COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
A Finite Element Approach to Resin Flow during the Resin Film Infusion
Process
M. Yang, S. L. Yan *
School of science, Wuhan University of Technology, Wuhan, 430070 China
Email: [email protected], [email protected]
Abstract A physically accurate and computationally effective finite element method (FEM) is presented to simulate
the isothermal resin infusing process. The FEM is based on conservation of resin mass at any instant of time and is
objective of resin film infusion (RFI) fiber impregnation and mold filling. Darcy’s low is invoked for the resin flow
velocity field, thereby forming a transient governing equation involving the pressure field and the resin saturation fill
factor which tracks the location of the resin front surface. Finite element approximations are then introduced for both
the fill factor and the pressure field, and the resulting transient discrete equations are solved in an iterative manner for
both the pressures and the fill factors for tracking the progression of the resin front in an Eulerian mold cavity. The
FEM primarily works with the finite element mesh geometry, and the control volume regions associated with nodes
need not be specified, so it is superior to the traditional finite element/control-volume (FE/CV) method. Based on the
FEM method, a computer code is developed to simulate the resin infusing visually during the resin film infusion
process. A numerical example presented here also demonstrated that compared with FE/CV method, FEM is
physically accurate and computationally efficient.
Key words: Resin film infusion, finite element method, control-volume/finite element
INTRODUCTION
Resin film infusion (RFI) is a new cost-effective fabrication technique being pursed by NASA and the Boeing
Company to develop wing structures for commercial transport aircraft [1]. The RFI process developed by Boeing
(formerly McDonnell Douglas) consists of an outer mold line tool, an epoxy resin film, a near net shape textile perform,
an inner mold line tool, and a reusable vacuum bag. Thick film plates of resin are placed on the outer mold line tool,
then the dry textile performs and inner mold line tools are placed on the top of the resin. The entire assembly is covered
with a reusable vacuum bag, and the part is placed inside an autoclave [1]. After the resin is melted, vacuum pressure is
used to infuse resin into the preform. Once infiltrated, the part is cured under pressure and temperature in the autoclave.
Dry performs are resin impregnated, consolidated and cured in a single step eliminating costly prepreg tape
manufacture and ply-by-ply layup [2-4].
The large numbers of material properties and processing parameters that must be specified and controlled during resin
infiltration make trial-and–error procedures of determining the processing cycle extremely inefficient. Analytical and
numerical models are clearly far superior alternatives for determination of optimal processing cycles.
The success of the RFI depends largely on the successful impregnating the preform by the resin. Prediction of the
location of the flow front and pressure distributing during the mold filling is useful in addressing several critical issues
in the process. It will allow the designer to optimize parameters such as compaction pressure, infusion time, perform
layup, etc. to achieve acceptable flow pattern before the mold is built and thus avoid potential problems.
Mold filling of resin film infusion process involves a moving flow front, and so presents a moving boundary problem.
The material inside the mold is constantly changing shape as it flows. This makes it necessary to redefine the geometry
of the domain in which the governing equations are to be solved after each successive time step. Approaches to solve
this problem can be classified into either fixed mesh or moving mesh [4]~[6]. The major method currently being used
to simulation polymer resin flow movement of resin film infusion is the finite element/control volume (FE/CV)
method, which does not need to re-mesh. For example, Joohyuk Park and Moon Koo Kang propose a numerical
algorithm for simulating RFI process for thin panels with stiffeners considering the effects of compaction pressure on
⎯ 625 ⎯
the perform based on the FE/CV method [7]; Alfored C. Loos and John D. MacRae etc. choose the finite element
control volume technique to develop a two-dimension resin flow model to determine the position of the resin flow
front and the pressure distribution [8]. The FE/CV approach makes the resin impregnation process to be regarded as a
quasi-steady process even though it is a transient process, assuming a steady state condition at each time step. The
selection of the time step increment for each of the quasi-steady state is based on the consideration that the time
increment used allows only one control volume region to be completely filled （This restriction of the time increment
ensures the stability of the quasi-steady-state approximation）.This may become cumbersome and time-consuming
when higher-order nodes are involved.
A pure finite element method (FEM) is proposed in this study to calculate the flow front movement for mold filling of
resin film infusion. FEM is based on the conservation of resin mass at any instant of time, so it is physically accurate.
Furthermore, this methodology primarily works with the finite element mesh geometry, and the control volume
regions associated with nodes need not be specified, which lead FEM is computationally efficient [4-6].
RESIN FLOW MODEL
An analysis impregnation is based on the conservation of the resin mass at any instant of time flowing through a porous
preform. A fill factor can be implicitly defined to present the amount of resin inside the preform and its distribution at
any time. Considering for a general Eulerian mold domain Ω , a fill factor ( φ ) of one means that the region is saturated
with resin, while a fill factor of zero means that the region contains no resin, regions with fill factors between zero and
one are partly filled. Since the mass conversation is maintained at any instant of time, the continuity equation based on
it can be written as [9]:
r r
∂ ( ρφ )
dΩ + ∫ ρφ (v ⋅ n )dΓ = 0
∂t
Ω
Γ
∫
(1)
r
r
Where ρ is the density of the polymer resin, n is the vector normal to the surface of the domain ( Γ ), v is the
velocity vector.
Expanding the Equation (1), we have
∂ρ
∂φ
r r
∫ ∂t φdΩ + ∫ ∂t ρdΩ + ∫ ρφ (v ⋅ n)dΓ = 0
Ω
Ω
(2)
Γ
If the density is assumed to be constant based on the incompressible condition, then equation (2) can be predigested:
∂φ
r r
∫ ∂t dΩ + ∫ φ (v ⋅ n)dΓ = 0
Ω
(3)
Γ
Employing the Gauss theorem:
r
r r
∫ ∇ ⋅ v dΩ = ∫ v ⋅ n d Γ
Ω
(4)
Γ
Equation (3) can be written as
∂φ
∫ ∂t dΩ + ∫ φ (∇ ⋅ u)dΩ = 0
Ω
(5)
Ω
For describing the polymer resin flow through fibrous reinforcements as in RFI, Darcy’s law is the most commonly
used equation, which can be expressed as:
r
[K ] ∇ P
v = −
(6)
μ
Substituting Eq. (6) into Eq. (5) yields:
∂φ
∫ ∂t dΩ = ∫ φ (∇ ⋅
Ω
Ω
[K ] ∇P)dΩ
(7)
μ
If the pressure gradients are assumed to be negligible in the unfilled and partially filled regions where
0 < φ < 1 ，then only the completely filled regions where φ = 1 are needed to be considered in the governing mass
⎯ 626 ⎯
balance equation. With this consideration, a differential form of the Eq. (7) is given by
[K ] ∇P)
∂φ
= ∇⋅(
μ
∂t
(8)
This is the governing equation for the present finite element formulations.
Solution of the governing equation requires specification of the boundary conditions. At any instant of time, the
pressure at the bottom of the preform (resin film/perform interface) is uniform:
P = P0
(9)
where P0 is the applied compaction pressure. At the flow front:
P=0
(10)
FINITE ELEMENT DISCRETIZATION
To determine the pressure field and the resin saturation fill factor, the mold cavity is discretized into finite elements.
The pressure and fill factor are approximated by the standard finite-element approximations given by:
P = ∑ N i Pi
φ = ∑ N iφ i
(11)
Where N i is the spatial shape function at each node and Pi and φi are the nodal values of the pressure and fill factor,
respectively.
Invoking the Galerkin weighted residual formulation, let Wi = N i ，with Eq.(11), then Eq. (8) is written as:
∫N N
i
j
dΩ
Ωe
∂φ i
[K ] ∇N )dΩ) P + [K ](∇P ⋅ nr ) N dΓ
= (− ∫ (∇N i
j
i
i
∫μ
μ
∂t
Ωe
Γe
(12)
or in a more concise form as:
[M ]e ⋅ ⎧⎨φ ⎫⎬ + [K ]e ⋅ {P} = { f }e
•
⎩ ⎭
(13)
where
[M ]e = ∫ N i N j dΩ
(14)
[K ]e = ∫ (∇N i ⋅ [K ] ∇N j )dΩ
(15)
Ω
Ω
e
μ
e
And
{f }e = ∫ [K ](∇P ⋅ nr) N i dΓ
Γe
μ
(16)
Introducing the finite difference approximation for the time derivative term in the Eq. (13), namely,
•
φ =
φ
n +1
−φ
Δt
n
(17)
Then Eq. (13) can be given by:
[M ]e ⋅ ({φ }n+1 − {φ}n ) + Δt [K ]e ⋅ {P} = Δt { f }e
(18)
The discretized finite element system is then obtained:
[M ]⋅ ({φ }n+1 − {φ}n ) + Δt [K ]⋅ {P} = Δt {f }
(19)
SOLUTION STRATEGY
At the beginning of the simulation the fill factors are set to equate one at the interface between resins and perform.
⎯ 627 ⎯
Then at each time step, values for the pressure field and the fill factor are iteratively computed until a mass
conservation is reached. The procedure is expressed as below:
(1) At the beginning of each time step,
{φ i }m n+1 = {φ i }n
(20)
where superscript n + 1 and n are the current and previous time step, respectively, and subscript refers to the m th
iteration.
(2) Apply the boundary conditions on [ K ] and solve the pressure from Eq. (19), which can be expressed as:
[K ]⋅ {P} = {f }+
[M ]⋅ ({φ }n − {φ }m n +1 )
(21)
Δt
(3) Update the nodal fill factor using the modified form of the Eq. (19),
{φ }m+1n+1 = {φ }n + [M ]−1 ⋅ (Δt { f }− Δt [K ]⋅ {P})
(22)
It must be noted that the fill factor can be either greater than one or less than zero.
(4) Check for convergence using
{φ }m n+1 − {φ }m+1 n +1 ≤ ε
(23)
(5) If convergence is not reached, reset the iteration counter m and the fill factor solution
{φ }m n+1 = {φ }m+1n+1
(24)
And back to step 2; if convergence is reached, go to next time step.
NUMERICAL EXAMPLE
In order to validate the present numerical developments and to demonstrate the applicability to RFI simulation
involving isothermal conditions, a blade-stiffened panel (see Fig. 1) is considered there. The evaluation criteria
involved predictions for the resin flow front and the pressure distribution in the fiber perform. The compaction
pressure, permeability of the preform and the viscosity data are given in Table 1.
Table 1 Material property parameters
Parameters
Magnitude
Compaction pressure, P (Pa)
1 × 105
2
Perform permeability (m )
Resin viscosity, μ (N/m2)
Figure1: Dimensions of the blade-stiffened panel
⎯ 628 ⎯
K x = 2.5 × 10 −10
K y = 1 × 10 −10
0.5
Figure 2: Mold mesh geometry
The numerical simulations are based on employing a two-dimensional quadrilateral mesh of 876 nodes and 747
elements as shown in Fig.2. The flow front contours and pressure distribution based on the present formulations with a
time step size of 1.0 are shown in Fig.3 and Fig.4, respectively.
Since the resin film is placed under the fiber performs, the fluid comes in at the bottom of perform and flows up
gradually. At shown in Fig.3, the resin flows far faster at the beginning than the latter, which is associated with the
pressure. During the resin infusing, pressure distribution keeps changing with time because of the moving of the flow
front. The front distribution shown in Fig.4 corresponds to the time near the completion of the resin infusing.
Figure 3: Flow front contours ( Δt = 1.0 second)
Figure 4: Pressure distribution ( Δt = 1.0 second)
In order to compare the present method with traditional FE/CV method, this example is simulated based on the FE/CV
formulation as well. The flow front location as computed by the pure finite element formulation is compared with the
finite element/control volume formulation as shown in Fig.5. From the figure, it is clear that the finite element
formulation gains the same outcomes as that’s by finite element/control volume. While the traditional finite
element/control volume method current commonly employed in the RFI simulations requires the restriction of time
step increment sizes based on the condition that only a single control volume region is filled at each time step, the
present formulations do not involve these computations and hence are computationally efficient. The computational
advantages of the present developments for the RFI simulations are clearly seen from Fig. 6.
Figure 5: Comparison with FE/CV
Figure 6: Comparison of computational times
The location of the flow fronts computed using different time step sizes employing the finite element formulations are
shown in Fig.7. This clearly suggests that the predicated locations of the flow fronts are independent of the time step
size.
Figure 7: Finite element comparisons
CONCLUSIONS
The paper described a pure finite element based methodology for RFI infiltrating simulations. We can predict the flow
front and pressure distribution in various times using the FE method. The FE methodology has the same accuracy and
⎯ 629 ⎯
significant computational advantages compared to the traditional FE/CV approach. The present method has no time
step increment restriction as in the FE/CV approach and works only with finite element mesh geometry, making its
application easy. The temporal flow front progression during resin infusing analysis simulations are independent of the
time step size employed in the analysis.
Acknowledgements
The support of National Science foundation of P.R. China (Grant No.50573060) is gratefully acknowledged.
REFERENCES
1. Dow MB, Dexter HB. Development of stitched, braided and woven composite structures in the act program and
at Langley research center (1985 to 1997). NASA/TP-97-206234.
2. Resin film infusion-composites cost reducer. Reinforced plastics. 2002, pp. 44-49.
3. Han NL, Suh SS, Yang JM, Hahn HT. Resin film infusion of stitched stiffened composite panels. Composites:
Part A, 2003; 34: 227-236.
4. Young WB, Han K, Fong LH. Flow simulation in molds with preplaced fiber mats. Polymer composites, 1991;
12(6): 391-403.
5. Bruschke MV, Adrvuni SG. A finite element/control volume approach to mold filling in antistrophic porous
media. Polymer composites, 1990; 11(6): 398-405.
6. Mohan RV, Ngo ND, Tamma KK. On a pure finite-element-based methodology for resin transfer mold filling
simulations. Polymer engineering and science, 1999; 39(1): 26-43.
7. Park Joohyuk, Kang Moon Koo. A numerical simulation of the resin film infusion process. Composite structures,
2003; 60: 431-437.
8. Loos AC, Mac Rae JD. A process simulation model for the manufacture of a blade-stiffed panel by the resin film
infusion process. Composites Science Technology, 1996; 56: 273-289.
9. Kong Xiangyan. Advanced seepage mechanics. Press of University of Science and Technology of China, 1999.
⎯ 630 ⎯