Three-dimensional reconstruction of resin flow using capacitance sensor data assimilation during a liquid composite molding process: A numerical study

Three-dimensional reconstruction of resin flow using capacitance sensor data assimilation during a liquid composite molding process: A numerical study

Composites: Part A 73 (2015) 1–10 Contents lists available at ScienceDirect Composites: Part A journal homepage: www.elsevier.com/locate/compositesa...

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Composites: Part A 73 (2015) 1–10

Contents lists available at ScienceDirect

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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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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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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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

7

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Time 500 s

Time 1000 s

Time 1500 s

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(a) True flow front and permeability distribution

Time 500 s

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Time 1500 s

Time 2500 s

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(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

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

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0

1000

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3000

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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

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0

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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.

References [1] Liu B, Bickerton S, Advani S. Modelling and simulation of resin transfer moulding (RTM) – gate control, venting and dry spot prediction. Compos Part A Appl Sci Manuf 1996;27:135–41. [2] Sánchez F, García Ja, Chinesta F, Gascón L, Zhang C, Liang Z, et al. A process performance index based on gate-distance and incubation time for the optimization of gate locations in liquid composite molding processes. Compos Part A Appl Sci Manuf 2006;37:903–12. [3] Trochu F, Ruiz E, Achim V, Soukane S. Advanced numerical simulation of liquid composite molding for process analysis and optimization. Compos Part A Appl Sci Manuf 2006;37:890–902. [4] Gokce A, Advani SG. Simultaneous gate and vent location optimization in liquid composite molding processes. Compos Part A Appl Sci Manuf 2004;35:1419–32. [5] Zhang F, Comas-Cardona S, Binetruy C. Statistical modeling of in-plane permeability of non-woven random fibrous reinforcement. Compos Sci Technol 2012;72:1368–79. [6] Zhang F, Cosson B, Comas-Cardona S, Binetruy C. Efficient stochastic simulation approach for RTM process with random fibrous permeability. Compos Sci Technol 2011;71:1478–85. [7] Bernstein JR, Wagner JW. Fiber optic sensors for use in monitoring flow front in vacuum resin transfer molding processes. Rev Sci Instrum 1997;68:2156–7. [8] Dunkers J, Lenhart J. Fiber optic flow and cure sensing for liquid composite molding. Opt Lasers Eng 2001;35:91–104. [9] Kueh SRM, Parnas RS, Advani SG. A methodology for using long-period gratings and mold-filling simulations to minimize the intrusiveness of flow sensors in liquid composite molding. Compos Sci Technol 2002;62:311–27. [10] Gupta N, Sundaram R. Fiber optic sensors for monitoring flow in vacuum enhanced resin infusion technology (VERITy) process. Compos Part A Appl Sci Manuf 2009;40:1065–70. [11] Eum SH, Kageyama K, Murayama H, Uzawa K, Ohsawa I, Kanai M, et al. Structural health monitoring using fiber optic distributed sensors for vacuumassisted resin transfer molding. Smart Mater Struct 2007;16:2627–35. [12] Dominauskas A, Heider D, Gillespie Jr JW. Electric time-domain reflectometry sensor for online flow sensing in liquid composite molding processing. Compos Part A Appl Sci Manuf 2003;34:67–74. [13] Dominauskas A, Heider D, Gillespie Jr JW. Electric time-domain reflectometry distributed flow sensor. Compos Part A Appl Sci Manuf 2007;38:138–46.

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

[14] Pandey G, Deffor H, Thostenson ET, Heider D. Smart tooling with integrated time domain reflectometry sensing line for non-invasive flow and cure monitoring during composites manufacturing. Compos Part A Appl Sci Manuf 2013;47:102–8. [15] Skordos AA, Karkanas PI, Partridge IK. A dielectric sensor for measuring flow in resin transfer moulding. Meas Sci Technol 2000;11:25–31. [16] Schwab S, Levy R, Glover G. Sensor system for monitoring impregnation and cure during resin transfer molding. Polym Compos 1996;17:1–5. [17] Luthy T, Ermanni P. Flow monitoring in liquid composite molding based on linear direct current sensing technique. Polym Compos 2003;24:249–62. [18] Danisman M, Tuncol G, Kaynar A, Sozer EM. Monitoring of resin flow in the resin transfer molding (RTM) process using point-voltage sensors. Compos Sci Technol 2007;67:367–79. [19] Yenilmez B, Murat Sozer E. A grid of dielectric sensors to monitor mold filling and resin cure in resin transfer molding. Compos Part A Appl Sci Manuf 2009;40:476–89. [20] Matsuzaki R, Kobayashi S, Todoroki A, Mizutani Y. Full-field monitoring of resin flow using an area-sensor array in a VaRTM process. Compos Part A Appl Sci Manuf 2011;42:550–9. [21] Di Fratta C, Klunker F, Ermanni P. A methodology for flow-front estimation in LCM processes based on pressure sensors. Compos Part A Appl Sci Manuf 2013;47:1–11. [22] Matsuzaki R, Kobayashi S, Todoroki A, Mizutani Y. Cross-sectional monitoring of resin impregnation using an area-sensor array in an RTM process. Compos Part A Appl Sci Manuf 2012;43:695–702.

[23] Kobayashi S, Matsuzaki R, Todoroki A. Multipoint cure monitoring of CFRP laminates using a flexible matrix sensor. Compos Sci Technol 2009;69:378–84. [24] Jeanmeure LFC, Dyakowski T, Zimmerman WBJ, Clark W. Direct flow-pattern identification using electrical capacitance tomography. Exp Therm Fluid Sci 2002;26:763–73. [25] Soleimani M, Lionheart WRB. Nonlinear image reconstruction for electrical capacitance tomography using experimental data. Meas Sci Technol 2005;16:1987–96. [26] Hirt C, Nichols B. Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 1981;39:201–25. [27] Simacek P, Advani SG. Desirable features in mold filling simulations for liquid composite molding processes. Polym Compos 2004;25:355–67. [28] Endruweit A, McGregor P, Long AC, Johnson MS. Influence of the fabric architecture on the variations in experimentally determined in-plane permeability values. Compos Sci Technol 2006;66:1778–92. [29] Spanos BPD, Ghanem R, Member S. Stochastic finite element expansion for random media. J Eng Mech 1989;115:1035–53. [30] Evensen G. Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J Geophys Res 1994;99:10143–62. [31] Tippett MK, Anderson JL, Bishop CH, Hamill TM, Whitaker JS. Ensemble square root filters. Mon Weather Rev 2003;131:1485–90. [32] Li J, Xiu D. On numerical properties of the ensemble Kalman filter for data assimilation. Comput Methods Appl Mech Eng 2008;197:3574–83.