Finite Element Simulation and Error Estimation of Polymer ... · the discretization method (Finite Elements, Finite Volumes, Boundary Elements, Spectral methods, etc.). This work

ISSN 1517-7076

Revista Matéria, v. 9, n. 4, pp. 453 – 464, 2004 http://www.materia.coppe.ufrj.br/sarra/artigos/artigo10620

Autor Responsável: M. Vaz Jr.

Finite Element Simulation and Error Estimation of Polymer Melt Flow M. Vaz Jr., E.L. Gaertner

Department of Mechanical Engineering, Centre for Technological Sciences State University of Santa Catarina

Campus Universitário, 89223-100 Joinville-SC Brazil e-mail: [email protected] , [email protected]

ABSTRACT

Accuracy analysis is fundamental to ascertain the reliability of a numerical solution of any industrial operation. The analysis is often complex due to different error sources. In this work, two strategies to assess the discretization errors is discussed: (i) Richardson extrapolation and (ii) a posteriori error estimation based on projecting/smoothing techniques. The former utilises three nested meshes whereas the latter is based on the post-processing of the problem solution over only one mesh. The model accounts for the full interaction between the thermal effects caused by viscous heating and the momentum diffusion effects dictated by a shear rate and temperature-dependent viscosity.

Keywords: Finite elements, error estimation, polymer melt flow.

1 INTRODUCTION Polymer technology and mould design have experienced great development in recent years with the

advent of new blends, forming processes and computational tools. Software packages aiming at simulation of polymer processing, such as extrusion and injection moulding, have been steadily developed, most of which able to handle applications ranging from common household objects to complex aerospace components. Due to the great variety of forming operations, allied to the complexity of such problems, most commercial programs attempt to combine practical rheological models with approximate computational modelling techniques. Such approach has hampered more comprehensive studies on the polymer behaviour and assessment of the interaction between the problem parameters. On the other hand, in academia, as early as the 50's and 60's (see Crochet et al. [4] and references therein), pioneering computational techniques, theoretical approaches to solid mechanics involving inelastic deformations (e.g. viscoelastic models used in polymer extrusion), and the generalised Newtonian model (fluid flow dominated by the shear viscosity frequently used in injection moulding) have been proposed, using, however, simplified description of the material properties. In recent years, advancements in computational power and modelling strategies have favoured numerical studies based on more realistic material models. The present work presents the numerical modelling of polymer melt flow in closed channels using the Finite Element method and discusses strategies for error estimation based on Richardson extrapolation and projection/smoothing techniques.

2 GOVERNING EQUATIONS AND POLYMER RHEOLOGY Since the pioneering work of Hieber and Shen [8], the Hele-Shaw model has been constantly

adopted to simulate two-dimensional injection moulding. This model approximates 3D polymer melt flows between two flat plates assuming that the gap thickness is much smaller then the channel or cavity characteristic length. This approach presents the great advantage of allowing the derivation of a flow governing equation based on a 2D pressure field over the mid-surface (and constant pressure at the cross-section), whose boundary conditions consist in imposing the inlet flow rate or pressure at the gate and zero pressure at the flow front. The success of this strategy can be measured by the number of works available in the literature and by its use in commercial packages as MOLDFLOW [9] and C-MOLD [3]. The drawback of the original Hele-Shaw model resides in the evaluation of the so-called fluidity (integral measure of the inverse of the viscosity over the channel cross-section), which is calculated based on the power law description of the viscosity combined with an in-plane computation of the shear strain rate. A more realistic formulation would require a viscosity description consistent with the rheological model and a point-wise evaluation of the shear strain rate and temperature over the gap thickness. Furthermore, the Hele-Shaw approximation is valid only for thin channels and cavities, which imposes restrictions on simulations of a

JR VAZ, M., GAERTNER, E.L., Revista Matéria, v. 9, n. 4, pp. 453 – 464, 2004.

wide range of plastic products and components. One of the objectives of the present work is to assess the melt flow behaviour in thick channels and cavities when the shear strain rate and temperature varies over the cross-section. Understanding of such dynamics will help future developments of more comprehensive computational models for injection moulding.

Numerical modelling of polymer melt flow in thick channels and moulds requires a fully coupled solution of mechanical and thermal problems. The mechanical problem consists of the simultaneous solution of the constitutive law,

γd &&& ),(2),( TγTγ ηη =τ = (1)

equilibrium laws,

[ ]zpwT∂∂

=∇),(div γη &

[ ]

(2)

and the thermal problem requires solution of the energy conservation law,

0div =+∇ γτTk &

&

(3)

where τ is the shear components of the stress tensor, d is the deformation rate, γ is the shear strain rate tensor, η is the apparent viscosity, T is temperature, ∂p/∂z is the pressure gradient, k is the thermal

conductivity, 2/1):2/1( γγ &&& =γ γη &=τ is the equivalent shear stress. is the equivalent shear strain rate, and

2.1 Polymer Rheology Equations (1)-(3) are complemented by defining the rheological behaviour of the polymer melt

through its shear viscosity. Attempts to model behaviour of different materials gave rise to a large spectrum of equations, amongst which the laws of Carreau (e.g. Bao [2]) and Cross (e.g. Verhoyen and Dupret [14]) are the most widely referred. In the present work, the relationship between viscosity, shear rate and temperature for Polyacetal POM-M90-44 is expressed with good accuracy using the Cross constitutive equation [7],

( ) [ ] )(10

)(1)(

TnγTT

−+

=&λ

,Tγ&η

η , (4)

where η0 is the Newtonian viscosity, n is the power-law index and λ is a material parameter, which in turn, are approximated through Arrhenius law.

⎟⎠⎞

⎜⎝⎛=

Ta

expa 21)T0 (η

, ⎟⎠⎞

⎜⎝⎛=

Tb

expb 21T )(λ

and ⎟⎠⎞

⎜⎝⎛−=

Tc

expcTn 21)(

, (5)

in which a1, a2, b1, b2, c1 and c2 are material constants.

3 FINITE ELEMENT METHOD The Finite Element Method - FEM is a numerical technique for obtaining approximate solutions

based either on variational principles or weighted residuals. The FEM solutions based on Galerkin weighted residuals have been given preference over variational methods due to the difficulty to express general heat transfer and fluid dynamic problems in a variational form. The Galerkin method assumes that the exact solution can be interpolated from “n” C0-continuous shape (or trial) functions Ni, so that the unknown

parameters, iφ , can be obtained by assuming that the weighted residual of the governing equations and corresponding boundary conditions is minimum [16]. Application to a general equation, div[Γφ ∇φ ]+ Sφ = 0, representing (2) and (3) yields

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[ ]( ) ( ) 0=−⋅∇ dAqΓ nφφdiv +Ω+∇ ∫∫Ω∂Ω

dSΓ δφφδφ φφ

(6)

where n is the outward normal vector to surface ∂Ω and q denotes prescribed heat transfer and shear stress for the energy and momentum equations respectively. The discrete equations are determined by applying the divergence theorem to the second term of Equation (6) and by assuming that both the problem solution, φ,

and weight functions, δφ and δφ

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧⎥⎦

⎤00

ff

WT

W

T

⎢⎣

⎡∇=T NK T

, are interpolated using the same trial functions (Galerkin approximation), so that.

⎢⎣

⎡K00K

W

T

, (7)

in which T and W are the nodal temperatures and velocities respectively, and KT and KW are the so-called stiffness matrices,

(∫Ω

Ω∇⎥⎦

⎤d

kk

N0

0

and

)( )∫

Ω

Ω∇⎥⎦

⎤⎢⎣

⎡∇= d

TT

W NNK T

γηγη

&

&

,00,

, (8)

and fT and fW are the force vectors.

∫Ω

Ω∂∂ d

zpN−=Wf

and ∫Ω

Ω= dγτT&Nf

, (9)

for the temperature and velocity problems respectively. As discussed in the previous sections, Equation (7) is coupled through the viscosity and viscous heating. In this work, a staggered approach is adopted, in which the temperature and velocity equations are solved in succession until a pre-defined residual is reached. This process is stable, however, greater convergence rates can be achieved using the Newton-Raphson iterative procedure.

4 ERROR ASSESSMENT

Error computation is one of the most important issues in numerical simulation of any industrial operation. Accuracy and convergence analyses are fundamental to give the package user a necessary confidence in the results. The most relevant contribution of this work is the comparative assessment of two error estimation strategies.

Every numerical solution is unavoidably subject to errors, which can be generally classified in three types: modelling errors, discretization errors and iteration errors. Modelling errors are due to the difference between the actual physical behaviour and the mathematical model. Discretization errors consist of the difference between the exact solution of the governing equations and the corresponding solution of the algebraic system. Iteration errors are defined as the difference between the iterative and exact solutions of the algebraic equation system [5]. This work is particularly concerned with the assessment of the discretization errors. Such errors are primarily associated with both the mesh (structured, unstructured, uniform, etc.) and the discretization method (Finite Elements, Finite Volumes, Boundary Elements, Spectral methods, etc.). This work utilises two strategies to evaluate the discretization errors: Richardson extrapolation [12] and a posteriori error estimation based on smoothing techniques [17]. Sections 4.1 and 4.2 present a brief summary and the reader is referred to Oberkampf and Trucano [10] and Zienkiewicz and Zhu [17] for further considerations.

4.1 Richardson extrapolation

One of the first attempts to increase the convergence rate of a numerical solution was proposed by Richardson [12], who ascertained that a higher order solution could be achieved assuming that the “exact” solution is described as.

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( )( ) ( ) ( ) ( ) 1+++ pp hOhαx=+= hhhexact φεφφ xxx (10)

in which φexact is the “exact” solution of the governing equations at a given point x, φ h is the discrete solution, ε h is the discretization error, h is the mesh size, α is a constant and p is the error order. A safe application of the Richardson extrapolation requires three assumptions: (i) the exact solution must be smooth enough that the Taylor series expansion for the error is justified; (ii) the formal convergence order, p, is known; and (iii) the mesh spacing is sufficiently small such that the leading-order error term dominates the total discretization error, i.e., the convergence is monotonic in the asymptotic range [10].

From Equation (10), it is possible to show that the discretization error, ε h, can be estimated using two different nested meshes, h1 and h2 (assuming that h2 / h1 > 1), as.

121

1 −

−≅− p

hhhexact

φ φ=h r

φ

*φp

φε φ

(11)

where r = h2 / h1 is the refinement ratio. The exact order of the discretization error is hardly known a priori,

specially in non-linear problems. Therefore, it is always recommended to use an estimate, , which can be easily determined by assuming asymptotic convergence (and verified) and by applying equation (11) to three nested meshes, h1 , h2 and h3 , using a constant refinement ratio, so that

( )rp* =φhh

hh loglog12

23

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−

φφφφ

and 1

2

2

3

hh

hh

r == . (12)

The global measure can be defined by the error norm as

( ) ∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=

nod

i ip

hhi r

1

22

121

φφ∑=

≈nod

ihh

1

2 εε φφ

, (13)

where nod represents the total number of nodes and φ corresponds to nodal velocities, φ = W, or nodal temperatures, φ = T . Although the technique described previously can handle only uniform meshes, extensions have also been developed for non-uniform grids [6,13]. Despite its simplicity and widespread use in simulations of Newtonian fluid flow, the Richardson extrapolation shows some shortcomings: the extrapolated solution is not conservative in the sense of the conservation laws and the accuracy of the extrapolation does not apply to high derivatives of the solution [13]. Furthermore, the presence of singularities or nonlinearities may compromise the monotonic rate of convergence, as further discussed in session 5.

4.2 A posteriori smoothing technique

Error estimation has become an important issue in computational mechanics since the late seventies when Babuška and Rheinboldt [1] proposed a posteriori error estimates for the Finite Element method using norms of Sobolev spaces. The topic gained momentum when Zienkiewicz and Zhu [17] introduced error estimates based on post-processing techniques of the Finite Element solutions, which could easily be used in association with mesh refinement strategies. Due to its relative simplicity, the original Zienkiewicz and Zhu’s scheme has been extended to a wide spectrum of applications, ranging from linear solid mechanics to the highly non-linear metal forming problems, such as forging, blanking and machining. The strategy has also been employed successfully by Wu et al. [15] for solving the incompressible (Newtonian) Navier-Stokes fluid flow around a cylinder using an error estimate based on the Energy norm,

( ) ( ) ( )( )[ ] Ω− dp hh ][tr][tr εε &&

& ε&h h& hε&

−−−= ∫Ω

phhE

2 γγττe &&-T

, (13)

where τ , p, γ and , are the deviatoric stresses, pressure, shear and normal strain rates of the “exact”

solution, τ , ph, γ and are those of the Finite Element solution, and tr[⋅] denotes the trace of a tensor. The authors emphasise that, for a continuous pressure field, the differences between p and ph are vanishingly

456


small and the last term of (13) may be disregarded. This technique can also be used in conjunction with other error norms [16]. In the present work, both the energy norm and the L2 norm associated with the equivalent shear stress and equivalent shear strain rate,

( )( )[ ] Ωdh−= ∫Ω

hE

γγττ && -2e

and ([ )( )] Ω−−= ∫

Ω

dhhL

ττττ2

2e

, (14)

have been used. The energy norm, as described in (14)a , has a special physical significance when compared to the energy conservation law (3). This error measure is intrinsically associated with the viscous heating, which in turn, can capture the effects of the velocity and temperature fields through the shear strain rate and viscosity.

The standard post-processing of the Finite Element solution provides the shear strain rate tensor at Gauss Points [16], so that

hγ&hh γ&& :21

=γ and ( )γTγh &&,ητ = . (15)

In addition, the key to this method is the estimation of a higher order approximation to the “exact” solution at the integration points based on the available Finite Element solution. The most used technique

comprises two steps: (i) obtaining nodal values, +τ +γ& and , from the post-processed Finite Element solution,

hτ and hγ& , (ii) followed by an interpolation to estimate

*hτ

*hγ and at the integration points. The former uses

either extrapolation, super-convergent patch recovery, local or global weighted residuals (or L2 projection), whereas the latter can be easily performed using the Finite Element shape functions [16,17]. In this work the Gauss point variables are extrapolated to nodes using polynomials, so that the estimated "exact" solution is:

{ } { }{ } { } { }{ } { }{ } { } { }hT

hT

CN

CN

γγ

ττ&& ][

][++

++

=

=T

h

Th

N

N

γ

τ& *

*

=

=

, (16)

in which +C is the Gauss-Point-to-node interpolation matrix. Therefore, the estimated global errors for

energy and L2 norms are evaluated as:

( )( )[ ] Ω−− jh

hh d* γγτ &&∑∫∑

= Ω=

=≈nel

jh

nel

jEjE

j1

*

1

22 τee

and

( )( )[ ] Ω−−nel

jh

hh

nel

d* τττ∑ ∫∑= Ω=

=≈j

hj

LjL

j1

*

1

22

22τee

, (17)

where Ωj represents the integral over an element j and nel the total number of elements. The previous scheme has the advantage of computing the error estimate using only one mesh.

Despite its extensive use in almost every field of computational mechanics, this technique has the disadvantage of estimating errors based on secondary measures (stresses and strain rates), i.e., the convergence rate of the actual unknowns (nodal velocities and nodal temperatures) are not directly accounted for. Bao [2] presents an alternative approach to error estimation based on Sobolev spaces for generalized Newtonian fluids using Carreau viscosity law. It is worthy to note that research on error estimation for polymer molten flow is still in its infancy and further investigation to assess accuracy and applications to complex flows are needed.

5 NUMERICAL EXAMPLES The most common approach to mould filling used in commercial simulation packages [3,9] is based

on the Hele-Shaw approximation. Introduced by Hieber and Shen [8], this model is derived from averaged measures of the equivalent shear strain rate and viscosity over the channel cross-section thereby restricting its application to thin channels and cavities. The present work presents a brief assessment of polymer melt flow in thick channels and error estimation using fully coupled governing equations. The example discusses the

457


Finite Element solution for a 5 x 5 mm channel using four-nodded quadrilateral elements and four integration points. The material and mesh data are presented in Table 1.

Table 1 - Material constants for Polyacetal POM-M90-44 and other simulation parameters.

Parameter Symbol Value

cross-section H x W 5 x 5 mm

Channel length L 250 mm

wall temperature Tw 493 K ( 220 °C )

Flow rate Q 3,69 cm3/s

Pressure drop pΔ 11,0 MPa

Thermal conductivity k 0,31 W/m.K

Specific mass ρ 1143,9 Kg/m3

a1 0,022603

a2 5003,01

b1 1,6425E-6

b2 3901,0

c1 1,3574

Viscosity parameters

c2 653,73

h1 h2 h3

9 x 9 5 x 5 3 x 3

17 x 17 9 x 9 5 x 5

33 x 33 17 x 17 9 x 9

65 x 65 33 x 33 17 x 17

Richardson meshes h1 < h2 < h3

129 x 129 65 x 65 33 x 33

5.1 Equivalent shear strain rate and polymer viscosity The formulation used in this work is able to provide a point-wise distribution of the shear strain rate

and polymer viscosity inside the channel. Figure 1(a) shows a significant variation of γ& over the channel cross-section, with maximum values at the channel walls and zero at its centre and corners. It is important to mention that the equivalent shear strain rate is associated with the second invariant, J2, of the shear strain rate

tensor, 2/1

22/1 )2/1(): J=γ&2/1(= γ&&γ , and therefore, independent of the co-ordinate system. The polymer

viscosity, as illustrated in Figure 1(b), is directly affected by the temperature and shear rate according to Equation (4). Low shear rates and temperatures yield higher viscosities. A better perception of this behaviour

is presented in Figure 2, which shows node values of γ& , T and η along the Y-Y' symmetry line. The simulation highlights the opposing effects of the shear rate and temperature on the polymer flow, i.e., at centre the temperature is maximum and the shear rate is zero, and at the side walls the temperature is minimum and the shear rate is maximum. The combination of both effects causes the viscosity to yield minimum values at approximately 1/4 of the distance between the walls and the centre (y ≈ 0,625 mm). These results shows sharp contrast to the Hele-Shaw assumptions and strongly suggests further investigation on injection moulding of thick-walled components using a fully coupled material model.

458


Figure 1: Equivalent shear strain rate and viscosity distribution.

Figure 2: Equivalent shear strain rate, temperature and viscosity along the Y-Y' symmetry line.

5.2 Error assessment A suitable quantification of the solution accuracy is crucial to establish confidence in the numerical

model. The instinctive assumption that by refining the mesh one obtains more accurate results is not always true. Some aspects on this issue were addressed by Pinto [11], who demonstrated that the pressure field provided by a well-known commercial package was not consistent with subsequent mesh refinement. Therefore, error assessment is a fundamental step during the development of a numerical model. This section addresses the discretization errors estimated using Richardson extrapolation and the Energy and L2 error norms described in sections 4.1 and 4.2 respectively.

459


Figure 3: Local error distribution using Richardson extrapolation.

Figure 3 illustrates local error distributions based on Richardson extrapolation using meshes h1 ⇒ 65 x 65, h2 ⇒ 33 x 33 and h3 ⇒ 17 x 17 nodes (r = 2). It is interesting to note that errors follow quite a distinctive pattern for velocities and temperatures. The former shows large errors near locations of minimum viscosities (see Figure 1b) whereas the latter presents large errors for high temperatures at the channel centre. Details of Richardson extrapolation can be observed in Figure 4, which presents the error estimates for velocities along the Y-Y’ symmetry line. The simulations show that mesh refinement shits the maximum errors from the centre towards the channel walls, near regions of low viscosities.

Figure 4: Richardson error along the Y-Y’ symmetry line.

It has been found that the coupled character of the problem and the high material nonlinearity can hinder the monotonic rate of convergence required by Richardson estimates. This difficulty was also observed by Oberkampf and Trucano [10] in the context of Newtonian flows using Finite Differences. The analyses show points of high convergence rates (p ≈ 5) close to points of virtual divergence (p ≈ -1) near the corners, which suggests greater care on the application of Richardson extrapolation to this class of problems.

460


However, despite the hindrances, this estimate becomes attractive due its capacity evaluate errors directly from primal variables.

Estimates using error norms are illustrated in Figure 5 for a 65 x 65 mesh. It can be observed that both norms provide very similar error patterns, i.e., large errors at the centre and near low viscosity regions. This behaviour may be credited to the combined effect, in special the minimum values, of the equivalent shear strain rate (Figure 1a) and viscosity (Figure 1b). The similarity of both error norms are highlighted in Figure 6 which shows the distributions along the Y-Y’ symmetry line. The larger gradients provided by the Energy norm recommends its use in combination with automatic re-meshing procedures.

Figure 5: Local error distributions for Energy and L2 norms.

Figure 6: Error norms along the Y-Y’ symmetry line.

Analysis of global convergence is important to establish both the robustness of the computational model and the asymptotic rate of convergence. Figure 7 presents the global convergence curves for both Richardson extrapolation (Equation 13) and Energy and L2 norms (Equation 17). The literature indicates that, in the absence of singularities, for linear analysis of solid materials [16] and Newtonian flows [15], the global discretization error using the classical FEM norms is proportional to hq, where h is the element size and q is

461


the polynomial order of the approximation. This example uses linear elements (q = 1) and simulations show that the convergence rate exhibited by the Energy and L2 norms follows the same rule, so that

0003,1

9926,0

011,28

0211,2

2h

h

L

E

=

=

e

e

(18)

Despite the shortcomings of Richardson estimate previously discussed, the corresponding global errors for velocity and temperature,

][8383

]/[44667944,0

0328,1

Kh

smmh

23,2 25,2* =Tpq

p

,0

,2

Th

Wh

=

=

ε

ε

(19)

Show a convergence rate similar to Energy and L2 norms. Furthermore, the global analysis of Richardson estimate suggests that the computational model is indeed a second-order scheme according to

Equation (12), with an average convergence order and for the velocity and temperature equations respectively. It is worth noting that h is associated with the asymptotic rate of convergence, defined as a global measure computed over the problem domain, whereas h is related to the order of the scheme, calculated locally at the nodes. The former estimates global discretization errors and the latter approaches local approximation errors.

* =Wp

Figure 7: Global rate of convergence.

6 CONCLUDING REMARKS

This work addresses aspects of numerical formulation and error estimation strategies for analysing polymer melt flow in thick channels and cavities. The governing equations are solved using the Finite Element method in association with linear quadrilateral elements. In contrast to the classical Hele-Shaw approximation, the fully coupled model used in the simulations is able to capture point-wise variations of the shear strain rate and viscosity over the channel cross-section.

Despite increasing use of commercial packages to simulate injection moulding processes, the literature is extremely poor on error analysis of this class of problems. Accuracy and convergence analyses are performed using error estimates based on Richardson extrapolation and Energy and L2 norms. The former has the advantage of evaluating errors directly form the primal variables (velocity and temperature), however the requirement of monotonic convergence can compromise analyses in highly nonlinear and coupled problems. Energy and L2 norms have been widely employed in computational mechanics and have constituted the natural choice to perform error assessment when using Finite Elements. Error predictions using error norms and Richardson extrapolation were found very similar in the present analysis. The maximum local errors estimated by Energy and L2 norms occur near regions of low viscosities and at the

462


channel centre. The estimates based on Richardson extrapolation presents different patterns for velocities and temperatures. The former indicates larger errors near regions of low viscosities and the latter at the channel centre. The global error analysis shows that the Finite Element model used in the simulations approaches a second-order scheme (as indicated by the Richardson extrapolation) and presents an asymptotic rate of convergence (as evinced by the error norms). The issues discussed in this paper are of paramount importance and strongly suggest further investigation on error and convergence analysis for complex polymer flows during the mould filling stage.

7 REFERENCES

[1] BABUŠKA, I., RHEINBOLDT, W.C., A Posteriori Error Estimates for the Finite Element Method, International Journal for Numerical Methods in Engineering, v.12, p.1597-1615, 1978.

[2] BAO, W., An Economical Finite Element Approximation of Generalized Newtonian Flows, Computer Methods in Applied Mechanics and Engineering, v.191, p.3637 –3648, 2002.

[3] C-MOLD. Design Guide. A Resource for Plastic Engineers. Ithaca: Advanced CAE Technology Inc., 1998.

[4] CROCHET, M.J., DAVIES, A.R., WALTERS,K., Numerical Simulation of Non-Newtonian Flow. Amsterdam: Elsevier, 1984.

[5] FERZIGER, J.H., PERIĆ, M., Computational Methods for Fluid Dynamics. 2nd edition. Heidelberg: Springer-Verlag, 1999.

[6] FERZIGER, J.H., PERIĆ, M., Further Discussion of Numerical Errors in CFD. International Journal for Numerical Methods in Engineering, v.23, 1263-1274, 1996.

[7] HERRMANN, M.H., Simulação de Processos de Injeção de Polímeros em Canais Circulares. 2001. 110 pages. Dissertation (Mestrado em Ciência e Engenharia de Materiais) - State University of Santa Catarina, Joinville, 2001.

[8] HIEBER, C.A., SHEN, S.F., A Finite-Element / Finite-Difference Simulation of the Injection-Moulding Filling Process. Journal of Non-Newtonian Fluid Mechanics, v.7, p.1-32, 1980.

[9] MOLDFLOW. Material Testing Overview. Wayland: Moldflow Corporation, 1995.

[10] OBERKAMPF, W.L., TRUCANO, T.G., Verification and Validation in Computational Fluid Dynamics, Progress in Aerospace Sciences, v.38, p.209-272, 2002.

[11] PINTO, M.A.G.A., Análise dos Resultados de Simulação de Injeção em Aplicativo Comercial, entre Modelos Matemáticos Baseados em Casca Média e Casca Externa. 2001. 111 pages. Dissertation (Mestrado em Ciência e Engenharia de Materiais) - State University of Santa Catarina, Joinville, 2001.

[12] RICHARDSON, L.F., The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations with an Application to the Stresses in a Masonry Dam, Transactions of the Royal Society of London, Series A, v.210, p.307-357, 1910.

[13] ROACHE, P.J., Quantification of Uncertainty in Computation Fluid Dynamics, Annual Review of Fluid Mechanics, v.29, p.123-160, 1997.

[14] VERHOYEN, O., DUPRET, F., A Simplified Meted for Introducing Viscosity Law in the Numerical Simulation of Hele-Shaw Flow, Journal of Non-Newtonian Fluid Mechanics, v.74, p.25-46, 1998.

[15] WU, J., ZHU, J.Z., SZMELTER, J., ZIENKIEWICZ, O.C., Error Estimation and Adaptivity in Navier-Stokes Incompressible Flows, Computational Mechanics, v.6, p.259-270, 1990.

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[16] ZIENKIEWICZ, O.C., TAYLOR, R.L., The Finite Element Method. 5th ed. London: Butterworth-Heinemann, 2000.

[17] ZIENKIEWICZ, O.C., ZHU, J.Z., A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis. International Journal for Numerical Methods in Engineering, v.24, p.337-357, 1987.

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Finite Element Simulation and Error Estimation of Polymer ... · the discretization method (Finite Elements, Finite Volumes, Boundary Elements, Spectral methods, etc.). This work