2007:168 civ master's thesis

2007:168 CIV

M A S T E R ' S T H E S I S

Inverse modelling of compression mouldingof sheet moulding compound using CFD

Sofia Ebermark

Luleå University of Technology

MSc Programmes in Engineering Engineering Physics

Department of Applied Physics and Mechanical EngineeringDivision of Fluid Mechanics

2007:168 CIV - ISSN: 1402-1617 - ISRN: LTU-EX--07/168--SE

iii

Preface

This work has been carried out at the Division of Fluid Mechanics at LuleaUniversity of Technology in Sweden as part of the Research Trainee Pro-gramme 2006/2007. This thesis constitutes the final part of the Master ofScience Programme in Engineering Physics. This work has been part of theFYS-project (Forbattrade ytor i SMC-produkter) that is lead by Scania andsponsored by VINNOVA and the participating industries.

First, I would like to thank my examiner and supervisor Professor StaffanLundstrom for guidance and support throughout the work and also my as-sistant supervisor Ph.D. Daniel Marjavaara for his valuable help during thiswork. I would also like to thank the people involved in the FYS-project.

Finally, I would like to thank the Division of Fluid Mechanics for alwaysmaking me feel part of the group and for a great atmosphere and the Re-search Trainee group of 2006/2007 for making this a great year.

Lulea, June 2007.Sofia Ebermark

v

Abstract

Sheet Moulding Compound (SMC) is primary used in compression mouldingwhich is a common manufacturing process for composite materials. Whenmanufacturing SMC, voids are likely to appear on the surfaces of the partscausing large defects after painting. The automotive industry often use SMCin their applications implying that the parts often must have a class A ap-pearance. When defects are discovered on the surfaces of the parts they needto be rejected or repaired which is a costly process. The formation of voidsis strongly related to the spatial distribution of the pressure. It is there-fore interesting to reveal the pressure distribution during moulding. If thepressure distribution is known critical areas can be localized. The purposeof the work is to investigate if an inverse modelling approach together withComputational Fluid Dynamics (CFD) can be used to predict the pressuredistribution during compression moulding of SMC.

The geometry in focus is a circular plate with a radius of 100 mm placed be-tween two heated mould halves. To be able to find the pressure distributionthe viscosity as a function of time and spatial coordinate needs to be knownsince it relates the deformation rate to stresses and hence the pressure. Theviscosity of the SMC is known to be dependent on many parameters such astemperature, shear rate, degree of curing, amount of fill-material and fibreorientation. Here, the viscosity is assumed to be a function of temperatureonly since it is difficult to derive an explicit expression for the viscosity asa function of all the parameters mentioned above. A way forward is insteadto do measurements of the pressure during moulding in a simple geometryand at the same time do numerical simulations that are fitted, by adjustingparameters in the viscosity model, to the experimental results by inversemodelling.

The results from the experiments show a rapid increase in pressure initially.After the pressure has reached its peak value it decreases almost linearlywith time. The same behaviour is captured by the viscosity model used inthe numerical simulations. By an optimization procedure it is possible tofind the parameters in the viscosity model that gives the best fit betweenthe experiments and the simulations. The optimization is performed at onespatial coordinate and the resulting pressure is similar to the one obtainedin the experiments. The pressure however differs when compared at an otherspatial coordinate. Comparisons between visualizations made previously ofthe in-mould flow and the simulations show however similar flow behaviour.In other words, the numerical model serve as a good start but needs furtherimprovements in order to capture the true mould flow behaviour.

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aim and Objective . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Description of project . . . . . . . . . . . . . . . . . . . . . . 21.5 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory 32.1 Fibre Reinforced Polymer Composites . . . . . . . . . . . . . 32.2 Compression Moulding . . . . . . . . . . . . . . . . . . . . . 32.3 Sheet Moulding Compound (SMC) . . . . . . . . . . . . . . . 3

2.3.1 Defects in SMC . . . . . . . . . . . . . . . . . . . . . . 32.4 Inverse Modelling . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Surrogate based optimization . . . . . . . . . . . . . . . . . . 5

2.5.1 Response Surface Methodology . . . . . . . . . . . . . 62.6 Computational Methods . . . . . . . . . . . . . . . . . . . . . 7

2.6.1 Governing equations . . . . . . . . . . . . . . . . . . . 72.7 Computational Fluid Dynamics (CFD) . . . . . . . . . . . . . 8

2.7.1 The Finite Volume Method . . . . . . . . . . . . . . . 92.7.2 Errors and accuracy . . . . . . . . . . . . . . . . . . . 92.7.3 Multiphase Flow . . . . . . . . . . . . . . . . . . . . . 102.7.4 Fluid Structure Interaction (FSI) . . . . . . . . . . . . 112.7.5 Transient simulation . . . . . . . . . . . . . . . . . . . 11

3 Methods 133.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . 133.2 Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 143.2.2 Grid generation . . . . . . . . . . . . . . . . . . . . . . 143.2.3 Physics definition . . . . . . . . . . . . . . . . . . . . . 153.2.4 Domain models . . . . . . . . . . . . . . . . . . . . . . 15

vii

viii CONTENTS

3.2.5 Boundary conditions . . . . . . . . . . . . . . . . . . . 163.3 Inverse Modelling and Rheology . . . . . . . . . . . . . . . . . 17

3.3.1 Viscosity model . . . . . . . . . . . . . . . . . . . . . . 173.3.2 Optimization . . . . . . . . . . . . . . . . . . . . . . . 17

4 Results 214.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 214.2 Results CFD-simulations . . . . . . . . . . . . . . . . . . . . . 21

4.2.1 Outer design . . . . . . . . . . . . . . . . . . . . . . . 234.2.2 Inner design points included . . . . . . . . . . . . . . . 234.2.3 Optimum for t > 0.5 s . . . . . . . . . . . . . . . . . . 274.2.4 Validation of optimum . . . . . . . . . . . . . . . . . . 274.2.5 Model Validation . . . . . . . . . . . . . . . . . . . . . 284.2.6 Results for different grids . . . . . . . . . . . . . . . . 304.2.7 Visualisation of the simulations . . . . . . . . . . . . . 30

5 Discussion 335.1 Assumptions and simplifications . . . . . . . . . . . . . . . . 335.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Bibliography 37

Chapter 1

Introduction

1.1 Background

Compression moulding is a common manufacturing process for compositematerials, for example Sheet Moulding Compound (SMC). During this pro-cess voids are likely to appear on the surface as flaws and great efforts mustbe spent on minimizing the number of them. This is especially importantin the automotive industries since their applications often must have a classA appearance. Surface voids can be transported out of the SMC or dissolveinto it and these mechanisms are strongly related to the spatial distributionof the pressure. Hence, if the pressure distribution is known, critical areaswhere voids appear can be discovered.

1.2 Aim and Objective

The purpose of the work is to investigate if inverse modelling together withComputational Fluid Dynamics (CFD) can be used to predict the pressuredistribution during compression moulding of SMC. The aim with this workis to implement a numerical model of the compression moulding process andto find a viscosity model that describes the pressure response of the material.

1.3 Previous work

Traditionally the formation of voids were investigated with experiments butsince the capacity of computers in the recent years has rapidly increased, full3D simulations of the mould filling process is possible to perform to get adeeper insight into the physical phenomena governing the process. One issueto be solved however is the modelling of the material i.e. the SMC-materialsince its dependence on several parameters is somewhat complex. Inversemodelling may be an alternative approach. In previous work done by P.T.Odenberger [1] experiments and numerical simulations of the compression

1

2 CHAPTER 1. INTRODUCTION

moulding process together with inverse modelling in a simple geometry wereestablished. The viscosity of the SMC was allowed to vary as a function oftime but was constant throughout the domain at each time.

1.4 Description of project

In this project the same arrangement as in previous work [1] will be used.A numerical model of the compression of a circular disc between two platesduring compression moulding will be implemented. In contrast to previouswork the viscosity will change as a function of temperature and the tem-perature in its turn will vary as a function of time and spatial coordinate.Hence, the equations governing the heat transfer need to be solved. By in-verse modelling the parameters in the viscosity model will be adjusted sothat the pressure in the numerical simulations agrees with the experiments.

1.5 Outline of this thesis

First, a short introduction to composites manufacturing is presented to givesome insight of the process. Then, the inverse modelling procedure will beexplained further and also some theory about how the governing equationsare solved numerically. After that, the experimental and numerical setup willbe presented followed by the results from the experiments and the inversemodelling. Finally the results will be discussed and some conclusions drawnand also some suggestions for future work will be given.

Chapter 2

Theory

2.1 Fibre Reinforced Polymer Composites

Fibre reinforced polymer composites are because of their high strength toweight ratio commonly used in the automotive industry to reduce weight andhence fuel consumption. A fibre reinforced polymer composite is a polymermatrix reinforced with fibres. These fibres are usually made from glass,carbon or aramid.

2.2 Compression Moulding

Compression moulding is a high-volume manufacturing method for polymercomposites and is suitable for manufacturing complex and high-strengthparts. In compression moulding the moulding material is placed in an openheated mould cavity, see figure 2.1. The mould is then closed and a pressureapplied to force the material to fill the cavity. Curing takes place and whenthe mould is opened the part is removed.

2.3 Sheet Moulding Compound (SMC)

Sheet moulding compound (SMC) or sheet moulding composite is a fibre-reinforced polymer material of typically 30% glass fibres primarily used incompression moulding. It is composed basically of four ingredients that isthermosetting resin, fibres, fillers and additives [2]. The automotive industryhas the highest level of consumption of SMC. For example some of theexterior parts on the Scania truck in figure 2.2 are made from SMC.

2.3.1 Defects in SMC

The parts made from SMC used in automotive applications such as hoodpanels must have high quality surfaces. When defects are discovered on

3

4 CHAPTER 2. THEORY

Figure 2.1: Schematic figure of the compression moulding process.

Figure 2.2: Scania truck with exterior parts made from SMC.

2.4. INVERSE MODELLING 5

the surfaces of the parts they need to be rejected or repaired which is acostly process. The defects originate from small voids on the surfaces of theparts. When the parts are painted, the air entrapped in these voids expandsresulting in large bubbles which finally collapse and cause large defects onthe surfaces.

2.4 Inverse Modelling

A system can be described by a set of model parameters. Some physicalrelation should exist between the model parameters and some measurableparameters. By inverse modelling the model parameters are tuned to bestfit the measurable parameters obtained in experiments [3]. Inverse mod-elling may be a helpful tool when dealing with complex materials whosedependence on several parameters is difficult to model.

2.5 Surrogate based optimization

Instead of performing time-consuming simulations for every parameter com-bination in a numerical model, a surrogate based optimization approach canbe used. The surrogate model is an approximate model that is intended tomimic the behaviour of the simulation model as closely as possible. Theexact function of the simulation code is not assumed to be known (or evenunderstood), solely the input-output behaviour is important. The code thusbehaves like a black-box. The aim with the surrogate modelling is to deter-mine a relationship between of a set of design variables x = (x1, x2, ..., xN )and their calculated response y for a set of data points.

The true response of a system is given by:

y = f(x), (2.1)

while the response of the surrogate model is given by:

y = g(x), (2.2)

such that:y = y + ε, (2.3)

where ε is the modelling error [4].

Surrogate modelling can be used to capture the global behaviour of thesystem but it can also be used to find an optimum. The surrogate basedoptimization (SBO) process can be divided into four steps;

1. Experimental Design

6 CHAPTER 2. THEORY

Figure 2.3: The surrogate model is a least square approximation as illus-trated here.

2. Numerical simulations at selected locations

3. Construction of surrogate model

4. Model validation

In the first step, the experimental design, the set of data points is chosenand these can be randomly distributed or form a pattern. In the secondstep, numerical simulations are performed for the chosen set of data pointsand then a surrogate model is fitted to the calculated data. Finally, themodel is used predict the response for given values of the design variablesin order to validate it [5].

2.5.1 Response Surface Methodology

One of the most common surrogate models is the polynomial response sur-faces (PRS) model in which polynomials are fitted to the calculated data.An example of a second-order response surface model in two variables is:

y = β0 + β1x1 + β2x2 + β3x21 + β4x

22 + β5x1x2 + ε. (2.4)

This is a multiple regression model and the parameters βi are the regressioncoeffients and the independent variables xi are the regressor variables. Moregenerally this regression model can be written as:

yi = β0 +k∑

j=1

βjxij +k∑

j=1

βjjx2j +

∑ k∑

j<i=2

βijxixj + εi (2.5)

or in matrix notation:

y = Xβ + ε

2.6. COMPUTATIONAL METHODS 7

The method of least squares selects the β’s so that the sum of squares ofthe errors ε is minimized. The least squares estimators of the regressioncoefficients are denoted b0, . . . ,bk.

b =(X′X

)−1 X′y (2.6)

and the polynomial response surface model can be written as [6]:

y(x) = xTb (2.7)

2.6 Computational Methods

2.6.1 Governing equations

Fluid motion is governed by the conservation of mass, momentum and en-ergy [7]. These conservation laws can be expressed in several forms e.g.differential and integral form. Here they will be presented in differentialform.

The differential form of the mass conservation equation is

∂ρ

∂t+

∂ (ρui)∂xi

= 0 (2.8)

where ui((i = 1, 2, 3) or (i = x, y, z)) are the components of the velocityvector in the direction of the coordinates xi and ρ is the density of the fluidand t is the time. This equation is known as the continuity equation.

By applying Newton’s second law of motion to an infinitesimal fluid ele-ment, the differential form of the conservation of momentum can be derivedyielding the Cauchy’s equation of motion

ρDui

Dt= ρgi +

∂τij

∂xj(2.9)

where ρgi and ∂τij

∂xjis the i-component of the body force and the surface force

per unit volume respectively. The stress tensor τij is a constitutive equationrelating the stress to the deformation and for a Newtonian fluid and it canbe written as

τij = −(

p +23µ∇ · u

)δij + 2µeij (2.10)

in which p denotes the static pressure, µ the dynamic viscosity, δij theKronecker delta and eij the strain rate tensor [8],

8 CHAPTER 2. THEORY

eij =12

(∂ui

∂xj+

∂uj

∂xi

). (2.11)

A substitution of the constitutive equation (eq. 2.10) into the Cauchy’sequation (eq. 2.9) results in the equation of motion for a Newtonian fluidthat can be expressed in the following way

ρDui

Dt= − ∂p

∂xi+ ρgi +

∂

∂xj

[2µeij − 2

3µ (∇ · u) δij

](2.12)

being the general form of the Navier-Stokes equation. The viscosity µ in thisequation can be a function of the temperature.

The conservation of energy is provided by the first law of thermodynamics.

ρD

Dt

(e +

12u2

i

)= ρgiui +

∂

∂xi(τijui)− ∂qi

∂xi(2.13)

in which the internal energy is represented by e and the heat flux per unitarea by qi. For a perfect gas e = CvT , where Cv is the specific heat atconstant volume. If the mechanical energy equation is subtracted from (eq.2.13) the thermal energy equation is obtained,

ρDe

Dt= −∇ · q− p (∇ · u) + φ. (2.14)

2.7 Computational Fluid Dynamics (CFD)

The set of partial differential equations (PDEs) describing fluid motion, re-ferred to as the Navier-Stokes equations, can only be solved analytically fora few special cases. Although known solutions are useful in helping to under-stand fluid flow, they can rarely be used directly in engineering analysis ordesign. Traditionally simplifications of the equations and experiments wereused to investigate fluid flow. For many types of flow these methods are notsufficient and an alternative or complementary method is needed. With thehelp of computers the partial differential equations of fluid mechanics canbe solved numerically by an iterative procedure. This is known as Compu-tational Fluid Dynamics (CFD).

In order to solve the differential equations numerically they need to be ap-proximated by a system of algebraic equations using a discretization method.The algebraic equations are solved at discrete locations in space and timein the computational domain [8].

2.7. COMPUTATIONAL FLUID DYNAMICS (CFD) 9

2.7.1 The Finite Volume Method

The Finite Volume Method (FVM) is used as a discretization method in mostCFD codes such as ANSYS-CFX. In this method, the computational domainis subdivided by a grid into a finite number of control volumes (CVs). Val-ues of the variables are calculated at the computational nodes being locatedat the centre of the control volumes. An integral form of the conservationequations are applied to each control volume and the integrals are approx-imated using variable values at the control volume surfaces [8]. Variablevalues at locations other than the computational nodes are approximatedin terms of the nodal values using a difference scheme. In order to preservethe accuracy, the order of the integral approximation must be at least thesame order as the difference scheme. Many difference schemes are based onseries expansion approximations such as the Taylor series. The more termsof the expansion used in the difference scheme, the more accurate the ap-proximation will be. More terms in the expansion will however increase thecomputational load. The order of the largest term in the truncated part ofthe series expansion determines the order of the scheme.

2.7.2 Errors and accuracy

Numerical solutions are approximations and they will always contain errorsarising from different parts of the process of obtaining the numerical solution.Three kinds of systematic errors are however always included in a numericalsolution [8]:

• Modelling errors

• Discretization errors

• Iteration errors

Modelling errors

The modelling errors are defined as the difference between the actual flowand the exact solution of the mathematical model of the problem. These er-rors originate from assumptions made when deriving the transport equationsfor the variables. Turbulent flows, multiphase flows, combustion and othercomplicated flows involve physical phenomena that are not perfectly de-scribed by current scientific theories and therefore modelling errors for thesetypes of flow may become very large. Simplifying the geometry and/or sim-plifying the boundary conditions may also introduce modelling errors. Themodelling errors can only be estimated by comparing numerical solutions,in which the discretization and iteration errors are negligible, with accurateexperimental data or data from more accurate models.

10 CHAPTER 2. THEORY

Discretization errors

The discretization errors are defined as the difference between the exactsolution of the conservation equations and the exact solution of the corre-sponding discretized equations. These errors should become small as thegrid spacing goes to zero. A way to estimate the discretization error is tocompare solutions from grids with different number of cells. If three gridsare available an estimate of the discretization error εd

h on a grid with gridspacing h can be obtained with:

εdh ≈

φh − φ2h

2p − 1(2.15)

where φ denotes the numerical solution and p is an exponent defined by:

p =log

(φ2h−φ4hφh−φ2h

)

log 2(2.16)

An approximation of the exact solution can then be obtained by adding theerror estimate εd

h to the solution on the finest grid φh. This is method isknown as Richardson extrapolation and is valid only when the convergenceis monotonic which can be expected only on sufficient fine grids.

Iteration errors

The system of algebraic equations obtained in the discretization processis usually non-linear and therefore needs to be linearized in order to besolved. This linear system is then solved by iterative methods in which astarting guess is successively improved. The iteration errors are defined asthe difference between the iterative and the exact solutions of the system ofalgebraic equations and it decreases as the number of iterations increases.The residual is a measure of iterative convergence as it relates to whetherthe equations have been solved.

2.7.3 Multiphase Flow

In multiphase flow more than one phase is present. Each phase may have itsown flow field or all phases may share a common flow field. In multiphaseflow the fluids are mixed on a macroscopic scale in contrast to multicompo-nent flow where they are mixed on a microscopic scale. There exist a numberof models for these types of flows. Two distinct multiphase flow models arethe Eulerian-Eulerian multiphase model and the Lagrangian Particel Track-ing multiphase model.

The flow can be either homogeneous or inhomogeneous. In the first, allthe fluids share the same velocity fields and other relevant fields such as

2.7. COMPUTATIONAL FLUID DYNAMICS (CFD) 11

temperature, turbulence and pressure. In the second case separate velocityfields and other relevant fields exist for each fluid. The pressure field ishowever shared by all fluids. The fluids interact via interphase transferterms [9].

2.7.4 Fluid Structure Interaction (FSI)

In Fluid Structure Interaction (FSI) simulations the solution fields in fluidand solid domains are coupled. The coupling between the solution fields canbe uni-directional, which means that one solution field strongly affects theother fields but is not being affected by the others [9].

2.7.5 Transient simulation

The time dependence of the flow characteristics is either steady state ortransient. In transient simulations the flow characteristics change with timeand real time information is required to determine the time intervals atwhich the flow field is calculated [9].

Courant number

The dimensionless Courant number is one of the key parameters in com-putational fluid dynamics and can be used to determine the time step in atransient simulation. The Courant number is defined as the ratio of the timestep ∆t to the characteristic convection time, which is the time required fora disturbance to be convected a typical length of a computational cell, ∆x[8]:

c =u∆t

∆x(2.17)

When diffusion is negligible the criterion for stability is:

c < 1 (2.18)

or

∆t <∆x

u(2.19)

i.e. the Courant number should be smaller than unity.

12 CHAPTER 2. THEORY

Chapter 3

Methods

3.1 Experimental Setup

To give input to the inverse modelling process measurements are performedduring pressing in a Fjellman 310 ton press. Charges are compressed in aplate to plate mould held at a constant temperature of 150 C. The pressure ismeasured with two Kistler pressure transducers mounted in the upper mouldat two spatial locations that is, at the centre and 37.5 mm away it, P0 andP1 respectively. The mould closing speed is set to 2 mm/s corresponding toreal production conditions. The height of the upper mould as a function oftime is also recorded. The SMC-charge has a circular diameter of 100 mmand consists of 6 layers of SMC (20 wt% fibre glass content) which gives atotal height of 15 mm.

Figure 3.1: Schematic figure of the experimental setup with the position ofthe pressure transducers. The area inside the dotted line is included in thenumerical model.

13

14 CHAPTER 3. METHODS

3.2 Numerical Setup

3.2.1 Geometry

The geometry in focus is a circular plate of radius 200 mm, which symbol-izes the region between two mould halves and it includes both the SMC-charge and the air surrounding it. In order to save computational timeaxi-symmetric conditions are applied in the θ-direction. The dimensions ofthe SMC-charge are taken from experiments and correspond to real produc-tion conditions. That is, a height of 15 mm and a radius of 50 mm. Thegeometry is created in ANSYS Workbench 10.0 and in order to facilitatethe grid generation process the centre of the plate is not included in thegeometry. Instead, it has an inner radius of 1.5mm.

Figure 3.2: The geometry used in the simulations. It has a radius of 200mm and a height of 15 mm.

3.2.2 Grid generation

The grid is generated in ANSYS ICEM CFD 10.0 using structured hexahe-dral elements. The elements are uniformly distributed in order to preservea good mesh quality during the entire simulation process. Three grids arecreated for the grid convergence study consisting of 2k, 4.2k and 8.5k ele-ments respectively. Due to the geometrically simple domain almost regularelements was achieved guaranteeing an almost 90 degree angle between el-ement sides. The aspect ratio which is a measure of the relative lengthbetween the sides of the element had a maximum value of 15.7 for the 4.2kmesh.

3.2. NUMERICAL SETUP 15

Figure 3.3: Part of the mesh at the outlet boundary. The elements areuniformly distributed and there is only one element in the theta-direction.

3.2.3 Physics definition

To obtain the resulting three-dimensional flow field from the fluid structureinteraction, a two-phase, transient simulation is performed with a high res-olution advection scheme. The flow field is assumed to be laminar due tothe relatively high viscosities. The commercial CFD code CFX-10.0 fromANSYS [9] is used for obtaining the solution. To solve the fluid structureinteraction, the mesh displacement equations are activated and the meshstiffness value is set to 1 to ensure that the mesh displacements are homoge-neously diffused throughout the mesh. Since this is a transient simulation atime step is set and this is done in an adaptive manner in order to preventthe root mean square value of the courant number to exceed 1 anywhere inthe the domain.

3.2.4 Domain models

An Eulerian-Eulerian homogeneous multiphase flow model is used in whichthe fluids share the same velocity field. The two phases present are SMCat a variety of temperatures and air at 25 C and they are both consideredcontinuous. Values for the SMC-material parameters are presented in table3.1.

All the parameters except the viscosity are assumed, for simplicity, not tobe a function of temperature, which is not true for some of them. Some ofthe material parameters are actually unique for each SMC-recipe but theyserve as an acceptable approximation.


Table 3.1: Parameter values for the SMC-materialMolar mass 7000 g·mol−1

Density 1.9 g·cm−3

Specific HeatCapacity

1500 J·kg−1·K−1

ThermalConductivity

0.8 W·m−1·K−1

Thermal Ex-pansivity

1.4·10−5 K−1

The SMC-charge is placed in the domain by setting its volume fraction to 1and the air to 0 for r < 50 mm and the other way around for

50 mm < r < 200 mm. The free surface model is used since the two phasesare assumed to be separated by a distinct surface. As a heat transfer model,the thermal energy model is used since kinetic effects are assumed to benegligible.

This is a fluid structure interaction (FSI) simulation where a solid wallis displacing a fluid. In this case the upper mould wall is displacing theSMC-charge. The mesh will therefore be deformed according to a presetdisplacement.

3.2.5 Boundary conditions

Mesh motion boundary conditions are specified on all boundaries in the do-main. They are applied on all nodes on the boundary region unlike otherboundary conditions, which are set on mesh faces. The upper wall is set tomove downwards with a velocity of 2 mm/s, the lower wall is specified asstationary and the other boundaries has an unspecified mesh displacement.

Outlet boundary conditions: At the outlet an opening boundary condition isset with atmospheric pressure and flow direction normal to the boundary.

Wall boundary conditions: The mould surfaces are assumed to be held atconstant temperature, 150 C and therefore the upper and lower wall hasisothermal boundary conditions with no slip. The wall at the inner radiusis a non slip adiabatic wall. On the two remaining boundaries, symmetryboundary conditions are applied.

3.3. INVERSE MODELLING AND RHEOLOGY 17

3.3 Inverse Modelling and Rheology

The aim with this thesis is to predict the pressure in the material duringmoulding. To be able to do that the viscosity as a function of time andspatial coordinate needs to be known since it relates the deformation rate tostresses and hence the pressure [10]. The relation between the deformationrate and stresses is non-linear for the SMC-material, which is the definitionof a non-Newtonian fluid. This non-linear relationship will however not beconsidered here. The viscosity of the SMC is known to be dependent onmany parameters such as temperature, shear rate, degree of curing, amountof fill-material and fibre orientation. It is not possible to derive an explicitexpression for the viscosity as a function of all these parameters. A wayforward is instead to do measurements of the pressure during moulding ina simple geometry and at the same time do numerical simulations that arefitted to the experimental results by inverse modelling.

3.3.1 Viscosity model

In this work the viscosity is assumed to be a function of the temperature inthe material according to equation 3.1 below,

µ = Ae−B(T−298) (3.1)

where A represents the value of the viscosity at room temperature and Brepresent the rate of decrease with temperature. Initial values of A and Bare taken from literature [11]. These variables are unique for each SMC-recipe but the chosen values serve as a good start and will be tuned by theinverse modelling process to best fit the SMC used in the experiments.

3.3.2 Optimization

The regression model chosen for this problem is as follows:

y = β0 + β1A + β2B + β3A2 + β4B

2 + β5AB + ε (3.2)

where A and B are the parameters in the viscosity model. The dependentvariable, y, in the regression model is defined as the sum of squares of thedifference between the measured and calculated value of the pressure,

y =32∑

i=1

(pexp − psim)2, (3.3)

where pexp is the pressure obtained in the experiments and psim is the cor-responding pressure from the simulations (see also figure 3.4). The pressuredifferences are calculated for every 0.1 second for the first 3 seconds at the


Figure 3.4: Schematic figure of the objective function y which is defined asthe sum of squares of the difference between the measured and calculatedvalue of the pressures, that is ε(t).

centre of the geometry (P0).

The aim is to find the values of A and B that minimizes the objective func-tion y. In other words, find the A and B that gives a pressure that is similarto the one obtained in the experiments.

The parameter A in the viscosity model represents the initial value of theviscosity at T = 298 K (25 C) and B represents the rate of change withtemperature. In other words, a large value of the parameter A will producea high initial viscosity and hence a large initial value of the pressure. If Bis large the viscosity will decrease rapidly with temperature. Values of theparameter A are limited to be between 1 · 105 Pa·s and 4 · 105 Pa·s and thevalues of B between 0.06 K−1 and 0.14 K−1, according to figure 3.5 below.These values are based on a few simulations.

3.3. INVERSE MODELLING AND RHEOLOGY 19

Figure 3.5: Schematic figure of the experimental design with the chosen datapoints.

Chapter 4

Results

4.1 Experimental results

In figure 4.1 the pressure, at the centre of the upper surface of the charge, ispresented for four mouldings with the same experimental setup, see section3.1. In order to prevent preheating of the charge, the upper mould halfbegins to move downwards as soon as the charge is placed on the lowermould half. Time t = 0 is defined as the time when the upper mould halfhits the SMC-charge and the pressing begins, which is in reality the pointwhere the pressure rises. When this happens the upper mould is a distanceof about 15 mm above its lowest position. This distance corresponds to theheight of the SMC-charge. Three of the pressings give similar results forthe pressure at P0 while the second pressing, test B, differs from the rest.Figure 4.2 shows the pressure a distance 37.5 mm away from the centre forthe same four pressings.

Figure 4.3 presents test C with the displacement of the upper mould in-cluded. It can be seen that at t = 0 when the pressing is assumed to beginthe upper mould is a distance of 15 mm above the lower mould, which is theheight of the SMC-charge.

4.2 Results CFD-simulations

As a second step in the response surface optimization procedure numericalsimulations are performed using the commercial CFD software CFX-10.0 fora variety of conditions representing selected locations in the design space.Since the simulations proved to be time-consuming for the finest grid, thecalculations are performed on the coarsest grid. Afterwards a grid conver-gence study is performed to estimate the grid error.

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22 CHAPTER 4. RESULTS

Figure 4.1: Pressure at the centre of the upper surface of the SMC-charge.

Figure 4.2: Pressure at a distance 37.5 mm away from the centre of thecharge.

4.2. RESULTS CFD-SIMULATIONS 23

Figure 4.3: The pressure at P0 and P1 and the displacement of the uppermould for test C.

4.2.1 Outer design

First, outer design points are chosen with values of A and B according tofigure 3.5 in section 3.3.2. The corresponding y-values and response surfacey(x) is plotted in figure 4.4. This response surface experiences no minimumbut the pressure curves in figure 4.5 shows that A=2.5 · 105 Pa·s gives apressure at P0 close to the measured value. The pressures from test Cin the experiments are chosen to represent the experimental values of thepressures at position P0.

4.2.2 Inner design points included

From the results in the previous section new values of A are selected between2 · 105 Pa·s and 3 · 105 Pa·s while the values of B remains between 0.1 K−1

and 0.14 K−1. Two additional design points are also included in order tofurther improve the response surface model (see figure 4.6). When all thedata points are included in the model the response surface in figure 4.7 isobtained. An optimum value of the response surface y(x), that is a minimumvalue, is found for A = 2.509 · 105 Pa·s and B = 0.151 K−1. A simulationfor this optimum is performed and the corresponding response, y, is addedto the response surface model.


Figure 4.4: Response surface plot for the outer data points.

Figure 4.5: Pressure as a function of time at position P0 for values of A andB according to the outer data points.


Figure 4.6: Modified experimental design with outer, inner and additionaldata points.

Figure 4.7: Response surface plot when all the data points are included.


Figure 4.8: Contour plot when all data points are included in the model.Optimum found for A = 2.509 · 105 Pa·s and B = 0.151 K−1

Figure 4.9: Pressures for the design points with the lowest calculated valueof y which means they are closest to the experimental values of the pressure.


4.2.3 Optimum for t > 0.5 s

The differences in pressure, that is (pexp − psim)2, between t = 0 s andt = 0.5 s will have large influence on the overall sum, while the differencesfor t > 0.5 s are more important for the visual fit. Therefore the summationwill be carried out for t > 0.5 s instead. Also, to find a good approximationof the response surface near its optimum, only the points closest to the pointwith the smallest calculated value of y are included in the model. A newoptimum value is then found for A = 2.521 · 105 Pa·s and B = 0.141 K−1.The corresponding calculated response is plotted in figure 4.10. The contourplot of the response surface is presented in figure 4.11.

Figure 4.10: Response surface plot near its optimum value. The pointmarked with a blue box in the surface plot represents the lowest calculatedvalue of y. A = 2.5 · 105 Pa·s and B = 0.14 K−1 for this point.

4.2.4 Validation of optimum

In the final step of the optimization process the surrogate model, y(x), ischecked to see whether it can be used to predict the response for given valuesof the design variables. To validate the optimum and hence the responsesurface model, a simulation is performed and the calculated and predictedvalues are presented in figure 4.12. The red box in the figure is the calculated


Figure 4.11: Contour plot of the response surface near its optimum. Thepoint in the middle represents the optimum.

value. It proves to be very close to the value predicted by the responsesurface (the green box), see also table 4.1. The values of A and B thatproduces the lowest calculated value of y gives the pressure that is presented,together with the measured value of the pressure, in figure 4.13 .

Table 4.1: Results near the optimum.A (Pa · s) B (K−1) y (Pa2)

Predicted value 2.521 · 105 0.141 5.36 · 1010

Calculated value 2.521 · 105 0.141 6.14 · 1010

Lowest calculated value 2.5 · 105 0.14 5.48 · 1010

4.2.5 Model Validation

To validate if the viscosity model obtained in the optimization process isvalid at other spatial locations, the pressure at a distance of 37.5 mm awayfrom the centre is compared between the simulations and the experiments.The result is presented in figure 4.14 below. As can be seen in the figure thepressure from the simulations differ quite a lot from the measured value ofthe pressure.


Figure 4.12: The calculated value of the objective function is marked by agreen box in the response surface plot and the corresponding predicted valueis marked with a blue box.

Figure 4.13: Pressure at P0 for the experiment and the simulation with thelowest calculated value of y.


Figure 4.14: Pressure at P1 for the experiment and the simulation with thelowest calculated value of y.

4.2.6 Results for different grids

A grid convergence study is performed to estimate the discretization error.Three grids with 2k, 4.2k and 8.5k elements are used. The values of A andB used in the viscosity model are the values obtained for optimum. Figure4.15 below show the result. The rise in pressure during the last 0.5 s maybe an indication of a diverging solution.

4.2.7 Visualisation of the simulations

To visualise the evolution of the flow front for the first three seconds of thesimulation the volume fraction of the SMC is plotted in figure 4.16. As canbe seen in the pictures the interface between the two phases becomes moreand more diffuse with time due to numerical diffusion.


Figure 4.15: Pressure for different grids. For the extrapolated value the griderror is added to the solution for the finest grid.

(a) t = 0 s (b) t = 1 s

(c) t = 2 s (d) t = 3 s

Figure 4.16: Pictures from the simulations showing the volume fraction ofthe SMC for the first three seconds of the pressing.

Chapter 5

Discussion

5.1 Assumptions and simplifications

As most mathematical models, the numerical model used in this thesis alsoinvolves some assumptions and simplifications. These simplifications limitthe possibility to solve the real flow field and some of them are presentedhere:

Axi-symmetric flow: This is probably not true for most mouldings since theSMC-material becomes anisotropic when the initially randomly distributedglass fibres are oriented by the flow. This orientation will probably pro-duce local differences in pressure that is not captured by the model. Thissimplification is made in order to simplify the model and hence reduce thesimulation time and the overall pressure distribution will probably be cap-tured.

Newtonian flow: The SMC-material is known to have a shear-thinning be-haviour, which means that the viscosity is dependent on the shear rate. Thisnon-Newtonian behaviour is not taken into account in this model.

Material parameters: The material parameters are constant (except the vis-cosity) and are assumed not to be a function of temperature. This is true ifthe temperature gradients are small, but is not true in this case where thetemperature goes from 150 C at the surface to 25 C at the centre. Also, thevalues of the constants are unique for every SMC-recipe. The values for theSMC used here are taken from literature not from actual measurements onthe material.

One phase only: The real SMC-material consists of many phases such asresin, fibres and fill-material. Here, the material is assumed continuous con-sisting of only one phase, the SMC-resin. The charge consists of 6-7 layers

33

34 CHAPTER 5. DISCUSSION

of SMC and air is actually entrapped between these layers while in the nu-merical model the charge is assumed to consist entirely of SMC.

Symmetric flow: In-real case moulding scenarios the SMC-charge experi-ences some degree of pre-heating before the pressing begins. This occurssince charge is placed on the lower mould half before it comes into contactwith the upper mould half.

5.2 Conclusions

The inverse modelling proved to be a successful method in the process offitting the numerical model to the experimental results, at least for the pres-sure at the centre of the disc. However, in order to get a viscosity modelthat predicts pressure in the entire disc, further improvements needs to bedone.

The viscosity model seems to capture the overall behaviour of the pressure.That is, a relatively high pressure initially which is then decreasing almostlinearly with time. By changing the constants in the viscosity model themagnitude of the initial pressure and the rate of change with time can beadjusted. The simulations show an immediate increase in pressure while thepressure in the experiments reaches its peak value after 0.5s. This transienteffect is probably a result of the initial air entrapment between the SMC-sheets. It takes some time before these layers are squeezed together in thepressings while in the simulations the charge is modelled as a homogenousmaterial.

When the data points in the response surface model were too far from theoptimum value, the pressure response could not be approximated with thesecond order response surface model. However, near its optimum this modelserved as a good approximation and the pressure response showed a secondorder behaviour.

Because compression moulding of SMC is relatively fast process, only thematerial closest to the mould wall will experience an increase in temperatureand hence a decrease in viscosity. The simulations show that this reductionin viscosity causes a secondary flow near the mould surface. This secondaryflow can only be captured if the elements closest to the wall are small enough.This is probably the reason why the pressure is so different between the dif-ferent grids used and is probably also one reason why the optimization onthe coarsest grid gave poor agreement for the pressure away from the centre.

5.3. FUTURE WORK 35

Poor agreement was achieved when comparing the pressure at another spa-tial point namely P1 between the experiments and the simulations. Thisindicates that the model has to be improved before it can be used to pre-dict the pressure distribution in the whole geometry. Comparisons betweenvisualisations made previously of the in-mould flow [1] and the simulationshowever show similar flow behaviour. In other words, the numerical modelserve as a good start to predict the pressure field but need further improve-ments in order to capture the true mould flow behaviour.

5.3 Future Work

In order to find a viscosity model that predicts the pressure distribution inthe entire disc some improvements of the numerical model and the optimiza-tion needs to be done. Some suggestions of improvements are presented here.

The viscosity model can be improved by using more than two adjustableparameters. For example, different exponential functions can be used in dif-ferent temperature intervals as a drastic change in viscosity is expected at acertain temperature. To improve the numerical model even further a modelfor the glass fibres can be included.

It is also important to have the correct values of the material parametersgoverning the heat transfer since the viscosity is strongly dependent on thetemperature in the material. An inverse modelling approach can be used forfinding these parameters too.

The optimization should be performed on a grid with small enough elementsclose to the wall to capture the effects from the decrease in viscosity. Anotheralternative is to include the error, obtained by using a coarser grid, in theobjective function y:

y =∑

(pexp(t)− (psim(t)− εgrid(t)))2. (5.1)

In order to obtain a better prediction of the pressure over the entire disc theoptimization should be performed at different spatial locations at the sametime. Then, the overall behaviour of the pressure is more likely capturedand a better prediction at locations other than those used in the optimiza-tion can be obtained. To improve the optimization even more results fromexperiments and simulations with different mould closing speeds can be in-cluded. It is aslo possible to optimize for different SMC-materials.

When a good enough prediction for the pressure distribution is achievedvoids can be included in the simulations to see how they move with the

36 CHAPTER 5. DISCUSSION

flow. Then areas with a large concentration of voids can be localized. Whenthis is known the formation of surface voids when manufacturing SMC canbe counteracted. The numerical model can then be used to predict criticalareas where voids are likely to appear when manufacturing SMC-parts.

Bibliography

[1] Odenberger P.T. Moulding of SMC: visualisation and inverse modelling.Licentiate thesis 2005:34, Lulea University of Technology, Sweden.

[2] Kia H.G. Sheet Moulding Compounds: Science and Technology. Hanser,Munich, 1993.

[3] Tarantola A. Inverse Problem Theory. Elsevier, Amsterdam, 1987.

[4] Marjavaara B.D. CFD Driven Optimization of Hydraulic Turbine DraftTubes using Surrogate Models. Phd thesis 2006:41, Lulea University ofTechnology, Sweden.

[5] Queipo N.V et al. Surrogate-based analysis and optimization. Progressin Aerospace Sciences, 41(1):1–28, 2005.

[6] Myers R.H. and Montgomery D.C. Response Surface Methodology. Wi-ley, New York, 2nd edition, 2002.

[7] Kundu P.K and Cohen I.M. Fluid Mechanics. Academic Press, SanDiego, 3rd edition, 2004.

[8] Ferziger J.H and Peric M. Computational Methods for Fluid Dynamics.Springer-Verlag, Berlin, Heidelberg, 3rd edition, 2002.

[9] CFX R© version 10.0 Copyright c© 1996-2005 ANSYS Europe Ltd.

[10] Barnes H.A. Hutton J.F. and K. Walters. An introduction to Rheology.Elsevier, Amsterdam, 1989.

[11] Lee C.C and Tucker C.L. Flow and heat transfer in compression moldfilling. Journal of Non-Newtonian Fluid Mechanics, 24:245–264, 1987.

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38 BIBLIOGRAPHY

2007:168 civ master's thesis

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