three-dimensional numerical flow simulation of resin...

THREE-DIMENSIONAL NUMERICAL FLOW SIMULATION OF RESIN TRANSFER

MOLDING PROCESS WITH DRAPING ANALYSIS

Sejin Han, *Mehran Ebrahimi, *Massimiliano Moruzzi, **Doug Kenik

Autodesk, 2353 North Triphammer Road, Ithaca, NY 14850, USA

* Autodesk, 210 King Street East, Toronto, ON M5A 1J7, Canada

**Autodesk, 203 S. 2nd St, Laramie, WY 82070, USA

Abstract

In this paper, the numerical flow simulation of

thermoset materials in resin transfer molding (RTM)

process with draping analysis is described. It gives an

introduction, theory and methods of analysis for the flow

and draping. Then, two example cases are shown. One is

a simple case where the accuracy of the solution can be

checked against analytical solutions. Another is the case

where the effect of using draping analysis in the RTM

flow simulation is shown. The simulation results in this

study are in good agreement with analytical solutions

where analytical solutions are available. They also verify

the significance of considering draping analysis results in

RTM flow simulations.

Introduction

The fiber composite materials have several

advantages over other materials (such as metals) in terms

of weight reduction, design flexibility, corrosion

resistance and reduced noise transmission [1]. The Resin

Transfer Molding (RTM) is one of the most popular

methods used in producing parts with fiber composite

materials. In the RTM process, the resin is forced to flow

through a cavity in which reinforcing preform (also called

fiber mat) is present [1]. The preforms (reinforcements)

are present in the mold as dry form. RTM process has the

advantages such as applicability to wide range of

components, and adjusting the fiber orientation to meet

the structural requirements and the use of lightweight

molds for the production [1]. This study is to analyze the

flow of thermoset materials in RTM process.

Nowadays, in the era of automation and advanced

manufacturing techniques, computer simulations imitating

real physical phenomena could help engineers to avoid

the time-consuming process of trial and error for creating

new designs. Consequently, developing more reliable

computer-aided-design (CAD) tools is a necessity in

today’s boom of new fabrication technologies. Thus,

exploiting numerical simulations, such as draping and

RTM simulations, in composite industries can extensively

reduce the manufacturing costs and accelerate the

processes.

In this study, three-dimensional numerical

simulations will be used to analyze the flow during RTM

processes. The three-dimensional analysis has the

following advantages compared to those using mid-plane

or dual-domain meshes. First, it can simulate anisotropy

of permeability in the thickness direction. Second, it can

handle complicated geometry cases better. The simulation

method developed in this study can also handle gravity

and venting effects.

The fiber mats undergo deformations as they are

shaped into complex geometries. The draping analysis is

utilized to analyze the deformation of the fiber mat [1].

For more accurate analysis of flow during RTM process,

the draping analysis needs to be used. In this study,

draping analysis is performed, and the result from the

draping analysis is used in the RTM flow simulation.

In the next sections, the simulation methods for the

flow and draping will be described, followed by some

example cases.

Governing Equations for Flow Analysis

To analyze the flow of resin during RTM process, we

need to solve the following set of equations [1, 2, 3].

First, we solve the continuity equation.

0)(

u

t

(1)

For the momentum equation, we solve the following

simplified equation (Darcy’s equation) [1].

pKu

1 (2)

In the above equation, u

is the superficial velocity, 𝜂

the viscosity, 𝐾 permeability and 𝑝 is the pressure [1].

SPE ANTEC® Anaheim 2017 / 787

The superficial velocity is related to the macroscopic

velocity (�⃗�) through the following equation:

�⃗� = �⃗⃗�/𝜙 (3)

where 𝜙 is the porosity. The permeability 𝐾 is a tensor

which can be represented in 3D as components in 𝑥, 𝑦 and

𝑧 directions as below:

𝐾 = [

𝐾𝑥𝑥 𝐾𝑥𝑦 𝐾𝑥𝑧

𝐾𝑦𝑥 𝐾𝑦𝑦 𝐾𝑦𝑧

𝐾𝑧𝑥 𝐾𝑧𝑦 𝐾𝑧𝑧

] (4)

The temperature will be solved using a similar

equation as that for reactive molding simulation, but some

terms are modified to account for the presence of fiber

mat and resin in the system. In this equation, the term

without a subscript r is for the composite, and the term

with subscript 𝑟 is for the resin.

tHTkTuC

t

TC rprrp

)(

(5)

The curing will be analyzed by solving the following

equation due to Kamal [4]:

)1)(( 21 QQDt

D (6)

where the terms Q1 and Q2 are described below:

))((

11

1

T

E

eAQ

))((

22

2

T

E

eAQ

(7)

A three-dimensional finite element method is used to

solve the above set of equations as in [3]. Tetrahedral

elements are used in the current simulation.

Draping Analysis

In this paper, a hybrid finite element-geometric

algorithm for draping simulation of woven fabric

composites over a triangulated 3D surface is used. In this

algorithm, a fabric, before draping, is considered as a

group of square cells, and each cell is modeled as four

side-springs and two diagonal springs connected together

(Figure 1). These assumptions cause a trade-off between

accuracy and speed of draping simulation. However, the

generated results could be trustworthy for the majority of

cases, and if more accurate modeling is required, other

techniques exploiting pure finite element analysis (FEA)

[5]–[9] could be implemented with considerably higher

computational costs.

Figure 1: Fabric cells represented by six springs used in

draping analysis.

This draping technique is built upon the fact that the

fabric draping is optimum when the amount of distortions

(wrinkles) in it is minimum and the magnitude of forces

applied to each fabric node after draping, due to the

change in spring lengths, is zero.

After draping, depending on the surface geometry,

the fabric can wrinkle and cells may distort and no longer

be a perfect square. Shear angle, defined as 𝛾 by Equation

(8), at each fabric node is used as a representation of

wrinkles in the fabric at that location (Figure 2). In other

words, the higher the shear angle is, the more severe the

wrinkles are at that location. In the following equation, 𝛼′

is the angle between 2 sides after the draping.

𝛾 =𝜋

2− 𝛼′ (8)

(a) (b)

Figure 2: (a) undistorted cell before draping and (b)

distorted cell after draping.

Unlike FEA-based methods that require an initial flat

(2D) fabric as an input, the algorithm used in this paper

does not initiate the simulation from a 2D pattern. This is

another advantage of this method over these techniques.

The process begins from a given starting point (seed

point) and propagation direction (local fiber orientation at

the seed point). Other required inputs are:


The triangulated surface to be covered by the

fabric

Fabric cell size along the sides

The ratio of spring constants along the sides and

diagonal directions. The higher this value, the

more resistant the fabric is against stretching

along the sides versus that along diagonals,

which results in shear.

Example Cases

In this section, two example cases of RTM will be

presented. One example is a simple case where draping

analysis is not needed. Another example is where draping

analysis is used.

1. Flat Plate Case

In the first example, a flat-plate case is used. The

geometry of the part used in this case is shown in

Figure 3. The part has dimension of 300 x 100 x 2

mm. The injection is done along one side of the part.

The fluid is Newtonian, and the flow is isothermal.

The viscosity of the fluid is constant at 0.4 Pa-sec. The

permeability is isotropic with a value of 1.0x10-9

[m2/sec]. The porosity is set at 0.5. This simple case is

chosen as an example because the simulation results

can be compared with analytical solutions.

Figure 3: Mesh used in the analysis of 3D RTM case.

The analysis was done for a constant flow rate

with and without venting analysis. Because of the

simple geometry, no draping analysis was done for

this case. For the case of constant flow rate without

venting analysis, the injection pressure (pressure at the

end of filling) can be calculated from the following

equation [1]:

𝑝 = 𝑄𝜂𝐿/𝐴𝐾 (9)

where 𝑄 is the flow rate, 𝜂 is the viscosity, , L is the flow

length, 𝐴 is the cross-sectional area and 𝐾 is the

permeability. A constant flow rate of 0.5 cm3/sec was

applied. By substituting the values for this case, we get

0.3 MPa as the injection pressure from the analytical

equation.

Figure 4 shows the pressure near the end of filling for

this case obtained from the simulation. This result shows

the pressure value of 0.299 MPa. The simulation result is

very close to the value from the analytical equation.

Figure 4: Pressure (in MPa) plot near the end of filling

obtained from a simulation for the case of a constant flow

rate of 0.5 cm3/sec.

Next, a venting analysis was done for the constant

flow rate of 0.5 cm3/sec. For this case, air pressure was

applied at the flow front during filling stage. The venting

pressure at the exit was set at -0.03 MPa. The negative

pressure means that a partial vacuum is applied. Figure 5

shows the venting pressure (which is the pressure in the

region not occupied by resin) obtained from the

simulation. Figure 6 is the pressure near the end of filling.

The value is 0.269 MPa. The analytical solution is 0.27

MPa. The calculation and the analytical solution show

good agreement.

Figure 5: Vent region pressure (in MPa) plot.

Figure 6: Pressure (in MPa) plot near the end of filling

obtained from a simulation for the case of a constant flow

rate of 0.5 cm3/sec with venting pressure of -0.03 MPa.


2. Hemisphere Case

The second case used in this study is for a part

with hemispherical geometry. The geometry of the

part used in this case is shown in Figure 7(a). The part

has a diameter of 100 mm and thickness of 3 mm. The

injection is done at the center of the hemisphere. The

resin is Arotran Q6055 from Ashland Chemical, and

the fluid is non-Newtonian. The flow is non-

isothermal with the initial melt temperature of 30oC

and the mold temperature of 90oC. The injection time

is 10 sec. The permeability is anisotropic with the

value of 3.1x10-11

[m2/sec] in the first principal

direction, 1.9x10-11

[m2/sec] in the second principal

direction and 5.7x10-11

[m2/sec] in the third principal

direction. The porosity is set at 0.43. The first and the

second principal directions are along planar directions,

and the third principal direction is in the thickness

direction.

For this case, draping analysis is performed. The

results from the draping analysis are shown in Figure

7. Figure 7(a) shows the triangulated surface for the

draping analysis. Figure 7(b) shows the draping results

and shear angle distribution after draping. As can be seen

from this plot, the shear angle is high along some of

the edges. Figure 7(c) shows the plot of triangulated

surface and draped composite together.

Flow analysis has been performed using the principal

directions calculated from the draping analysis. As for the

effect of shear angle, analyses were performed with and

without the effect of shear angle. This is to see the effects

of shear angle on the flow results. The effect of shear

angle on flow has been considered in the analysis by

changing the fiber volume fraction and the permeability

as in [10].

Flow simulation results are shown in Figures 8 - 11.

Figure 8 shows the fiber-mat orientation calculated from

draping analysis. Figure 8(a) is for the first principal

direction, and Figure 8(b) is for the second principal

direction. The value is the component value in y-

direction. Figure 9 shows the shear angle calculated from

draping analysis. It shows that the maximum shear angle

is about 60 degrees.

Figure 10 shows the fill time results obtained from

the flow analysis. Figure 10(a) is with the effect of shear

angle, and 10(b) is without. As can be seen, the shear

angle changes the filling pattern where the shear angle is

high.

Figure 11 shows the pressure results near the end of

filling obtained from the flow analysis. Figure 11(a) is

with the effect of shear angle, and 11(b) is without. As

can be seen, including the effect of shear angle increases

the pressure for the current case.

(a)

(b)

(c)

Figure 7: (a) A triangulated hemisphere, (b) shear angle

distribution of the draped composite and (c) the

hemisphere and draped composite together.


(a)

(b)

Figure 8: (a) Mat orientation in the first principal

direction and (b) the second principal direction.

Figure 9: Shear angle calculated from draping analysis.

(a)

(b)

Figure 10: (a) Fill time calculated with the effect of shear

angle and (b) without the effect of shear angle.

(a)


(b)

Figure 11: (a) Pressure near the end of fill calculated with

the effect of shear angle and (b) without the effect of

shear angle.

Conclusion

In this paper, the method of solving the flow during

RTM process with draping analysis has been described.

Two test cases were used for the verification of RTM

simulation. The results from a case show good agreement

between the simulation and the analytical results. This

paper also showed the effect of including the draping

analysis in the RTM flow simulation.

Nomenclature

A Cross sectional area for the flow

, A1, A2, E1, E2 Curing kinetics parameters

PC Specific heat

K Permeability

k Thermal conductivity

L Flow length

p Pressure

Q Flow rate

T Temperature

t Time

u

Superficial velocity vector

�⃗� Macroscopic velocity vector

Degree of cure

’ Angle between 2 sides after the draping

e Expansivity

Porosity

Shear angle

Shear rate

Viscosity of the resin

Density

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