Finite Elements in Analysis and Design 47 (2011) 764–770

Contents lists available at ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/finel

3D numerical simulation of filling and curing processes in non-isothermalRTM process cycle

Fei Shi n, Xianghuai Dong

Department of Plasticity Technology, Shanghai Jiaotong University, Shanghai 200030, China

a r t i c l e i n f o

Article history:

Received 10 June 2009

Received in revised form

19 December 2010

Accepted 27 February 2011Available online 2 April 2011

Keywords:

3D numerical simulation

Unstructured tetrahedron mesh

Non-isothermal RTM

4X/$ - see front matter Crown Copyright & 2

016/j.finel.2011.02.007

esponding author.

ail address: sh_fi@sjtu.edu.cn (F. Shi).

a b s t r a c t

It is important to simulate non-isothermal RTM process cycle due to the high speed of filling mold

during the filling stage and the long curing period during the curing stage. In this paper, we present

numerical formulations for resin flow based on the concept of quasi-steady state situation at the filling

stage and for resin cure at the curing stage. To make sure the applicability to complex product shapes,

we use the four-node unstructured tetrahedron mesh, based on which the numerical formulation of

temperature and cure convection is developed. The validity of our method is established in the case

where flexible meshes are used. The results show that the numerical procedure, tested on known data,

provides numerically valid and reasonably accurate predictions.

Crown Copyright & 2011 Published by Elsevier B.V. All rights reserved.

1. Introduction

The Resin Transfer Molding (RTM) process cycle consists oftwo sequential stages, i.e. the filling and cure stages. The simula-tion of non-isothermal RTM process is very difficult because theflow, heat transfer, and resin cure are highly interrelated at thefilling and curing stages. The flow pattern and cure process arestrongly affected by heat transfer and resin cure since theviscosity changes as a function of temperature and conver-sion [18,19], which in turn are determined by the flow and cure.In order to model the process accurately, the flow, heat transfer,and resin cure must be incorporated correctly.

In most of the previous researches [5,11,3,15,14,24], the simula-tions were simplified by calculating the flow under the two-dimen-sional (2D) setting and assuming the heat diffusion to be only in theplane. Because of the deeply decreased computation cost, this methodwas widely applied even though the generated results could beinapplicable for the actual process. Some previous researches [7,9,13]developed 2.5-dimentional (2.5D) non-isothermal models, whichinclude the flow in plane and heat diffusion in the thickness for thethin parts based on the Control Volume/Finite Element Method (CV/FEM). In Ref. [1], the authors presented a numerical simulation topredict the flow pattern, extent of reaction, and temperature changeduring filling and curing in a thin rectangular mold. In practice, 2D or2.5D model is in general not valid, especially for thicker parts. Theflow in the thickness must be carefully taken into account and theheat transfer and resin cure must be computed in three-dimensional(3D) space. To address this issue, in Ref. [16], the authors considered

011 Published by Elsevier B.V. All

flow within each nodal volume as a one-dimensional (1D) flowregardless of the number of upstream or downstream nodes based onthe Lagrangian interpretation of the first order upwind scheme. Insome other researches [22,23], the 3D simulation of filling wasperformed during the RTM using the CV/FEM based on regularstructured mesh. There are also previous studies, say Refs. [4,10],focusing on applying the 3D Galerkin finite element methods tosimulate the filling stage or the curing stage. In these researches, thestructured hexahedral mesh was used and hence the flexibility ofdealing with complicated shapes was not sufficient. Therefore, theywere limited to regular parts and simple flow fields. When calculatingcomplicated shape parts, the results are usually not satisfactory. Inthe presented paper, by using the four-node unstructured tetrahedronmesh that fits complicated shapes and flow fields, we derive thediscretization schemes of the energy equation and the chemicalequation based on upwind scheme in the three directions. Hence,our method can be widely applied to different kinds of parts andsignificantly improve the calculation precision.

The rest of the paper is organized as follows. In Section 2, weprovide the basic formulation used in the paper. Following it,in Sections 3 and 4, we discuss the derivation of numericalformulations based on four-node unstructured tetrahedron meshand calculation procedure, which is used during developing code.We then present a case study to verify our method on knowndataset in Section 5.

2. Control equations

In this section, we present the basic control equations. For thenon-isothermal RTM process under investigation, we make the

rights reserved.

F. Shi, X. Dong / Finite Elements in Analysis and Design 47 (2011) 764–770 765

following assumptions:

(1)

The resin is an incompressed liquid and its viscosity is afunction of the temperature and curing ratio.

(2)

The temperature of the resin and fiber is identical at eachposition.

(3)

There is no resin before the flow front and the control volumeis fully saturated after the flow front.

(4)

There is a quasi-steady time window, in which the viscositydissipation energy can be neglected.

(5)

Resin stops flow at the curing stage.

2.1. Flow control equation

At filling stage, Darcy’s equation, which describes how theresin flows in the fiber, is still applicable. In the case, Darcy’s law,which describes the flow within the mold, can be written asfollows:

v¼�1

mKrp ð1Þ

where m is the Darcy velocity vector, K is the permeability tensor,rp is pressure gradient, and m is the resin viscosity, which is afunction of the resin temperature and degree of cure given below

m¼ m0eEm=RT ag

ag�a

� �d1þd2að2Þ

where m0 is the initial degree of viscosity, Em is the activationenergy of resin, ag is the degree of cure of the resin gel point, d1

and d2 are two constants, R is the gas constant, and T and a are thetemperature and the degree of cure, respectively.

Based on the quasi-steady state assumption, the mass balanceequation at each node is given by

�rUv¼ 0: ð3Þ

Substituting Eq. (1) into Eq. (3), the flow control equation canbe expressed as

ZZS

1

m nx,ny,nz

� �K

@p@x@p@y

@p@z

2664

3775dS¼ 0 ð4Þ

where nx, ny, nz is the normal tensor n in x, y, z directions of thesurface of the integrated volume, and S is the surface in which thecontrol volume is enclosed.

The possible boundary conditions are:

(1)

p¼p0 for constant pressure injection or u¼u0 for constantvelocity injection at the inlet;

(2)

p¼0 at the flow front; (3) qp/qn¼0 at the mold wall.

Fig. 1. Control volume composed of the tetrahedron element.

2.2. Energy equation

For the 3D simultaneous equations, it is supposed that the heatconvection of the resin and fiber occurs at the same time, that isto say, Tf¼Tr¼T. In the filling stage, the energy equation of thebalance mode is written as

frrcprþð1�fÞrf cpf

h i @T

@tþrrcpr vx

@T

@xþvy

@T

@yþvz

@T

@z

� �

¼ kL@2T

@x2þ@2T

@y2þ@2T

@z2

!þf_s ð5Þ

where f is the porosity, rr and rf are the resin density and thefiber density, respectively, cpr and cpf are the resin specific heatand the fiber specific heat, respectively, vx, vy, vz is the velocity inx, y, z directions, kL is the whole heat conductivity of fiber andresin, which can be expressed as

kL ¼krkf

krWf þkf Wr,

where

Wr ¼frf

�frf

þ1�frr

!, Wf ¼ 1�Wr

_s denotes the quantity of the heat during the resin curing process,which can be expressed as _s ¼DHGða,TrÞ, where DH is the heat ofreaction and the curing kinetics equation G(a,Tr) (based onKamal’s model [20,8,12] is expressed as follows:

Gða,TrÞ ¼ ðA1e�E1=RTrþA2e�E2=RTram1 Þ 1�að Þm2

where A1, A2, E1, E2, m1, m2 are all constants, and t is the timevariable. In Section 5, we will glimpse at curves of G vs. t and a vs.t at variable temperatures in Fig. 4.

In the above equation, the first part of the left hand sidedescribes energy change of the resin and fiber, and the secondpart corresponds to the heat convection of resin. The first part ofright hand describes the heat conduction of resin and fiber, andthe second corresponds to heat release due to curing of resin.

The boundary conditions are given as

(1)

T¼T0 at the inlet gate; (2) T¼Tf or kL@T=@n9ff ¼ ð1�fÞrf Cpf vnðTf0

�TÞ at the flow front [2];

(3) T¼Tm at the mold wall.

Clearly, the boundary condition imposed at flow front may beexpressed either by a constant temperature equal to the tem-perature of the fiber mat or a heat balance equation. Comparisonof different boundary conditions provided by Antonucci et al. [2]shows that both the constant temperature and the heat balanceequation lead to similar temperature results at the flow frontespecially at the end of mold filling. We use both the constanttemperature and the heat balance equation given earlier asthermal boundary conditions in our numerical simulation.

When filling is over, the process is changed into curing and theresin stops flow in the fiber. So, in Eq. (5), vx¼vy¼vz¼0.

2.3. Chemical equation

In the resin curing process, it is assumed that moleculardiffusion of macromolecules is negligible. At the filling stage,the resin chemical reaction follows the chemical equation that is

F. Shi, X. Dong / Finite Elements in Analysis and Design 47 (2011) 764–770766

provided in Ref. [21]:

f@a@tþvx

@a@xþvy

@a@yþvz

@a@y¼fGða,TÞ ð6Þ

The boundary conditions are given by

(1)

a¼a0 at the inlet; (2) (d/dt($a)¼$G(a, T)) at the flow front,

where $ is filling coefficient of control volume.

Fig. 2. CV P and adjacent CV E among unstructured meshes.

Calculate the m

Calculate p

Calculate v i

Start

Input da

Calculate number of the sub

Calculate T and α

Calculate μ

Update the flo

N Converg

All nodes f

Y

Y

Output the re

Cure proc

Solve T and α

ConvergencT andα

Cure finish

Y

Y

Y

Fig. 3. Flow

In the presented work, the changes in physical properties ofthe resin, i.e., the density, the specific heat capacity, and thethermal conductivity are ignored due to resin polymerization. Infact, it has been observed that ignoring these changes in thesimulation does not affect the results [17].

The same as Eq. (5), when the filling is over and the curingstarts, vx¼vy¼vz¼0.

3. Numerical formulation

Before the differential equations are used to simulate the non-isothermal RTM process, it must be integrated in the controlvolume. Throughout the filling mold process, the shapes of theresin saturated field and the flow front vary instantaneously.While the finite element method can be used, much computationtime will be wasted since this computation is not needed formany nodes during most calculation steps. Therefore, researchershave developed the Volume Of Fluid (VOF), which possesses manyadvantages in solving moving boundary problems [11].

odel data

in Eq.(8)

n Eq.(1)

ta

-time step in Eq.(11)

in Eq.(10)

in Eq.(2)

w front

Sub-time step

N

e?

illed?

sults

ess?

in Eq(10)

e in ?

N

ed? N

N

chart.

F. Shi, X. Dong / Finite Elements in Analysis and Design 47 (2011) 764–770 767

In our method, each tetrahedron element, connected by theirvertices, is further divided by joining the midpoints of its fouredges and centroids of three planes to its centroid. The controlvolume associated with a node is then defined as the union of allsubdivided volumes that are connected to that node, as shown inFig. 1. Control volume of node Q is encircled by plane O14, O15,O35, O36, O26, O24, Q14, Q15, Q35, Q36, Q26, and Q24 in theelement. Similar to the elements, the control volume spansthe entire computational domain without overlapping. In trackingthe flow front, the coefficient $ is used to indicate whether theassociated control volume is filled entirely, partly, or not at all. Inparticular,$¼1, if the associated control volume is filled entirely,$¼0, if it is not filled at all, and 0o$o1, if it is filled partly.

Based on the finite element method, the pressure in theelement is expressed as follows:

P¼X4

k ¼ 1

Nkpk ð7Þ

where Nk is the shape function and pk is the pressure.Substituting Eq. (7) into Eq. (4) and assuming that the

principal axis of perform fiber is consistent with that of the resinflow, the flow control equation is written as

XNe

Z ¼ 1

8>>>><>>>>:

1

6DVmX3

x ¼ 1

fso14½nx,ny,nz�o14þso15½nx,ny,nz�o15gx

�

kxx 0 0

0 kyy 0

0 0 kzz

0B@

1CA

bi bj bm bl

ci cj cm cl

di dj dm dl

264

375

pi

pj

pm

pl

266664

377775

9>>>>=>>>>;

Z

¼ 0 ð8Þ

Table 1Parameter of resin and fiber used in simulation [23].

Parameter Value Parameter Value

rf/(kg/m3) 2560 m0/(Pa s) 2.78�10�4

rr/(kg/m3) 1100 Em/(J/mol) 18,000

kf/(W/m K) 0.0335 ag 0.14

kr/(W/m K) 0.168 d1 1.5

cpf/(J/kg K) 670 d2 1.0

cpr/(J/kg K) 1680 A1/(s�1) 3.7833�105

kxx¼kyy¼kzz/(m2) 2�10�9

A2/(s�1) 6.7833�105

f 0.7 E1/(J/mol) 54,418

m1 0.3 E2/(J/mol) 50,232

m2 1.7 DH/(J/m3) 1.54�108

Fig. 4. Curing kinetics curves: (a) Reaction rate

where Ne is the number of elements that includes node Q, S is thearea of control surface, DV is volume of control volume, and kxx,kyy, and kzz are components of the permeability tensor, bi, bj, bm, bl,ci, cj, cm, cl, di, dj, dm, and dl are element variables. pi, pj, pm, and pl

are node pressure of elements.For Eqs. (5) and (6), the discretized equations are derived using

differential scheme based on unstructured tetrahedron mesh.Taking into account the flowing direction, the upwind scheme isapplied and the calculation results are free of vibration. As aresult, it has been widely used for a long time. On the other hand,the differential equations raise the instantaneous problems. Thetime domain must be discretized. There are three commonmethods for the discretization: the explicit method, the implicitmethod, and the Crank–Nicolson method. Because the implicitmethod is stable under any conditions, it has been applied widely.

Eqs. (5) and (6) can be unified by

k1@x@tþk2divðvxÞ ¼ k3divðgradxÞþsPC ð9Þ

where k1, k2, and k3 are coefficients from Eqs. (5) or (6) and x isthe variable of temperature T and cure convection a. Eq. (9) isdiscretized in the entire calculation field and integrated in thecontrol volume based on unstructured elements. There is aninterface at least for two adjacent node control volumes. To anytwo adjacent control volumes, we can show in Fig. 2. In Fig. 2, n isthe area normal vector of interface, m is the Flow velocity, and k isthe vector between two adjacent nodes. So

n¼ nxiþnyjþnzk,

v¼ vxiþvyjþvzk,

k¼ PE,¼ ðxE�xPÞiþðyE�yPÞjþðzE�zPÞk

Therefore, different parts of Eq. (9) can be expressed as [6]ZDt

ZDV

k1@x@t

dVdt¼ k1DVðx�x0ÞDt,

ZDt

ZDV

k2divðvxÞdVdt¼ k2xvUn� �

Dt,

ZDt

ZDV

k3divðgradxÞdVdt¼ k3nUk

9k92Dt,

ZDt

ZDV

sPCdVdt¼ ðsCþsPxÞDVDt,

where Dt is the time step, sPC¼sCþsPx, with sC and sPr0 beingtwo constants to linearize sPC.

s vs. time and (b) Degree of cure vs. time.

F. Shi, X. Dong / Finite Elements in Analysis and Design 47 (2011) 764–770768

Finally, Eq. (9) is rearranged as follows:

aQxQ ¼XNs

Z ¼ 1

aZxZþbQ ð10Þ

where: aQ ¼SNs

Z ¼ 1aZþðk1DVÞ0=Dt�sPDV , 0 is the result from thelast iteration, Ns is the number of adjacent nodes, xQ is thetemperature or degree of cure node Q, xZ is the temperature ordegree of cure of adjacent nodes.

aZ ¼DZþmaxð0,�FZÞ, DZ and FZ are the diffusive and convec-tive parts given by DZ¼k3n . k/9k92 and FZ¼k2vUn, respectively,and k is a vector between adjacent nodes.

bQ ¼ðk1xDVÞ0

DtþsCDV

For implicit time integration method, if large time steps areused in the flow equation, one can end up with large truncationerrors. To overcome this deficiency, it is more appropriate todivide the time steps into smaller sub intervals. The number ofsub intervals may be obtained by the following expression:

N¼ Intcmax

g

� �ð11Þ

where the constant value g should be between 0.7 and 1.1 for thebest accuracy. cmax is the largest Courant number in all filledelements.

cmax ¼Maxðc1,c2,c3:::ckÞ

Fig. 5. Part shape and tetrahedron mesh.

Fig. 6. Temperature vs.time at selected point A and C: (a) Simulation and experi

In an individual element, this number is expressed by

ck¼

9v9Dt

Dl

where Dl is the smallest side of the element and Dt is thecalculation time step.

4. Calculation procedure

We have provided the basis of non-isothermal RTM process,and have obtained Eqs. (8) and (9), which are used to calculatepressure, temperature, and cure convection. Now we are readyto present our computation procedure. The flow chart is shownin Fig. 3. Applying the control equations combined with theboundary conditions we simulate the process on a computer with1.8 G Intel Pentium D Processor, 1 GB Memory, running windowsXP operation system. We calibrate our model based on ourpractice. Finally, we develop codes to analyze the filling andcuring processes.

5. Numerical examples

There is a case provided in Ref. [23] with a lot of data, whichare shown in Table 1. According to parameters of resin in thiscase, we can depict the curing kinetics equation in Fig. 4 at 45, 60,and 75 1C.

ment during filling and (b) Simulation and experiment during process cycle.

Fig. 7. Degree of cure at point A and C during process cycle.

F. Shi, X. Dong / Finite Elements in Analysis and Design 47 (2011) 764–770 769

As shown in Fig. 4(b), the degree of cure increases when timeis passed at same temperature. At the same time, while tempera-ture increases, it does too. Fig. 4(a) shows relation of the curingreaction rates vs. time is different from that of the degree of curevs. time. While starting, the reaction rates increase over time.When they transit the peak, the situation is changed. In addition,the peak appears earlier when temperature increases. So, it isimportant to choose the fit temperature of mold and fiber, and thepeak does not appear at the filling stage.

In this case, the resin is injected at 20 1C and the mold walltemperature is kept constant at 60 1C. The initial fiber mattemperature is equal to the mold wall temperature. The resin isinjected under constant flow rate of 6�10�6 m3/s through aninlet located at the top of the mold. The part shape is shown in theFig. 5. And the location of point A and point C are the same to

Fig. 8. Predicted pressure history at the inlet.Fig. 10. Predicted temperature slice and curing slice at the end of process cycle:

(a) Temperature slice in X-direction and (b) Curing slice in X-direction.

Fig. 9. Predicted flow pattern: (a) t¼0.017 s, (b) t¼1.64 s, (c) t¼6.4 s and (d) t¼10.05 s.

F. Shi, X. Dong / Finite Elements in Analysis and Design 47 (2011) 764–770770

Ref. [23]. The geometry is discretized into 474 nodes and 1372four-node tetrahedron elements.

Fig. 6(a) and (b) shows the temperature curve predicted by ourcomputer code and obtained from the experiment [23] at twoselected points within the mold cavity. It is clear that they arevery similar and therefore, our code is proper for non-isothermalRTM to be simulated. The filling time-consume is about 11 s, andthe process cycle is 600 s.

Fig. 7 shows the curing curve depicted by our computer codeat two selected points. As shown, the reaction was mainlycompleted at the curing stage.

Fig. 8 is the predicted pressure history at the inlet. Theinjection pressure is lower than 0.2 MPa in the filling.

Fig. 9 is the predicted flow pattern. Accounting for flowpattern, the outlet can be set up at the place where the resinflow ends, namely, the left and right side of the part. Or else thedry spot will appear at the end of part in that case.

Fig. 10 is the predicted temperature slice and curing slice atend of curing in the X-direction. As shown, for temperature ofinterior higher than in exterior, the degree of cure of interior isbigger. Analyzing the two figures, we find that the difference ofthe temperature and the cure between the interior and theexterior is bigger, and it is higher than 20 1C at some locations.Therefore, when a part is taken out from mold, the residual stressdeforms it due to the release of stress. This issue must be takeninto account by designers.

6. Conclusions

Using the unstructured tetrahedron mesh, the numericalsimulation for 3D non-isothermal RTM process cycle is a usefultool to predict the molding process. In particular, we conclude thefollowing

(1)

The unstructured four-node tetrahedron mesh has strongeradaptability to part shapes and complicated flow field thanstructured mesh. The applicability of our program is shownfor the flexible mesh.

(2)

The 3D numerical simulation provides more information thanthe 2D simulation. Only a middle plane is studied in the 2Dsimulation. using 3D simulation, information at any locationin the part can be obtained. It is helpful for the followingprocesses to be carefully designed before a mold is built.

(3)

The examples presented in this paper illustrate the flexibilityand effectiveness of the developed computer code for thesimulation of complex mold cavity.

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