean, x t , as fðlÞ

fðlÞ

ft þ dxt . The update scheme of the ensemble mean and xt ¼ x the deviation is

et ¼ x ft þ Kt ½ct hðx ft Þ; x

ð27Þ

and eðlÞ

dxt

fðlÞ e t Ht dxfðlÞ ¼ dxt K t

ðl ¼ 1; ; LÞ;

ð28Þ

et is the ensemble mean of estimated state vectors, dxeðlÞ where x is t the deviation from the ensemble mean, Kt is the Kalman gain given by Eq. (23), and

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 !T qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ1 : Ht Pft HTt þ R Ht Pft HTt þ R þ R

e t ¼ Pf HT K t t

ð29Þ

ð24Þ

ðlÞ

ðlÞ ðlÞ ft Þ þ Ht ðxfðlÞ ft Þ et ¼ ct þ v t ½hðx x t

The estimated state vectors are obtained by uniting the ensemble eðlÞ et þ dxeðlÞ ¼x t .

mean and the deviation as xt

2.3.4. Deterministic state sampling based on cubature rules In addition to the deterministic update scheme in the EnSRF that avoids random sampling of observations, a method to reduce sampling errors of state vectors has been proposed [32], where the Monte Carlo sampling of the state vectors is replaced by optimal cubature rules. Consider an n-dimensional vector, z, and an arbitrary function, g(z), of z. The expected value of gðzÞ with respect to the probability density function, pðzÞ, can be approximated by numerical integration:

Z

1

gðzÞpðzÞdz

1

and et is a measurement residual vector deﬁned as

ðlÞ

ð26Þ

The above steps are iterated in the EnKF and the unobservable state vector, xt , is estimated sequentially; i.e., the time-dependent resin distribution, f t , and the permeability ﬁeld approximated by the coefﬁcient vector, bt , are estimated simultaneously.

E½gðzÞ ¼

is the error covariance matrix for the forecast

Pft ¼

L 1X eðlÞ x : L l¼1 t

eðlÞ wt1

where is a forecast of the state vector. Here, = 0 since system noise is not considered. The ensemble mean of the forecast vectors is

ft ¼ x

et ¼ x

ð25Þ

where v t is a realization of the observation noise; i.e., a random vector from the normal distribution with zero-mean and covariance R. Note that the observation equation is linearized at the ensemble ft . This approach of adding a realizamean of the forecast vectors, x tion of the observation noise to the measurement vector is called the perturbed observation method. et , is The ensemble mean of the estimated state vectors, x

X wk gðzðkÞ Þ;

ð30Þ

k

where zðkÞ is an integration point, and wk is an integration weight at zðkÞ . In the cubature rule called a degree-2 formula, when the probability density function, p(z), is the standard multivariate normal distribution, i.e., p(z) = N(0, In), the number of integration points is n + 1 and integration weights are wk ¼ 1=ðn þ 1Þ. Furthermore, a set of n+1 integration points, ðk ¼ 1; ; n þ 1Þ, can be deﬁned as

pﬃﬃﬃ 2rðk 1Þp ; 2 cos nþ1 r ¼ 1; 2; ; ½n=2; ðkÞ

z2r1 ¼

ðkÞ

z2r ¼

ðkÞ

ðkÞ

ðkÞ T

zðkÞ ¼ ðz1 ; z2 ; ; zn Þ

pﬃﬃﬃ 2rðk 1Þp ; 2 sin nþ1 ð31Þ

where [n/2] is the greatest integer less than n/2, and if n is odd zn ¼ ð1Þk . ðkÞ

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M. Murata et al. / Composites: Part A 73 (2015) 1–10

The cubature rule-based EnKF and EnSRF use the above set of integration points for deterministic sampling of state vectors. In this study, we use them for sampling state vectors in an initial ensemble, more speciﬁcally for sampling the initial set of d-dimeneðlÞ

sional coefﬁcient vectors, b0 (l ¼ 1; ; L), associated with the standard multivariate normal distribution. Hence, the number of ensemble members is set to L = d + 1.

The applicability of the proposed method is investigated by numerical experiments, where the permeability distribution of a simulation model to be estimated is different for each experiment. 3.1. Experiment 1: True permeability distribution is characterized by a stochastic process modeled in a stochastic simulation of resin ﬂow 3.1.1. Numerical simulation model and experimental procedure Fig. 2 shows the analysis model for simulating the resin ﬂow and electrical capacitance measurement during an LCM process. This model to be estimated is referred as the true model. Liquid resin is injected from the resin inlet into the glass fabrics placed in the mold cavity. Meanwhile, capacitances are measured by combinations of a total of eight electrodes arranged on the molding tools. Four of the electrodes are arrayed on the inner surface of the top molding tool, and the remaining four electrodes are on the inner surface of the bottom molding tool. The spatial distribution of permeability in fabric region of the true model is randomly generated by the method described in Section 2.2.2 and illustrated in Fig. 3. The color of the contour in Fig. 3 indicates that permeability at that point is a times the mean permeability, K. The

Resin inlet

30

Glass fabrics

z y x

15

Mean of the stochastic process Variance parameter Correlation length

a ra2 gx = gy (mm) gz (mm)

Dimension of the coefﬁcient vector

d

1.0 0.302 90 15 40

Table 2 Process parameters for LCM ﬂow simulation.

3. Numerical experiment

180

Table 1 Parameters for generation of the true permeability distribution.

Mean permeability

K x ¼ K y (m2)

3.6 1011 9.2 1013

Resin viscosity Porosity Injection pressure Vent pressure

K z (m2) l (Pa s) / pInj (Pa) pVent (Pa)

0.36 0.41 1.0 105 0.0

parameters characterizing the statistical properties of the generated permeability ﬁeld are summarized in Table 1. Time evolution of the resin ﬂow at constant injection pressure was computed in this model by the ﬁnite element method and time-series of the true state vector of the resin ﬂow, fttrue, was obtained. Process parameters for LCM ﬂow simulation, such as the mean permeability and injection pressure, are given in Table 2. The mean permeability of woven glass fabrics shows out-of-plane anisotropy. The capacitances measured by combinations of the electrodes were also computed at constant time intervals, and the observation vectors, ct, were obtained. Dielectric properties of the resin-unﬁlled glass fabric region and the resinﬁlled region used in computing the capacitance are given in Table 3. The resin ﬂow was estimated from the observation vectors by the sequential data assimilation method described in Section 2.3. Initial ensemble members were generated by the deterministic sampling method for a stochastic simulation, and the measured capacitance vectors were assimilated into the stochastic simulation of an LCM process by the EnSRF update scheme. Parameters used in the LCM ﬂow reconstruction by the data assimilation method are listed in Table 4. 3.1.2. Results and discussion Fig. 4(a) illustrates the normalized square error between the true resin distribution and the estimated resin distribution, true jjf f e jj2 =N. Fig. 4(b) shows the normalized square error

unit: mm

Fig. 2. Analysis model of an LCM process for resin ﬂow and electrical capacitance simulation. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

t

t

between the true permeability distribution and the estimated

Table 3 Dielectric constants of resin unﬁlled region and ﬁlled region.

α z

y

1.6

Unﬁlled region

In-plane Out-of-plane

ex = ey ez

2.36 1.63

Resin-ﬁlled region

In-plane Out-of-plane

ex = ey ez

7.42 7.35

1.25 1.0

x

0.75 0.5 0.3

Fig. 3. Permeability distribution of the true model in experiment 1. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Table 4 Parameters in LCM ﬂow monitoring. Mean of stochastic process Variance parameter Correlation length

a ra2 gx = gy (mm) gz (mm)

Dimension of the coefﬁcient vector Number of ensemble members Observation error covariance matrix Data assimilation interval

d L R (pF2) DtDA (s)

1.0 0.102 90 15 149 150 0.0012I 50

M. Murata et al. / Composites: Part A 73 (2015) 1–10

0.0014

Normalized square error of αt

Normalized square error of ft

6

0.0012 0.001 0.0008 0.0006 0.0004 0.0002 0

0

1000

2000

3000

4000

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

1000

2000

Time, t [s]

Time, t [s]

(a)

(b)

3000

4000

Fig. 4. Normalized square error of (a) resin distribution and (b) permeability distribution in experiment 1.

Time 1000 s

Time 2000 s

Time 3000 s

(a) True flow front

Time 1000 s

Time 2000 s

Time 3000 s

(b) Estimated flow front Fig. 5. Comparison of (a) the true and (b) the estimated ﬂow front in experiment 1. Flow front is illustrated by a meshed surface. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

et jj2 =N. The permeability distripermeability distribution, jjatrue a t bution initially generated in the stochastic simulation was different from the true permeability distribution, as indicated by the normalized error of permeability distribution at t = 0 in Fig. 4(b). In this case, the normalized error in the resin distribution would have gradually increased if the stochastic simulation had been conducted individually without data assimilation. In Fig. 4(a), however, the normalized error in the resin distribution not only increased but also decreased, which means that the true resin distribution can be tracked by the estimated resin distribution. Additionally, the estimated permeability distribution steadily converged with the true permeability distribution, as shown in Fig. 4(b). Flow front advancement of the true model and the estimated model is visualized in Fig. 5. The ﬂow fronts are deﬁned as curved surfaces where the ﬁll fraction, f, is 0.5 and is illustrated by meshed surfaces. The true and estimated ﬂow fronts had a similar appearance, and resin ﬂow estimation by the proposed method worked well. Fig. 6(a) and (b) illustrate the true and estimated ﬂow front advancement and permeability distribution, respectively. The ﬂow fronts are illustrated by meshed surfaces, as in Fig. 5, and the permeability is illustrated by colored surfaces indicating the coefﬁcient of permeability. The estimated permeability distribution was not similar to the true permeability distribution in appearance initially, but it gradually converged onto the true permeability with time, which is consistent with the quantitative evaluation in Fig. 4(b).

3.2. Experiment 2: True permeability distribution is unrelated to a stochastic process assumed in a stochastic simulation of resin ﬂow 3.2.1. Numerical simulation model and experimental procedure As shown in the previous section, it was conﬁrmed that resin ﬂow and permeability distribution can be estimated by the proposed method under the condition that the true permeability distribution is characterized by a stochastic process assumed in the stochastic simulation. This section examines the performance of the proposed method under the condition that the true permeability distribution differs substantially from the stochastic process assumed in the data assimilation method. Fig. 7 shows two analysis models for simulating resin ﬂow and electrical capacitance measurement. A low-permeability region is located at the center of the fabric region (Fig. 7(a)) and on the front side of the fabric region (Fig. 7(b)). The permeability in the region is 0.3 times that in the other region, K. Thus, the permeability distribution of each model differs substantially from the stochastic process assumed in the stochastic simulation of resin ﬂow. The other settings of the analysis models, for example, electrode conﬁguration, are the same as those in the true model shown in Fig. 2. The numerical experiments performed using the analysis models in Fig. 7(a) and (b) are referred to as 2-a and 2-b, respectively. In each experiment, resin ﬂow and capacitance measurement were simulated, and then, resin ﬂow and permeability distribution were

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Time 500 s

Time 1000 s

Time 1500 s

Time 2000 s

Time 2500 s

Time 3500 s

(a) True flow front and permeability distribution

Time 500 s

Time 2000 s

Time 1000 s

Time 1500 s

Time 2500 s

Time 3500 s

(b) Estimated flow front and permeability distribution 0.5

0.75

1.0

1.25

1.5

α

1.6

0.3

Fig. 6. Comparison of (a) the true and (b) the estimated ﬂow front advancement and permeability distribution in experiment 1. Flow front is illustrated by a meshed surface, and permeability distribution is illustrated by colored contours. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Low-permeability region Resin inlet

15

30

z

Resin inlet

z

y x

y x

180

unit: mm

(a) True model for experiment 2-a

Low-permeability region

(b) True model for experiment 2-b

Fig. 7. Analysis models of an LCM process for resin ﬂow and electrical capacitance simulation in experiment 2. A low-permeability region is located (a) at the center of the fabric region in experiment 2-a and (b) on the front side of the fabric region in experiment 2-b. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

estimated by assimilating the measured time-series of capacitance into the stochastic simulation of resin ﬂow. The parameters used in the numerical experiments were the same as those listed in Tables 2–4, except for the observation error matrix, R, which was increased ﬁve times from that in Table 4. This is because the larger the difference between the true model and the model assumed in

the stochastic simulation, the larger is the linearization error in the linearized observation Eq. (16). 3.2.2. Results and discussion Fig. 8(a) illustrates the normalized square error of the resin distrue tribution, jjf f e jj2 =N, and Fig. 8(b) shows the normalized t

t

Normalized square error of αt

M. Murata et al. / Composites: Part A 73 (2015) 1–10

Normalized square error of ft

8

0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0

0

1000

2000

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4000

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

1000

Time, t [s]

2000

3000

4000

Time, t [s]

(b)

(a)

Normalized square error of αt

Normalized square error of ft

Fig. 8. Normalized square error of (a) resin distribution and (b) permeability distribution in experiment 2-a.

0.004 0.0035 0.003 0.0025 0.002 0.0015 0.001 0.0005 0

0

1000

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4000

0

1000

2000

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Time, t [s]

(a)

(b)

3000

4000

Fig. 9. Normalized square error of (a) resin distribution and (b) permeability distribution in experiment 2-b.

decreased, though it was larger than that in the experiment in the previous section. In Figs. 8(b) and 9(b), the error between the true and the estimated permeability distribution gradually decreased; i.e., the estimated permeability distribution gradually converged onto the true permeability distribution. Fig. 10 shows (a) the true and (b) the estimated ﬂow front advancement and permeability distribution in experiment 2-a,

et jj2 =N, in squared error of the permeability distribution, jjatrue a t experiment 2-a with the true model of Fig. 7(a). Fig. 9(a) illustrates the normalized square error of the resin distribution, and Fig. 9(b) illustrates the normalized squared error of the permeability distribution in experiment 2-b with the true model of Fig. 7(b). In both Figs. 8(a) and 9(a), the error between the true and the estimated resin distribution did not just increase, but sometimes

Time 1000 s

Time 2000 s

Time 3500 s

(a) True flow front and permeability distribution

Time 1000 s

Time 2000 s

Time 3500 s

(b) Estimated flow front and permeability distribution 0.5 0.3

0.75

1.0

1.25

1.5

1.6

α

Fig. 10. Comparison of (a) the true and (b) the estimated ﬂow front advancement and permeability distribution in experiment 2-a. A low-permeability region is located at the center of the fabric in the true model. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

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Time 1000 s

Time 2000 s

Time 3500 s

(a) True flow front and permeability distribution

Time 1000 s

Time 2000 s

Time 3500 s

(b) Estimated flow front and permeability distribution 0 .5

0 .7 5

0.3

1 .0

1.25

1.5

1.6

α

Fig. 11. Comparison of (a) the true and (b) the estimated ﬂow front advancement and permeability distribution in experiment 2-b. A low-permeability region is located on the front side of the fabric in the true model. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

where a low-permeability region is located at the center of the fabric in the true model. Fig. 11 shows these results for experiment 2b, where a low-permeability region is on the front side of the fabric in the true model. Flow fronts are visualized by meshed surfaces, and permeability distribution is illustrated by colored surfaces. In Fig. 10(a), the true ﬂow front curved considerably when it permeated the low-permeability region at the center of the fabric. The estimated ﬂow front in Fig. 10(b) also curved at the center of the simulation model, and the estimated permeability distribution became gradually similar to the true permeability distribution. Similarly in Fig. 11, the true ﬂow front and permeability distribution were estimated simultaneously. These results indicate that the resin ﬂow and permeability distribution can be estimated using the proposed method, despite a considerable difference between the permeability distribution in the true model and that initially assumed in the stochastic simulation. The validity of the proposed method for use in three-dimensional ﬂow reconstruction is demonstrated by the results presented in Sections 3.1 and 3.2. However, in the numerical experiments, the difference between the actual capacitance measurement and the numerical model results was not investigated. Because the proposed method relies on the accuracy of the observation model, which relates resin distribution to capacitance, it remains a challenge to minimize the differences to obtain sufﬁciently accurate ﬂow reconstruction in actual experiments. 4. Conclusions We proposed a three-dimensional ﬂow reconstruction method without embedding sensors into the composite during an LCM process. Electrical capacitance values measured from pairs of electrodes arrayed on molding tools were integrated into a stochastic simulation of resin ﬂow by a sequential data assimilation method based on the ensemble Kalman ﬁlter (EnKF), and then, three-dimensional resin ﬂow and permeability distribution were estimated simultaneously. The validity of the proposed method was investigated in numerical experiments. In the ﬁrst case considered, the true permeability distribution was characterized by a stochastic process similar to that assumed in a stochastic simulation of resin

ﬂow. In the second case, the true permeability distribution was unrelated to the stochastic process assumed in a stochastic simulation. From both numerical experiments, it was conﬁrmed that resin ﬂow and permeability distribution can be reconstructed by the proposed method, whether or not the true permeability distribution corresponds to the permeability distribution initially modeled in the stochastic simulation.

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