Prof Jo Seung

Kim

Suleiman Ola

Dina Ahmad

Akhrass Al

Yi Hou

Quyhn Nguyen

Renata Troian

Lara Abouorm

Laura Dubius

Yeonhee Jung

An efficient analysis of resin transfer molding process using extended finite element method

Keywords: RTM Process, XFEM, Level Set Method, Numerical Analysis, Localization method

unforgettable memories. I could concentrate on my work and overcome loneliness in the foreign country because of their friendships.

III

Symbols

List of Figures

For the manufacturing of composite structures by RTM process, it is important to predict weld lines and to avoid the formation of air voids, which could cause critical defects in the structure. The some defects by incorrect filling of the mold are shown in the figure 1.3. These defects could lead to modify the mold. That is expensive and time-consuming. Furthermore, to set up the mold filling stage during RTM process, various variables should be considered such as injection and vent pressure, injection gate and vent numbers, and their locations. The design of these variables is sometimes expensive and time-consuming since the motion of the resin flow is impossible to check in a closed mold while the mold filling.

That's why the numerical simulation could be very interesting so as to save the cost and time.

During RTM process, the resin flow in the mold is affected by resin viscosity and permeability of fiber preform. The resin viscosity is changed during injection process due to the change of temperature. However, the viscosity could be assumed to be isothermal and constant during the filling process if the filling time is shorter than curing time or the change of temperature is small. Permeability is related to fabric of fiber preform. When the preform is closed fabric, the resin flow slowly moves. On the contrary, when the preform is opened fabric, the resin passes fast. On macroscopic scale, the preform could be assumed as nondeformable condition, therefore, it could be considered to be constant during the mold filling.

The resin flow in the mold represents a moving interface problem since the resin flow front moves through the preform placed in the mold during the filling process. The numerical methods for the moving interface problem could be divided into two groups: moving grid and fixed grid. The fixed grid approach has been more widely used because the moving grid approach needs relatively expensive and difficult computation [Park CH et al., 2003-2004;Kang MK et al., 2000].

In the industry, the structures manufactured by RTM process generally have complex shapes with holes and curvatures. However, the simulation results could be inaccurate when applying traditional numerical approaches for analysis of the complex shape structures. Even though it is possible to acquire appropriate results by using very fine mesh models, the computation time becomes longer. Therefore, many researchers have investigated to increase the accuracy and efficiency of RTM process simulation.

Literature Review

RTM process has become a popular method of manufacturing middle/large structures of fiber reinforced composite materials because of its capabilities including non-expensive process equipment, excellent control on mechanical properties, closed mold process, low filling pressure, and so on [Trochu F et al., 1992;Morren G et al.,2008;Lin R et al.,1991]. However, the design of mold filling equipment is quite difficult because the reinforcement consisting of several layers of fiber mats inside the closed mold should be perfectly impregnated by the resin injected through several injection gates [Rouison D et al., 2006;Danisman M et al., 2007]. Therefore, many researchers have investigated for numerical simulation of mold filling stage in RTM process [START_REF] Boccard | Model for determining the vent locations and the fill time of resin transfer molds[END_REF][START_REF] Jiang | Optimum arrangement of gate and vent locations for RTM process design using a mesh distance-based approach[END_REF].

Designers have used the simulation to predict pressure distribution, resin flow pattern, void formation, and other phenomena during mold filling [CH Park et al., 2003-2004]. Nevertheless, the simulation has been restrictedly utilized because the simulation could not give an accurate prediction of the mold filling stage for manufacturing complex shape structures. Actually, early researches had analyzed a simple shape structures due to the difficulty to analyze the moving resin fronts [Bruschke MV et al., 1990;Kang MK et al., 1999]. Still now, many researchers have investigated to develop more proper methods for the analysis of complex shape structures [Liu XL, 2000;Shojaei A et al., 2002].

The Volume of Fluid (VOF) method was popularly used to track the resin flow front for RTM process simulation [Trouchu F et al., 2006;Young WB et al., 1991].

The VOF method is one of the representative methods to track moving interfaces [Hirt CW et al., 1981]. However, because the moving interface is defined by the fractional fluid volume in the element or control volume according to the used method, it could be inaccurate for the cases to track complex movements of the resin front [Kim MS et al., 2003 In RTM process simulation, the most frequently used numerical methods during the last two decades are the Control Volume Finite Element Method (CVFEM)

and the nonconforming FEM [Dong C, 2008;[START_REF] Jiang | PCG solver and its computational complexity for implicit control-volume finite-element method of RTM mold filling simulation[END_REF]. CVFEM was the most frequently developed numerical method for RTM process simulation until the beginning of the years 2000, because CVFEM could efficiently satisfy the mass conservation of the resin between two adjacent elements, with regard to the standard finite element approach [Estacio KC et al.,2007;Feerick R, 1997].

However, the construction of control volumes in the whole computational domain could make the computation complex [Shojaei A et al., 2003]. Trochu et al.

proposed to apply nonconforming FEM to RTM process simulation. Since the nonconforming FEM calculates the pressure at the inter-elements boundaries, the flow rate at the element boundaries can be obtained without any additional treatment, from which the filling factor is derived. However, the filling factor is not the nodal value but the element one [Trochu F, 1993]. That is why these methods could not properly approximate the pressure field in the elements intersected by the resin flow front because of the discontinuity of the pressure gradient field near the flow front. This point is very important in the case of industrial problems having complex shapes. presented XFEM for the multi-dimensional Stefan problems. And, Chessa et al.

[2003] applied XFEM to two-phase immiscible flow problems having a discontinuity of the pressure gradient at the interface between two fluids.

Scope of Study

One of the important points to simulate the resin flow in the mold is to provide accurate approximation of the moving resin flow front during the mold filling process. Moreover, it is necessary to increase the computational efficiency so as to be able to treat industrial problems within an acceptable computing time.

This study suggests to apply eXtended Finite Element Method (XFEM) combined with level set method to analyze the mold filling stage during RTM process. The main advantage of this approach is more accurate approximation of the pressure distribution in the mold and resin flow front location at each time step. And, this study attempts the formulation compels the coefficient matrix to be symmetric and positive definite in order to apply a more efficient linear solver. In addition, this study proposes to apply some techniques, such as localization of XFEM and level set method and usage of an effective solver, for increasing the computational efficiency.

The enriched shape functions of XFEM describing the pressure field are defined by the level set values. The pressure field is divided into two regions: normal and enriched. The enriched region is defined only near the resin flow front; therefore, the size of the linear system slightly increases. Since XFEM formulation done by this study makes the coefficient matrix to be symmetric and positive definite, the multi frontal solver proposed by IPSAP [Kim JH, 2005;Chung SW, 2003], an effective approach for solving the equilibrium systems, is used in order to reduce the computing time.

The level set method is composed of two procedures. there is explanation about the numerical methods traditionally used for RTM process simulation and applied in this study.

Chapter 3 presents in detail the advanced numerical analysis proposed by this work. At first, it is explained about mathematical modeling of RTM process. Next, the numerical formulations by XFEM and level set method are presented. This section is divided by two parts: the formulations for the pressure calculation (XFEM) and the transport of resin flow front (level set method). Furthermore, the whole computational procedures are minutely depicted. Finally, this study is validated by using some examples. Especially, the analysis result by this study is compared with an analytical result.

Chapter 4 goes on to trace the computing efficiency. At first, localization of XFEM and level set method are explained in this chapter. Furthermore, the computational efficiency by the localization is investigated.

Chapter 5 shows the applications. The plate with complex edge is analyzed to show. In addition, the race tracking effect and the macro void formation in the mold are analyzed by the thin and complex shaped structure. The analysis for manufacture of the wind turbine blade is presented. Finally, the pre and post processors of Diamond/RTM are described.

In Chapter 6, the conclusions of this research are summarized with suggestion for future work.

Chapter 2. Analytical Models and Numerical Methods

In this chapter, the analytical models and numerical methods of RTM process simulation are described. First of all, in the section 2.1, the process models and process parameters are introduced for the manufacturing operation of RTM process. In the section 2.2, VOF method and Level set method are presented for tacking the resin flow front. And, in the section 2.3, CVFEM and XFEM are described. These methods are used to calculate the pressure distribution in the mold during the mold filling.

Mathematical models

In the analysis of RTM process, the mathematical models are derived from the physical laws of fluid flow. When considering fluid flow with velocity, 𝒖, and density, 𝜌, the conservation of mass in fluid mechanics is written as:

𝜕𝜌 𝜕𝑡 + ∇ • (𝜌𝒖) = 0, (2.1)
where t is time. In RTM process, because the resin is assumed as an incompressible fluid, the density, 𝜌, is constant. Therefore, the equation (2.1) becomes:

∇ • 𝒖 = 0. (2.
2)

The equation of momentum conservation is written as:

ρ 𝐷𝒖 𝐷𝑡 = -∇𝑝 + ∇ • 𝝉 + 𝐅 𝐵 , (2.3)
where 𝑝, 𝝉, and 𝐅 𝐵 are the pressure, viscous stress and body force, respectively.

The inertia term, the left hand side term of the equation (2.3), is important if the velocity is high and the viscosity is low. It's why this inertia term is in general neglected for RTM process. The term 𝐅 𝐵 of the equation (2.3) means the body forces due to the gravity and might be negligible when the gravity effect is insignificant. The term ∇ • 𝝉 means the damping forces due to the viscosity. In the model for fluid flow through the porous medium, this term can be expressed by fluid viscosity, fluid velocity, and permeability of porous media as:

∇ • 𝝉 = - μ 𝑲 𝒖, (2.4)
where μ and 𝑲 are the viscosity and permeability. The fluid flow in RTM process can be described as fluid flow through the porous medium. Therefore, for RTM process, the equation (2.3) can be simplified as:

0 = -∇𝑝 - μ 𝑲 𝒖 + 𝐅 𝐵 . (2.5)
permeability is small in closed fabric conditions and the resin flows slowly with small permeability as shown in the figure 2.1. The permeability has been developed by a function of the fiber volume fraction. Carman-Kozeny equation is one of the most representative models:

𝑲 = A (1 -𝑉 𝑓 ) 3 𝑉 𝑓 2 , (2.9)
where V f , K, and A are the fiber preform, permeability, and Kozeny constant (empirical value), respectively. For RTM process simulation, the permeability could be putted as a constant value with the assumption that the fiber arrangement is non-deformable during the mold filling.

dα d𝑡 = 𝐶 1 exp(- 𝐸 𝑅𝑇 )(1 -𝛼) 𝑛 , (2.10)
where α, t, and R are the degree of cure, time, and universal gas constant, respectively. And, C 1 , E, and n are constants to be determined experimentally [Advani SG et al., 2003]. As mentioned earlier, the degree of cure could be neglected if the analysis target is the mold filling stage and the mold filling is completed before the curing reaction.

The temperature influences the viscosity and degree of cure. The temperature of the resin and the fiber mixture are generally governed by a heat transfer model, such as:

𝜌𝑐 ∂𝑇 ∂𝑡 + 𝜌 𝑟 𝑐 𝑟 (𝒖 • ∇𝑇) = 𝑘∇ 2 𝑇 + ɸ∆𝐻𝐺 ̇, (2.11)
where ρ, c and k are the density, specific heat, and thermal conductivity of the resin and fiber mixture, respectively. The subscript r denotes the resin. And, ɸ, ∆H, 𝐺 ̇ are the porosity, heat of reaction, and reaction rate, respectively [Advani SG et al., 2003]. According to the analysis target, this model could be negligible with the assumption that the process is under the isothermal condition. After all, the interactions between resin flow, heat transfer, and curing reaction processes is described in the figure 2.3. The resin flow is affected by the change of viscosity which is dependent on the temperature and degree of cure. And, the heat is convected through the resin flow and generated during the curing reaction. The cure is convected through movement of the resin and affected by heat transfer because the temperature changes the degree of cure. However, these models is often modified or simplified according to the study targets. For example, when the target is focused on its quick computation on the structure with complex geometry, the resin flow motion could be assumed under isothermal and no-curing reaction conditions. And, the resin flow model could be used with constant viscosity and permeability for numerical simulation.

Methods for Transport of Resin Front

A. VOF (Volume of Fraction) Method VOF method has been frequently used to solve moving boundary problems in a fixed mesh. And, it is the representative of the boundary tracking method. For the numerical simulation of RTM process, the resin flow front position in the mold is obtained by VOF method [Hirt CW et al., 1981]. Following the methodology proposed by [Peng D et al., 1999], the differential equation for re-initialization of the level set values is defined as:

𝜕𝑑 𝜕𝜏 + 𝑠𝑖𝑔𝑛(𝑑 0 )(|∇d| -1) = 0, (2.15)
where d is the signed distance function from the front, the imaginary time within the re-initialization procedure, and sign(d 0 ) the sign function of the initial value d 0 .

Additionally, the sign(d) function is defined as

𝑠𝑖𝑔𝑛(𝑑) = 𝑑 √𝑑 2 + ℎ 2 , (2.16)
where h is the mesh size [Peng D et al., 1999].

The initial value of d is defined as:

𝑑(𝒙, 0) = 𝑑 0 (𝒙) = 𝜑(𝒙, 𝑡).

(2.17)

After the calculation, the level set function φ is re-initialized as 𝜑(𝒙, 𝑡) = 𝑑(𝒙, 𝜏).

(2.18)

The re-initialization procedure is repeated for several times at each time step of the equation (2.13). The pressure gradient at the flow front should be calculated carefully. The gradient of the approximated pressure function on the enriched region is obtained as [Chessa J et al., 2003]:

Methods for Pressure Calculation

∇𝑝 ℎ = ∑ 𝑝 𝐼 𝐼∈𝑛 ∇𝑁 𝐼 + ∑ 𝑎 𝐽 [∇𝑁 𝐽 (|𝜑| -|𝜑 𝐽 |) + 𝑁 𝐽 𝑠𝑖𝑔𝑛(φ)∇𝜑] 𝐽∈𝑛 𝑒 . (2.21)
The jump in the approximated pressure function across the resin front is given as [Chessa J et al., 2003]:

⟦∇𝑝 ℎ ⟧ = 2 ∑ 𝑁 𝐽 𝑎 𝐽 𝐽∈𝑛 𝑒
∇𝜑 on the resin front. Chapter 3.

Analysis of RTM Process

This chapter delineates our work on the analysis of RTM process using XFEM and level set method in detail. At first, the resin flow during mold filling is mathematically modeled. Here, there are explanations about our assumption for modeling and the mathematical models divided into two parts: the transport of resin flow front and the pressure calculation in mold. In the section 3.2, the numerical formulations of the two parts are explained in detail. After that, a flow chart of global computational procedures is shown in the section 3.3. Finally, our methods are validated through some simple examples.

Modeling

A. Assumptions

For the analysis of mold filling stage, the resin flow in the mold is simplified with the following assumptions. The resin is assumed as an incompressible fluid. And, the mold filling could be carried on under an isothermal condition if the mold filling is finished before the curing reaction. With these assumptions, the resin viscosity could be considered as constant during the mold filling. The permeability of the fiber preform in the mold could be constant with the assumption that the fiber preform is considered as uniform and non-deformable during the mold filling.

These assumptions have been used by many researchers so as to simplify the resin flow motion in the mold as a quasi-steady state problem [Young WB et al., 1991;Feerick R et al., 1997;[START_REF] Jiang | PCG solver and its computational complexity for implicit control-volume finite-element method of RTM mold filling simulation[END_REF]. When the resin slowly moves, the inertia effect could be negligible and the resin flow could be assumed as a succession of a steady state flow. After obtaining the pressure field at the instance t, the flow velocity is calculated and the resin flow front is transported by the velocity. In other words, the mold filling stage is mathematically modeled into two governing equations: transport of resin flow front and pressure calculation in the mold.

B. Mathematical Models

The figure 3.1 shows a domain for analysis of RTM process. The domain (Ω)

indicates that the dry preform is impregnated by the resin entered through injection gates.

∇ • (-(𝐊 𝜇 ⁄ )∇𝑝) = 0 in Ω. (3.3)
The boundary conditions are defined as the below equation (3.4). At injection gates, resin is injected by constant pressure or flow rate in the mold. Thus, the boundary conditions are set to injection pressure at injection gates as:

𝑝 = 𝑝 𝑖𝑛𝑗𝑒𝑐𝑡𝑖𝑜𝑛 in ∂Ω in . (3.4 -a)
In addition, the pressure at vents is same with the air pressure as:

𝑝 = 𝑝 𝑎𝑖𝑟 in ∂Ω v . (3.4 -b)
At mold walls, the resin flow cannot pass through mold walls; therefore, the partial derivative of pressure with respect to the normal direction of mold walls is equal to zero as:

𝜕𝑝 𝜕𝑛 = 0 in ∂Ω w . ( 3
∂φ/∂t + u•∇φ=0 in Ω, (3.5)

Formulation

A. Transport of Flow Front

The differential equations for transport of flow front are solved by the implicit characteristic Galerkin FEM [Lin CL et al., 2005]. The implicit characteristic Galerkin FEM is employed to stabilize the non-self adjoint term. The time derivative term is discretized by the implicit characteristic method as [Zienkiewicz OC et al., 2000]:

𝜑 𝑛+1 -𝜑 𝑛 = -∆𝑡[𝐮 • ∇𝜑] 𝑛 + ∆𝑡 2 2 [∇ • ((𝐮⨂𝐮) • ∇𝜑)] 𝑛+1 , (3.6)
where ∆t is the size of the time step. For the discretization of the spatial differential terms in the equation (3.6), the standard Galerkin FEM is applied as:

∫ (𝜑 𝑛+1 -𝜑 𝑛 )𝑤𝑑𝑉 Ω = -Δ𝑡 ∫ (𝐮 • ∇𝜑 𝑛 )𝑤𝑑𝑉 Ω - ∆𝑡 2 2 ∫ ((𝐮⨂𝐮) • ∇𝜑 𝑛+1 ) • ∇𝑤𝑑𝑉 Ω , (3.7)
where w is the weight function and Ω is the whole space in the mold. The velocity u for transport of the front is calculated by the equation (3.3).

The re-initialization equation is also discretized by the procedures of the equation (3.6) and (3.7) as:

∫ (𝑑 𝑛+1 -𝑑 𝑛 )𝑤𝑑𝑉 Ω = -Δ𝜏 ∫ 𝑠𝑖𝑔𝑛(𝑑 0 )|∇𝑑 𝑛 |𝑤𝑑𝑉 Ω - ∆𝜏 2 2 ∫ ∇𝑑 𝑛+1 • ∇𝑤𝑑𝑉 Ω +Δ𝜏 ∫ 𝑠𝑖𝑔𝑛(𝑑 0 )wdV Ω , (3.8)
where ∆ is the size of the time step in the re-initialization procedure.

In the equation (3.8), the second term of the right-hand side could be derived by the next procedure:

𝑠𝑖𝑔𝑛(𝑑 0 )|∇𝑑| = 𝑠𝑖𝑔𝑛(𝑑 0 ) ∇𝑑 |∇𝑑| • ∇𝑑 , (𝑠𝑖𝑔𝑛(𝑑 0 ) ∇𝑑 |∇𝑑| ) 𝑠𝑖𝑔𝑛(𝑑 0 )|∇𝑑| = ∇𝑑.
(3.9)

The coefficient matrices of systems obtained by the formulations are symmetric and positive definite; therefore, the multi-frontal solver of IPSAP [Kim JH, 2005] can be applied.

B. Pressure Calculation

The differential equation for the pressure calculation is solved by using XFEM.

XFEM formulation is similar with the general FEM. The weak form of the equation (3.3) is obtained as:

∫ (∇ • (- 𝐊 𝜇 ∇𝑝))𝑤𝑑𝑉 Ω = 0, (3.10)
where w is the weight function. Applying the Green-Gauss theorem, the equation (3.10) is transformed as:

∫ ∇w • (- 𝐊 𝜇 ∇𝑝) 𝑑𝑉 Ω -∫ w (- 𝐊 𝜇 ∇𝑝) • 𝐧𝑑𝑆 𝜕Ω = 0, (3.11)
where n is the outward normal vector. By XFEM, the pressure and weight functions are discretized as the equation (2.19). In other words, the approximation of the solution is done by adding the additional terms for the enrichment in the terms of the standard Galerkin FEM. The permeability K of fiber preform is considered as constant in the whole domain. However, the viscosity μ of resin is different at each region. At the filled region by resin, the resin viscosity is applied. And, the empty region has air viscosity. The viscosity of the region intersected by resin flow front should have an averaged value of resin and air. Therefore, the fluid viscosity is defined as:

𝜇 = 𝐻(𝜑)𝜇 𝑟 + (1 -𝐻(𝜑))𝜇 𝑎 , (3.12)
where μ  is the resin viscosity; μ  the air viscosity, andφ the level set value. Here, H(φ) is defined as

𝐻(𝜑) = { 1 1 2 (1 - 𝜑 ℎ ) 0 𝑖𝑓 𝜑 < -ℎ 𝑖𝑓 |𝜑| ≤ ℎ 𝑖𝑓 𝜑 > ℎ , (3.13)
where h is the mesh size.

The whole computation scheme is shown in the figure 3

Validation

Our methods implemented in C++ are verified in this section. At first, the pressure calculation part is tested on a simple square plate quarter-filled with resin. Its object is to check whether XFEM can correctly calculate the pressure field, in particular, near the flow front. After, the level set method is tested with a square plate with an injection gate at the center of the plate. At the end, the numerical results obtained by our methods on plates are compared with the analytical results

for simple channel and radial flows.

XFEM is changed near the front, contrary to the one by FEM. Therefore, it is possible to conclude that the enrichment by XFEM is well-working in this computation.

Transport of Flow Front

The transport of flow front is verified through simple models. The geometry and mesh information of the models is shown in the figure 3.6. The model is a thin square plate which has the width and height of 0.1 m. And the thickness is 1 mm.

The type of element is three-node triangle. The number of elements and nodes are 2048 and 1089, respectively. The mesh condition is a little fine in order to well show the round shape. The analytic mold filling time for the channel flow, when the injection pressure is identical to the width direction, is given as:

KP L t filling 2 2   , (3.15)
where L means the position of the flow front in the length direction. And, K, and P are the viscosity, porosity, permeability, and pressure, respectively.

The analytical total filling time by the equation (3.15) is equal to 1250 seconds.

The numerical total filling time obtained in this study is equal to 1268.42 seconds.

The difference between the analytical and numerical results is less than 1.5%. In comparison with the numerical solution (difference of 3%) proposed by S. For the localization of level set method, a cut-off function is necessary to prevent numerical oscillations at the band boundary. The role of a cut-off function is described in the figure 4.2. As shown in the figure (a), the level set function, marked by red, ought to be strongly discontinuous when it is transported only in the local domain. Therefore, a cut-off function is applied to make the level set function continuous as shown in the figure (b). The cut-off function is defined as [Peng D et al., 1999]:

                   1 ln 2 4
𝑐(𝜑) = { 1 (|𝜑| -𝛾) 2 (2|𝜑| + 𝛾 -3𝛽)/(𝛾 -𝛽) 3 0 𝑖𝑓 |𝜑| ≤ 𝛽 𝑖𝑓 𝛽 < |𝜑| ≤ 𝛾 𝑖𝑓 |𝜑| > 𝛾 , (4.1)
where φ is the level set value, γ six times of the mesh size, β three times of the mesh size. By applying the cut-off function in the differential equation From the efficiency test, we could know that the effect by localization of Level set method is bigger than that by the localization of XFEM. The reaon might be that the computing amount of Level set method is greater than that of XFEM for pressure calculation. Nevertheless, the use of both localizations of Level set method and XFEM minimizes the computing time.

Chapter 5. Applications

Three examples are treated in this chapter to illustrate industrial applications. The first one is to analyze the filling motion of a complex edge mold with a hole. This application is used to show that our methods are well-fitted to complex shape structures. The numerical result on the filling time is compared with the experimental one. The second one shows a race tracking effect and macro void formation, which are practical phenomena frequently occurred during manufacturing. The last one concerns the numerical simulation of a composite wind turbine blade made by RTM process. This is a real industrial example with real material properties and could be used for benchmarking.

Plate with Complex Edge

A plate having complex shaped edges and a hole is chosen, and the numerical when the localization method is applied and is down 37.7%, compared with the computation without localization.

Thin and Complex Shape Structure

A thin and complex shaped structure is used to analyze the race-tracking effect and macro void formation in the mold. The numerical simulations of the racetracking effect and macro void formation in the mold could be useful to correctly design process variables, such as the injection/vent locations and their numbers. If there is no special treatment, the race-tracking effect appears in the actual injection procedure due to small space between the mold and the fiber preform edges. Macro void formation in the mold can occur due to the incorrect design of the variables such as the injection/vent positions and their numbers. The type of elements used in this analysis is 3-nodes triangle.

Race Tracking Effect

As shown in the figure 5.4, to produce the race-tracking effect, a narrow band at the edge, of which the width is constant and equal to 10 mm, has a high

Macro Void Formation

To show the formation of macro voids during the mold filling, the hole is removed at the upper side. And without the race-tracking effect, the permeability of the narrow band at the edges is as same as the value in the inside of the mold. The

Wind Turbine Blade

As real industrial example, a composite wind turbine blade is treated in this section. Manufacturing of composite wind turbine blades by RTM process offers less labor cost and higher fiber volume fraction than the hand lay-up process. If the size of blades is bigger, it might be essential to predict the resin flow in the mold during the filling so as to avoid problems due to large scale. For example, an engineer will simulate the resin flow in the mold when he or she wants to quickly complete the injection before curing reaction of the resin by adding more injection gates and vent.

However, it might be difficult to get reasonable results by using the traditional numerical methods because the resin flow is quite complex due to the resin injected from multiple gates and the various permeability values by mixed fabric lay-ups. Thus, the analysis methods applied in this study could help to fulfill more accurate analysis of this example. The purpose of this section is to offer a good reference, an industrial example of RTM process analysis. For that, the modeling information is minutely delineated in this section.

The computations are performed with the localization method in order to reduce the computation time.

A. Geometry and Material

Chapter 6. Conclusions

In this study, the analysis of RTM process was established using XFEM combined with the level set method, so as to track more accurately the resin flow front during the mold filling. The most important advantage of this approach is the possibility to describe the discontinuity of the pressure gradient at the resin flow fronts. The enriched shape functions of XFEM were defined by the level set function and reflect the pressure distribution according to the flow front's position.

Furthermore, the number of DOF was slightly increased because the enrichment was just applied to near the flow front. The position of the resin front was determined by solving the transport equation written with the level set function.

The transport equation of flow front and reinitialization of the level set value were solved by an implicit characteristic Galerkin FEM. A parallel MFS solver was used to solve the equilibrium systems to reduce the computing time. Moreover, a GUI was developed and implanted to link this RTM process simulation with Diamond program (Diamond/RTM). This method was validated through several simple examples in comparison with analytical solutions found in the literature.

The difference between the obtained results and the analytical solutions was very small.

Furthermore, a localization method was used to XFEM and level set method so as to increase computing efficiency. The localization of XFEM was newly proposed in this study for RTM process simulation. In particular, smaller is the flow front, the efficiency of this method becomes higher.

Through several examples having complex geometry, the separating and merging of the resin flow front were illustrated with the present method. Moreover, our method could well simulate race tracking effects and macro void formation. For the industrial application, the composite wind turbine blade was treated to simulate the mold filling stage during RTM process.

Through this study, we proved that XFEM combined with the level set method is very efficient to simulate the resin flow motion in the mold during the mold filling.

In the future, these methods could be adapted to an optimization of the RTM process parameters, injection pressure, number and position of injection and vent gates, etc., because they could make the process simulation faster with a good accuracy.

This study is limited to the problem finished the filling before the curing reaction.

If an engineer wants to simulate a large scale structure with a single injection gate, the gel time of resin should be considered. Therefore, if the analysis for heat transfer and curing reaction is included, it will be possible to enlarge the application examples of this study.

In this study, XFEM was used for the analysis of RTM Process. XFEM is still an efficient numerical method for other physical problems. But, the numerical integration might be carried on very carefully on the fully and partially enriched regions so as to maximize the efficiency of the method. For example, XFEM might be a good method for the analysis of inter-tow flows to capture micro-voids, and for various moving interface problems such as the explosion of fuel tank as well as the car crash. La méthode développée est validée à travers quelques exemples en 2D en comparaison avec les résultats analytiques ou expérimentaux trouvés dans la littérature [32][33][34][35]. Quelques exemples de plaque 3D sont traités pour simuler l'effet du bord et la formation de macro-vides [36]. En outre, un programme intégrant la méthode présentée a été développé avec un outil d'interface graphique convivial pour les pré-et post-traitements de la simulation.

Modèles mathématiques

Pour la simulation du procédé RTM, l'écoulement de la résine dans le moule est modélisé avec les hypothèses suivantes : l'écoulement de la résine est considéré comme incompressible et iso-thermique. Par conséquent, la loi de conservation de masse est donnée :

∇ • 𝐮 = 0 (1)
EA(F) -5 où u est la vitesse de résine. Et, l'équation d'équilibre d'inertie est donnée : où ρ est la masse volumique, t le temps, p la pression de la résine, K le tenseur de perméabilité, et μ la viscosité de la résine.

Avec l'hypothèse iso-thermique, la viscosité de la résine peut être constante durant le remplissage du moule. La perméabilité de la préforme dans le moule peut être constante due à l'hypothèse dans laquelle la préforme est considérée comme uniforme et indéformable pendant le remplissage du moule. Si l'écoulement de la résine est suffisamment lent, l'effet d'inertie peut être négligé et l'écoulement de la résine peut être considéré comme une succession d'états stationnaires de l'écoulement. Par conséquent, après l'obtention du champ de pression et de la vitesse du fluide dans le moule, le front d'écoulement de la résine est déplacé pour le pas de temps suivant S.G. Advani et al. [27]. L'équation de Darcy dans l'écoulement stationnaire est dérivée de l'équation (2) comme :

𝐮 = -(𝐊 𝜇 ⁄ )∇𝑝 (3)
En mettant l'équation (3) dans l'équation (1), l'équation d'équilibre pour le champ de pression devient :

∇ • (-(𝐊 𝜇 ⁄ )∇𝑝) = 0 (4)
Les conditions frontières sont données :

𝑝 𝑖𝑛 =
Le transport du front d'écoulement pour le pas de temps suivant s'effectue par la méthode Level set. Le détail de cette méthode est expliqué dans le chapitre suivant.

𝜌 D𝐮 D𝑡 ⁄ = -∇𝑝 -(𝜇 𝐊 ⁄ )𝐮 (2)
EA(F) -7

𝜕𝜑 𝜕𝑡 ⁄ + 𝑠𝑖𝑔𝑛(𝑑 0 )(‖∇𝑑‖ -1) = 0, 𝑑(𝒙, 0) = 𝑑 0 (𝒙) = 𝜑(𝒙, 𝑡)

où d est la fonction de distance avec signe du front,  le temps fictif pour la procédure de réinitialisation, et sign(d 0 ) le signe de la valeur initiale d 0 . En plus, la fonction sign(d 0 ) est définie par :

𝑠𝑖𝑔𝑛(𝑑) = 𝑑/ √ 𝑑 2 + ℎ 2 (9)
où h est la taille de maillage. La fonction Level set est réinitialisée pour la condition initiale en résolvant l'équation (8). La procédure de réinitialisation s'effectue à chaque pas de temps.

Méthode d'éléments finis étendue (XFEM)

La pression dans le moule est obtenue par la méthode XFEM. L'avantage principal de cette méthode est d'obtenir le champ de pression d'une manière précise sans remailler sur la zone du front d'écoulement.

Figure 2 : la définition des noeuds enrichis et le type de régions.

EA(F) -8

Le domaine de calcul est divisé en deux régions : normale et enrichie. La région enrichie est discrétisée par des éléments ayant des noeuds enrichis. Ces noeuds sont ajoutés aux éléments intersectés par le front d'écoulement (Figure 2). La région normale est discrétisée par des éléments sans aucun noeud enrichi.

L'enrichissement signifie que les degrés de liberté (ddl) additionnels sont ajoutés dans un élément fini classique. La fonction de forme pour cet enrichissement est définie par :

𝑝 ℎ = ∑ 𝑁 𝐼 𝑝 𝐼 + 𝐼∈𝑛 ∑ 𝑁 𝐽 𝑒 𝑎 𝐽 𝐽∈𝑛 𝑒 (10)
où p h est la fonction de la pression, p I la valeur de la pression au noeud I, a J le paramètre nodal additionnel au noeud enrichi J, N I la fonction de forme, 𝑁 𝐽 𝑒 la fonction de forme enrichie, n le nombre de noeuds, et n e le nombre de noeuds enrichis. La fonction de forme enrichie est construite comme : Le gradient de pression au front d'écoulement doit être calculé précisément. Le gradient de la fonction de pression approximative dans la région enrichie s'obtient :

𝑁 𝐽 𝑒 = 𝑁 𝐽 (|𝜑| -|𝜑 𝐽 |) (11)
∇𝑝 ℎ = ∑ 𝑝 𝐼 𝐼∈𝑛 ∇𝑁 𝐼 + ∑ 𝑎 𝐽 [∇𝑁 𝐽 (|𝜑| -|𝜑 𝐽 |) + 𝑁 𝐽 𝑠𝑖𝑔𝑛(φ)∇𝜑] 𝐽∈𝑛 𝑒 (12)
Le saut de la fonction de pression approximative à travers le front de la résine est défini par :

⟦∇𝑝 ℎ ⟧ = 2 ∑ 𝑁 𝐽 𝑎 𝐽 𝐽∈𝑛 𝑒 ∇𝜑 on the resin front (13)
EA(F) -10

Il y a deux types d'enrichissement comme illustré dans la figure 2 :

enrichissement complet ou partiel. L'enrichissement complet concerne les éléments n'ayant que des noeuds enrichis. L'enrichissement partiel concernent ceux ayant des noeuds enrichis et normaux. Cet enrichissement partiel est nécessaire pour maintenir la continuité du champ de pression.

Formulations numériques

Transport du front d'écoulement

Une méthode de Galerkin implicite est utilisée pour stabiliser le phénomène d'oscillations numériques lors de la résolution de l'équation (6) [22]. Le terme dérivé par rapport au temps est discrétisé par [28] :

𝜑 𝑛+1 -𝜑 𝑛 = -∆𝑡[𝐮 • ∇𝜑] 𝑛 + ∆𝑡 2 2 ⁄ [∇ • ((𝐮⨂𝐮) • ∇𝜑)] 𝑛+1 (14)
où ∆t est le pas de temps. Pour la discrétisation des termes différentiels spatiaux de l'équation (14), la méthode de Galerkin standard est utilisée :

∫ (𝜑 𝑛+1 -𝜑 𝑛 )𝑤𝑑𝑉 Ω = -Δ𝑡 ∫ (𝐮 • ∇𝜑 𝑛 )𝑤𝑑𝑉 Ω -∆𝑡 2 2 ⁄ ∫ ((𝐮⨂𝐮) • ∇𝜑 𝑛+1 ) • ∇𝑤𝑑𝑉 Ω (15)
où w est la fonction de pondération et Ω l'espace complet dans le moule. La vitesse u pour le transport du front d'écoulement est calculée par l'équation (3).

La viscosité de la résine est définie par :

𝜇 = 𝐻(𝜑)𝜇 𝑟 + (1 -𝐻(𝜑))𝜇 𝑎 (16)
où μ r  est la viscosité de la résine et μ a  celle de l'air, et H(φ) la fonction Level set.

H(φ) est défini par :

EA(F) -11

𝐻(𝜑) = { 1 1 2 (1 -𝜑 ℎ ⁄ ) 0 𝑖𝑓 𝜑 < -ℎ 𝑖𝑓 |𝜑| ≤ ℎ 𝑖𝑓 𝜑 > ℎ (17)
L'équation de réinitialisation est aussi discrétisée par les équations ( 14) et (15).

∫ (𝑑 𝑛+1 -𝑑 𝑛 )𝑤𝑑𝑉 Ω = -Δ𝜏 ∫ 𝑠𝑖𝑔𝑛(𝑑 0 )|∇𝑑 𝑛 |𝑤𝑑𝑉 Ω -∆𝜏 2 2 ⁄ ∫ ∇𝑑 𝑛+1 • ∇𝑤𝑑𝑉 Ω + Δ𝜏 ∫ 𝑠𝑖𝑔𝑛(𝑑 0 )wdV Ω (18)
où ∆ est le pas de temps dans la procédure de réinitialisation.

Champ de pression

L'équation (4) est résolue par la méthode XFEM [14]. Suivant la méthode d'éléments finis standard, la forme faible de l'équation (4) s'écrit par :

∫ (∇ • (-𝐊 𝜇 ⁄ ∇𝑝))𝑤𝑑𝑉 Ω = 0 (19)
où w est une fonction de pondération. En appliquant le théorème de Green-Gauss, l'équation (19) devient :

∫ ∇𝑤 • (-𝐊 𝜇 ⁄ ∇𝑝)𝑑𝑉 Ω -∫ 𝑤(-𝐊 𝜇 ⁄ ∇𝑝) • 𝐧𝑑𝑆 𝜕Ω = 0 (20)
où n est le vecteur normal sortant. Par la méthode XFEM, la pression et les fonctions de pondération sont discrétisées comme l'équation (12).

Les éléments interceptés par le front d'écoulement sont divisés en deux parties : région remplie et celle non remplie. Ces deux régions sont séparément intégrées car la dérivée de la fonction de pression est légèrement discontinue au front d'écoulement [29,30]. La viscosité de la résine des sous-domaines, entrecoupés par le front d'écoulement, est définie par : EA(F) -13

EA(F) -12 𝜇 = 𝜇 𝑟 2 ⁄ (1 -𝑠𝑖𝑔𝑛(𝜑)) + 𝜇 𝑎 2 ⁄ (1 + 𝑠𝑖𝑔𝑛(𝜑)) (21)

Procédures de calcul

Amélioration de l'efficacité de calcul par localisation

La localisation pour la méthode XFEM couplée avec la méthode Level set permet d'améliorer l'efficacité de calcul en minimisant le nombre de calcul [37]

Localisation pour la méthode XFEM couplée avec la méthode Level Set

Cet algorithme, utilisé pour la localisation avec la méthode Level set, s'est basé sur l'algorithme proposé par Peng et al. [23]. Nous avons modifié cet algorithme en vue de l'appliquer à la méthode de Galerkin implicite.

EA(F) -14 pression du point de fuite à la frontière de la bande de l'air, car la variation de la pression dans l'air est quasiment négligeable. Dans la figure 8, la distribution de la pression est obtenue par la localisation de la méthode XFEM. La répartition est comparable à celle obtenue sans localisation (voir la figure 6).

Efficacité de la localisation

Nous prenons l'exemple de l'écoulement laminaire dans une plaque rectangulaire (Figure 9). Pour montrer l'efficacité de l'algorithme de localisation, nous d'utiliser cette méthode pour des cas industriels qui nécessitent un temps de calcul très important.

Validation

Quelques exemples sont traités pour valider ces méthodes en comparaison avec les résultats analytiques ou expérimentaux trouvés dans la littérature [32][33][34][35]. Tout d'abord, deux exemples en écoulement 2D simple de type laminaire ou radial sont pris pour comparer nos résultats avec la solution analytique. Ensuite, un exemple de plaque avec une géométrie complexe est retenu pour comparer nos résultats avec les résultats expérimentaux publiés par A. Shojaei et al. [34]. Enfin, un exemple 3D de type boitier est pris pour montrer les effets du bord et la formation de macro-vides [36].

Exemples d'écoulement simple

Deux exemples d'écoulement de type laminaire ou radial simple sont traités pour valider les présentes méthodes avec les solutions analytiques. Le temps de remplissage total obtenu par nos méthodes est comparé avec celui analytique proposé par Z. Cai [32]. Le temps de remplissage total obtenu par notre méthode est égal à 123 seconds.

Celui de [34]

𝜌 D𝐮 D𝑡 ⁄ = -∇𝑝 -(𝜇 𝐊 ⁄ )𝐮 (1)
EA(K) -6

시간 간격마다 계산한 후, 다음 시간으로 유동 전면부를 이송시킨다 [27]. 정상상태에서 Darcy 의 식은 다음과 같이 유도된다:

𝐮 = -(𝐊 𝜇 ⁄ )∇𝑝 (3)
식 (3)을 식 (1)에 대입하면, 압력장을 구하기 위한 지배방적식이 유도된다: ∇ • (-(𝐊 𝜇 ⁄ )∇𝑝) = 0 (4)
경계조건은 다음과 같이 주어진다:

𝑝 𝑖𝑛 = 주입압력 주입구에서 𝑝 𝑣𝑒𝑛𝑡 = 배출 압력 배출구에서 𝜕𝑝 𝜕𝑛 ⁄ = 0 금형 벽에서 (5)

Abstract:

Numerical simulation for Resin Transfer Molding (RTM) manufacturing process is attempted by using the eXtended Finite Element Method (XFEM) combined with the level set method. XFEM allows to obtaining a good numerical precision of the pressure near the resin flow front, where its gradient is discontinuous. The enriched shape functions of XFEM are derived by using the level set values so as to correctly describe the interpolation with the resin flow front. In addition, the level set method is used to transport the resin flow front at each time step during the mold filling. The level set values are calculated by an implicit characteristic Galerkin FEM. The multi-frontal solver of IPSAP is adopted to solve the system. This work is validated by comparing the obtained results with analytic solutions.

Moreover, a localization method of XFEM and level set method is proposed to increase the computing efficiency. The computation domain is reduced to the small region near the resin flow front. Therefore, the total computing time is strongly reduced by it. The efficiency test is made with simple channel or radial flow models.

Several application examples are analyzed to demonstrate ability of this method. A wind turbine blade is also treated as industrial application. Finally, a Graphic User Interface (GUI) tool is developed so as to make easy the pre/post-processing of the simulation.

Figure 1 .

 1 Figure 1.1. Applications of the RTM process in the industry of transport.

1 .

 1 Figure 1.1. Applications of the RTM process in the industry of transport.

Figure 1 . 2 .

 12 Figure 1.2. Manufacturing procedures of the RTM process.

Figure 1 . 3 .

 13 Figure 1.3. Defects of structures manufactured by the RTM process.

Figure 2 . 1 .

 21 Figure 2.1. Permeability depending on the fiber volume fraction.

Figure 2 . 2 .

 22 Figure 2.2. The degree of cure as a function of the time and temperature.

Figure 2 . 3 .

 23 Figure 2.3. Interactions between resin flow, heat transfer, and curing reaction processes.

Figure 2 . 4 .

 24 Figure 2.4. Filling factors at filled, front, or empty region (the red color means the region filled by resin).

Figure 2 . 5 .Figure 2 . 6 .

 2526 Figure 2.5. Level set values composed by the filled region, the front, and the unfilled region Figure 2.6. An ideal case of the level set function.

Figure 2 . 7 .

 27 Figure 2.7. Control volume into three-node triangular elements.

Figure 2 . 8 .

 28 Figure 2.8. Definition of enriched nodes.

Figure 2 . 9 .

 29 Figure 2.9. Enriched shape functions of one-dimensional linear element.

Figure 2 . 10 .

 210 Figure 2.10. Types of enrichment.

Figure 3 . 1 .

 31 Figure 3.1. Computational domain for analysis of RTM process.

Figure 3 . 2 .

 32 Figure 3.2. Subdivision in elements intersected by flow front.

Figure 3 . 3 .

 33 Figure 3.3. Flow chart of the computational procedures.

Figure 3 . 4 .

 34 Figure 3.4. A simple square plate.

Figure 3 . 5 .VFigure 3 . 6 .

 3536 Figure 3.5. Pressure contours & graphs of a simple square plate quarter-filled by resin.

Figure 3 . 7 .

 37 Figure 3.7. A model for the verification of the re-initialization procedure.

Figure 3 . 8 .

 38 Figure 3.8. Change of level set values by the re-initialization procedure.

Figure 3 . 9 .

 39 Figure 3.9. Thin square plate with the uniformly expanded fluid flow.

Figure 3 . 10 .

 310 Figure 3.10. Change of level set values by the level set calculation procedure including the transport of flow front and the re-initialization.

Figure 3 . 11 .

 311 Figure 3.11. Plate for uniform expansion flow with two fronts.

Figure 3 . 12 .

 312 Figure 3.12. Merging of two flow fronts.

Figure 3 . 13 .Figure 3 . 14 .

 313314 Figure 3.13. Geometry and mesh of the channel flow rectangular plate Figure 3.14. Geometry and mesh of the radial flow model.

Figure 4 . 1 .

 41 Figure 4.1. Computational domain for localization of the level set method.

Figure 4 . 2 .

 42 Figure 4.2. Change of a level-set function by applying a cut-off function.

Figure 4 . 3 .

 43 Figure 4.3. Level set contours obtained without (Top) and with (Bottom) localization of the level set method.

Figure 4 . 4 .

 44 Figure 4.4. Pressure contour of the channel flow model.

Figure 4 . 5 .

 45 Figure 4.5. Localized computational domain (Localization of XFEM).

Figure 4 . 6 .

 46 Figure 4.6. Pseudo-code to make a node list on the boundary.

Figure 4 . 7 .

 47 Figure 4.7. Pressure contours during the mold filling.

Figure 5 . 1 .

 51 Figure 5.1. Geometry and mesh of the plate.

Figure 5 . 2 .

 52 Figure 5.2. Comparison of results on 2D complex geometry plate(Position of flow front at every 15 sec.).

Figure 5 . 3 .

 53 Figure 5.3. Geometry of the thin and complex shape structure.

Figure 5 . 4 .

 54 Figure 5.4. Two different permeability regionsfor the race-tracking effect.

Figure 5 . 5 .

 55 Figure 5.5. Results of the race tracking effect model (Position of flow front at every 5 sec.).

Figure 5 . 6 .

 56 Figure 5.6. Formation of macro voids (Position of flow front at every 5 sec.).

Figure 5 . 7 .

 57 Figure 5.7. Insertion of vents (Position of flow front at every 5 sec.).

Figure 5 . 8 .

 58 Figure 5.8. AOC 15/50 wind turbine blade.

Figure 5 . 9 .

 59 Figure 5.9. Three parts and seven sections of the composite wind turbine blade.

Figure 5 . 10 .Figure 5 . 11 .

 510511 Figure 5.10. Injection gates & vents on the top skin.

Figure 5 . 12 .

 512 Figure 5.12. Injection gates & vents on the spar.

Figure 5 . 13 .

 513 Figure 5.13. Resin flow pattern during the mold filling of the spar.

Figure 5 . 14 .

 514 Figure 5.14. Injection gates & vents on the bottom skin.

Figure 5 . 15 .

 515 Figure 5.15. Flow pattern during the mold filling of the bottom skin.

Figure 1 . 2 .

 12 Figure 1.2. Manufacturing procedures of RTM process.

  The first one is to transport the flow front, and the second is the reinitialization to keep the level set function close to the signed distance function. An implicit characteristic Galerkin Finite Element Method (FEM) is used to stabilize numerical oscillations caused by the convective term in the governing equations. This formulation also makes the coefficient matrix to be symmetric and positive definite. This study is validated through several examples in comparison with analytical or experimental results found in the literature. Additionally, complex shaped plate examples are used to simulate race-tracking effect and the formation of macro voids. The wind turbine blade example is also simulated. The exclusive program for RTM process simulation (Diamond/RTM) is implanted in Diamond program, a general finite element program having a Graphic User Interface (GUI).

Figure 2 . 2 .

 22 Figure 2.2. The degree of cure as a function of the time and temperature.

Figure 2 . 3 .

 23 Figure 2.3. Interactions between resin flow, heat transfer, and curing reaction processes.

Figure 2 . 4 . 2 . 5 .Figure 2 . 6 .

 242526 Figure 2.4. Filling factors in the filled, front, or empty region (the red color means the region filled by resin).

Figure 2 . 10 .

 210 Figure 2.10. Types of enrichment.

  .4 -c) Then, the pressure field and fluid velocity in the mold are solved at each time as steady-state problem, the resin flow front is transported for the next time step. The resin flow front is transported by solving the following equation: where t is the time, φ the level set function, and u the fluid velocity [Sethin JA, 1999]. By solving the equation (3.5), the level set values are updated at each time step. The details to derive the equation (3.5) are explained in the section 2.2. To correctly transport the flow front, re-initialization of level set values should be performed after solving the equation (3.5). The details for the re-initialization are also presented in the section 2.2.

(a) Case 1 (b) Case 2 Figure 3 . 2 .

 1232 Figure 3.2. Subdivision in elements intersected by flow front.

. 3 .

 3 The details are explained as follows. The first step is to read the input data, such as the mesh information, material properties, and boundary conditions, and to initialize the pressure field and level set values. At the second step, the fluid velocity is calculated from the pressure field by using the Darcy's equation, explained in the section 3.1. The level set values are calculated in order to transport the resin front.The re-initialization of the level set values is performed at each time step. Then, the index of the nodes is set by the level set distribution: enriched nodes by 1 and normal nodes by 0 to denote the corresponding shape functions. The second and third steps are continued until the mold filling is complete. The multi-frontal solver of IPSAP[Kim JK et al., 2005] was applied to solve the discretized equilibrium systems. The multi-frontal solver of IPSAP is presented in Appendix A. This method and a GUI (Graphic User Interface) are implemented in DIAMOND program for analysis of RTM process. DIAMOND program, developed by Aerospace Structures Laboratory in Seoul National University, is presented in Appendix C [Kim SJ, 2010].

Figure 3 . 6 .

 36 Figure 3.6. Geometry and mesh information of a thin square plate for verifying level set calculation. At first, the verification of re-initialization procedure is performed by using the model described in the figure 3.7. The model is filled with resin until the circle region which is 0.025 meters in radius. The circle region is placed at the middle of the square. The region except the one surrounded by the circle is filled with air. In order to set the initial level set values into incorrect values, the initial condition of level set is given as: |∇𝜑| = |∇𝑑 0 | = 2. (3.14)

Figure 3 . 9 .Figure 3 . 10 .

 39310 Figure 3.9. Thin square plate with the uniformly expanded fluid flow.

Figure 3 . 11 .Figure 3 . 12 .

 311312 Figure 3.11. Plate for uniform expansion flow with two fronts.

3 .

 3 Figure 3.13. Geometry and mesh of the channel flow rectangular plate

  soukane and F.Trochu [2006], which treated the same example, our program is more accurate.B. 2D Radial Flow Example in a Circular PlateThe figure3.14 shows the radial flow model with a thin circular disk. Because of its axisymmetric conditions, a quarter of the disk is modeled. The radius of the disk is 1 m. And the thickness is equal to 1 mm. The resin is injected through a hole located at the center, of which the radius is equal to 0.2 m. The numbers of nodes and elements are 2082 and 3987, respectively. The dashed line at the outer side of the plate means the vent location. The value of resin viscosity, preform permeability, porosity, and injection pressure are the same as in the previous example (2D channel flow example).

Figure 3 . 14 .

 314 Figure 3.14. Geometry and mesh of the radial flow model.

  r is the radius of the plate and 𝑟 0 the radius of the hole. The analytical filling time of the radial flow example is equal to 1411 seconds. The numerical filling time obtained in this study is 1436 seconds. The difference between the two results is 1.8 %. Our program is more accurate in comparison with Mark Lin et al.[1998] which give 4% of difference with the analytical solution.

  (3.6) to calculate the level set values, the governing equation is modified as: 𝜕𝜑 𝜕𝑡 + 𝑐(𝜑)𝐮 • ∇φ = 0. (4.2) The level set values, obtained by the equation (4.2), are correct in the space within β range from the resin flow front. In the space between γ and β from the front, the values are modified by the cut-off function. Outside of the band of γ range from the resin flow front, the level set values are not updated. 56 1 among the nodes composing each element. The elements are included in a list of the local domain when the number of nodes indexed to 1 is equal to the number of nodes composing each element.

Figure 4 . 6 .

 46 Figure 4.6. Pseudo-code to make a node list on the boundary.

  result is compared with the experimental. Since the flow pattern is affected by the structure geometry during mold filling, it is important to determine the flow front position every time in complex geometry. The experimental result and the modeling information are acquired from the research of Akbar Shojaei et al. [2003].

Figure 5 . 1 .

 51 Figure 5.1. Geometry and mesh of the plate.

  (a) Obtained by methods. (b) Given by experiment.

Figure 5 . 2 .

 52 Figure 5.2. Comparison of results on 2D complex geometry plate (Position of flow front at every 15 sec.).

Figure 5 . 3 .

 53 Figure 5.3. Geometry of the thin and complex shape structure.

  to 6.21x10 -10 m 2 . The meshed model has 3684 nodes and 7027 elements.

Figure 5 . 4 .

 54 Figure 5.4. Two different permeability regions for the race-tracking effect.

Figure 5 . 5 .

 55 Figure 5.5. Results of the race tracking effect model (Position of flow front at every 5 sec.).

  other geometry and injection/vent gates are same with the figure5.3. This model has 3812 nodes and 7328 3-node elements.

Figure 5 . 6 . 5 . 7 .

 5657 Figure 5.6. Formation of macro voids (Position of flow front at every 5 sec.).

Figure 5 . 7 .

 57 Figure 5.7. Insertion of vents (Position of flow front at every 5 sec.).

  PropertiesA composite blade for the Atlantic Orient Corporation (AOC) 15/50 wind turbine[Cairns DS et al., 1999; MCKittrick LR, 2001] is treated to simulate the resin flow during RTM process. The modeling data such as the geometry and material properties are obtained from the report of Cairns and Rossell [2004]. The geometry of the composite blade is shown in the figure 5.8. The length of the composite blade is about 5.76 m. The node coordinates to draw the blade model are in Appendix B. The blade is composed of five components: web, spar cap, flange, leading edge and trailing edge. The each component has different fiber permeability because the lay-up is different.

Figure 5 . 8 .Figure 5 . 11 .

 58511 Figure 5.8. AOC 15/50 wind turbine blade.

Figure 5 . 13 .

 513 Figure 5.13. Resin flow pattern during the mold filling of the spar.

  où φ J est la valeur Level set du noeud enrichi J. La discontinuité au front d'écoulement peut être représentée par les fonctions composées des fonctions et valeurs Level set. Comme exemple, les fonctions de forme enrichies d'un élément unidimensionnel sont montrées dans la figure 3. Le caractère significatif des fonctions de forme enrichies est que la position du front d'écoulement est prise en compte dans les fonctions de forme enrichies. Il est important de noter que les valeurs des fonctions de forme enrichies doivent être égales à 0 aux noeuds. EA(F) -9

Figure 3 :

 3 Figure 3 : les fonctions de forme enrichies d'un élément linéaire unidirectionnel.

Figure 4 :

 4 Figure 4 : Schéma global du calcul.

Figure 5 : 1 (

 51 Figure 5 : Domain de calcul pour la localisation avec la méthode Level set.

Figure 6 .

 6 Figure 6. Pression dans le moule d'une plaque en écoulement laminaire.

Figure 7 :

 7 Figure 7 : Domaine de calcul pour la localisation de la méthode XFEM.

Figure 8 :

 8 Figure 8 : Pression obtenue par la localisation de la méthode XFEM.

6. 1 . 1 .Figure 9 . 6 . 1 . 2 .

 119612 Figure 9. Géométrie et maillage de la plaque rectangulaire.

Figure 11 .

 11 Figure 11. Géométrie et maillage de la plaque en 2D avec une géométrie complexe.

Figure 12 :

 12 Figure 12 : Résultats obtenus par notre méthode sur la plaque 2D avec une géométrie complexe : le front d'écoulement tracé à tous les 15 seconds.

Figure 13 : 24 6. 3 . 1 .Figure 15 :

 13243115 Figure 13 : Résultats publiés par A. Shojaei et al. [34] sur la plaque 2D avec une géométrie complexe : le front d'écoulement tracé à tous les 15 seconds.

Figure 16 : 6 . 3 . 2 .

 16632 Figure 16 : Résultats du modèle avec l'effet du bord. Les fronts d'écoulement tracés à tous les 5 seconds.

Figure 17 :

 17 Figure 17 : Position du front d'écoulement de la résine à tous les 5 seconds et formation de macro-vides comme marqué en noir aux coins des pieds.

Figure 18 :

 18 Figure 18 : Position du front d'écoulement de la résine à tous les 5 seconds après avoir ajouté deux points de fuite.

제 4 -

 4 장. 수식화 제 1 절. 유동 전면부의 이송식 (6)은 외연적 특성 Galerkin 유한 요소 법을 적용하여 계산한다[22].시간에 대한 미분항은 외연적 특성 기법에 의해 이산화 된다[28]:𝜑 𝑛+1 -𝜑 𝑛 = -∆𝑡[𝐮 • ∇𝜑] 𝑛 + ∆𝑡 2 2 ⁄ [∇ • ((𝐮⨂𝐮) • ∇𝜑)] 𝑛+1(14) 여기서 ∆t 는 시간간격이다. 공간에 대한 미분항의 이산화에는 Galerkin 유한 요소 법을 적용한다: 여기서 w 는 가중함수이고 Ω 는 전체 계산 영역이다. 전면 이송 속도 u 은 식 (3)을 이용하여 계산한다. 그리고 점성은 다음과 같이 정의한다: EA(K) -12 𝜇 = 𝐻(𝜑)𝜇 𝑟 + (1 -𝐻(𝜑))𝜇 𝑎 (16) 여기서, μ  는 수지 점성, μ  는공기점성φ 는 Level set 값이다. H(φ)는 아래와 같이 정의된다: 재초기화를 위한 식 또한 식 (14)와 (15)의 과정을 따라 이산화 된다: ∫ (𝑑 𝑛+1 -𝑑 𝑛 )𝑤𝑑𝑉 Ω = -Δ𝜏 ∫ 𝑠𝑖𝑔𝑛(𝑑 0 )|∇𝑑 𝑛 |𝑤𝑑𝑉 Ω 이론을 적용하면 식 (19)는 아래와 같이 된다: 여기서 n 은 법선 벡터이다. 확장 유한 요소 법을 적용하여 압력과 가중 함수는 식 (12)와 같이 이산화된다. 한편, 확장 지역에서 점성은 아래와 같이 정의 된다: EA(K) -14

그림 9 .

 9 채널 유동을 위한 직사각 평판 의 형상과 그물망 물성과 경계조건은 다음과 같다: 수지 점성 = 0.1 Pa•s, 투과성 = 10 -9 m 2 , porosity = 0.5, 주입 압력 = 20 KPa, 배출구 압력 = 0. 채널 유동을 위한 이론적 충진 시간은 다음 식으로 계산된다: 𝑡 𝑓𝑖𝑙𝑙𝑖𝑛𝑔 = 𝐿

  ; Lin CL, 2005]. The level set method is the representative of methods to capture moving interfaces [Malladi R et al., 1995; Peter DM et al., 2005]. The level set method is often used for RTM process simulations since it predicts intuitively the flow front's position and is very efficient for multiple flow fronts occurred by complex shapes of the mold or by multiple injection gates [Gantois R, 2010; Soukane S and Trochu F, 2006].

  The eXtended Finite Element Method (XFEM) is widely used to treat various discontinuity problems such as crack or interface problems[Belytschko T et al.,

	2009; Sukumar N et al., 2000-2001]. The early works about XFEM were focused
	on the simulation of fracture and went towards the development of XFEM for
	fracture mechanics [Moes N and Belytschko T, 2002]. Furthermore, in the field of
	contact mechanics, XFEM techniques for powder compaction problems are
	presented by Khoei et al. [2006]. Moes et al. [2003] used XFEM to solve scales
	with complex geometry in multiscale analysis of components. Chessa et al. [2002]

Table 4 .

 4 The forth column is the percentage of the reduced time, as compared with the computation without the localization.

		Computing Time
	No. of Node			Reduction
		Without Localization (a)	With Localization (b)
			Level-Set	29.0 sec.	22.7 %
	861	37.5 sec.
			XFEM + Level-Set 24.1 sec.	35.7 %
			Level-Set	154.0 sec. 27.2 %
	2556	211.4 sec.
			XFEM + Level-Set 118.0 sec. 44.2 %
			Level-Set	742.5 sec. 40.0 %
	7381	1236.8 sec.
			XFEM + Level-Set 561.1 sec. 54.6 %
	The first column indicates the number of node (DOF). The data of the second
	column is the computing time by the original code which does not apply the
	localization algorithm. The third column is the time with the localization. The
	calculation time by applying the localization of level set method is the upper data.

1. Comparison of computing times (A channel flow model).

And, the below date is the time with localization of XFEM and level set method.

Table 4 .

 4 2. Comparison of computing times (A radial flow model).

		Computation Time
	No. of Node	Without Localization(a)	With Localization (b)	Reduction
		Level-Set	139.8 sec.	24.5 %
	2082	185.2 sec.
		XFEM + Level-Set	117.0 sec.	36.8 %
		Level-Set	943.4 sec.	30.1 %
	7295	1349.5 sec.
		XFEM + Level-Set	782.8 sec.	42.0 %
		Level-Set	1965.9 sec.	37.0 %
	12480	3122.6 sec.
		XFEM + Level-Set	1665.2 sec.	46.7 %
	The second example is the radial flow model shown in the figure 3.15. The table 2
	shows the comparison of computing times. The numbers of nodes for three cases
	are 2082, 7295, and 8040 nodes, respectively. The reductions of the computing
	times with localization of level set method are approximately 25%, 30%, and 37%,
	respectively. The reduction rates with localization of XFEM and level set method
	are 37%, 42%, and 40%, respectively.
	The reduction tendency of the computing time looks similar in the channel flow
	and the radial flow cases. It means that we could save more computing time as the

problem size becomes bigger. It might be very interesting when we treat industrial problems or optimization ones which could be very time consuming by iterations.

  non conformes calculent la pression aux interfaces des éléments adjacents, la vitesse de l'écoulement à la frontière des éléments peut être obtenue sans aucun traitement supplémentaire, de laquelle le taux de remplissage du moule est dérivé.Dans cette étude, les fonctions de forme enrichies de la XFEM décrivant le champ de pression sont définies par les valeurs Level set suivant[14]. Puisque la XFEM met le système d'équilibre symétrique et défini positif, une méthode de résolution en parallèle classique, comme la méthode multi-frontale, peut être utilisée afin de réduire le temps de calcul. La méthode Level set est composée de deux étapes. La première est de déplacer le front découlement, et la seconde est la réinitialisation pour maintenir la fonction Level set près de la fonction de distance avec signe.Une méthode de Galerkin à caractère implicite est utilisée pour stabiliser les oscillations numériques dues au terme convectif dans les équations d'équilibre[22]. Le champ de pression est divisé en deux régions : normale et enrichie. La zone enrichie est définie seulement près du front d'écoulement de la résine afin de réduire la taille du système d'équilibre à résoudre. Puisque la zone enrichie n'est

	Cairns DS, McKittrick LR, Combs DS, Mandell JF, Rabern DA, and VanLuchene RD, Malladi R, Sethian JA, and Vemuri BC, Shape modeling with front propagation: a level Peter DM and Spelt, A level-set approach for simulations of flows with multiple moving Tan H and Pillai KM, Numerical simulation of reactive flow in liquid composite molding 0.1954 0.1871 0.0000 0.5552 0.3858 1.7363 0.0000 0.3000 3.7351 9 0.5004 0.3059 5.0923 1. Introduction
	Design/manufacturing synthesis of a composite blade for the AOC 15/50 wind turbine, 18th ASME Wind Energy Symposium, P. 58-65 (1999) set approach, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 2, P. 158-175 (1995) contact lines with hysteresis, Journal of Computational Physics, Vol. 207, No. 2, P. 389-404 (2005) using flux-corrected transport (FCT) based finite element/control volume (FE/CV) 0.1354 0.1803 0.0000 0.7493 0.3517 1.7363 0.1097 0.3537 3.7351 9 0.3904 0.2927 5.0923 method, International Journal of Heat and Mass Transfer, Vol. 53, No. 9-10, P. 2256-2271 (2010) 0.0000 0.3000 0.3983 0.6221 0.3391 1.7363 0.1800 0.3645 3.7351 9 0.2960 0.2737 5.0923 Le procédé RTM (Resin Transfer Molding) est un des procédés de fabrication des
	Chang CJ, Hsieh JW, Chen YS, and Hu WF, Tracking multiple moving boundary objects using a level set method, International Journal of Pattern Recognition and Artificial Intelligence, Vol. 18, No. 2, P. 101-125 (2004) Chessa J and Belytschko T, An extended finite element method and level sets for axisymmetric two-phase flow with surface tension, International Journal for Numerical Methods in Engineering, Vol. 58, No. 13, P. 2041-2064 (2003) Chessa J and Belytschko T, An extended finite element method for two-phase fluids, Mathur R, Fink B, and Advani SG, Use of genetic algorithms to optimize gate and vent locations for the resin transfer molding process, Polymer Composites, Vol. 20, No. 2, P. 167-178 (1999) McKittrick LR, Analysis of a composite blade design for the AOC 15/50 wind turbine using a finite element model, Sandia National Laboratories (2001) Moes N and Belytschko T, Extended finite element method for cohesive crack growth, Engineering Fracture Mechanics, Vol. 69, No. 7, P. 813-833 (2002) Rouison D, Sain M, and Couturier M, Resin transfer molding of hemp fiber composites: optimization of the process and mechanical properties of the materials, Composite Science and Technology, Vol. 66, No. 7-8, P. 895-906 (2006) Sethian JA, Level set methods and fast marching methods 2nd edition, Cambridge University Press (1999) Shojaei A and Ghaffarian SR, Modeling and simulation approaches in the resin transfer molding process: A review, Polymer Composites, Vol. 24, No. 4, P. 525-544 (2003) 0.1317 0.3991 0.3983 0.3177 0.2374 1.7363 0.2098 0.3680 3.7351 9 0.1954 0.2619 5.0923 structures composites les plus largement utilisés dans l'industrie de transport, par Cependant, le taux de remplissage n'est pas une valeur nodale, mais celle de Trochu F and Gauvin R, Limitations of a boundary-fitted finite difference method for the 0.1982 0.4130 0.3983 0.2272 0.2065 1.7363 0.3141 0.3664 3.7351 9 0.1648 0.2637 5.0923 exemple, l'automobile, la construction navale, l'aéronautique et l'espace. Pour la l'élément [5]. C'est pourquoi ces méthodes ne pouvaient pas donner une simulation of the resin transfer molding process, Journal of Reinforced Plastics and Composites, Vol. 11, No. 7, P. 772-786 (1992) Trochu F, Gauvin R, and Gao DM, Numerical-Analysis of the Resin Transfer Molding Process by the Finite-Element Method. Advances in Polymer Technology, Vol. 12, No. 4, P. 329-342 (1993) 0.2264 0.4177 0.3983 0.3178 0.4180 0.3983 0.4989 0.4057 0.3983 0.6477 0.3514 0.3983 0.1962 0.1992 1.7363 0.1263 0.2031 1.7363 0.0000 0.3000 2.4103 1.2230 0.3690 2.4103 0.4576 0.3418 3.7351 0.6071 0.3213 3.7351 0.4946 0.3099 3.7351 0.3128 0.2578 3.7351 9 0.0910 0.2715 fabrication des structures composites par le procédé RTM, il est indispensable de approximation propre du champ de pression dans les éléments intersectés par le 5.0923 10 0.0000 0.3000 simuler l'écoulement de la résine dans le moule en vue de prédire les lignes de front d'écoulement de la résine à cause de la discontinuité du gradient de pression 5.7573 10 0.0899 0.3345 soudure et d'éviter la formation de poches d'air, qui pourraient être l'origine des près du front d'écoulement. Ce point est très important dans le cas des problèmes 5.7573 10 0.1583 0.3412 5.7573 défauts critiques dans la structure. Un des points importants est, lors de la industriels ayant des formes géométriques complexes.
	Journal of Applied Mechanics, Vol. 70, No. 1, P. 10-17 (2003) Chessa J, Smolinski P, and Belyschko T, The extended finite element method (XFEM) for Moes N, Cloirec M, Cartraud P, and Remacle JF, A computational approach to handle complex microstructure geometries, and Engineering, Vol. 192, No. 28-30, P. 3163-3177 Shojaei A, Ghaffarian SR and Karimian SMH, Numerical analysis of controlled injection strategies in resin transfer molding, Journal of Composite Materials, Vol. 37, No. 11, P. Trochu F, Ruiz E, Achim V, and Soukane S, Advanced numerical simulation of liquid 0.5429 0.3143 0.3983 0.1892 0.3837 2.4103 0.2078 0.2404 3.7351 10 0.1899 0.3414 5.7573 composite molding for process analysis and optimization, Composites Part A: applied science and manufacturing, Vol. 37, No. 6, P. 890-902 (2006) 0.3142 0.2244 0.3983 0.3215 0.3902 2.4103 0.1773 0.2421 3.7351 10 0.2921 0.3354 5.7573
	solidification problems, International journal for numerical methods in engineering, Vol. (2003) 1011-1035 (2003) 0.2264 0.1971 0.3983 0.5281 0.3701 2.4103 0.1078 0.2542 3.7351 10 0.3643 0.3198 5.7573 53, No. 8, P. 1959-1977 (2002) Mohammadi S, Extended finite element method for fracture analysis of structures, Shojaei A, Ghaffarian SR, and Karimian SMH, Numerical simulation of three-0.1954 0.1905 0.3983 0.7173 0.3417 2.4103 0.0000 0.3000 4.4263 10 0.4445 0.3046 5.7573
	Chung SW and Kim SJ, Parallel computation of a damage localization problem using Blackwell Publish Ltd. (2008) dimensional mold filling process in resin transfer molding using quasi-steady state and 0.1305 0.1849 0.3983 0.5254 0.3206 2.4103 0.1035 0.3458 4.4263 10 0.3741 0.2983 5.7573
	parallel multifrontal solver, Computational Mechanics, Vol. 33, No. 1, P. 42-51 (2003) Morren G, Bossuyt S, and Hugo S, 2D permeability tensor identification of fibrous partial saturation formulations, Composites Science and Technology, Vol. 62, No. 6, P. 861-879 (2002) 0.0000 0.3000 1.0633 0.3228 0.2531 2.4103 0.1707 0.3564 4.4263 10 0.2915 0.2847 5.7573
	Danisman M, Tuncol G, Kaynar A, and Sozer ME, Monitoring of resin flow in the resin reinforcements for RTM using an inverse method, Composites Part A: Applied Science 0.1281 0.3882 1.0633 0.2204 0.2209 2.4103 0.2017 0.3598 4.4263 10 0.1893 0.2758 5.7573
	transfer molding (RTM) process using point-voltage sensors, Composite Science and Technology, Vol. 67, No. 3-4, P. 367-379 (2007) and Manufacturing, Vol.39, No. 9 (2008) Smolianski A, Finite element level set operator splitting (FELSOS) approach for computing two-fluid unsteady flows with free moving interfaces, International Journal for 0.1960 0.4050 1.0633 0.1897 0.2158 2.4103 0.3074 0.3579 4.4263 10 0.1584 0.2746 5.7573
	Mousavi SE and Sukumar N, Generalized Gaussian quadrature rules for discontinuities Numerical Methods in Fluids, Vol 48, No. 3, P. 231-269 (2005) 0.2265 0.4091 1.0633 0.1230 0.2217 2.4103 0.4464 0.3328 4.4263 10 0.0892 0.2768 5.7573
	Dong C, A modified rule of mixture for the vacuum-assisted resin transfer moulding process simulation, Composite Science and Technology, Vol. 68, No. 9 P. 2125-2133 and crack singularities in the extended finite element method, Computer Methods in Applied Mechanics and Engineering, Vol. 199, No. 49-52, P. 3237-3249 (2010) Soukane S and Trochu F, Application of the level set method to the simulation of resin 0.3279 0.4112 1.0633 0.0000 0.3000 3.0439 0.5537 0.3134 4.4263
	(2008) transfer molding, Composites Science and Technology, Vol. 66, No. 7-8, P. 1067-1080 0.5989 0.3858 1.0633 0.1162 0.3605 3.0439 0.4725 0.3056 4.4263
	Estacio KC and Mangiavacchi N, Simplified model for mould filling simulations using CVFEM and unstructured meshes, Numerical Methods in Biomedical Engineering, Vol. 23, No. 5, P. 345-361 (2007) Phelan FR., Simulation of injection process in resin transfer molding, Polymer Composites, Vol. 18, No. 4, P. 460-476 (1997) Gantois R, Cantarel A, Dusserre G, J.-N. Felices, and F. Schmidt, Numerical simulation of resin transfer molding using BEM and level set method, International Journal of Material Forming, Vol. 3, Suppl 1, P. 635-638 (2010) Lin R, Lee LJ, and Liou M., Non-isothermal mold filling and curing simulation in thin cavities with preplaced fiber mats, International Polymer Processing, Vol. 4. No.4, P. 356-369 (1991) Liu B, Bickerton S, and Advani SG, Modeling and simulation of resin transfer moulding (RTM)?gate control, venting and dry spot prediction, Composite Part A: Applied Science and Manufacturing, Vol. 27, No. 2, P. 89-163 (1996) Pacquaut G, Bruchon J, Moulin1N, and Drapier S, Combining a level-set method and a mixed stabilized P1/P1 formulation for coupling Stokes?Darcy flows, International Journal for Numerical Methods in Fluids, Vol. 69, No. 2, P. 459-480 (2012) Park CH, Lee WI, Han WS, and Vautrin A, Simultaneous optimization of composite structures considering mechanical performance and manufacturing cost, Composite Structures, Vol. 65, No. 1, P. 117-127 (2004) Park CH, Lee WI, Han WS, and Vautrin A, Weight minimization of composite laminated plates with multiple constraints, Composites Science and Technology, Vol. 63, No.7, P.1015-1026 (2003) (2006) Subbiah S, Trafford DL, and Guceri SI, Non-isothermal flow of polymers into two-dimensional, thin cavity molds: a numerical grid generation approach, International Journal of Heat and Mass Transfer, Vol. 32, No. 3, P. 415-434 (1989) Sukumar N, Chopp DL, Moos N, and Belytschko T, Modeling holes and inclusions by level sets in the extended finite-element method, Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 46-47, P. 6183-6200 (2001) Sukumar N, Moes N, Moran B, and Belytschko T, Extended finite element method for three-dimensional crack modeling, International Journal for Numerical Methods in Engineering, Vol. 48, No. 11, P. 1549-1570 (2000) Zhao HK, Chan T, Merriman B, and Osher S, A variational level set approach to multiphase motion, Journal of Computational Physics, Vol. 127, P. 179-195 (1996) Zheng LL and Zhang H, An adaptive level set method for moving boundary problems: application to droplet spreading and solidification, Numerical Heat Transfer, Part B, Vol. 37, No. 4, P. 437-454 (2000) Zienkiewicz OC and Taylor RL, The finite element method volume 3: Fluid dynamics, Butterworth Heinemann, 5th edition (2000) 0.3266 0.2411 1.0633 0.2254 0.2035 1.0633 0.1953 0.1978 1.0633 0.1271 0.1942 1.0633 0.7214 0.3516 1.0633 0.5907 0.3267 1.0633 0.0000 0.3000 1.7363 0.1273 0.3803 1.7363 0.2272 0.3997 1.7363 0.1855 0.3745 3.0439 0.2151 0.3779 3.0439 0.3047 0.3801 3.0439 0.4685 0.3593 3.0439 0.6579 0.3304 3.0439 0.5530 0.3233 3.0439 0.3035 0.2504 3.0439 0.2144 0.2309 3.0439 0.1851 0.2273 3.0439 0.3060 0.2660 4.4263 0.2013 0.2514 Les méthodes numériques les plus souvent exploitées pendant les deux dernières 4.4263 0.1697 0.2542 décennies sont la méthode d'éléments finis avec contrôle de volume (CVFEM, pouvait être imprécis pour les cas où on trace des mouvements complexes du front 4.4263 0.1037 0.2640 Control Volume Finite Element Method) et les éléments finis non conformes. La d'écoulement de la résine [22-25]. La méthode Level set est une des plus 4.4263 0.0000 0.3000 méthode CVFEM a été la plus fréquemment développée pour la simulation du représentatives pour capturer les interfaces mobiles [26]. Cette méthode est 5.0923 0.0918 0.3399 procédé RTM jusqu'au début des années 2000 [1-12], car la méthode CVFEM souvent utilisée pour la simulation du procédé RTM car elle prédit intuitivement 5.0923 0.1659 0.3477 pouvait satisfaire efficacement la conservation de masse de la résine entre des la position du front d'écoulement et est très efficace pour les fronts d'écoulement 5.0923 0.1959 0.3503 éléments voisins par rapport à celle des éléments finis classique [2]. En revanche, multiples dus à la forme géométrique complexe du moule ou à plusieurs points 5.0923 0.2991 0.3471 la construction des volumes de contrôle dans le domaine de calcul entier pouvait d'injection [24, 25]. Dans cette étude, une méthode de type Level set est retenue 5.0923 410-438 (1999) 0.3212 0.4033 1.7363 0.1145 0.2387 3.0439 0.3931 0.3263 5.0923 pour calculer le front d'écoulement de la résine.
	84 88 B -2 B -4	B -3 B -5

Gerstenberger A and Wall WA, An eXtended Finite Element Method/Lagrange multiplier based approach for fluid?structure interaction, Computer Methods in Applied Mechanics and Engineering, Vol. 197, No.

19

-20, P. 1699-1714 (2008) Liu XL, Isothermal flow simulation of liquid composite molding. Composites Part A: applied science and manufacturing, Vol. 31, No. 12, P. 1295-1302 (2000). Peng D, Merriman B, Stanley Osher, Hongkai Zhao, and Myungjoo Kang, A PDE-based fast local level set method, Journal of Computational Physics, Vol. 155, No. 2, P.Ventura G, On the elimination of quadrature sub-cells for discontinuous functions in the extended finite-element method, International Journal for Numerical Methods in Engineering, Vol. 66, No. 5, P. 761-791 (2006) Young WB, Development of a helicopter landing gear prototype using resin infusion molding, Journal of Reinforced Plastics and Composites, Vol. 28, No. 7, P. 833-839 (2009) Young WB, Gate location optimization in liquid composite molding using genetic algorithms, Journal of Composite Materials, Vol. 20, No. 2, P. 1098-1113 (1999)

Young WB, Rupel K, Han K, Lee LJ, and Liou MJ, Analysis of resin injection molding in molds with preplaced fiber mats. Ⅱ : Numerical Simulation and Experiments of Mold Filling, Polymer Composites, Vol. 12, No.1, P.

30-38 (1991)

simulation de l'écoulement de la résine, d'assurer une approximation correcte de la position du front d'écoulement à chaque instant pendant le processus de remplissage du moule. En outre, il est nécessaire de paralléliser le calcul afin de pouvoir traiter des problèmes de taille industrielle dans un temps de calcul acceptable.

L'écoulement de la résine dans le moule représente un problème d'interface mobile à travers la préforme placée a priori dans le moule. Les méthodes numériques utilisées pour la simulation de l'écoulement de la résine peuvent être regroupées en deux groupes : maillage mobile et maillage fixe. L'approche avec le maillage fixe est plus largement utilisé car celle du maillage mobile nécessite un calcul relativement plus cher et complexe

[1][2][3][4][5][6][7][8]

. mettre le calcul complexe

[3, 5]

. Trochu et al a proposé d'appliquer les éléments finis non conformes à la simulation du procédé RTM

[5]

. Puisque les éléments EA(F) -3 finis La méthode d'éléments finis étendue (XFEM, eXtended Finite Element Method) est largement utilisée pour traiter de divers problèmes de discontinuité tels que la rupture ou les problèmes d'interface

[13][14][15][16][17][18][19]

. Cesse et al a appliqué la méthode XFEM aux problèmes d'écoulement immiscible biphasé qui ont une discontinuité sur le gradient de pression à l'interface entre deux fluides

[14]

. Cette méthode a été retenue dans cette étude en vue d'augmenter la précision numérique du champ de gradient de pression (champ de vitesses) près du front d'écoulement.

Quant à l'approche avec le maillage fixe, la méthode VOF (Volume Of Fluid) est une des méthodes les plus représentatives pour tracer les interfaces mobiles

[20, 21]

. Cette méthode a été couramment utilisée pour la simulation du procédé RTM

[1][2][3][4][5][6][7][8][9]

. Cependant, puisque l'interface mobile est définie par le volume de fluide fractionnel dans l'élément ou le volume de contrôle suivant la méthode utilisée, il EA(F) -4 définie que près du front d'écoulement de la résine, la taille du système d'équilibre est légèrement augmentée. Le solveur multi-frontal en parallèle proposé par ISAP

[31]

est retenu pour résoudre le système d'équilibre discrétisé.

La localisation avec la méthode XFEM et la méthode Level set sert à augmenter l'efficacité de calcul.

  . La localisation signifie la définition d'une région en bande près du front d'écoulement de la résine. La simulation de l'écoulement de la résine dans le moule peut être traduite par la description du mouvement du front d'écoulement de la résine. Par conséquent, les valeurs Level set exceptées la région près du front d'écoulement sont insignifiantes. C'est pourquoi il est possible d'augmenter l'efficacité de calcul si le domaine est raisonnablement réduit pour obtenir seulement les valeurs Level set sans aucune influence sur les résultats. En général, la plupart du temps de calcul est consommé par la résolution du système d'équilibre du problème. C'est pourquoi si plus petite est la taille du système (nombre de DDL), le temps de calcul peut certainement être plus réduit. Même si une procédure complémentaire est nécessaire pour localiser le domaine de calcul, le temps de calcul total devait être réduit d'une manière importante car le temps

	de localisation est relativement très court. Dans ce chapitre, un algorithme de
	localisation pour la méthode XFEM couplée avec la méthode Level set est
	présenté. En outre, l'efficacité de calcul par localisation est prouvée à travers
	quelques exemples.

  est de 117 seconds en expérimental et de 126 seconds en numérique.

	La différence entre le résultat expérimental et le nôtre est égale à 4,3 %. En
	comparaison avec la solution numérique de [34] dont la différence s'étend à 7,7 %,
	notre résultat est plus proche au résultat expérimental.

EA(F) -22

  여기서, φ J 는 확장절점 J 의 Level set 값이다. 확장 형상 함수에 대해 쉽게 설명하기 위해 그림 3 에서는 1 차원 선형요소의 확장 형상 함수의 생성 과정을 그래프로 보여준다. 확장형상함수의 중요한 특징은 요소 내의 전면부의 위치가 반영이 되어 있다는 것이다. 그리고 절점에서 확정형상 함수의 값이 0 인 것이 특징이다. 에서 볼 수 있듯이, 확장 지역은 두 가지 유형으로 분류된다 (전체 혹은 부분 확장 지역). 여기서 전체 확장 지역은 확장 절점으로만 구성되어 있고 부분 확장 지역은 일반 그리고 확장 절점으로 이루어 진다. 부분 확장 지역은 확장과 일반 지역 사이의 압력 장을 연속적으로 유지하기 위해 필요하다.

	계산 영역은 일반과 확장 지역으로 나누어 진다. 확장 지역은 그림 2 에
	서 볼 수 있듯이 전면부가 지나가는 요소를 구성하는 절점이 포함되어
	있는 곳이다. 확장이란 추가적으로 DOF 를 더하는 것을 의미하므로
	압력의 근사는 아래의 식과 같이 정의된다:
	𝑝 ℎ = ∑ 𝑁 𝐼 𝑝 𝐼 +	∑ 𝑁 𝐽 𝑒 𝑎 𝐽	(10)
	𝐼∈𝑛	𝐽∈𝑛 𝑒
	여기서, p 𝑁 𝐽 𝑒 = 𝑁 𝐽 (|𝜑| -|𝜑 𝐽 |)	(11)
	매 시간간격마다 유동 전면부의 이송은 Level set 기법을 이용하여
	계산한다. 자세한 사항은 다음 장에서 설명된다.
	제 3 장. 수치기법
	제 1 절. Level set 기법
	그림 1 에 나타나듯이 Level set 값은 유동 전면으로부터의 부호 거리
	값이다. 그래서 Level set 값이 전면에서는 0 이고, 빈 공간에서는 음수,
	수지가 찬 부분에서는 양수이다.
		EA(K) -9

h 는 압력 함수, p I 절점 I 에서 압력, a J 는 확장 절점 J 에서 추가 절점 변수, N I 는 일반형상함수, 𝑁 𝐽 𝑒 는 확장형상함수, n 은 절점의 개수, n e 확장 절점의 개수 이다. 그 확장형상함수는 다음과 같이 정의된다: EA(K) -11 그림 2

확장유한요소법과 레벨셋 기법의 국소화 레벨셋

  제 5 장. 기역화 기법에 의한 효율성 향상 본 연구는 확장유한요소법과 레벨셋 기법의 지역화를 통해 계산효율을 증가시킨다 [37]. 여기서 국소화는 유동 전면부 근처의 작은 지역을 계산 영역으로 할당하는 것을 의미한다. 금형 내의 수지 유동을 시뮬레이션 하는 것은 엄밀히 수지 유동 전면부의 움직임을 묘사하는 것이다. 그러므로 유동 전면부 이외 지역의 레벨셋 값은 불필요하다. 기법의 국소화 알고리즘은 Peng[23]등이 제안한 방법을 기초로 하고 있다. 본 연구에서는 그들의 알고리즘을 외연적 특성 Galerkin 유한요소법에 적용하기 위해 수정한다. 결과를 비교한다 [32-35]. 먼저, 이차원 채널 혹은 방사상 유동을 계산하고 이론치와 비교한다. 다음으로 Shojaei 등 [34]이 제안한 복잡한 형상 예제를 이용하여 해석하고 그 결과를 실험치와 비교한다. 마지막에는 삼차원 평판 모델을 이용하여 Race-Tracking 효과와 기공형성을 보여준다 [36]. 는 직사각형 평판의 이차원 채널 유동을 위한 예제이다. 이 예제의 길이, 넓이, 두께는 각각 1 m, 0.5 m, 1 mm 이다. 수지는 실선으로 표시된 왼쪽 면으로부터 주입되고 점선으로 표시된 오른쪽 면을 통해서 빠져나간다. 선형 삼각 평판 요소가 이용되고 전체 노드와 요소의 개수는 각각 861, 1600 개이다.

	Extended Abstract in Korean
	표 1. 채널 유동 모델의 계산 시간 제 6 장. 검증
	본 연구에서는 문헌으로부터 얻은 몇 가지 예제를 계산하고 이론치 계산 시간 절점 개수 감소율 국소화 미적용 국소화 적용 혹은 실험치와 그
	861	37.5 초	레벨셋 확장유한요소법 + 레벨셋	29.0 초 24.1 초	22.7 % 35.7 %
	2556	211.4 초	레벨셋 확장유한요소법 + 레벨셋	154.0 초 118.0 초	27.2 % 44.2 %
	7381	1236.8 초	레벨셋 확장유한요소법 + 레벨셋	742.5 초 561.1 초	40.0 % 54.6 %
	표 1 은 채널유동 모델의 계산시간 측정 결과이다. 감소율은
	계산 시간의 감소율은 절점의 개수가 증가할수록 커졌다. 그 이유는
	절점이 많은 모델일수록 계산량의 감소 비율이 크기 때문이다. 그리고
	레벨셋의 부분의 계산양이 압력을 구하는 부분의 계산 양보다 많기
	때문에 레벨셋의 국소화가 확장유한요소법의 국소화보다 계산시간의
	i7 CPU 950 (3.07 GHz, 쿼드코어)이고 사용운영체제는 윈도우 7 단축에 더 큰 영향을 주었지만, 확장유한요소법의 국소화가 레벨셋의
	프로페셔널 K (64 bit) 이다. 국소화와 더불어 계산시간의 단축을 극대화 하였다고 볼 수 있다. 계산
	국소화기법을 사용한 경우 계산 시간이 감소하는 양을 백분율로 나타낸 시간 테스트를 통해 국소화 기법의 적용은 계산 효율의 증가에 크게
	것이다. 레벨셋의 국소화만 적용한 경우에는 절점수가 다른 세가지 기여 한다는 것을 확인할 수 있었다.
	모델에서 각각 23%, 27%, 40%의 비율로 계산 시간이 감소하였다.
	그리고 확장유한요소법과 레벨셋의 국소화를 같이 적용하였을 때는
	각각 36%, 44%, 55%의 비율로 계산 시간이 감소하였다.
			EA(K) -19 EA(K) -20

만약, 결과에 대한 영향 없이 필요한 레벨셋 값 만을 구하도록 영역이 축소시킬 수 있다면, 계산 효율은 증가할 것이다. 일반적으로 편미분방정식을 풀 때, 선형시스템을 계산하는 부분에서 가장 많은 시간이 소요된다. 그러므로 시스템 크기가 적어지면 계산시간은 확연히 줄어든다. 이번 장에서는 확장유한요소법과 레벨셋 기법의 국소화의 알고리즘을 설명하고 국소화에 의한 효율성을 몇 가지 예제를 통해 보여준다. 제 1 절. EA(K) -18 그림 8. 확장유한요소법 국소화를 이용하여 계산한 압력 제 2 절. 국소화의 효율성 국소화의 효율성을 테스트하기 위해 그림 9 의 채널 유동 모델을 이용한다. 국소화 기법으로 얼마나 시간이 단축되는 지를 테스트 하기 위해 국소화 기법을 사용하지 않은 경우, 레벨셋의 국소화만 사용하는 경우, 레벨셋과 확장유한요소법의 국소화를 같이 사용하는 경우의 계산 시간을 측정하였다. 계산에 사용된 컴퓨터의 사양은 인텔(R) 코어(TM) 제 1 절. 단순 유동 모델 단순 채널 모델의 결과는 Cai[32]가 제안한 이론치를 이용해 비교한다. 1. 직사각형 평판의 이차원 채널 유동 그림 9 EA(K) -21

  여기서 L 은 길이 방향에서 유동 전면부의 위치이다. 그리고K, P 는 각각 점성, 공극률, 투과성, 압력을 의미한다. 이 모델의 이론적 충진 시간은 1250 초이고, 우리의 해석 결과는 1268.42 초이다. 두 결과의 차이는 1.5 %이다. 같은 예제를 계산한 Soukane 과 Trochu [24]의 수치 결과가 이론치와 3%의 차이가 나는 것과 비교할 때, 우리의 계산이 이론치와 더 비슷함을 알 수 있다. 여기서 r 은 평판의 반지름, 𝑟 0 은 주입구의 반지름이다 [32, 35]. 이론적 주입시간은 1411 초이고 우리의 해석 시간은 1436 초이다. 두 결과 사이의 차이는 1.8%이다. Lin 등 [33]의 수치해석계산에서 이론식과 비교해서 4%의 차이가 있는 것을 볼 때 우리의 프로그램이 더 효율적이다. 그림 11 의 이차원 복잡한 형상 평판을 이용하여 Shojaei 등[34]이 수행한 실험결과를 우리의 해석 결과와 비교한다. 복잡한 형상의 금형을 충진하는 동안 수지의 유동은 복잡한 모양으로 흘러간다. 각 방향의 길이와 두께는 각각 0.4 m 와 6.5 mm 이다. 주입구의 위치는 가운데 빨간 색 점으로 표시되어 있다. 주입구의 지름은 4 mm 이다. 배출구는 파란색으로 표시된 세 점에 위치해 있다. Shojaei 등[34]은 금형의 모든 모서리에 실리콘을 도포하여 모서리 효과를 예방했다. 그물망은 2204 개의 절점과 4111 개의 3 절점 삼각 요소로 이루어져 있다. 주입되는 SAE 40 모터 오일의 점성은 상온에서 0.3 Pa•s 이다 (25 ℃). 투과성과 공극율은 각각 6.83x10 -9 m 2 과 0.81 이고 주입 압력은 35 KPa 이다. 본 연구를 에서는 RTM 공정을 좀 더 정밀하게 해석하기 위해서 레벨셋 기법과 결합된 확장유한요소법을 이용하였다. 이 기법 적용의 가장 큰 장점은 수지 전면부의 비연속적인 압력 구배를 수치해석에 반영할 수 있는 것이다. 여기서 사용되는 확장형상함수는 레벨셋 함수를 이용해서 전면부의 위치를 반영할 수 있도록 만들었다. 그리고 확장유한요소법을 적용할 경우 유동 전면부 근처의 절점만 확장영역에 포함되므로 해석에 필요한 전체적인 자유도는 아주 적게 증가하였다. 그리고 본 연구에서는 전면부의 위치를 계산하기 위해 레벨셋 함수로 이루어진 이송방정식을 이용하였다. 그 이송방정식과 레벨셋 값의 재초기화 식을 풀기 위해 내연적 특성 Galerkin 유한요소법을 적용하였다. 본 연구의 검증을 위해 해석 결과를 문헌에서 얻은 이론값과 비교하였다. 수행한 해석 결과는 이론 값과 비교했을 때 차이가 거의 없었다. 본 연구는 효율성 향상 방안으로 확장 유한 요소법과 레벨셋 기법의 국소화를 적용하였다. XFEM 의 국소화는 level set 기법의 국소화에서 힌트를 얻어 본 연구에서 처음으로 적용하였는데, 그 국소화 적용에 의해 계산 시간이 눈에 띄게 줄어듦을 확인할 수 있었다. 가장자리가 복잡한 모양의 평판 해석에서는 수지 전면부의 갈라지고 합쳐지는 현상이 잘 시뮬레이션 되었다. 그리고 얇고 복잡한 모양의 평판 해석에서는 race tracking 과 기공 형성 현상을 잘 묘사할 수 있었다. 본 연구를 통해서, 우리는 확장유한요소법이 RTM 공정에서 몰드내의 수지 유동 해석에 유용하게 쓰일 수 있는 것을 확인하였다. 본 연구에서 제안한 기법들은 해석을 더 효율적이고 정밀하게 만들어 주었다. 더 나아가, 본 연구가 RTM 공정의 설계 변수를 최적화하기 위해 사용된다면 높은 효율성과 정확성 같은 장점이 더욱 부각될 수 있을 것이라 예상된다. 그리고 본 연구에 경화 반응과 열전달 과정이 추가 된다면 그 사용 범위를 더 확장할 수 있을 것이라 생각된다.

	Extended Abstract in Korean Extended Abstract in Korean Extended Abstract in Korean
		de Saint-Étienne
	NNT : 2013 EMSE 0701
	Prénom NOM : Yeonhee JUNG
	DISSERTATION TITLE
	An efficient analysis of resin transfer molding process using extended finite element
	method
	Speciality: Mechanical Engineering
	참고문헌	2 𝜇/2𝐾𝑃	(24)
		EA(K) -23 EA(K) -31

제 2 절. 이차원 복잡한 형상 평판 EA(K) -24 그림 11. 이차원 복잡한 모양 평판의 형상과 그물망 우리 프로그램으로 계산한 주입시간은 123 초이고 Shojaei 등[34]의 실험과 해석 결과는 각각 117 초와 126 초이다. 실험치와의 차이는 우리의 해석치는 4.3 %이고 Shojaei 등 [34]의 해석치는 7.7 %이다. 그러므로 본 연구의 해석 결과가 실험 결과와 더 비슷하다. 그림 12 은 우리의 계산결과이고 충진 동안 수지 전면부의 위치를 15 초 마다 그린 것이다. 그리고 그림 13 는 실험으로 측정된 전면부의 위치를 15 초 마다 그린 것이다. EA(K) -30 제 7 장. 결론 Keywords: RTM Process, XFEM, Level Set Method, Numerical Analysis, Localization method.

  analyse efficace du procédé RTM à l'aide de la méthode XFEM Spécialité: Mécanique et Ingénierie Mots clefs : Procédé RTM, XFEM, Level set, Simulation numérique, Méthode de localisation. Le procédé de fabrication par RTM (Resin Transfer Molding) a été étudié numériquement à l'aide de la méthode XFEM (eXtended Finite Element Method) combinée avec la méthode Level set. La méthode XFEM permet d'obtenir une bonne précision numérique de la pression près du front d'écoulement, où son gradient est discontinu. Les fonctions de forme enrichies de la méthode XFEM sont proposées à l'aide des valeurs de Level set en vue de décrire correctement l'interpolation avec le front d'écoulement. En plus, la méthode de Level set est utilisée pour transporter le front d'écoulement à chaque pas de temps durant le remplissage du moule. Les valeurs de Level set sont calculées à l'aide d'une méthode de Galerkin implicite. Le solveur multi-frontal d'IPSAP a été utilisé pour la résolution du système. Cette étude a été validée en comparaison avec les solutions analytiques. En outre, une méthode de localisation avec XFEM et la méthode Level set a été proposée afin d'améliorer l'efficacité de calcul. Elle permet de réduire le domaine de calcul près du front d'écoulement. Par conséquent, le temps de calcul est fortement réduit grâce à cette méthode. Un test d'efficacité a été fait avec des modèles simples en écoulement laminaire ou radial. Quelques exemples d'application sont présentés pour illustrer la capacité de cette méthode.

	École Nationale Supérieure des Mines
	de Saint-Étienne
	NNT : 2013 EMSE 0701
	Prénom NOM : Yeonhee JUNG
	TITRE DE LA THÈSE

Une

Résumé :

Une pale d'éolienne a également traitée comme application industrielle. Enfin, une interface d'utilisateur graphique a été développée en vue de fournir une facilité des pré-et post-processus.

Department MPE gave me a lot of help for my research and stable school life.

My thesis was done by dual degree program between Ecole Nationale

 Table 4.1. Calculation time of the channel flow mode.

Chapter 4. Efficiency of Localization

Method

This study presents a localization method of level-set method and XFEM so as to improve computational efficiency. The localization means to minimize the whole computational amounts through reallocating the computational domain to a small area at every time step. For the analysis of mold filling, the important data is the level set values in the region near the flow front and the pressure/velocity values in the resin region. If it is possible to calculate the data of only the small region, the computational time could be dramatically decreased. However, the localization is quite difficult to apply because the level set values easily oscillate at the boundaries of the local domain. And also, when calculating the pressure values, the boundary location is changed every time step. Therefore, the key points in the localization method are how to prevent the oscillation and detect the boundary location at the every time step. In the section 4.1, the method to prevent the oscillation is introduced as well as the method to reallocate the local domain for calculating the level set values. And, the section 4.2 is focused on how to detect the boundary location with the method to reallocate the local domain for the pressure calculation. The efficiency of the localization method is explained in the section 4.3.

A -1

Appendix A.

Multi-frontal solver of IPSAP & Diamond/RTM

Diamond/RTM

The present method for numerical simulation of the RTM process is implanted in

Key-points of cross-sections for the composite wind turbine blade model

The composite wind turbine blade model is drawn by 10 cross-sections. The keypoints of the 10 cross-sections are given in the table B. The cross-section number is started from the root end.

Parallel computer memory architectures

Parallel programming

The speedup of a program using multiple processors in parallel computing is limited by the sequential fraction of the program. The speedup is calculated by Amdahl's law: